Subgroups — C₆₇×#60374 #30739625

Order 17152.

4782 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#577×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3488	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3013	C₂	1	6	yes	maximal, complemented
5	C₆₇×#3010	C₂	1	6	yes	maximal, complemented
6	C₆₇×#3006	C₂	1	3	yes	maximal, complemented
7	C₆₇×#3004	C₂	1	3	yes	maximal, complemented
8	C₆₇×#3002	C₂	1	3	yes	maximal, complemented
9	C₂×#448×C₆₇	C₂	1	3	yes	maximal, complemented
10	C₆₇×#2898	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2880	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₆₇	—	2	16	no	—
13	C₂×2₋¹⁺⁴×C₆₇	—	2	1	no	—
14	C₂×2₋¹⁺⁴×C₆₇	—	2	3	no	—
15	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	2	yes	—
16	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	6	yes	complemented
17	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	1	yes	characteristic, complemented
18	C₂²×#41×C₆₇	—	2	3	no	—
19	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×#41×C₆₇	C₂²	1	3	yes	complemented
21	C₂³×Q₈×C₆₇	—	2	1	no	—
22	C₂³×Q₈×C₆₇	C₂²	1	3	yes	complemented
23	#577×C₆₇	—	2	16	no	—
24	C₂×#137×C₆₇	C₂²	1	6	yes	complemented
25	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	6	yes	complemented
26	C₂×#135×C₆₇	C₂²	1	6	yes	complemented
27	C₂²×SD₁₆×C₆₇	C₂²	1	6	yes	complemented
28	C₂×#131×C₆₇	C₂²	1	2	yes	complemented
29	#550×C₆₇	—	2	1	no	—
30	#548×C₆₇	—	2	1	no	—
31	#546×C₆₇	—	2	3	no	—
32	D₄²×C₆₇	—	2	3	no	—
33	#537×C₆₇	C₂²	1	3	yes	complemented
34	#535×C₆₇	C₂²	1	3	yes	complemented
35	#525×C₆₇	C₂²	1	3	yes	—
36	C₂×#121×C₆₇	C₂²	1	3	yes	—
37	#520×C₆₇	C₂²	1	1	yes	characteristic, complemented
38	#452×C₆₇	C₂²	1	6	yes	complemented
39	#451×C₆₇	C₂²	1	6	yes	complemented
40	#451×C₆₇	C₂²	1	3	yes	complemented
41	#450×C₆₇	C₂²	1	1	yes	characteristic, complemented
42	#449×C₆₇	C₂²	1	6	yes	complemented
43	#449×C₆₇	C₂²	1	12	yes	complemented
44	#448×C₆₇	C₂²	1	6	yes	complemented
45	#448×C₆₇	C₂²	1	12	yes	complemented
46	#448×C₆₇	C₂²	1	3	yes	complemented
47	#448×C₆₇	C₂²	1	6	yes	complemented
48	#447×C₆₇	C₂²	1	3	yes	complemented
49	#419×C₆₇	C₂²	1	6	yes	complemented
50	#417×C₆₇	C₂²	1	6	yes	complemented
51	#416×C₆₇	C₂²	1	6	yes	complemented
52	C₂×#103×C₆₇	C₂²	1	6	yes	complemented
53	#408×C₆₇	C₂²	1	6	yes	complemented
54	#407×C₆₇	C₂²	1	3	yes	complemented
55	C₂×C₂²⋊C₈×C₆₇	C₂²	1	3	yes	complemented
56	2₋¹⁺⁴×C₆₇	—	2	48	no	—
57	2₋¹⁺⁴×C₆₇	—	2	16	no	—
58	2₋¹⁺⁴×C₆₇	—	2	8	no	—
59	2₋¹⁺⁴×C₆₇	—	2	24	no	—
60	2₋¹⁺⁴×C₆₇	—	2	24	no	—
61	2₋¹⁺⁴×C₆₇	D₄	1	16	yes	complemented
62	H(5,𝔽₂)×C₆₇	—	2	8	no	—
63	C₂×#41×C₆₇	—	2	48	no	—
64	C₂×#41×C₆₇	—	2	8	no	—
65	C₂×#41×C₆₇	—	2	24	no	—
66	C₂×#41×C₆₇	—	2	1	no	—
67	C₂×#41×C₆₇	—	2	6	no	—
68	C₂×#41×C₆₇	—	2	12	no	—
69	C₂×#41×C₆₇	—	2	6	no	—
70	C₂×#41×C₆₇	—	2	3	no	—
71	C₂×#41×C₆₇	—	2	3	no	—
72	C₂×#41×C₆₇	D₄	1	3	yes	—
73	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
74	C₂×#41×C₆₇	C₂³	1	6	yes	—
75	C₂×#41×C₆₇	C₂³	1	3	yes	—
76	C₂×#41×C₆₇	C₂³	1	6	yes	—
77	C₂×#41×C₆₇	C₂³	1	3	yes	—
78	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
79	C₂²×Q₈×C₆₇	—	2	16	no	—
80	C₂²×Q₈×C₆₇	—	2	2	no	—
81	C₂²×Q₈×C₆₇	—	2	6	no	—
82	C₂²×Q₈×C₆₇	—	2	24	no	—
83	C₂²×Q₈×C₆₇	—	2	3	no	—
84	C₂²×Q₈×C₆₇	—	2	3	no	—
85	C₂²×Q₈×C₆₇	—	2	3	no	—
86	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
87	C₂²×Q₈×C₆₇	C₂³	1	3	yes	—
88	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
89	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
90	C₂²×D₄×C₆₇	—	2	3	no	—
91	C₂²×D₄×C₆₇	—	2	3	no	—
92	C₂²×D₄×C₆₇	—	2	3	no	—
93	C₂²×D₄×C₆₇	D₄	1	3	yes	—
94	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
95	C₂²×D₄×C₆₇	C₂³	1	3	yes	—
96	C₂³×C₄×C₆₇	—	2	3	no	—
97	C₂³×C₄×C₆₇	—	2	3	no	—
98	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
99	#137×C₆₇	—	2	24	no	—
100	AGL(1,ℤ/8)×C₆₇	—	2	24	no	—
101	#135×C₆₇	—	2	24	no	—
102	C₂×Q₁₆×C₆₇	C₂³	1	6	yes	—
103	C₂×SD₁₆×C₆₇	—	2	24	no	—
104	C₂×SD₁₆×C₆₇	C₂³	1	6	yes	—
105	C₂×SD₁₆×C₆₇	C₂³	1	12	yes	—
106	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
107	C₂×D₈×C₆₇	C₂³	1	6	yes	—
108	#131×C₆₇	—	2	8	no	—
109	C₂×M₁₆×C₆₇	C₂³	1	6	yes	—
110	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
111	#127×C₆₇	—	2	3	no	—
112	#124×C₆₇	—	2	3	no	—
113	#123×C₆₇	—	2	3	no	—
114	C₂²⋊Q₈×C₆₇	—	2	3	no	—
115	C₂²⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
116	C₂²⋊Q₈×C₆₇	C₂³	1	3	yes	—
117	#121×C₆₇	—	2	6	no	—
118	#121×C₆₇	—	2	3	no	—
119	#121×C₆₇	—	2	3	no	—
120	#121×C₆₇	—	2	3	no	—
121	#121×C₆₇	C₂³	1	3	yes	—
122	#121×C₆₇	C₂³	1	6	yes	—
123	#121×C₆₇	C₂³	1	3	yes	—
124	#120×C₆₇	—	2	3	no	—
125	#120×C₆₇	—	2	6	no	—
126	C₄×Q₈×C₆₇	—	2	1	no	—
127	C₄×D₄×C₆₇	—	2	3	no	—
128	C₄×D₄×C₆₇	—	2	3	no	—
129	C₄×D₄×C₆₇	—	2	3	no	—
130	#117×C₆₇	C₂³	1	3	yes	—
131	C₂×C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
132	C₂×C₂²⋊C₄×C₆₇	—	2	3	no	—
133	#103×C₆₇	C₂³	1	6	yes	—
134	#103×C₆₇	C₂³	1	12	yes	—
135	#103×C₆₇	C₂³	1	6	yes	—
136	#102×C₆₇	C₂³	1	2	yes	—
137	#102×C₆₇	C₂³	1	6	yes	—
138	C₂²⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
139	C₂²⋊C₈×C₆₇	C₂³	1	6	yes	—
140	C₂²⋊C₈×C₆₇	C₂³	1	3	yes	—
141	C₂²⋊C₈×C₆₇	C₂³	1	6	yes	—
142	C₂⁴×C₆₇	—	2	1	no	—
143	C₂⁴×C₆₇	—	2	3	no	—
144	C₂⁴×C₆₇	C₂×D₄	1	3	yes	—
145	#41×C₆₇	—	2	96	no	—
146	#41×C₆₇	—	2	24	no	—
147	#41×C₆₇	—	2	48	no	—
148	#41×C₆₇	—	2	24	no	—
149	#41×C₆₇	—	2	4	no	—
150	#41×C₆₇	—	2	12	no	—
151	#41×C₆₇	—	2	48	no	—
152	#41×C₆₇	—	2	12	no	—
153	#41×C₆₇	—	2	12	no	—
154	#41×C₆₇	—	2	12	no	—
155	#41×C₆₇	—	2	12	no	—
156	#41×C₆₇	—	2	12	no	—
157	#41×C₆₇	—	4	4	no	—
158	#41×C₆₇	C₂×D₄	1	8	yes	—
159	#41×C₆₇	C₂×D₄	1	24	yes	complemented
160	C₂×Q₈×C₆₇	—	2	24	no	—
161	C₂×Q₈×C₆₇	—	2	48	no	—
162	C₂×Q₈×C₆₇	—	2	24	no	—
163	C₂×Q₈×C₆₇	—	2	24	no	—
164	C₂×Q₈×C₆₇	—	2	4	no	—
165	C₂×Q₈×C₆₇	—	2	16	no	—
166	C₂×Q₈×C₆₇	—	2	12	no	—
167	C₂×Q₈×C₆₇	—	2	24	no	—
168	C₂×Q₈×C₆₇	—	2	6	no	—
169	C₂×Q₈×C₆₇	—	2	3	no	—
170	C₂×Q₈×C₆₇	—	2	6	no	—
171	C₂×Q₈×C₆₇	—	2	12	no	—
172	C₂×Q₈×C₆₇	C₂×D₄	1	3	yes	—
173	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₂×Q₈×C₆₇	C₂×D₄	1	24	yes	complemented
175	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₂×Q₈×C₆₇	C₂⁴	1	6	yes	—
177	C₂×D₄×C₆₇	—	2	24	no	—
178	C₂×D₄×C₆₇	—	2	6	no	—
179	C₂×D₄×C₆₇	—	2	6	no	—
180	C₂×D₄×C₆₇	—	2	12	no	—
181	C₂×D₄×C₆₇	—	2	4	no	—
182	C₂×D₄×C₆₇	—	2	3	no	—
183	C₂×D₄×C₆₇	—	2	3	no	—
184	C₂×D₄×C₆₇	—	2	3	no	—
185	C₂×D₄×C₆₇	—	2	6	no	—
186	C₂×D₄×C₆₇	—	2	1	no	—
187	C₂×D₄×C₆₇	—	2	12	no	—
188	C₂×D₄×C₆₇	—	2	3	no	—
189	C₂×D₄×C₆₇	—	4	12	no	—
190	C₂×D₄×C₆₇	—	8	6	no	—
191	C₂×D₄×C₆₇	C₂×D₄	1	6	yes	—
192	C₂×D₄×C₆₇	C₂×D₄	1	3	yes	—
193	C₂×D₄×C₆₇	C₂×D₄	1	3	yes	—
194	C₂×D₄×C₆₇	C₂⁴	1	3	yes	—
195	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
196	C₂²×C₄×C₆₇	—	2	24	no	—
197	C₂²×C₄×C₆₇	—	2	24	no	—
198	C₂²×C₄×C₆₇	—	2	3	no	—
199	C₂²×C₄×C₆₇	—	2	6	no	—
200	C₂²×C₄×C₆₇	—	2	6	no	—
201	C₂²×C₄×C₆₇	—	2	6	no	—
202	C₂²×C₄×C₆₇	—	2	6	no	—
203	C₂²×C₄×C₆₇	—	2	12	no	—
204	C₂²×C₄×C₆₇	—	2	3	no	—
205	C₂²×C₄×C₆₇	—	2	3	no	—
206	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
207	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
208	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
209	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
210	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
211	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
212	Q₁₆×C₆₇	—	2	12	no	—
213	SD₁₆×C₆₇	—	2	12	no	—
214	SD₁₆×C₆₇	—	2	24	no	—
215	SD₁₆×C₆₇	—	2	4	no	—
216	D₈×C₆₇	—	2	12	no	—
217	M₁₆×C₆₇	—	2	12	no	—
218	C₂×C₈×C₆₇	—	2	12	no	—
219	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
220	C₂×C₈×C₆₇	C₂⁴	1	6	yes	—
221	C₄⋊C₄×C₆₇	—	2	3	no	—
222	C₄⋊C₄×C₆₇	—	2	1	no	—
223	C₄⋊C₄×C₆₇	—	2	3	no	—
224	C₄⋊C₄×C₆₇	C₂⁴	1	3	yes	—
225	C₄⋊C₄×C₆₇	C₂⁴	1	1	yes	characteristic
226	C₂²⋊C₄×C₆₇	—	2	3	no	—
227	C₂²⋊C₄×C₆₇	—	2	1	no	—
228	C₂²⋊C₄×C₆₇	—	2	3	no	—
229	C₂²⋊C₄×C₆₇	—	2	3	no	—
230	C₂²⋊C₄×C₆₇	—	2	6	no	—
231	C₂²⋊C₄×C₆₇	—	2	3	no	—
232	C₄²×C₆₇	—	2	3	no	—
233	C₂³×C₆₇	—	2	4	no	—
234	C₂³×C₆₇	—	2	6	no	—
235	C₂³×C₆₇	—	2	6	no	—
236	C₂³×C₆₇	—	2	2	no	—
237	C₂³×C₆₇	—	2	3	no	—
238	C₂³×C₆₇	—	4	3	no	—
239	C₂³×C₆₇	—	4	2	no	—
240	C₂³×C₆₇	—	4	2	no	—
241	C₂³×C₆₇	—	4	6	no	—
242	C₂³×C₆₇	—	8	3	no	—
243	C₂³×C₆₇	—	8	3	no	—
244	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
245	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
246	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
247	Q₈×C₆₇	—	2	24	no	—
248	Q₈×C₆₇	—	2	12	no	—
249	Q₈×C₆₇	—	2	6	no	—
250	Q₈×C₆₇	—	2	24	no	—
251	Q₈×C₆₇	—	2	6	no	—
252	Q₈×C₆₇	—	2	2	no	—
253	Q₈×C₆₇	#120	1	4	yes	—
254	Q₈×C₆₇	C₂²×D₄	1	12	yes	—
255	D₄×C₆₇	—	2	24	no	—
256	D₄×C₆₇	—	2	4	no	—
257	D₄×C₆₇	—	2	24	no	—
258	D₄×C₆₇	—	2	12	no	—
259	D₄×C₆₇	—	2	6	no	—
260	D₄×C₆₇	—	2	2	no	—
261	D₄×C₆₇	—	2	6	no	—
262	D₄×C₆₇	—	4	6	no	—
263	D₄×C₆₇	—	4	6	no	—
264	D₄×C₆₇	—	4	6	no	—
265	D₄×C₆₇	—	8	3	no	—
266	D₄×C₆₇	—	8	6	no	—
267	D₄×C₆₇	#120	1	12	yes	—
268	D₄×C₆₇	C₂²×D₄	1	4	yes	—
269	C₂×C₄×C₆₇	—	2	12	no	—
270	C₂×C₄×C₆₇	—	2	24	no	—
271	C₂×C₄×C₆₇	—	2	24	no	—
272	C₂×C₄×C₆₇	—	2	24	no	—
273	C₂×C₄×C₆₇	—	2	6	no	—
274	C₂×C₄×C₆₇	—	2	6	no	—
275	C₂×C₄×C₆₇	—	2	6	no	—
276	C₂×C₄×C₆₇	—	2	24	no	—
277	C₂×C₄×C₆₇	—	2	6	no	—
278	C₂×C₄×C₆₇	—	2	2	no	—
279	C₂×C₄×C₆₇	—	2	6	no	—
280	C₂×C₄×C₆₇	—	2	1	no	—
281	C₂×C₄×C₆₇	—	2	3	no	—
282	C₂×C₄×C₆₇	—	2	1	no	—
283	C₂×C₄×C₆₇	—	4	6	no	—
284	C₂×C₄×C₆₇	—	8	3	no	—
285	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
286	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
288	C₂×C₄×C₆₇	C₂²×D₄	1	12	yes	—
289	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
290	C₈×C₆₇	—	2	2	no	—
291	C₈×C₆₇	—	2	6	no	—
292	C₂²×C₆₇	—	2	1	no	—
293	C₂²×C₆₇	—	2	3	no	—
294	C₂²×C₆₇	—	2	1	no	—
295	C₂²×C₆₇	—	2	3	no	—
296	C₂²×C₆₇	—	2	3	no	—
297	C₂²×C₆₇	—	2	3	no	—
298	C₂²×C₆₇	—	2	4	no	—
299	C₂²×C₆₇	—	2	3	no	—
300	C₂²×C₆₇	—	4	12	no	—
301	C₂²×C₆₇	—	4	8	no	—
302	C₂²×C₆₇	—	4	2	no	—
303	C₂²×C₆₇	—	4	2	no	—
304	C₂²×C₆₇	—	4	3	no	—
305	C₂²×C₆₇	—	4	3	no	—
306	C₂²×C₆₇	—	4	3	no	—
307	C₂²×C₆₇	—	16	6	no	—
308	C₂²×C₆₇	C₂×#120	1	6	yes	—
309	C₂²×C₆₇	#577	1	2	yes	complemented
310	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
311	C₄×C₆₇	—	2	12	no	—
312	C₄×C₆₇	—	2	3	no	—
313	C₄×C₆₇	—	2	1	no	—
314	C₄×C₆₇	—	4	1	no	—
315	C₄×C₆₇	—	4	1	no	—
316	C₄×C₆₇	—	4	3	no	—
317	C₄×C₆₇	C₂×#120	1	6	yes	—
318	C₄×C₆₇	C₂³×D₄	1	2	yes	—
319	#60374	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
320	C₂×C₆₇	—	2	2	no	—
321	C₂×C₆₇	—	2	6	no	—
322	C₂×C₆₇	—	4	3	no	—
323	C₂×C₆₇	—	4	4	no	—
324	C₂×C₆₇	—	8	3	no	—
325	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
326	C₂×C₆₇	C₂×#577	1	1	yes	characteristic
327	C₂×C₆₇	#3586	1	1	yes	characteristic
328	C₂²×2₋¹⁺⁴	C₂×C₆₇	1	1	yes	characteristic, complemented
329	C₂×#577	C₂×C₆₇	1	2	yes	complemented
330	#3488	C₂×C₆₇	1	1	yes	characteristic, complemented
331	#3013	C₂×C₆₇	1	6	yes	complemented
332	#3010	C₂×C₆₇	1	6	yes	complemented
333	#3006	C₂×C₆₇	1	3	yes	complemented
334	#3004	C₂×C₆₇	1	3	yes	complemented
335	#3002	C₂×C₆₇	1	3	yes	complemented
336	C₂×#448	C₂×C₆₇	1	3	yes	complemented
337	#2898	C₂×C₆₇	1	2	yes	complemented
338	#2880	C₂×C₆₇	1	1	yes	characteristic, complemented
339	C₆₇	#60374	1	1	yes	characteristic, direct product factor, Sylow
340	C₂×2₋¹⁺⁴	—	2	16	no	—
341	C₂×2₋¹⁺⁴	—	2	1	no	—
342	C₂×2₋¹⁺⁴	—	2	3	no	—
343	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	2	yes	—
344	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	6	yes	complemented
345	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	1	yes	characteristic, complemented
346	C₂²×#41	—	2	3	no	—
347	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
348	C₂²×#41	C₂²×C₆₇	1	3	yes	complemented
349	C₂³×Q₈	—	2	1	no	—
350	C₂³×Q₈	C₂²×C₆₇	1	3	yes	complemented
351	#577	—	2	16	no	—
352	C₂×#137	C₂²×C₆₇	1	6	yes	complemented
353	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	6	yes	complemented
354	C₂×#135	C₂²×C₆₇	1	6	yes	complemented
355	C₂²×SD₁₆	C₂²×C₆₇	1	6	yes	complemented
356	C₂×#131	C₂²×C₆₇	1	2	yes	complemented
357	#550	—	2	1	no	—
358	#548	—	2	1	no	—
359	#546	—	2	3	no	—
360	D₄²	—	2	3	no	—
361	#537	C₂²×C₆₇	1	3	yes	complemented
362	#535	C₂²×C₆₇	1	3	yes	complemented
363	#525	C₂²×C₆₇	1	3	yes	—
364	C₂×#121	C₂²×C₆₇	1	3	yes	—
365	#520	C₂²×C₆₇	1	1	yes	characteristic, complemented
366	#452	C₂²×C₆₇	1	6	yes	complemented
367	#451	C₂²×C₆₇	1	3	yes	complemented
368	#451	C₂²×C₆₇	1	6	yes	complemented
369	#450	C₂²×C₆₇	1	1	yes	characteristic, complemented
370	#449	C₂²×C₆₇	1	6	yes	complemented
371	#449	C₂²×C₆₇	1	12	yes	complemented
372	#448	C₂²×C₆₇	1	3	yes	complemented
373	#448	C₂²×C₆₇	1	6	yes	complemented
374	#448	C₂²×C₆₇	1	12	yes	complemented
375	#448	C₂²×C₆₇	1	6	yes	complemented
376	#447	C₂²×C₆₇	1	3	yes	complemented
377	#419	C₂²×C₆₇	1	6	yes	complemented
378	#417	C₂²×C₆₇	1	6	yes	complemented
379	#416	C₂²×C₆₇	1	6	yes	complemented
380	C₂×#103	C₂²×C₆₇	1	6	yes	complemented
381	#408	C₂²×C₆₇	1	6	yes	complemented
382	#407	C₂²×C₆₇	1	3	yes	complemented
383	C₂×C₂²⋊C₈	C₂²×C₆₇	1	3	yes	complemented
384	2₋¹⁺⁴	—	2	48	no	—
385	2₋¹⁺⁴	—	2	16	no	—
386	2₋¹⁺⁴	—	2	8	no	—
387	2₋¹⁺⁴	—	2	24	no	—
388	2₋¹⁺⁴	—	2	24	no	—
389	2₋¹⁺⁴	D₄×C₆₇	1	16	yes	complemented
390	H(5,𝔽₂)	—	2	8	no	—
391	C₂×#41	—	2	48	no	—
392	C₂×#41	—	2	8	no	—
393	C₂×#41	—	2	24	no	—
394	C₂×#41	—	2	1	no	—
395	C₂×#41	—	2	6	no	—
396	C₂×#41	—	2	12	no	—
397	C₂×#41	—	2	6	no	—
398	C₂×#41	—	2	3	no	—
399	C₂×#41	—	2	3	no	—
400	C₂×#41	D₄×C₆₇	1	3	yes	—
401	C₂×#41	D₄×C₆₇	1	1	yes	characteristic
402	C₂×#41	C₂³×C₆₇	1	3	yes	—
403	C₂×#41	C₂³×C₆₇	1	6	yes	—
404	C₂×#41	C₂³×C₆₇	1	6	yes	—
405	C₂×#41	C₂³×C₆₇	1	3	yes	—
406	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
407	C₂²×Q₈	—	2	16	no	—
408	C₂²×Q₈	—	2	2	no	—
409	C₂²×Q₈	—	2	6	no	—
410	C₂²×Q₈	—	2	24	no	—
411	C₂²×Q₈	—	2	3	no	—
412	C₂²×Q₈	—	2	3	no	—
413	C₂²×Q₈	—	2	3	no	—
414	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
415	C₂²×Q₈	C₂³×C₆₇	1	3	yes	—
416	C₂²×Q₈	C₂³×C₆₇	1	6	yes	—
417	C₂²×Q₈	C₂³×C₆₇	1	6	yes	—
418	C₂²×D₄	—	2	3	no	—
419	C₂²×D₄	—	2	3	no	—
420	C₂²×D₄	—	2	3	no	—
421	C₂²×D₄	D₄×C₆₇	1	3	yes	—
422	C₂²×D₄	C₂³×C₆₇	1	3	yes	—
423	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
424	C₂³×C₄	—	2	3	no	—
425	C₂³×C₄	—	2	3	no	—
426	C₂³×C₄	C₂³×C₆₇	1	3	yes	—
427	#137	—	2	24	no	—
428	AGL(1,ℤ/8)	—	2	24	no	—
429	#135	—	2	24	no	—
430	C₂×Q₁₆	C₂³×C₆₇	1	6	yes	—
431	C₂×SD₁₆	—	2	24	no	—
432	C₂×SD₁₆	C₂³×C₆₇	1	6	yes	—
433	C₂×SD₁₆	C₂³×C₆₇	1	12	yes	—
434	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
435	C₂×D₈	C₂³×C₆₇	1	6	yes	—
436	#131	—	2	8	no	—
437	C₂×M₁₆	C₂³×C₆₇	1	6	yes	—
438	C₂²×C₈	C₂³×C₆₇	1	6	yes	—
439	#127	—	2	3	no	—
440	#124	—	2	3	no	—
441	#123	—	2	3	no	—
442	C₂²⋊Q₈	—	2	3	no	—
443	C₂²⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
444	C₂²⋊Q₈	C₂³×C₆₇	1	3	yes	—
445	#121	—	2	6	no	—
446	#121	—	2	3	no	—
447	#121	—	2	3	no	—
448	#121	—	2	3	no	—
449	#121	C₂³×C₆₇	1	3	yes	—
450	#121	C₂³×C₆₇	1	6	yes	—
451	#121	C₂³×C₆₇	1	3	yes	—
452	#120	—	2	3	no	—
453	#120	—	2	6	no	—
454	C₄×Q₈	—	2	1	no	—
455	C₄×D₄	—	2	3	no	—
456	C₄×D₄	—	2	3	no	—
457	C₄×D₄	—	2	3	no	—
458	#117	C₂³×C₆₇	1	3	yes	—
459	C₂×C₄⋊C₄	C₂³×C₆₇	1	3	yes	—
460	C₂×C₂²⋊C₄	—	2	3	no	—
461	#103	C₂³×C₆₇	1	6	yes	—
462	#103	C₂³×C₆₇	1	12	yes	—
463	#103	C₂³×C₆₇	1	6	yes	—
464	#102	C₂³×C₆₇	1	2	yes	—
465	#102	C₂³×C₆₇	1	6	yes	—
466	C₂²⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
467	C₂²⋊C₈	C₂³×C₆₇	1	3	yes	—
468	C₂²⋊C₈	C₂³×C₆₇	1	6	yes	—
469	C₂²⋊C₈	C₂³×C₆₇	1	6	yes	—
470	C₂⁴	—	2	1	no	—
471	C₂⁴	—	2	3	no	—
472	C₂⁴	C₂×D₄×C₆₇	1	3	yes	—
473	#41	—	2	96	no	—
474	#41	—	2	24	no	—
475	#41	—	2	48	no	—
476	#41	—	2	24	no	—
477	#41	—	2	4	no	—
478	#41	—	2	12	no	—
479	#41	—	2	48	no	—
480	#41	—	2	12	no	—
481	#41	—	2	12	no	—
482	#41	—	2	12	no	—
483	#41	—	2	12	no	—
484	#41	—	2	12	no	—
485	#41	—	4	4	no	—
486	#41	C₂×D₄×C₆₇	1	8	yes	—
487	#41	C₂×D₄×C₆₇	1	24	yes	complemented
488	C₂×Q₈	—	2	24	no	—
489	C₂×Q₈	—	2	48	no	—
490	C₂×Q₈	—	2	24	no	—
491	C₂×Q₈	—	2	24	no	—
492	C₂×Q₈	—	2	4	no	—
493	C₂×Q₈	—	2	16	no	—
494	C₂×Q₈	—	2	12	no	—
495	C₂×Q₈	—	2	24	no	—
496	C₂×Q₈	—	2	6	no	—
497	C₂×Q₈	—	2	3	no	—
498	C₂×Q₈	—	2	6	no	—
499	C₂×Q₈	—	2	12	no	—
500	C₂×Q₈	C₂×D₄×C₆₇	1	3	yes	—
501	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
502	C₂×Q₈	C₂×D₄×C₆₇	1	24	yes	complemented
503	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
504	C₂×Q₈	C₂⁴×C₆₇	1	6	yes	—
505	C₂×D₄	—	2	24	no	—
506	C₂×D₄	—	2	6	no	—
507	C₂×D₄	—	2	6	no	—
508	C₂×D₄	—	2	12	no	—
509	C₂×D₄	—	2	4	no	—
510	C₂×D₄	—	2	3	no	—
511	C₂×D₄	—	2	3	no	—
512	C₂×D₄	—	2	3	no	—
513	C₂×D₄	—	2	6	no	—
514	C₂×D₄	—	2	1	no	—
515	C₂×D₄	—	2	12	no	—
516	C₂×D₄	—	2	3	no	—
517	C₂×D₄	—	4	12	no	—
518	C₂×D₄	—	8	6	no	—
519	C₂×D₄	C₂×D₄×C₆₇	1	6	yes	—
520	C₂×D₄	C₂×D₄×C₆₇	1	3	yes	—
521	C₂×D₄	C₂×D₄×C₆₇	1	3	yes	—
522	C₂×D₄	C₂⁴×C₆₇	1	3	yes	—
523	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
524	C₂²×C₄	—	2	24	no	—
525	C₂²×C₄	—	2	24	no	—
526	C₂²×C₄	—	2	3	no	—
527	C₂²×C₄	—	2	6	no	—
528	C₂²×C₄	—	2	6	no	—
529	C₂²×C₄	—	2	6	no	—
530	C₂²×C₄	—	2	6	no	—
531	C₂²×C₄	—	2	12	no	—
532	C₂²×C₄	—	2	3	no	—
533	C₂²×C₄	—	2	3	no	—
534	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
535	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
536	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
537	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
538	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
539	C₂²×C₄	C₂⁴×C₆₇	1	3	yes	—
540	Q₁₆	—	2	12	no	—
541	SD₁₆	—	2	12	no	—
542	SD₁₆	—	2	24	no	—
543	SD₁₆	—	2	4	no	—
544	D₈	—	2	12	no	—
545	M₁₆	—	2	12	no	—
546	C₂×C₈	—	2	12	no	—
547	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
548	C₂×C₈	C₂⁴×C₆₇	1	6	yes	—
549	C₄⋊C₄	—	2	3	no	—
550	C₄⋊C₄	—	2	1	no	—
551	C₄⋊C₄	—	2	3	no	—
552	C₄⋊C₄	C₂⁴×C₆₇	1	3	yes	—
553	C₄⋊C₄	C₂⁴×C₆₇	1	1	yes	characteristic
554	C₂²⋊C₄	—	2	3	no	—
555	C₂²⋊C₄	—	2	1	no	—
556	C₂²⋊C₄	—	2	3	no	—
557	C₂²⋊C₄	—	2	3	no	—
558	C₂²⋊C₄	—	2	6	no	—
559	C₂²⋊C₄	—	2	3	no	—
560	C₄²	—	2	3	no	—
561	C₂³	—	2	4	no	—
562	C₂³	—	2	6	no	—
563	C₂³	—	2	6	no	—
564	C₂³	—	2	2	no	—
565	C₂³	—	2	3	no	—
566	C₂³	—	4	3	no	—
567	C₂³	—	4	2	no	—
568	C₂³	—	4	2	no	—
569	C₂³	—	4	6	no	—
570	C₂³	—	8	3	no	—
571	C₂³	—	8	3	no	—
572	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
573	C₂³	C₂²×D₄×C₆₇	1	3	yes	—
574	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
575	Q₈	—	2	24	no	—
576	Q₈	—	2	12	no	—
577	Q₈	—	2	6	no	—
578	Q₈	—	2	24	no	—
579	Q₈	—	2	6	no	—
580	Q₈	—	2	2	no	—
581	Q₈	#120×C₆₇	1	4	yes	—
582	Q₈	C₂²×D₄×C₆₇	1	12	yes	—
583	D₄	—	2	24	no	—
584	D₄	—	2	4	no	—
585	D₄	—	2	24	no	—
586	D₄	—	2	12	no	—
587	D₄	—	2	6	no	—
588	D₄	—	2	2	no	—
589	D₄	—	2	6	no	—
590	D₄	—	4	6	no	—
591	D₄	—	4	6	no	—
592	D₄	—	4	6	no	—
593	D₄	—	8	6	no	—
594	D₄	—	8	3	no	—
595	D₄	#120×C₆₇	1	12	yes	—
596	D₄	C₂²×D₄×C₆₇	1	4	yes	—
597	C₂×C₄	—	2	12	no	—
598	C₂×C₄	—	2	24	no	—
599	C₂×C₄	—	2	24	no	—
600	C₂×C₄	—	2	24	no	—
601	C₂×C₄	—	2	6	no	—
602	C₂×C₄	—	2	6	no	—
603	C₂×C₄	—	2	6	no	—
604	C₂×C₄	—	2	24	no	—
605	C₂×C₄	—	2	6	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	6	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	3	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	4	6	no	—
612	C₂×C₄	—	8	3	no	—
613	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
614	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
615	C₂×C₄	C₂²×D₄×C₆₇	1	3	yes	—
616	C₂×C₄	C₂²×D₄×C₆₇	1	12	yes	—
617	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
618	C₈	—	2	2	no	—
619	C₈	—	2	6	no	—
620	C₂²	—	2	3	no	—
621	C₂²	—	2	1	no	—
622	C₂²	—	2	3	no	—
623	C₂²	—	2	1	no	—
624	C₂²	—	2	3	no	—
625	C₂²	—	2	3	no	—
626	C₂²	—	2	3	no	—
627	C₂²	—	2	4	no	—
628	C₂²	—	4	12	no	—
629	C₂²	—	4	8	no	—
630	C₂²	—	4	2	no	—
631	C₂²	—	4	2	no	—
632	C₂²	—	4	3	no	—
633	C₂²	—	4	3	no	—
634	C₂²	—	4	3	no	—
635	C₂²	—	16	6	no	—
636	C₂²	C₂×#120×C₆₇	1	6	yes	—
637	C₂²	#577×C₆₇	1	2	yes	complemented
638	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
639	C₄	—	2	12	no	—
640	C₄	—	2	3	no	—
641	C₄	—	2	1	no	—
642	C₄	—	4	3	no	—
643	C₄	—	4	1	no	—
644	C₄	—	4	1	no	—
645	C₄	C₂×#120×C₆₇	1	6	yes	—
646	C₄	C₂³×D₄×C₆₇	1	2	yes	—
647	C₂	—	2	2	no	—
648	C₂	—	2	6	no	—
649	C₂	—	4	3	no	—
650	C₂	—	4	4	no	—
651	C₂	—	8	3	no	—
652	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic
653	C₂	C₂×#577×C₆₇	1	1	yes	characteristic
654	C₂	C₆₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×#41×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×2₋¹⁺⁴×C₆₇	C₂	semidirect
2. C₂×#577×C₆₇	C₂	semidirect
3. C₆₇×#3488	C₂	semidirect
4. C₆₇×#3013	C₂	semidirect
5. C₆₇×#3010	C₂	semidirect
6. C₆₇×#3006	C₂	semidirect
7. C₆₇×#3004	C₂	semidirect
8. C₆₇×#3002	C₂	semidirect
9. C₂×#448×C₆₇	C₂	semidirect
10. C₆₇×#2898	C₂	semidirect
11. C₆₇×#2880	C₂	semidirect
16. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
17. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
20. C₂²×#41×C₆₇	C₂²	semidirect
22. C₂³×Q₈×C₆₇	C₂²	semidirect
24. C₂×#137×C₆₇	C₂²	semidirect
25. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
26. C₂×#135×C₆₇	C₂²	semidirect
27. C₂²×SD₁₆×C₆₇	C₂²	semidirect
28. C₂×#131×C₆₇	C₂²	semidirect
33. #537×C₆₇	C₂²	semidirect
34. #535×C₆₇	C₂²	semidirect
37. #520×C₆₇	C₂²	semidirect
38. #452×C₆₇	C₂²	semidirect
39. #451×C₆₇	C₂²	semidirect
40. #451×C₆₇	C₂²	semidirect
41. #450×C₆₇	C₂²	semidirect
42. #449×C₆₇	C₂²	semidirect
43. #449×C₆₇	C₂²	semidirect
44. #448×C₆₇	C₂²	semidirect
45. #448×C₆₇	C₂²	semidirect
46. #448×C₆₇	C₂²	semidirect
47. #448×C₆₇	C₂²	semidirect
48. #447×C₆₇	C₂²	semidirect
49. #419×C₆₇	C₂²	semidirect
50. #417×C₆₇	C₂²	semidirect
51. #416×C₆₇	C₂²	semidirect
52. C₂×#103×C₆₇	C₂²	semidirect
53. #408×C₆₇	C₂²	semidirect
54. #407×C₆₇	C₂²	semidirect
55. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
61. 2₋¹⁺⁴×C₆₇	D₄	semidirect
159. #41×C₆₇	C₂×D₄	semidirect
174. C₂×Q₈×C₆₇	C₂×D₄	semidirect
309. C₂²×C₆₇	#577	semidirect
319. #60374	C₆₇	direct (N × H)
328. C₂²×2₋¹⁺⁴	C₂×C₆₇	semidirect
329. C₂×#577	C₂×C₆₇	semidirect
330. #3488	C₂×C₆₇	semidirect
331. #3013	C₂×C₆₇	semidirect
332. #3010	C₂×C₆₇	semidirect
333. #3006	C₂×C₆₇	semidirect
334. #3004	C₂×C₆₇	semidirect
335. #3002	C₂×C₆₇	semidirect
336. C₂×#448	C₂×C₆₇	semidirect
337. #2898	C₂×C₆₇	semidirect
338. #2880	C₂×C₆₇	semidirect
339. C₆₇	#60374	direct (N × H)
344. C₂×2₋¹⁺⁴	C₂²×C₆₇	semidirect
345. C₂×H(5,𝔽₂)	C₂²×C₆₇	semidirect
348. C₂²×#41	C₂²×C₆₇	semidirect
350. C₂³×Q₈	C₂²×C₆₇	semidirect
352. C₂×#137	C₂²×C₆₇	semidirect
353. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
354. C₂×#135	C₂²×C₆₇	semidirect
355. C₂²×SD₁₆	C₂²×C₆₇	semidirect
356. C₂×#131	C₂²×C₆₇	semidirect
361. #537	C₂²×C₆₇	semidirect
362. #535	C₂²×C₆₇	semidirect
365. #520	C₂²×C₆₇	semidirect
366. #452	C₂²×C₆₇	semidirect
367. #451	C₂²×C₆₇	semidirect
368. #451	C₂²×C₆₇	semidirect
369. #450	C₂²×C₆₇	semidirect
370. #449	C₂²×C₆₇	semidirect
371. #449	C₂²×C₆₇	semidirect
372. #448	C₂²×C₆₇	semidirect
373. #448	C₂²×C₆₇	semidirect
374. #448	C₂²×C₆₇	semidirect
375. #448	C₂²×C₆₇	semidirect
376. #447	C₂²×C₆₇	semidirect
377. #419	C₂²×C₆₇	semidirect
378. #417	C₂²×C₆₇	semidirect
379. #416	C₂²×C₆₇	semidirect
380. C₂×#103	C₂²×C₆₇	semidirect
381. #408	C₂²×C₆₇	semidirect
382. #407	C₂²×C₆₇	semidirect
383. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
389. 2₋¹⁺⁴	D₄×C₆₇	semidirect
487. #41	C₂×D₄×C₆₇	semidirect
502. C₂×Q₈	C₂×D₄×C₆₇	semidirect
637. C₂²	#577×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×2₋¹⁺⁴×C₆₇	2
C₂×#577×C₆₇	2
C₆₇×#3488	2
C₆₇×#3013	2
C₆₇×#3010	2
C₆₇×#3006	2
C₆₇×#3004	2
C₆₇×#3002	2
C₂×#448×C₆₇	2
C₆₇×#2898	2
C₆₇×#2880	2
#60374	67

Subgroups — C67×#60374 #30739625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60374 #30739625