Subgroups — C₆₇×#60370 #30739621

Order 17152.

4746 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#585×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#137×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#536×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3011	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3009	C₂	1	8	yes	maximal, complemented
6	C₆₇×#3007	C₂	1	8	yes	maximal, complemented
7	C₆₇×#2896	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2879	C₂	1	1	yes	maximal, characteristic, complemented
9	#585×C₆₇	—	2	16	no	—
10	C₂×2₋¹⁺⁴×C₆₇	—	2	1	no	—
11	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	4	yes	—
12	C₂×H(5,𝔽₂)×C₆₇	—	2	2	no	—
13	C₂×H(5,𝔽₂)×C₆₇	—	2	1	no	—
14	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	4	yes	—
15	C₂²×#41×C₆₇	—	2	4	no	—
16	C₂²×#41×C₆₇	C₂²	1	4	yes	—
17	C₂²×#41×C₆₇	C₂²	1	2	yes	—
18	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
19	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
20	C₂×#137×C₆₇	—	2	16	no	—
21	C₂×#137×C₆₇	C₂²	1	8	yes	complemented
22	C₂×#137×C₆₇	C₂²	1	8	yes	complemented
23	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	complemented
24	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
25	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
26	#536×C₆₇	—	2	4	no	—
27	#536×C₆₇	C₂²	1	8	yes	complemented
28	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
29	C₂×#124×C₆₇	—	2	1	no	—
30	C₂×#123×C₆₇	—	2	2	no	—
31	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
32	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	#452×C₆₇	C₂²	1	32	yes	complemented
34	#451×C₆₇	C₂²	1	16	yes	complemented
35	#450×C₆₇	C₂²	1	16	yes	complemented
36	#419×C₆₇	C₂²	1	4	yes	complemented
37	#418×C₆₇	C₂²	1	4	yes	complemented
38	#417×C₆₇	C₂²	1	8	yes	complemented
39	#416×C₆₇	C₂²	1	8	yes	complemented
40	#408×C₆₇	C₂²	1	8	yes	complemented
41	#407×C₆₇	C₂²	1	4	yes	complemented
42	2₋¹⁺⁴×C₆₇	—	2	16	no	—
43	2₋¹⁺⁴×C₆₇	—	2	16	no	—
44	2₋¹⁺⁴×C₆₇	—	2	16	no	—
45	H(5,𝔽₂)×C₆₇	—	2	16	no	—
46	H(5,𝔽₂)×C₆₇	—	2	32	no	—
47	H(5,𝔽₂)×C₆₇	—	2	16	no	—
48	H(5,𝔽₂)×C₆₇	—	2	16	no	—
49	C₂×#41×C₆₇	—	2	64	no	—
50	C₂×#41×C₆₇	—	2	32	no	—
51	C₂×#41×C₆₇	—	2	4	no	—
52	C₂×#41×C₆₇	—	2	8	no	—
53	C₂×#41×C₆₇	—	2	4	no	—
54	C₂×#41×C₆₇	—	2	4	no	—
55	C₂×#41×C₆₇	—	2	2	no	—
56	C₂×#41×C₆₇	—	2	4	no	—
57	C₂×#41×C₆₇	—	2	2	no	—
58	C₂×#41×C₆₇	—	2	8	no	—
59	C₂×#41×C₆₇	—	2	8	no	—
60	C₂×#41×C₆₇	D₄	1	16	yes	—
61	C₂×#41×C₆₇	D₄	1	4	yes	—
62	C₂×#41×C₆₇	C₂³	1	8	yes	—
63	C₂×#41×C₆₇	C₂³	1	4	yes	—
64	C₂×#41×C₆₇	C₂³	1	4	yes	—
65	C₂×#41×C₆₇	C₂³	1	4	yes	—
66	C₂²×Q₈×C₆₇	—	2	4	no	—
67	C₂²×Q₈×C₆₇	—	2	1	no	—
68	C₂²×Q₈×C₆₇	—	2	2	no	—
69	C₂²×Q₈×C₆₇	—	2	8	no	—
70	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
71	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
72	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
73	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
74	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
75	C₂²×D₄×C₆₇	—	2	4	no	—
76	C₂²×D₄×C₆₇	—	2	2	no	—
77	C₂²×D₄×C₆₇	—	2	8	no	—
78	C₂²×D₄×C₆₇	—	2	4	no	—
79	C₂²×D₄×C₆₇	D₄	1	2	yes	—
80	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
81	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
82	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
83	C₂³×C₄×C₆₇	—	2	4	no	—
84	C₂³×C₄×C₆₇	—	2	2	no	—
85	C₂³×C₄×C₆₇	—	2	1	no	—
86	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
87	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
88	#137×C₆₇	—	2	32	no	—
89	#137×C₆₇	—	2	32	no	—
90	C₂×Q₁₆×C₆₇	—	2	16	no	—
91	C₂×Q₁₆×C₆₇	C₂³	1	16	yes	—
92	C₂×SD₁₆×C₆₇	—	2	16	no	—
93	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	—
94	C₂×M₁₆×C₆₇	—	2	8	no	—
95	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
96	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
97	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
98	C₄⋊Q₈×C₆₇	—	2	8	no	—
99	#124×C₆₇	—	2	8	no	—
100	#123×C₆₇	—	2	16	no	—
101	C₂²⋊Q₈×C₆₇	—	2	8	no	—
102	C₂²⋊Q₈×C₆₇	C₂³	1	16	yes	—
103	#117×C₆₇	—	2	2	no	—
104	#117×C₆₇	C₂³	1	4	yes	—
105	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
108	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
109	C₂×C₄²×C₆₇	—	2	1	no	—
110	#103×C₆₇	C₂³	1	16	yes	—
111	#102×C₆₇	C₂³	1	16	yes	—
112	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	—
113	C₂⁴×C₆₇	—	2	4	no	—
114	C₂⁴×C₆₇	—	2	2	no	—
115	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
116	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
117	#41×C₆₇	—	2	32	no	—
118	#41×C₆₇	—	2	64	no	—
119	#41×C₆₇	—	2	32	no	—
120	#41×C₆₇	—	2	32	no	—
121	#41×C₆₇	—	2	32	no	—
122	#41×C₆₇	—	2	16	no	—
123	#41×C₆₇	—	2	16	no	—
124	#41×C₆₇	—	2	16	no	—
125	#41×C₆₇	—	2	16	no	—
126	#41×C₆₇	—	2	16	no	—
127	#41×C₆₇	—	2	8	no	—
128	#41×C₆₇	—	2	8	no	—
129	#41×C₆₇	—	2	8	no	—
130	#41×C₆₇	—	2	8	no	—
131	#41×C₆₇	C₂×D₄	1	16	yes	—
132	#41×C₆₇	C₂×D₄	1	16	yes	—
133	C₂×Q₈×C₆₇	—	2	32	no	—
134	C₂×Q₈×C₆₇	—	2	16	no	—
135	C₂×Q₈×C₆₇	—	2	4	no	—
136	C₂×Q₈×C₆₇	—	2	4	no	—
137	C₂×Q₈×C₆₇	—	2	2	no	—
138	C₂×Q₈×C₆₇	—	2	4	no	—
139	C₂×Q₈×C₆₇	—	2	2	no	—
140	C₂×Q₈×C₆₇	—	2	4	no	—
141	C₂×Q₈×C₆₇	—	2	8	no	—
142	C₂×Q₈×C₆₇	—	2	8	no	—
143	C₂×Q₈×C₆₇	—	2	8	no	—
144	C₂×Q₈×C₆₇	—	4	8	no	—
145	C₂×Q₈×C₆₇	—	4	4	no	—
146	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
147	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
148	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
149	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
150	C₂×D₄×C₆₇	—	2	32	no	—
151	C₂×D₄×C₆₇	—	2	64	no	—
152	C₂×D₄×C₆₇	—	2	32	no	—
153	C₂×D₄×C₆₇	—	2	16	no	—
154	C₂×D₄×C₆₇	—	2	8	no	—
155	C₂×D₄×C₆₇	—	2	8	no	—
156	C₂×D₄×C₆₇	—	2	4	no	—
157	C₂×D₄×C₆₇	—	2	4	no	—
158	C₂×D₄×C₆₇	—	2	16	no	—
159	C₂×D₄×C₆₇	—	2	4	no	—
160	C₂×D₄×C₆₇	—	2	4	no	—
161	C₂×D₄×C₆₇	—	2	2	no	—
162	C₂×D₄×C₆₇	—	2	4	no	—
163	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
164	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
165	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
166	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₆₇	—	2	32	no	—
168	C₂²×C₄×C₆₇	—	2	4	no	—
169	C₂²×C₄×C₆₇	—	2	2	no	—
170	C₂²×C₄×C₆₇	—	2	16	no	—
171	C₂²×C₄×C₆₇	—	2	4	no	—
172	C₂²×C₄×C₆₇	—	2	8	no	—
173	C₂²×C₄×C₆₇	—	2	4	no	—
174	C₂²×C₄×C₆₇	—	2	8	no	—
175	C₂²×C₄×C₆₇	—	2	2	no	—
176	C₂²×C₄×C₆₇	—	2	2	no	—
177	C₂²×C₄×C₆₇	—	2	4	no	—
178	C₂²×C₄×C₆₇	—	4	4	no	—
179	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
180	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
181	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
184	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
185	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
186	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
187	Q₁₆×C₆₇	—	2	32	no	—
188	SD₁₆×C₆₇	—	2	32	no	—
189	M₁₆×C₆₇	—	2	8	no	—
190	M₁₆×C₆₇	—	2	8	no	—
191	C₂×C₈×C₆₇	—	2	8	no	—
192	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
193	C₄⋊C₄×C₆₇	—	2	16	no	—
194	C₄⋊C₄×C₆₇	—	2	2	no	—
195	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
196	C₂²⋊C₄×C₆₇	—	2	4	no	—
197	C₂²⋊C₄×C₆₇	—	2	16	no	—
198	C₄²×C₆₇	—	2	4	no	—
199	C₂³×C₆₇	—	2	16	no	—
200	C₂³×C₆₇	—	2	4	no	—
201	C₂³×C₆₇	—	2	4	no	—
202	C₂³×C₆₇	—	2	8	no	—
203	C₂³×C₆₇	—	4	2	no	—
204	C₂³×C₆₇	—	4	4	no	—
205	C₂³×C₆₇	—	8	4	no	—
206	C₂³×C₆₇	—	8	8	no	—
207	C₂³×C₆₇	#120	1	4	yes	—
208	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
209	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
210	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
211	Q₈×C₆₇	—	2	16	no	—
212	Q₈×C₆₇	—	2	8	no	—
213	Q₈×C₆₇	—	2	8	no	—
214	Q₈×C₆₇	—	2	4	no	—
215	Q₈×C₆₇	—	2	8	no	—
216	Q₈×C₆₇	—	4	4	no	—
217	Q₈×C₆₇	—	4	8	no	—
218	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
219	D₄×C₆₇	—	2	32	no	—
220	D₄×C₆₇	—	2	16	no	—
221	D₄×C₆₇	—	2	16	no	—
222	D₄×C₆₇	—	2	16	no	—
223	D₄×C₆₇	—	2	16	no	—
224	D₄×C₆₇	—	2	8	no	—
225	D₄×C₆₇	—	2	8	no	—
226	D₄×C₆₇	—	2	4	no	—
227	D₄×C₆₇	C₂²×D₄	1	8	yes	—
228	C₂×C₄×C₆₇	—	2	16	no	—
229	C₂×C₄×C₆₇	—	2	32	no	—
230	C₂×C₄×C₆₇	—	2	8	no	—
231	C₂×C₄×C₆₇	—	2	4	no	—
232	C₂×C₄×C₆₇	—	2	4	no	—
233	C₂×C₄×C₆₇	—	2	16	no	—
234	C₂×C₄×C₆₇	—	2	8	no	—
235	C₂×C₄×C₆₇	—	2	16	no	—
236	C₂×C₄×C₆₇	—	2	8	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	4	no	—
239	C₂×C₄×C₆₇	—	4	4	no	—
240	C₂×C₄×C₆₇	—	4	4	no	—
241	C₂×C₄×C₆₇	—	4	8	no	—
242	C₂×C₄×C₆₇	#120	1	8	yes	—
243	C₂×C₄×C₆₇	#120	1	4	yes	—
244	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
245	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
246	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
248	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
249	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
250	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
251	C₈×C₆₇	—	2	8	no	—
252	C₂²×C₆₇	—	2	2	no	—
253	C₂²×C₆₇	—	2	4	no	—
254	C₂²×C₆₇	—	2	8	no	—
255	C₂²×C₆₇	—	2	4	no	—
256	C₂²×C₆₇	—	2	8	no	—
257	C₂²×C₆₇	—	4	4	no	—
258	C₂²×C₆₇	—	4	8	no	—
259	C₂²×C₆₇	—	8	4	no	—
260	C₂²×C₆₇	—	8	8	no	—
261	C₂²×C₆₇	—	8	16	no	—
262	C₂²×C₆₇	C₂×#120	1	4	yes	—
263	C₂²×C₆₇	C₂×#120	1	2	yes	—
264	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
265	C₄×C₆₇	—	2	8	no	—
266	C₄×C₆₇	—	2	4	no	—
267	C₄×C₆₇	—	4	4	no	—
268	C₄×C₆₇	—	4	4	no	—
269	C₄×C₆₇	C₂×#120	1	2	yes	—
270	C₄×C₆₇	C₂×#120	1	4	yes	—
271	C₄×C₆₇	C₂³×D₄	1	2	yes	—
272	#60370	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
273	C₂×C₆₇	—	2	2	no	—
274	C₂×C₆₇	—	2	4	no	—
275	C₂×C₆₇	—	4	4	no	—
276	C₂×C₆₇	—	4	8	no	—
277	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
278	C₂×C₆₇	#3586	1	2	yes	—
279	C₂×#585	C₂×C₆₇	1	1	yes	characteristic
280	C₂²×#137	C₂×C₆₇	1	2	yes	complemented
281	C₂×#536	C₂×C₆₇	1	1	yes	characteristic, complemented
282	#3011	C₂×C₆₇	1	8	yes	complemented
283	#3009	C₂×C₆₇	1	8	yes	complemented
284	#3007	C₂×C₆₇	1	8	yes	complemented
285	#2896	C₂×C₆₇	1	2	yes	complemented
286	#2879	C₂×C₆₇	1	1	yes	characteristic, complemented
287	C₆₇	#60370	1	1	yes	characteristic, direct product factor, Sylow
288	#585	—	2	16	no	—
289	C₂×2₋¹⁺⁴	—	2	1	no	—
290	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	4	yes	—
291	C₂×H(5,𝔽₂)	—	2	2	no	—
292	C₂×H(5,𝔽₂)	—	2	1	no	—
293	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	4	yes	—
294	C₂²×#41	—	2	4	no	—
295	C₂²×#41	C₂²×C₆₇	1	4	yes	—
296	C₂²×#41	C₂²×C₆₇	1	2	yes	—
297	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
298	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
299	C₂×#137	—	2	16	no	—
300	C₂×#137	C₂²×C₆₇	1	8	yes	complemented
301	C₂×#137	C₂²×C₆₇	1	8	yes	complemented
302	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	complemented
303	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
304	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
305	#536	—	2	4	no	—
306	#536	C₂²×C₆₇	1	8	yes	complemented
307	C₂×C₄⋊Q₈	—	2	1	no	—
308	C₂×#124	—	2	1	no	—
309	C₂×#123	—	2	2	no	—
310	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
311	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic, complemented
312	#452	C₂²×C₆₇	1	32	yes	complemented
313	#451	C₂²×C₆₇	1	16	yes	complemented
314	#450	C₂²×C₆₇	1	16	yes	complemented
315	#419	C₂²×C₆₇	1	4	yes	complemented
316	#418	C₂²×C₆₇	1	4	yes	complemented
317	#417	C₂²×C₆₇	1	8	yes	complemented
318	#416	C₂²×C₆₇	1	8	yes	complemented
319	#408	C₂²×C₆₇	1	8	yes	complemented
320	#407	C₂²×C₆₇	1	4	yes	complemented
321	2₋¹⁺⁴	—	2	16	no	—
322	2₋¹⁺⁴	—	2	16	no	—
323	2₋¹⁺⁴	—	2	16	no	—
324	H(5,𝔽₂)	—	2	16	no	—
325	H(5,𝔽₂)	—	2	32	no	—
326	H(5,𝔽₂)	—	2	16	no	—
327	H(5,𝔽₂)	—	2	16	no	—
328	C₂×#41	—	2	64	no	—
329	C₂×#41	—	2	32	no	—
330	C₂×#41	—	2	4	no	—
331	C₂×#41	—	2	8	no	—
332	C₂×#41	—	2	4	no	—
333	C₂×#41	—	2	4	no	—
334	C₂×#41	—	2	2	no	—
335	C₂×#41	—	2	4	no	—
336	C₂×#41	—	2	2	no	—
337	C₂×#41	—	2	8	no	—
338	C₂×#41	—	2	8	no	—
339	C₂×#41	D₄×C₆₇	1	16	yes	—
340	C₂×#41	D₄×C₆₇	1	4	yes	—
341	C₂×#41	C₂³×C₆₇	1	8	yes	—
342	C₂×#41	C₂³×C₆₇	1	4	yes	—
343	C₂×#41	C₂³×C₆₇	1	4	yes	—
344	C₂×#41	C₂³×C₆₇	1	4	yes	—
345	C₂²×Q₈	—	2	4	no	—
346	C₂²×Q₈	—	2	1	no	—
347	C₂²×Q₈	—	2	2	no	—
348	C₂²×Q₈	—	2	8	no	—
349	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
350	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
351	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
352	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
353	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
354	C₂²×D₄	—	2	4	no	—
355	C₂²×D₄	—	2	2	no	—
356	C₂²×D₄	—	2	8	no	—
357	C₂²×D₄	—	2	4	no	—
358	C₂²×D₄	D₄×C₆₇	1	2	yes	—
359	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
360	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
361	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
362	C₂³×C₄	—	2	4	no	—
363	C₂³×C₄	—	2	2	no	—
364	C₂³×C₄	—	2	1	no	—
365	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
366	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
367	#137	—	2	32	no	—
368	#137	—	2	32	no	—
369	C₂×Q₁₆	—	2	16	no	—
370	C₂×Q₁₆	C₂³×C₆₇	1	16	yes	—
371	C₂×SD₁₆	—	2	16	no	—
372	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	—
373	C₂×M₁₆	—	2	8	no	—
374	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
375	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
376	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
377	C₄⋊Q₈	—	2	8	no	—
378	#124	—	2	8	no	—
379	#123	—	2	16	no	—
380	C₂²⋊Q₈	—	2	8	no	—
381	C₂²⋊Q₈	C₂³×C₆₇	1	16	yes	—
382	#117	—	2	2	no	—
383	#117	C₂³×C₆₇	1	4	yes	—
384	C₂×C₄⋊C₄	—	2	4	no	—
385	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
386	C₂×C₂²⋊C₄	—	2	1	no	—
387	C₂×C₂²⋊C₄	—	2	4	no	—
388	C₂×C₄²	—	2	1	no	—
389	#103	C₂³×C₆₇	1	16	yes	—
390	#102	C₂³×C₆₇	1	16	yes	—
391	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	—
392	C₂⁴	—	2	4	no	—
393	C₂⁴	—	2	2	no	—
394	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
395	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
396	#41	—	2	32	no	—
397	#41	—	2	64	no	—
398	#41	—	2	32	no	—
399	#41	—	2	32	no	—
400	#41	—	2	32	no	—
401	#41	—	2	16	no	—
402	#41	—	2	16	no	—
403	#41	—	2	16	no	—
404	#41	—	2	16	no	—
405	#41	—	2	16	no	—
406	#41	—	2	8	no	—
407	#41	—	2	8	no	—
408	#41	—	2	8	no	—
409	#41	—	2	8	no	—
410	#41	C₂×D₄×C₆₇	1	16	yes	—
411	#41	C₂×D₄×C₆₇	1	16	yes	—
412	C₂×Q₈	—	2	32	no	—
413	C₂×Q₈	—	2	16	no	—
414	C₂×Q₈	—	2	4	no	—
415	C₂×Q₈	—	2	4	no	—
416	C₂×Q₈	—	2	2	no	—
417	C₂×Q₈	—	2	4	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	—	2	4	no	—
420	C₂×Q₈	—	2	8	no	—
421	C₂×Q₈	—	2	8	no	—
422	C₂×Q₈	—	2	8	no	—
423	C₂×Q₈	—	4	8	no	—
424	C₂×Q₈	—	4	4	no	—
425	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
426	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
427	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
428	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
429	C₂×D₄	—	2	32	no	—
430	C₂×D₄	—	2	64	no	—
431	C₂×D₄	—	2	32	no	—
432	C₂×D₄	—	2	16	no	—
433	C₂×D₄	—	2	8	no	—
434	C₂×D₄	—	2	8	no	—
435	C₂×D₄	—	2	4	no	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	2	16	no	—
438	C₂×D₄	—	2	4	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
443	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
444	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
445	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
446	C₂²×C₄	—	2	32	no	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	16	no	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	2	8	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	8	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	4	no	—
457	C₂²×C₄	—	4	4	no	—
458	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
459	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
460	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
462	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
463	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
464	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
465	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
466	Q₁₆	—	2	32	no	—
467	SD₁₆	—	2	32	no	—
468	M₁₆	—	2	8	no	—
469	M₁₆	—	2	8	no	—
470	C₂×C₈	—	2	8	no	—
471	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
472	C₄⋊C₄	—	2	16	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
475	C₂²⋊C₄	—	2	4	no	—
476	C₂²⋊C₄	—	2	16	no	—
477	C₄²	—	2	4	no	—
478	C₂³	—	2	16	no	—
479	C₂³	—	2	4	no	—
480	C₂³	—	2	4	no	—
481	C₂³	—	2	8	no	—
482	C₂³	—	4	2	no	—
483	C₂³	—	4	4	no	—
484	C₂³	—	8	4	no	—
485	C₂³	—	8	8	no	—
486	C₂³	#120×C₆₇	1	4	yes	—
487	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
488	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
489	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
490	Q₈	—	2	16	no	—
491	Q₈	—	2	8	no	—
492	Q₈	—	2	8	no	—
493	Q₈	—	2	4	no	—
494	Q₈	—	2	8	no	—
495	Q₈	—	4	4	no	—
496	Q₈	—	4	8	no	—
497	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
498	D₄	—	2	32	no	—
499	D₄	—	2	16	no	—
500	D₄	—	2	16	no	—
501	D₄	—	2	16	no	—
502	D₄	—	2	16	no	—
503	D₄	—	2	8	no	—
504	D₄	—	2	8	no	—
505	D₄	—	2	4	no	—
506	D₄	C₂²×D₄×C₆₇	1	8	yes	—
507	C₂×C₄	—	2	16	no	—
508	C₂×C₄	—	2	32	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	16	no	—
513	C₂×C₄	—	2	8	no	—
514	C₂×C₄	—	2	16	no	—
515	C₂×C₄	—	2	8	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	4	4	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	#120×C₆₇	1	8	yes	—
522	C₂×C₄	#120×C₆₇	1	4	yes	—
523	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
524	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
525	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
526	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
527	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
528	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
529	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
530	C₈	—	2	8	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	4	no	—
533	C₂²	—	2	8	no	—
534	C₂²	—	2	4	no	—
535	C₂²	—	2	8	no	—
536	C₂²	—	4	4	no	—
537	C₂²	—	4	8	no	—
538	C₂²	—	8	4	no	—
539	C₂²	—	8	8	no	—
540	C₂²	—	8	16	no	—
541	C₂²	C₂×#120×C₆₇	1	4	yes	—
542	C₂²	C₂×#120×C₆₇	1	2	yes	—
543	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
544	C₄	—	2	8	no	—
545	C₄	—	2	4	no	—
546	C₄	—	4	4	no	—
547	C₄	—	4	4	no	—
548	C₄	C₂×#120×C₆₇	1	2	yes	—
549	C₄	C₂×#120×C₆₇	1	4	yes	—
550	C₄	C₂³×D₄×C₆₇	1	2	yes	—
551	C₂	—	2	2	no	—
552	C₂	—	2	4	no	—
553	C₂	—	4	4	no	—
554	C₂	—	4	8	no	—
555	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic
556	C₂	C₆₇×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#585×C₆₇	C₂
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
2. C₂²×#137×C₆₇	C₂	semidirect
3. C₂×#536×C₆₇	C₂	semidirect
4. C₆₇×#3011	C₂	semidirect
5. C₆₇×#3009	C₂	semidirect
6. C₆₇×#3007	C₂	semidirect
7. C₆₇×#2896	C₂	semidirect
8. C₆₇×#2879	C₂	semidirect
19. C₂³×Q₈×C₆₇	C₂²	semidirect
21. C₂×#137×C₆₇	C₂²	semidirect
22. C₂×#137×C₆₇	C₂²	semidirect
23. C₂²×Q₁₆×C₆₇	C₂²	semidirect
24. C₂²×SD₁₆×C₆₇	C₂²	semidirect
25. C₂²×M₁₆×C₆₇	C₂²	semidirect
27. #536×C₆₇	C₂²	semidirect
31. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
32. C₂×#117×C₆₇	C₂²	semidirect
33. #452×C₆₇	C₂²	semidirect
34. #451×C₆₇	C₂²	semidirect
35. #450×C₆₇	C₂²	semidirect
36. #419×C₆₇	C₂²	semidirect
37. #418×C₆₇	C₂²	semidirect
38. #417×C₆₇	C₂²	semidirect
39. #416×C₆₇	C₂²	semidirect
40. #408×C₆₇	C₂²	semidirect
41. #407×C₆₇	C₂²	semidirect
272. #60370	C₆₇	direct (N × H)
280. C₂²×#137	C₂×C₆₇	semidirect
281. C₂×#536	C₂×C₆₇	semidirect
282. #3011	C₂×C₆₇	semidirect
283. #3009	C₂×C₆₇	semidirect
284. #3007	C₂×C₆₇	semidirect
285. #2896	C₂×C₆₇	semidirect
286. #2879	C₂×C₆₇	semidirect
287. C₆₇	#60370	direct (N × H)
298. C₂³×Q₈	C₂²×C₆₇	semidirect
300. C₂×#137	C₂²×C₆₇	semidirect
301. C₂×#137	C₂²×C₆₇	semidirect
302. C₂²×Q₁₆	C₂²×C₆₇	semidirect
303. C₂²×SD₁₆	C₂²×C₆₇	semidirect
304. C₂²×M₁₆	C₂²×C₆₇	semidirect
306. #536	C₂²×C₆₇	semidirect
310. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
311. C₂×#117	C₂²×C₆₇	semidirect
312. #452	C₂²×C₆₇	semidirect
313. #451	C₂²×C₆₇	semidirect
314. #450	C₂²×C₆₇	semidirect
315. #419	C₂²×C₆₇	semidirect
316. #418	C₂²×C₆₇	semidirect
317. #417	C₂²×C₆₇	semidirect
318. #416	C₂²×C₆₇	semidirect
319. #408	C₂²×C₆₇	semidirect
320. #407	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#585×C₆₇	2
C₂²×#137×C₆₇	2
C₂×#536×C₆₇	2
C₆₇×#3011	2
C₆₇×#3009	2
C₆₇×#3007	2
C₆₇×#2896	2
C₆₇×#2879	2
#60370	67

Subgroups — C67×#60370 #30739621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60370 #30739621