Subgroups — C₆₇×#61053 #30740304

Order 17152.

2666 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#578×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3486	C₂	1	2	yes	maximal, complemented
4	D₄×Q₁₆×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3285	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3270	C₂	1	8	yes	maximal, complemented
7	C₆₇×#3156	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#3059	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3050	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#3044	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3011	C₂	1	2	yes	maximal, complemented
12	C₆₇×#3004	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2950	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#2929	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	2	yes	—
16	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
17	#578×C₆₇	—	2	8	no	—
18	C₂×#137×C₆₇	C₂²	1	2	yes	—
19	C₂×#137×C₆₇	C₂²	1	2	yes	—
20	C₂×#137×C₆₇	C₂²	1	4	yes	complemented
21	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#135×C₆₇	C₂²	1	2	yes	complemented
23	C₂²×Q₁₆×C₆₇	—	2	8	no	—
24	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
25	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
26	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
27	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	complemented
30	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
31	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	D₄×Q₈×C₆₇	—	2	4	no	—
33	D₄×Q₈×C₆₇	C₂²	1	8	yes	—
34	#548×C₆₇	—	2	4	no	—
35	#548×C₆₇	C₂²	1	8	yes	—
36	#536×C₆₇	—	2	4	no	—
37	#536×C₆₇	C₂²	1	2	yes	—
38	#536×C₆₇	C₂²	1	2	yes	—
39	#525×C₆₇	—	2	4	no	—
40	#525×C₆₇	C₂²	1	2	yes	—
41	#525×C₆₇	C₂²	1	1	yes	characteristic
42	#519×C₆₇	C₂²	1	2	yes	—
43	#497×C₆₇	C₂²	1	4	yes	complemented
44	#495×C₆₇	C₂²	1	2	yes	complemented
45	#494×C₆₇	C₂²	1	2	yes	complemented
46	#470×C₆₇	C₂²	1	4	yes	complemented
47	#467×C₆₇	C₂²	1	2	yes	—
48	#467×C₆₇	C₂²	1	4	yes	complemented
49	#466×C₆₇	C₂²	1	2	yes	—
50	#465×C₆₇	C₂²	1	4	yes	—
51	#464×C₆₇	C₂²	1	8	yes	complemented
52	#462×C₆₇	C₂²	1	8	yes	complemented
53	#452×C₆₇	C₂²	1	8	yes	complemented
54	#451×C₆₇	C₂²	1	8	yes	—
55	#451×C₆₇	C₂²	1	8	yes	complemented
56	#448×C₆₇	C₂²	1	8	yes	—
57	#441×C₆₇	C₂²	1	4	yes	complemented
58	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	complemented
59	#435×C₆₇	C₂²	1	4	yes	—
60	C₈×D₄×C₆₇	C₂²	1	4	yes	—
61	#427×C₆₇	C₂²	1	1	yes	characteristic
62	#424×C₆₇	C₂²	1	1	yes	characteristic
63	#419×C₆₇	C₂²	1	2	yes	—
64	#419×C₆₇	C₂²	1	2	yes	—
65	#416×C₆₇	C₂²	1	2	yes	—
66	#408×C₆₇	C₂²	1	1	yes	characteristic
67	#407×C₆₇	C₂²	1	1	yes	characteristic
68	#405×C₆₇	C₂²	1	1	yes	characteristic
69	2₋¹⁺⁴×C₆₇	—	2	8	no	—
70	2₋¹⁺⁴×C₆₇	—	4	4	no	—
71	C₂×#41×C₆₇	—	2	4	no	—
72	C₂×#41×C₆₇	—	2	2	no	—
73	C₂×#41×C₆₇	C₂³	1	2	yes	—
74	C₂×#41×C₆₇	C₂³	1	4	yes	—
75	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₆₇	—	2	2	no	—
77	C₂²×Q₈×C₆₇	—	2	2	no	—
78	C₂²×Q₈×C₆₇	—	2	4	no	—
79	C₂²×Q₈×C₆₇	—	2	8	no	—
80	C₂²×Q₈×C₆₇	—	4	4	no	—
81	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
82	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
83	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
84	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
85	C₂³×C₄×C₆₇	—	2	2	no	—
86	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
87	#137×C₆₇	—	2	8	no	—
88	#137×C₆₇	—	2	16	no	—
89	#137×C₆₇	—	4	4	no	—
90	#135×C₆₇	—	2	8	no	—
91	#135×C₆₇	—	2	4	no	—
92	C₂×Q₁₆×C₆₇	—	2	8	no	—
93	C₂×Q₁₆×C₆₇	—	2	16	no	—
94	C₂×Q₁₆×C₆₇	—	2	16	no	—
95	C₂×Q₁₆×C₆₇	—	2	8	no	—
96	C₂×Q₁₆×C₆₇	—	2	4	no	—
97	C₂×Q₁₆×C₆₇	D₄	1	8	yes	complemented
98	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
99	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
100	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
102	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
103	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
104	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
105	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
106	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
107	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
108	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
109	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
110	#131×C₆₇	—	2	4	no	—
111	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
112	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
113	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
114	C₂²×C₈×C₆₇	—	2	4	no	—
115	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
116	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
117	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
118	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
119	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
120	C₄⋊Q₈×C₆₇	—	2	2	no	—
121	C₄⋊Q₈×C₆₇	—	2	8	no	—
122	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
123	#124×C₆₇	—	2	2	no	—
124	#124×C₆₇	—	2	8	no	—
125	#124×C₆₇	C₂³	1	4	yes	—
126	#123×C₆₇	—	2	4	no	—
127	#123×C₆₇	—	2	2	no	—
128	#123×C₆₇	—	2	4	no	—
129	#123×C₆₇	D₄	1	2	yes	—
130	C₂²⋊Q₈×C₆₇	—	2	8	no	—
131	C₂²⋊Q₈×C₆₇	—	2	4	no	—
132	C₂²⋊Q₈×C₆₇	—	2	8	no	—
133	C₂²⋊Q₈×C₆₇	—	2	8	no	—
134	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
135	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
136	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
137	#121×C₆₇	—	2	8	no	—
138	#121×C₆₇	D₄	1	2	yes	—
139	#121×C₆₇	C₂³	1	4	yes	—
140	#120×C₆₇	—	2	2	no	—
141	#120×C₆₇	D₄	1	2	yes	—
142	C₄×Q₈×C₆₇	—	2	4	no	—
143	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
144	C₄×D₄×C₆₇	—	2	8	no	—
145	C₄×D₄×C₆₇	—	2	4	no	—
146	C₄×D₄×C₆₇	—	2	8	no	—
147	C₄×D₄×C₆₇	C₂³	1	4	yes	—
148	#117×C₆₇	—	2	4	no	—
149	#117×C₆₇	C₂³	1	2	yes	—
150	#117×C₆₇	C₂³	1	1	yes	characteristic
151	#107×C₆₇	C₂³	1	2	yes	—
152	#106×C₆₇	C₂³	1	2	yes	—
153	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
154	#103×C₆₇	C₂³	1	8	yes	—
155	#103×C₆₇	C₂³	1	8	yes	—
156	#103×C₆₇	C₂³	1	4	yes	—
157	#102×C₆₇	C₂³	1	4	yes	—
158	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
159	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
160	#97×C₆₇	C₂³	1	2	yes	—
161	C₄×C₈×C₆₇	C₂³	1	2	yes	—
162	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
163	#41×C₆₇	—	2	2	no	—
164	#41×C₆₇	—	2	2	no	—
165	#41×C₆₇	—	2	8	no	—
166	#41×C₆₇	—	2	4	no	—
167	#41×C₆₇	—	2	4	no	—
168	#41×C₆₇	—	4	4	no	—
169	#41×C₆₇	—	4	16	no	—
170	#41×C₆₇	—	4	8	no	—
171	C₂×Q₈×C₆₇	—	2	2	no	—
172	C₂×Q₈×C₆₇	—	2	8	no	—
173	C₂×Q₈×C₆₇	—	2	2	no	—
174	C₂×Q₈×C₆₇	—	2	4	no	—
175	C₂×Q₈×C₆₇	—	2	4	no	—
176	C₂×Q₈×C₆₇	—	2	16	no	—
177	C₂×Q₈×C₆₇	—	2	8	no	—
178	C₂×Q₈×C₆₇	—	2	4	no	—
179	C₂×Q₈×C₆₇	—	2	4	no	—
180	C₂×Q₈×C₆₇	—	4	8	no	—
181	C₂×Q₈×C₆₇	—	4	8	no	—
182	C₂×Q₈×C₆₇	—	4	4	no	—
183	C₂×Q₈×C₆₇	—	4	16	no	—
184	C₂×Q₈×C₆₇	—	4	8	no	—
185	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
186	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
187	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
188	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
189	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
190	C₂×D₄×C₆₇	—	2	4	no	—
191	C₂×D₄×C₆₇	—	2	2	no	—
192	C₂×D₄×C₆₇	—	2	4	no	—
193	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
194	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
195	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
196	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
197	C₂²×C₄×C₆₇	—	2	2	no	—
198	C₂²×C₄×C₆₇	—	2	2	no	—
199	C₂²×C₄×C₆₇	—	2	2	no	—
200	C₂²×C₄×C₆₇	—	2	8	no	—
201	C₂²×C₄×C₆₇	—	2	4	no	—
202	C₂²×C₄×C₆₇	—	2	2	no	—
203	C₂²×C₄×C₆₇	—	2	4	no	—
204	C₂²×C₄×C₆₇	—	2	2	no	—
205	C₂²×C₄×C₆₇	—	4	8	no	—
206	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
207	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
210	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
211	Q₁₆×C₆₇	—	2	16	no	—
212	Q₁₆×C₆₇	—	2	8	no	—
213	Q₁₆×C₆₇	—	2	4	no	—
214	Q₁₆×C₆₇	—	2	4	no	—
215	Q₁₆×C₆₇	—	2	2	no	—
216	Q₁₆×C₆₇	—	4	4	no	—
217	Q₁₆×C₆₇	C₂×D₄	1	8	yes	—
218	Q₁₆×C₆₇	C₂×D₄	1	8	yes	—
219	SD₁₆×C₆₇	—	2	8	no	—
220	SD₁₆×C₆₇	—	2	4	no	—
221	SD₁₆×C₆₇	—	4	4	no	—
222	D₈×C₆₇	—	2	2	no	—
223	M₁₆×C₆₇	—	2	2	no	—
224	M₁₆×C₆₇	—	2	2	no	—
225	M₁₆×C₆₇	—	2	4	no	—
226	C₂×C₈×C₆₇	—	2	2	no	—
227	C₂×C₈×C₆₇	—	2	4	no	—
228	C₂×C₈×C₆₇	—	2	4	no	—
229	C₂×C₈×C₆₇	—	2	4	no	—
230	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
231	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
232	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
233	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
234	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
235	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
236	C₄⋊C₄×C₆₇	—	2	8	no	—
237	C₄⋊C₄×C₆₇	—	2	8	no	—
238	C₄⋊C₄×C₆₇	—	2	8	no	—
239	C₄⋊C₄×C₆₇	—	2	4	no	—
240	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
241	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
242	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
243	C₂²⋊C₄×C₆₇	—	2	2	no	—
244	C₂²⋊C₄×C₆₇	—	2	4	no	—
245	C₂²⋊C₄×C₆₇	—	2	8	no	—
246	C₂²⋊C₄×C₆₇	—	2	8	no	—
247	C₂²⋊C₄×C₆₇	—	2	4	no	—
248	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
249	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
250	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
251	C₄²×C₆₇	—	2	8	no	—
252	C₄²×C₆₇	—	2	2	no	—
253	C₄²×C₆₇	C₂⁴	1	2	yes	—
254	C₂³×C₆₇	—	2	2	no	—
255	C₂³×C₆₇	—	2	4	no	—
256	C₂³×C₆₇	—	2	2	no	—
257	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
258	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
259	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
260	Q₈×C₆₇	—	2	8	no	—
261	Q₈×C₆₇	—	2	2	no	—
262	Q₈×C₆₇	—	2	4	no	—
263	Q₈×C₆₇	—	2	2	no	—
264	Q₈×C₆₇	—	2	4	no	—
265	Q₈×C₆₇	—	2	4	no	—
266	Q₈×C₆₇	—	2	2	no	—
267	Q₈×C₆₇	—	4	16	no	—
268	Q₈×C₆₇	—	4	8	no	—
269	Q₈×C₆₇	—	4	4	no	—
270	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
271	D₄×C₆₇	—	2	4	no	—
272	D₄×C₆₇	—	2	8	no	—
273	D₄×C₆₇	—	2	2	no	—
274	D₄×C₆₇	—	2	2	no	—
275	D₄×C₆₇	—	2	2	no	—
276	D₄×C₆₇	—	4	8	no	—
277	D₄×C₆₇	—	4	4	no	—
278	D₄×C₆₇	—	4	8	no	—
279	C₂×C₄×C₆₇	—	2	2	no	—
280	C₂×C₄×C₆₇	—	2	8	no	—
281	C₂×C₄×C₆₇	—	2	4	no	—
282	C₂×C₄×C₆₇	—	2	4	no	—
283	C₂×C₄×C₆₇	—	2	2	no	—
284	C₂×C₄×C₆₇	—	2	2	no	—
285	C₂×C₄×C₆₇	—	2	2	no	—
286	C₂×C₄×C₆₇	—	2	2	no	—
287	C₂×C₄×C₆₇	—	2	4	no	—
288	C₂×C₄×C₆₇	—	2	4	no	—
289	C₂×C₄×C₆₇	—	2	2	no	—
290	C₂×C₄×C₆₇	—	4	8	no	—
291	C₂×C₄×C₆₇	—	4	16	no	—
292	C₂×C₄×C₆₇	—	4	8	no	—
293	C₂×C₄×C₆₇	—	4	4	no	—
294	C₂×C₄×C₆₇	—	4	4	no	—
295	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
296	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
297	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
298	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
300	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
301	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
302	C₈×C₆₇	—	2	2	no	—
303	C₈×C₆₇	—	2	2	no	—
304	C₈×C₆₇	—	2	2	no	—
305	C₈×C₆₇	C₂²×D₄	1	2	yes	—
306	C₈×C₆₇	C₂²×D₄	1	2	yes	—
307	C₂²×C₆₇	—	2	8	no	—
308	C₂²×C₆₇	—	2	2	no	—
309	C₂²×C₆₇	—	2	2	no	—
310	C₂²×C₆₇	—	2	2	no	—
311	C₂²×C₆₇	—	2	2	no	—
312	C₂²×C₆₇	—	2	4	no	—
313	C₂²×C₆₇	—	4	2	no	—
314	C₂²×C₆₇	—	4	4	no	—
315	C₂²×C₆₇	D₄²	1	2	yes	—
316	C₂²×C₆₇	#578	1	4	yes	complemented
317	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
318	C₄×C₆₇	—	2	2	no	—
319	C₄×C₆₇	—	2	8	no	—
320	C₄×C₆₇	—	2	2	no	—
321	C₄×C₆₇	—	2	4	no	—
322	C₄×C₆₇	—	4	2	no	—
323	C₄×C₆₇	—	4	4	no	—
324	C₄×C₆₇	—	4	4	no	—
325	C₄×C₆₇	D₄²	1	2	yes	—
326	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
327	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
328	#61053	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
329	C₂×C₆₇	—	2	2	no	—
330	C₂×C₆₇	—	2	4	no	—
331	C₂×C₆₇	—	4	2	no	—
332	C₂×C₆₇	—	8	2	no	—
333	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
334	C₂×C₆₇	C₂×#578	1	2	yes	—
335	C₂×#578	C₂×C₆₇	1	1	yes	characteristic, complemented
336	C₂³×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
337	#3486	C₂×C₆₇	1	2	yes	complemented
338	D₄×Q₁₆	C₂×C₆₇	1	4	yes	complemented
339	#3285	C₂×C₆₇	1	4	yes	complemented
340	#3270	C₂×C₆₇	1	8	yes	complemented
341	#3156	C₂×C₆₇	1	1	yes	characteristic, complemented
342	#3059	C₂×C₆₇	1	1	yes	characteristic, complemented
343	#3050	C₂×C₆₇	1	1	yes	characteristic, complemented
344	#3044	C₂×C₆₇	1	2	yes	complemented
345	#3011	C₂×C₆₇	1	2	yes	complemented
346	#3004	C₂×C₆₇	1	2	yes	complemented
347	#2950	C₂×C₆₇	1	1	yes	characteristic, complemented
348	#2929	C₂×C₆₇	1	1	yes	characteristic, complemented
349	C₆₇	#61053	1	1	yes	characteristic, direct product factor, Sylow
350	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	2	yes	—
351	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
352	#578	—	2	8	no	—
353	C₂×#137	C₂²×C₆₇	1	2	yes	—
354	C₂×#137	C₂²×C₆₇	1	2	yes	—
355	C₂×#137	C₂²×C₆₇	1	4	yes	complemented
356	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
357	C₂×#135	C₂²×C₆₇	1	2	yes	complemented
358	C₂²×Q₁₆	—	2	8	no	—
359	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
360	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
361	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
362	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	—
363	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
364	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	complemented
365	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
366	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
367	D₄×Q₈	—	2	4	no	—
368	D₄×Q₈	C₂²×C₆₇	1	8	yes	—
369	#548	—	2	4	no	—
370	#548	C₂²×C₆₇	1	8	yes	—
371	#536	—	2	4	no	—
372	#536	C₂²×C₆₇	1	2	yes	—
373	#536	C₂²×C₆₇	1	2	yes	—
374	#525	—	2	4	no	—
375	#525	C₂²×C₆₇	1	2	yes	—
376	#525	C₂²×C₆₇	1	1	yes	characteristic
377	#519	C₂²×C₆₇	1	2	yes	—
378	#497	C₂²×C₆₇	1	4	yes	complemented
379	#495	C₂²×C₆₇	1	2	yes	complemented
380	#494	C₂²×C₆₇	1	2	yes	complemented
381	#470	C₂²×C₆₇	1	4	yes	complemented
382	#467	C₂²×C₆₇	1	2	yes	—
383	#467	C₂²×C₆₇	1	4	yes	complemented
384	#466	C₂²×C₆₇	1	2	yes	—
385	#465	C₂²×C₆₇	1	4	yes	—
386	#464	C₂²×C₆₇	1	8	yes	complemented
387	#462	C₂²×C₆₇	1	8	yes	complemented
388	#452	C₂²×C₆₇	1	8	yes	complemented
389	#451	C₂²×C₆₇	1	8	yes	—
390	#451	C₂²×C₆₇	1	8	yes	complemented
391	#448	C₂²×C₆₇	1	8	yes	—
392	#441	C₂²×C₆₇	1	4	yes	complemented
393	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	complemented
394	#435	C₂²×C₆₇	1	4	yes	—
395	C₈×D₄	C₂²×C₆₇	1	4	yes	—
396	#427	C₂²×C₆₇	1	1	yes	characteristic
397	#424	C₂²×C₆₇	1	1	yes	characteristic
398	#419	C₂²×C₆₇	1	2	yes	—
399	#419	C₂²×C₆₇	1	2	yes	—
400	#416	C₂²×C₆₇	1	2	yes	—
401	#408	C₂²×C₆₇	1	1	yes	characteristic
402	#407	C₂²×C₆₇	1	1	yes	characteristic
403	#405	C₂²×C₆₇	1	1	yes	characteristic
404	2₋¹⁺⁴	—	2	8	no	—
405	2₋¹⁺⁴	—	4	4	no	—
406	C₂×#41	—	2	4	no	—
407	C₂×#41	—	2	2	no	—
408	C₂×#41	C₂³×C₆₇	1	2	yes	—
409	C₂×#41	C₂³×C₆₇	1	4	yes	—
410	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
411	C₂²×Q₈	—	2	2	no	—
412	C₂²×Q₈	—	2	2	no	—
413	C₂²×Q₈	—	2	4	no	—
414	C₂²×Q₈	—	2	8	no	—
415	C₂²×Q₈	—	4	4	no	—
416	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
417	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
418	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
419	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
420	C₂³×C₄	—	2	2	no	—
421	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
422	#137	—	2	8	no	—
423	#137	—	2	16	no	—
424	#137	—	4	4	no	—
425	#135	—	2	8	no	—
426	#135	—	2	4	no	—
427	C₂×Q₁₆	—	2	8	no	—
428	C₂×Q₁₆	—	2	16	no	—
429	C₂×Q₁₆	—	2	16	no	—
430	C₂×Q₁₆	—	2	8	no	—
431	C₂×Q₁₆	—	2	4	no	—
432	C₂×Q₁₆	D₄×C₆₇	1	8	yes	complemented
433	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
434	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
435	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
436	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
437	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
438	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
439	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
440	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
441	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
442	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
443	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
444	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
445	#131	—	2	4	no	—
446	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
447	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
448	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
449	C₂²×C₈	—	2	4	no	—
450	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
451	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
452	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
453	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
454	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
455	C₄⋊Q₈	—	2	2	no	—
456	C₄⋊Q₈	—	2	8	no	—
457	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
458	#124	—	2	2	no	—
459	#124	—	2	8	no	—
460	#124	C₂³×C₆₇	1	4	yes	—
461	#123	—	2	4	no	—
462	#123	—	2	2	no	—
463	#123	—	2	4	no	—
464	#123	D₄×C₆₇	1	2	yes	—
465	C₂²⋊Q₈	—	2	8	no	—
466	C₂²⋊Q₈	—	2	4	no	—
467	C₂²⋊Q₈	—	2	8	no	—
468	C₂²⋊Q₈	—	2	8	no	—
469	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
470	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
471	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
472	#121	—	2	8	no	—
473	#121	D₄×C₆₇	1	2	yes	—
474	#121	C₂³×C₆₇	1	4	yes	—
475	#120	—	2	2	no	—
476	#120	D₄×C₆₇	1	2	yes	—
477	C₄×Q₈	—	2	4	no	—
478	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
479	C₄×D₄	—	2	8	no	—
480	C₄×D₄	—	2	4	no	—
481	C₄×D₄	—	2	8	no	—
482	C₄×D₄	C₂³×C₆₇	1	4	yes	—
483	#117	—	2	4	no	—
484	#117	C₂³×C₆₇	1	2	yes	—
485	#117	C₂³×C₆₇	1	1	yes	characteristic
486	#107	C₂³×C₆₇	1	2	yes	—
487	#106	C₂³×C₆₇	1	2	yes	—
488	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
489	#103	C₂³×C₆₇	1	4	yes	—
490	#103	C₂³×C₆₇	1	8	yes	—
491	#103	C₂³×C₆₇	1	8	yes	—
492	#102	C₂³×C₆₇	1	4	yes	—
493	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
494	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
495	#97	C₂³×C₆₇	1	2	yes	—
496	C₄×C₈	C₂³×C₆₇	1	2	yes	—
497	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
498	#41	—	2	2	no	—
499	#41	—	2	2	no	—
500	#41	—	2	8	no	—
501	#41	—	2	4	no	—
502	#41	—	2	4	no	—
503	#41	—	4	4	no	—
504	#41	—	4	16	no	—
505	#41	—	4	8	no	—
506	C₂×Q₈	—	2	2	no	—
507	C₂×Q₈	—	2	8	no	—
508	C₂×Q₈	—	2	2	no	—
509	C₂×Q₈	—	2	4	no	—
510	C₂×Q₈	—	2	4	no	—
511	C₂×Q₈	—	2	16	no	—
512	C₂×Q₈	—	2	8	no	—
513	C₂×Q₈	—	2	4	no	—
514	C₂×Q₈	—	2	4	no	—
515	C₂×Q₈	—	4	8	no	—
516	C₂×Q₈	—	4	8	no	—
517	C₂×Q₈	—	4	4	no	—
518	C₂×Q₈	—	4	16	no	—
519	C₂×Q₈	—	4	8	no	—
520	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
521	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
522	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
523	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
524	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
525	C₂×D₄	—	2	4	no	—
526	C₂×D₄	—	2	2	no	—
527	C₂×D₄	—	2	4	no	—
528	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
529	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
530	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
531	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
532	C₂²×C₄	—	2	2	no	—
533	C₂²×C₄	—	2	2	no	—
534	C₂²×C₄	—	2	2	no	—
535	C₂²×C₄	—	2	8	no	—
536	C₂²×C₄	—	2	4	no	—
537	C₂²×C₄	—	2	2	no	—
538	C₂²×C₄	—	2	4	no	—
539	C₂²×C₄	—	2	2	no	—
540	C₂²×C₄	—	4	8	no	—
541	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
542	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
543	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
544	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
545	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
546	Q₁₆	—	2	16	no	—
547	Q₁₆	—	2	8	no	—
548	Q₁₆	—	2	4	no	—
549	Q₁₆	—	2	4	no	—
550	Q₁₆	—	2	2	no	—
551	Q₁₆	—	4	4	no	—
552	Q₁₆	C₂×D₄×C₆₇	1	8	yes	—
553	Q₁₆	C₂×D₄×C₆₇	1	8	yes	—
554	SD₁₆	—	2	8	no	—
555	SD₁₆	—	2	4	no	—
556	SD₁₆	—	4	4	no	—
557	D₈	—	2	2	no	—
558	M₁₆	—	2	2	no	—
559	M₁₆	—	2	2	no	—
560	M₁₆	—	2	4	no	—
561	C₂×C₈	—	2	2	no	—
562	C₂×C₈	—	2	4	no	—
563	C₂×C₈	—	2	4	no	—
564	C₂×C₈	—	2	4	no	—
565	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
566	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
567	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
568	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
569	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
570	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
571	C₄⋊C₄	—	2	8	no	—
572	C₄⋊C₄	—	2	8	no	—
573	C₄⋊C₄	—	2	8	no	—
574	C₄⋊C₄	—	2	4	no	—
575	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
576	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
577	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
578	C₂²⋊C₄	—	2	2	no	—
579	C₂²⋊C₄	—	2	4	no	—
580	C₂²⋊C₄	—	2	8	no	—
581	C₂²⋊C₄	—	2	8	no	—
582	C₂²⋊C₄	—	2	4	no	—
583	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
584	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
585	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
586	C₄²	—	2	8	no	—
587	C₄²	—	2	2	no	—
588	C₄²	C₂⁴×C₆₇	1	2	yes	—
589	C₂³	—	2	2	no	—
590	C₂³	—	2	4	no	—
591	C₂³	—	2	2	no	—
592	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
593	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
594	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
595	Q₈	—	2	8	no	—
596	Q₈	—	2	2	no	—
597	Q₈	—	2	4	no	—
598	Q₈	—	2	2	no	—
599	Q₈	—	2	4	no	—
600	Q₈	—	2	4	no	—
601	Q₈	—	2	2	no	—
602	Q₈	—	4	16	no	—
603	Q₈	—	4	8	no	—
604	Q₈	—	4	4	no	—
605	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
606	D₄	—	2	4	no	—
607	D₄	—	2	8	no	—
608	D₄	—	2	2	no	—
609	D₄	—	2	2	no	—
610	D₄	—	2	2	no	—
611	D₄	—	4	8	no	—
612	D₄	—	4	4	no	—
613	D₄	—	4	8	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	8	no	—
616	C₂×C₄	—	2	4	no	—
617	C₂×C₄	—	2	4	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	4	no	—
623	C₂×C₄	—	2	4	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	4	8	no	—
626	C₂×C₄	—	4	16	no	—
627	C₂×C₄	—	4	8	no	—
628	C₂×C₄	—	4	4	no	—
629	C₂×C₄	—	4	4	no	—
630	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
631	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
632	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
633	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
634	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
635	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
636	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
637	C₈	—	2	2	no	—
638	C₈	—	2	2	no	—
639	C₈	—	2	2	no	—
640	C₈	C₂²×D₄×C₆₇	1	2	yes	—
641	C₈	C₂²×D₄×C₆₇	1	2	yes	—
642	C₂²	—	2	8	no	—
643	C₂²	—	2	2	no	—
644	C₂²	—	2	2	no	—
645	C₂²	—	2	2	no	—
646	C₂²	—	2	4	no	—
647	C₂²	—	2	2	no	—
648	C₂²	—	4	4	no	—
649	C₂²	—	4	2	no	—
650	C₂²	D₄²×C₆₇	1	2	yes	—
651	C₂²	#578×C₆₇	1	4	yes	complemented
652	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
653	C₄	—	2	4	no	—
654	C₄	—	2	2	no	—
655	C₄	—	2	8	no	—
656	C₄	—	2	2	no	—
657	C₄	—	4	4	no	—
658	C₄	—	4	2	no	—
659	C₄	—	4	4	no	—
660	C₄	D₄²×C₆₇	1	2	yes	—
661	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
662	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
663	C₂	—	2	2	no	—
664	C₂	—	2	4	no	—
665	C₂	—	4	2	no	—
666	C₂	—	8	2	no	—
667	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
668	C₂	C₂×#578×C₆₇	1	2	yes	—

	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 ◁ #525×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#578×C₆₇	C₂	semidirect
2. C₂³×Q₁₆×C₆₇	C₂	semidirect
3. C₆₇×#3486	C₂	semidirect
4. D₄×Q₁₆×C₆₇	C₂	semidirect
5. C₆₇×#3285	C₂	semidirect
6. C₆₇×#3270	C₂	semidirect
7. C₆₇×#3156	C₂	semidirect
8. C₆₇×#3059	C₂	semidirect
9. C₆₇×#3050	C₂	semidirect
10. C₆₇×#3044	C₂	semidirect
11. C₆₇×#3011	C₂	semidirect
12. C₆₇×#3004	C₂	semidirect
13. C₆₇×#2950	C₂	semidirect
14. C₆₇×#2929	C₂	semidirect
20. C₂×#137×C₆₇	C₂²	semidirect
22. C₂×#135×C₆₇	C₂²	semidirect
29. C₂²×Q₁₆×C₆₇	C₂²	semidirect
43. #497×C₆₇	C₂²	semidirect
44. #495×C₆₇	C₂²	semidirect
45. #494×C₆₇	C₂²	semidirect
46. #470×C₆₇	C₂²	semidirect
48. #467×C₆₇	C₂²	semidirect
51. #464×C₆₇	C₂²	semidirect
52. #462×C₆₇	C₂²	semidirect
53. #452×C₆₇	C₂²	semidirect
55. #451×C₆₇	C₂²	semidirect
57. #441×C₆₇	C₂²	semidirect
58. C₄×Q₁₆×C₆₇	C₂²	semidirect
97. C₂×Q₁₆×C₆₇	D₄	semidirect
316. C₂²×C₆₇	#578	semidirect
328. #61053	C₆₇	direct (N × H)
335. C₂×#578	C₂×C₆₇	semidirect
336. C₂³×Q₁₆	C₂×C₆₇	semidirect
337. #3486	C₂×C₆₇	semidirect
338. D₄×Q₁₆	C₂×C₆₇	semidirect
339. #3285	C₂×C₆₇	semidirect
340. #3270	C₂×C₆₇	semidirect
341. #3156	C₂×C₆₇	semidirect
342. #3059	C₂×C₆₇	semidirect
343. #3050	C₂×C₆₇	semidirect
344. #3044	C₂×C₆₇	semidirect
345. #3011	C₂×C₆₇	semidirect
346. #3004	C₂×C₆₇	semidirect
347. #2950	C₂×C₆₇	semidirect
348. #2929	C₂×C₆₇	semidirect
349. C₆₇	#61053	direct (N × H)
355. C₂×#137	C₂²×C₆₇	semidirect
357. C₂×#135	C₂²×C₆₇	semidirect
364. C₂²×Q₁₆	C₂²×C₆₇	semidirect
378. #497	C₂²×C₆₇	semidirect
379. #495	C₂²×C₆₇	semidirect
380. #494	C₂²×C₆₇	semidirect
381. #470	C₂²×C₆₇	semidirect
383. #467	C₂²×C₆₇	semidirect
386. #464	C₂²×C₆₇	semidirect
387. #462	C₂²×C₆₇	semidirect
388. #452	C₂²×C₆₇	semidirect
390. #451	C₂²×C₆₇	semidirect
392. #441	C₂²×C₆₇	semidirect
393. C₄×Q₁₆	C₂²×C₆₇	semidirect
432. C₂×Q₁₆	D₄×C₆₇	semidirect
651. C₂²	#578×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#578×C₆₇	2
C₂³×Q₁₆×C₆₇	2
C₆₇×#3486	2
D₄×Q₁₆×C₆₇	2
C₆₇×#3285	2
C₆₇×#3270	2
C₆₇×#3156	2
C₆₇×#3059	2
C₆₇×#3050	2
C₆₇×#3044	2
C₆₇×#3011	2
C₆₇×#3004	2
C₆₇×#2950	2
C₆₇×#2929	2
#61053	67

Subgroups — C67×#61053 #30740304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61053 #30740304