Subgroups — C₆₇×#61410 #30740661

Order 17152.

1958 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3508	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3400	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3397	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3394	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3392	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3154	C₂	1	2	yes	maximal, complemented
8	C₂×#492×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3043	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3042	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3039	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#3038	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#2949	C₂	1	2	yes	maximal, complemented
15	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#137×C₆₇	C₂²	1	2	yes	—
17	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
18	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
21	#559×C₆₇	C₂²	1	4	yes	—
22	#557×C₆₇	C₂²	1	4	yes	complemented
23	#556×C₆₇	C₂²	1	4	yes	complemented
24	#552×C₆₇	—	2	4	no	—
25	#550×C₆₇	C₂²	1	4	yes	—
26	#546×C₆₇	—	2	4	no	—
27	#541×C₆₇	—	2	2	no	—
28	#540×C₆₇	—	2	2	no	—
29	#536×C₆₇	C₂²	1	2	yes	—
30	#535×C₆₇	C₂²	1	2	yes	—
31	#533×C₆₇	—	2	2	no	—
32	#533×C₆₇	C₂²	1	1	yes	characteristic
33	#532×C₆₇	—	2	2	no	—
34	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
36	#519×C₆₇	C₂²	1	2	yes	—
37	#518×C₆₇	C₂²	1	2	yes	—
38	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
40	#497×C₆₇	C₂²	1	4	yes	complemented
41	#496×C₆₇	C₂²	1	4	yes	complemented
42	#495×C₆₇	C₂²	1	4	yes	complemented
43	#492×C₆₇	C₂²	1	4	yes	—
44	#492×C₆₇	C₂²	1	4	yes	—
45	#492×C₆₇	C₂²	1	4	yes	complemented
46	#464×C₆₇	C₂²	1	8	yes	complemented
47	#463×C₆₇	C₂²	1	8	yes	complemented
48	#462×C₆₇	C₂²	1	4	yes	complemented
49	#461×C₆₇	C₂²	1	4	yes	complemented
50	#461×C₆₇	C₂²	1	8	yes	complemented
51	#460×C₆₇	C₂²	1	4	yes	—
52	#460×C₆₇	C₂²	1	8	yes	complemented
53	#459×C₆₇	C₂²	1	4	yes	—
54	#446×C₆₇	C₂²	1	4	yes	complemented
55	C₈×Q₈×C₆₇	C₂²	1	4	yes	complemented
56	#440×C₆₇	C₂²	1	8	yes	complemented
57	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	complemented
58	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
59	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
60	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
61	C₄×D₈×C₆₇	C₂²	1	2	yes	complemented
62	#427×C₆₇	C₂²	1	1	yes	characteristic
63	#424×C₆₇	C₂²	1	2	yes	—
64	#423×C₆₇	C₂²	1	1	yes	characteristic
65	#419×C₆₇	C₂²	1	2	yes	—
66	#418×C₆₇	C₂²	1	2	yes	—
67	#416×C₆₇	C₂²	1	1	yes	characteristic
68	#416×C₆₇	C₂²	1	1	yes	characteristic
69	#405×C₆₇	C₂²	1	2	yes	—
70	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
71	C₂×#41×C₆₇	—	2	4	no	—
72	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
75	C₂²×D₄×C₆₇	—	2	2	no	—
76	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
77	#137×C₆₇	—	4	4	no	—
78	AGL(1,ℤ/8)×C₆₇	—	4	4	no	—
79	#135×C₆₇	—	2	4	no	—
80	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
82	C₂×SD₁₆×C₆₇	—	2	8	no	—
83	C₂×SD₁₆×C₆₇	—	4	4	no	—
84	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
85	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
86	C₂×SD₁₆×C₆₇	C₂³	1	1	yes	characteristic
87	C₂×SD₁₆×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
89	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
90	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
91	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×D₈×C₆₇	C₂³	1	4	yes	—
93	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
94	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₄⋊Q₈×C₆₇	—	2	1	no	—
97	C₄⋊Q₈×C₆₇	—	2	1	no	—
98	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
99	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
100	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
101	#127×C₆₇	—	2	1	no	—
102	#127×C₆₇	—	2	1	no	—
103	#127×C₆₇	C₂³	1	4	yes	—
104	#127×C₆₇	C₂³	1	4	yes	—
105	#126×C₆₇	—	2	8	no	—
106	#125×C₆₇	—	2	4	no	—
107	#125×C₆₇	—	2	8	no	—
108	#125×C₆₇	D₄	1	2	yes	—
109	#124×C₆₇	—	2	4	no	—
110	#124×C₆₇	C₂³	1	4	yes	—
111	#124×C₆₇	C₂³	1	4	yes	—
112	#123×C₆₇	—	2	8	no	—
113	#123×C₆₇	—	2	4	no	—
114	C₂²⋊Q₈×C₆₇	—	2	4	no	—
115	C₂²⋊Q₈×C₆₇	—	2	4	no	—
116	C₂²⋊Q₈×C₆₇	D₄	1	4	yes	—
117	#121×C₆₇	—	2	4	no	—
118	#121×C₆₇	—	2	4	no	—
119	#121×C₆₇	—	2	8	no	—
120	#120×C₆₇	—	2	4	no	—
121	C₄×Q₈×C₆₇	—	2	4	no	—
122	C₄×Q₈×C₆₇	—	2	4	no	—
123	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
124	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
125	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
126	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
127	C₄×D₄×C₆₇	—	2	2	no	—
128	C₄×D₄×C₆₇	—	2	2	no	—
129	C₄×D₄×C₆₇	—	2	4	no	—
130	C₄×D₄×C₆₇	—	2	4	no	—
131	C₄×D₄×C₆₇	C₂³	1	2	yes	—
132	C₄×D₄×C₆₇	C₂³	1	2	yes	—
133	C₄×D₄×C₆₇	C₂³	1	8	yes	—
134	#117×C₆₇	—	2	4	no	—
135	#117×C₆₇	—	2	4	no	—
136	#117×C₆₇	—	2	1	no	—
137	#117×C₆₇	—	2	1	no	—
138	#117×C₆₇	C₂³	1	1	yes	characteristic
139	#117×C₆₇	C₂³	1	1	yes	characteristic
140	#117×C₆₇	C₂³	1	2	yes	—
141	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
143	C₂×C₄²×C₆₇	—	2	1	no	—
144	C₂×C₄²×C₆₇	—	2	1	no	—
145	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
146	#107×C₆₇	C₂³	1	2	yes	—
147	#106×C₆₇	C₂³	1	2	yes	—
148	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
149	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
150	#103×C₆₇	C₂³	1	2	yes	—
151	#103×C₆₇	C₂³	1	8	yes	—
152	#103×C₆₇	C₂³	1	2	yes	—
153	#102×C₆₇	C₂³	1	8	yes	—
154	#102×C₆₇	C₂³	1	2	yes	—
155	#102×C₆₇	C₂³	1	2	yes	—
156	#97×C₆₇	C₂³	1	4	yes	—
157	C₄×C₈×C₆₇	C₂³	1	2	yes	—
158	C₄×C₈×C₆₇	C₂³	1	2	yes	—
159	C₄×C₈×C₆₇	C₂³	1	4	yes	—
160	C₂⁴×C₆₇	—	2	2	no	—
161	#41×C₆₇	—	2	2	no	—
162	#41×C₆₇	—	2	2	no	—
163	#41×C₆₇	—	2	2	no	—
164	#41×C₆₇	—	2	2	no	—
165	#41×C₆₇	—	8	8	no	—
166	C₂×Q₈×C₆₇	—	2	2	no	—
167	C₂×Q₈×C₆₇	—	2	4	no	—
168	C₂×Q₈×C₆₇	—	2	4	no	—
169	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
170	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
171	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
172	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
173	C₂×D₄×C₆₇	—	2	1	no	—
174	C₂×D₄×C₆₇	—	2	1	no	—
175	C₂×D₄×C₆₇	—	2	2	no	—
176	C₂×D₄×C₆₇	—	2	4	no	—
177	C₂×D₄×C₆₇	—	2	8	no	—
178	C₂×D₄×C₆₇	—	2	4	no	—
179	C₂×D₄×C₆₇	—	2	4	no	—
180	C₂×D₄×C₆₇	—	2	4	no	—
181	C₂×D₄×C₆₇	—	8	4	no	—
182	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
184	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₆₇	—	2	2	no	—
186	C₂²×C₄×C₆₇	—	2	2	no	—
187	C₂²×C₄×C₆₇	—	2	1	no	—
188	C₂²×C₄×C₆₇	—	2	1	no	—
189	C₂²×C₄×C₆₇	—	2	2	no	—
190	C₂²×C₄×C₆₇	—	2	8	no	—
191	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
192	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
193	Q₁₆×C₆₇	—	2	2	no	—
194	Q₁₆×C₆₇	—	4	4	no	—
195	SD₁₆×C₆₇	—	2	8	no	—
196	SD₁₆×C₆₇	—	2	8	no	—
197	SD₁₆×C₆₇	—	2	2	no	—
198	SD₁₆×C₆₇	—	2	2	no	—
199	SD₁₆×C₆₇	—	4	8	no	—
200	SD₁₆×C₆₇	—	4	4	no	—
201	SD₁₆×C₆₇	—	4	4	no	—
202	D₈×C₆₇	—	2	2	no	—
203	D₈×C₆₇	—	4	4	no	—
204	M₁₆×C₆₇	—	2	4	no	—
205	C₂×C₈×C₆₇	—	2	2	no	—
206	C₂×C₈×C₆₇	#41	1	4	yes	complemented
207	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
208	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
209	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
210	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
211	C₄⋊C₄×C₆₇	—	2	4	no	—
212	C₄⋊C₄×C₆₇	—	2	4	no	—
213	C₄⋊C₄×C₆₇	—	2	2	no	—
214	C₄⋊C₄×C₆₇	—	2	2	no	—
215	C₄⋊C₄×C₆₇	—	2	8	no	—
216	C₄⋊C₄×C₆₇	—	2	8	no	—
217	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
218	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
219	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
220	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
221	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
222	C₂²⋊C₄×C₆₇	—	2	1	no	—
223	C₂²⋊C₄×C₆₇	—	2	1	no	—
224	C₂²⋊C₄×C₆₇	—	2	8	no	—
225	C₂²⋊C₄×C₆₇	—	2	2	no	—
226	C₂²⋊C₄×C₆₇	—	2	2	no	—
227	C₂²⋊C₄×C₆₇	—	2	8	no	—
228	C₂²⋊C₄×C₆₇	—	2	8	no	—
229	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
230	C₄²×C₆₇	—	2	1	no	—
231	C₄²×C₆₇	—	2	1	no	—
232	C₄²×C₆₇	—	2	2	no	—
233	C₄²×C₆₇	—	2	2	no	—
234	C₄²×C₆₇	—	2	8	no	—
235	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
236	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
237	C₄²×C₆₇	C₂⁴	1	2	yes	—
238	C₄²×C₆₇	C₂⁴	1	4	yes	—
239	C₂³×C₆₇	—	2	1	no	—
240	C₂³×C₆₇	—	2	4	no	—
241	C₂³×C₆₇	—	2	1	no	—
242	C₂³×C₆₇	—	4	4	no	—
243	C₂³×C₆₇	—	8	4	no	—
244	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
245	Q₈×C₆₇	—	2	8	no	—
246	Q₈×C₆₇	—	2	4	no	—
247	Q₈×C₆₇	—	2	1	no	—
248	Q₈×C₆₇	—	2	1	no	—
249	Q₈×C₆₇	—	2	1	no	—
250	Q₈×C₆₇	—	2	4	no	—
251	Q₈×C₆₇	—	2	1	no	—
252	Q₈×C₆₇	—	4	4	no	—
253	D₄×C₆₇	—	2	1	no	—
254	D₄×C₆₇	—	2	1	no	—
255	D₄×C₆₇	—	2	4	no	—
256	D₄×C₆₇	—	2	1	no	—
257	D₄×C₆₇	—	2	4	no	—
258	D₄×C₆₇	—	2	1	no	—
259	D₄×C₆₇	—	4	2	no	—
260	D₄×C₆₇	—	4	2	no	—
261	D₄×C₆₇	—	4	4	no	—
262	D₄×C₆₇	—	4	8	no	—
263	D₄×C₆₇	—	8	4	no	—
264	D₄×C₆₇	—	8	8	no	—
265	C₂×C₄×C₆₇	—	2	4	no	—
266	C₂×C₄×C₆₇	—	2	2	no	—
267	C₂×C₄×C₆₇	—	2	1	no	—
268	C₂×C₄×C₆₇	—	2	2	no	—
269	C₂×C₄×C₆₇	—	2	4	no	—
270	C₂×C₄×C₆₇	—	2	1	no	—
271	C₂×C₄×C₆₇	—	4	4	no	—
272	C₂×C₄×C₆₇	—	4	2	no	—
273	C₂×C₄×C₆₇	—	4	2	no	—
274	C₂×C₄×C₆₇	—	4	4	no	—
275	C₂×C₄×C₆₇	—	4	4	no	—
276	C₂×C₄×C₆₇	—	8	8	no	—
277	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
278	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
279	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
281	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
282	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
283	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
284	C₈×C₆₇	—	2	2	no	—
285	C₈×C₆₇	—	2	4	no	—
286	C₈×C₆₇	C₂×#41	1	2	yes	—
287	C₈×C₆₇	C₂×#41	1	2	yes	—
288	C₂²×C₆₇	—	2	2	no	—
289	C₂²×C₆₇	—	4	1	no	—
290	C₂²×C₆₇	—	4	1	no	—
291	C₂²×C₆₇	—	4	4	no	—
292	C₂²×C₆₇	—	4	2	no	—
293	C₂²×C₆₇	—	4	2	no	—
294	C₂²×C₆₇	—	4	8	no	—
295	C₂²×C₆₇	—	16	4	no	—
296	C₂²×C₆₇	#550	1	2	yes	—
297	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
298	C₄×C₆₇	—	2	2	no	—
299	C₄×C₆₇	—	2	8	no	—
300	C₄×C₆₇	—	4	1	no	—
301	C₄×C₆₇	—	4	1	no	—
302	C₄×C₆₇	—	4	4	no	—
303	C₄×C₆₇	—	4	2	no	—
304	C₄×C₆₇	—	4	2	no	—
305	C₄×C₆₇	#550	1	2	yes	—
306	C₄×C₆₇	#577	1	4	yes	—
307	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
308	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
309	#61410	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
310	C₂×C₆₇	—	2	2	no	—
311	C₂×C₆₇	—	8	1	no	—
312	C₂×C₆₇	—	8	1	no	—
313	C₂×C₆₇	—	8	4	no	—
314	C₂×C₆₇	C₂×#550	1	1	yes	characteristic
315	C₂×C₆₇	C₂×#577	1	2	yes	—
316	#3509	C₂×C₆₇	1	1	yes	characteristic, complemented
317	#3508	C₂×C₆₇	1	1	yes	characteristic, complemented
318	#3400	C₂×C₆₇	1	4	yes	complemented
319	#3397	C₂×C₆₇	1	4	yes	complemented
320	#3394	C₂×C₆₇	1	4	yes	complemented
321	#3392	C₂×C₆₇	1	4	yes	complemented
322	#3154	C₂×C₆₇	1	2	yes	complemented
323	C₂×#492	C₂×C₆₇	1	1	yes	characteristic, complemented
324	#3043	C₂×C₆₇	1	2	yes	complemented
325	#3042	C₂×C₆₇	1	2	yes	complemented
326	#3039	C₂×C₆₇	1	1	yes	characteristic, complemented
327	#3038	C₂×C₆₇	1	1	yes	characteristic, complemented
328	#2960	C₂×C₆₇	1	1	yes	characteristic, complemented
329	#2949	C₂×C₆₇	1	2	yes	complemented
330	C₄×#135	C₂×C₆₇	1	1	yes	characteristic, complemented
331	C₆₇	#61410	1	1	yes	characteristic, direct product factor, Sylow
332	C₂×#137	C₂²×C₆₇	1	2	yes	—
333	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
334	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
335	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
336	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
337	#559	C₂²×C₆₇	1	4	yes	—
338	#557	C₂²×C₆₇	1	4	yes	complemented
339	#556	C₂²×C₆₇	1	4	yes	complemented
340	#552	—	2	4	no	—
341	#550	C₂²×C₆₇	1	4	yes	—
342	#546	—	2	4	no	—
343	#541	—	2	2	no	—
344	#540	—	2	2	no	—
345	#536	C₂²×C₆₇	1	2	yes	—
346	#535	C₂²×C₆₇	1	2	yes	—
347	#533	—	2	2	no	—
348	#533	C₂²×C₆₇	1	1	yes	characteristic
349	#532	—	2	2	no	—
350	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
351	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
352	#519	C₂²×C₆₇	1	2	yes	—
353	#518	C₂²×C₆₇	1	2	yes	—
354	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
355	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
356	#497	C₂²×C₆₇	1	4	yes	complemented
357	#496	C₂²×C₆₇	1	4	yes	complemented
358	#495	C₂²×C₆₇	1	4	yes	complemented
359	#492	C₂²×C₆₇	1	4	yes	—
360	#492	C₂²×C₆₇	1	4	yes	—
361	#492	C₂²×C₆₇	1	4	yes	complemented
362	#464	C₂²×C₆₇	1	8	yes	complemented
363	#463	C₂²×C₆₇	1	8	yes	complemented
364	#462	C₂²×C₆₇	1	4	yes	complemented
365	#461	C₂²×C₆₇	1	4	yes	complemented
366	#461	C₂²×C₆₇	1	8	yes	complemented
367	#460	C₂²×C₆₇	1	4	yes	—
368	#460	C₂²×C₆₇	1	8	yes	complemented
369	#459	C₂²×C₆₇	1	4	yes	—
370	#446	C₂²×C₆₇	1	4	yes	complemented
371	C₈×Q₈	C₂²×C₆₇	1	4	yes	complemented
372	#440	C₂²×C₆₇	1	8	yes	complemented
373	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	complemented
374	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
375	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
376	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
377	C₄×D₈	C₂²×C₆₇	1	2	yes	complemented
378	#427	C₂²×C₆₇	1	1	yes	characteristic
379	#424	C₂²×C₆₇	1	2	yes	—
380	#423	C₂²×C₆₇	1	1	yes	characteristic
381	#419	C₂²×C₆₇	1	2	yes	—
382	#418	C₂²×C₆₇	1	2	yes	—
383	#416	C₂²×C₆₇	1	1	yes	characteristic
384	#416	C₂²×C₆₇	1	1	yes	characteristic
385	#405	C₂²×C₆₇	1	2	yes	—
386	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
387	C₂×#41	—	2	4	no	—
388	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
389	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
390	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
391	C₂²×D₄	—	2	2	no	—
392	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
393	#137	—	4	4	no	—
394	AGL(1,ℤ/8)	—	4	4	no	—
395	#135	—	2	4	no	—
396	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
397	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
398	C₂×SD₁₆	—	2	8	no	—
399	C₂×SD₁₆	—	4	4	no	—
400	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
401	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
402	C₂×SD₁₆	C₂³×C₆₇	1	1	yes	characteristic
403	C₂×SD₁₆	C₂³×C₆₇	1	1	yes	characteristic
404	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
405	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
406	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
407	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
408	C₂×D₈	C₂³×C₆₇	1	4	yes	—
409	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
410	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
411	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
412	C₄⋊Q₈	—	2	1	no	—
413	C₄⋊Q₈	—	2	1	no	—
414	C₄⋊Q₈	D₄×C₆₇	1	2	yes	—
415	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
416	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
417	#127	—	2	1	no	—
418	#127	—	2	1	no	—
419	#127	C₂³×C₆₇	1	4	yes	—
420	#127	C₂³×C₆₇	1	4	yes	—
421	#126	—	2	8	no	—
422	#125	—	2	4	no	—
423	#125	—	2	8	no	—
424	#125	D₄×C₆₇	1	2	yes	—
425	#124	—	2	4	no	—
426	#124	C₂³×C₆₇	1	4	yes	—
427	#124	C₂³×C₆₇	1	4	yes	—
428	#123	—	2	8	no	—
429	#123	—	2	4	no	—
430	C₂²⋊Q₈	—	2	4	no	—
431	C₂²⋊Q₈	—	2	4	no	—
432	C₂²⋊Q₈	D₄×C₆₇	1	4	yes	—
433	#121	—	2	4	no	—
434	#121	—	2	4	no	—
435	#121	—	2	8	no	—
436	#120	—	2	4	no	—
437	C₄×Q₈	—	2	4	no	—
438	C₄×Q₈	—	2	4	no	—
439	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
440	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
441	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
442	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
443	C₄×D₄	—	2	2	no	—
444	C₄×D₄	—	2	2	no	—
445	C₄×D₄	—	2	4	no	—
446	C₄×D₄	—	2	4	no	—
447	C₄×D₄	C₂³×C₆₇	1	2	yes	—
448	C₄×D₄	C₂³×C₆₇	1	2	yes	—
449	C₄×D₄	C₂³×C₆₇	1	8	yes	—
450	#117	—	2	4	no	—
451	#117	—	2	4	no	—
452	#117	—	2	1	no	—
453	#117	—	2	1	no	—
454	#117	C₂³×C₆₇	1	1	yes	characteristic
455	#117	C₂³×C₆₇	1	1	yes	characteristic
456	#117	C₂³×C₆₇	1	2	yes	—
457	C₂×C₄⋊C₄	—	2	2	no	—
458	C₂×C₂²⋊C₄	—	2	4	no	—
459	C₂×C₄²	—	2	1	no	—
460	C₂×C₄²	—	2	1	no	—
461	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
462	#107	C₂³×C₆₇	1	2	yes	—
463	#106	C₂³×C₆₇	1	2	yes	—
464	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
465	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
466	#103	C₂³×C₆₇	1	2	yes	—
467	#103	C₂³×C₆₇	1	2	yes	—
468	#103	C₂³×C₆₇	1	8	yes	—
469	#102	C₂³×C₆₇	1	2	yes	—
470	#102	C₂³×C₆₇	1	2	yes	—
471	#102	C₂³×C₆₇	1	8	yes	—
472	#97	C₂³×C₆₇	1	4	yes	—
473	C₄×C₈	C₂³×C₆₇	1	2	yes	—
474	C₄×C₈	C₂³×C₆₇	1	2	yes	—
475	C₄×C₈	C₂³×C₆₇	1	4	yes	—
476	C₂⁴	—	2	2	no	—
477	#41	—	2	2	no	—
478	#41	—	2	2	no	—
479	#41	—	2	2	no	—
480	#41	—	2	2	no	—
481	#41	—	8	8	no	—
482	C₂×Q₈	—	2	2	no	—
483	C₂×Q₈	—	2	4	no	—
484	C₂×Q₈	—	2	4	no	—
485	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
486	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
487	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
488	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	1	no	—
491	C₂×D₄	—	2	2	no	—
492	C₂×D₄	—	2	4	no	—
493	C₂×D₄	—	2	8	no	—
494	C₂×D₄	—	2	4	no	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	2	4	no	—
497	C₂×D₄	—	8	4	no	—
498	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
499	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
500	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
501	C₂²×C₄	—	2	2	no	—
502	C₂²×C₄	—	2	2	no	—
503	C₂²×C₄	—	2	1	no	—
504	C₂²×C₄	—	2	1	no	—
505	C₂²×C₄	—	2	2	no	—
506	C₂²×C₄	—	2	8	no	—
507	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
508	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
509	Q₁₆	—	2	2	no	—
510	Q₁₆	—	4	4	no	—
511	SD₁₆	—	2	8	no	—
512	SD₁₆	—	2	8	no	—
513	SD₁₆	—	2	2	no	—
514	SD₁₆	—	2	2	no	—
515	SD₁₆	—	4	8	no	—
516	SD₁₆	—	4	4	no	—
517	SD₁₆	—	4	4	no	—
518	D₈	—	2	2	no	—
519	D₈	—	4	4	no	—
520	M₁₆	—	2	4	no	—
521	C₂×C₈	—	2	2	no	—
522	C₂×C₈	#41×C₆₇	1	4	yes	complemented
523	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
524	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
525	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
526	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
527	C₄⋊C₄	—	2	4	no	—
528	C₄⋊C₄	—	2	4	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	2	2	no	—
531	C₄⋊C₄	—	2	8	no	—
532	C₄⋊C₄	—	2	8	no	—
533	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
534	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
535	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
536	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
537	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
538	C₂²⋊C₄	—	2	1	no	—
539	C₂²⋊C₄	—	2	1	no	—
540	C₂²⋊C₄	—	2	8	no	—
541	C₂²⋊C₄	—	2	2	no	—
542	C₂²⋊C₄	—	2	2	no	—
543	C₂²⋊C₄	—	2	8	no	—
544	C₂²⋊C₄	—	2	8	no	—
545	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
546	C₄²	—	2	1	no	—
547	C₄²	—	2	1	no	—
548	C₄²	—	2	2	no	—
549	C₄²	—	2	2	no	—
550	C₄²	—	2	8	no	—
551	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
552	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
553	C₄²	C₂⁴×C₆₇	1	2	yes	—
554	C₄²	C₂⁴×C₆₇	1	4	yes	—
555	C₂³	—	2	1	no	—
556	C₂³	—	2	4	no	—
557	C₂³	—	2	1	no	—
558	C₂³	—	4	4	no	—
559	C₂³	—	8	4	no	—
560	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
561	Q₈	—	2	8	no	—
562	Q₈	—	2	4	no	—
563	Q₈	—	2	1	no	—
564	Q₈	—	2	1	no	—
565	Q₈	—	2	1	no	—
566	Q₈	—	2	4	no	—
567	Q₈	—	2	1	no	—
568	Q₈	—	4	4	no	—
569	D₄	—	2	1	no	—
570	D₄	—	2	1	no	—
571	D₄	—	2	4	no	—
572	D₄	—	2	1	no	—
573	D₄	—	2	4	no	—
574	D₄	—	2	1	no	—
575	D₄	—	4	2	no	—
576	D₄	—	4	2	no	—
577	D₄	—	4	4	no	—
578	D₄	—	4	8	no	—
579	D₄	—	8	4	no	—
580	D₄	—	8	8	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	4	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	4	no	—
591	C₂×C₄	—	4	4	no	—
592	C₂×C₄	—	8	8	no	—
593	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
594	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
595	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
596	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
597	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
598	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
599	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
600	C₈	—	2	2	no	—
601	C₈	—	2	4	no	—
602	C₈	C₂×#41×C₆₇	1	2	yes	—
603	C₈	C₂×#41×C₆₇	1	2	yes	—
604	C₂²	—	2	2	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	8	no	—
608	C₂²	—	4	1	no	—
609	C₂²	—	4	1	no	—
610	C₂²	—	4	4	no	—
611	C₂²	—	16	4	no	—
612	C₂²	#550×C₆₇	1	2	yes	—
613	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
614	C₄	—	2	2	no	—
615	C₄	—	2	8	no	—
616	C₄	—	4	2	no	—
617	C₄	—	4	2	no	—
618	C₄	—	4	1	no	—
619	C₄	—	4	1	no	—
620	C₄	—	4	4	no	—
621	C₄	#550×C₆₇	1	2	yes	—
622	C₄	#577×C₆₇	1	4	yes	—
623	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
624	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
625	C₂	—	2	2	no	—
626	C₂	—	8	4	no	—
627	C₂	—	8	1	no	—
628	C₂	—	8	1	no	—
629	C₂	C₂×#550×C₆₇	1	1	yes	characteristic
630	C₂	C₂×#577×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 ◁ #533×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3509	C₂	semidirect
2. C₆₇×#3508	C₂	semidirect
3. C₆₇×#3400	C₂	semidirect
4. C₆₇×#3397	C₂	semidirect
5. C₆₇×#3394	C₂	semidirect
6. C₆₇×#3392	C₂	semidirect
7. C₆₇×#3154	C₂	semidirect
8. C₂×#492×C₆₇	C₂	semidirect
9. C₆₇×#3043	C₂	semidirect
10. C₆₇×#3042	C₂	semidirect
11. C₆₇×#3039	C₂	semidirect
12. C₆₇×#3038	C₂	semidirect
13. C₆₇×#2960	C₂	semidirect
14. C₆₇×#2949	C₂	semidirect
15. C₄×#135×C₆₇	C₂	semidirect
22. #557×C₆₇	C₂²	semidirect
23. #556×C₆₇	C₂²	semidirect
40. #497×C₆₇	C₂²	semidirect
41. #496×C₆₇	C₂²	semidirect
42. #495×C₆₇	C₂²	semidirect
45. #492×C₆₇	C₂²	semidirect
46. #464×C₆₇	C₂²	semidirect
47. #463×C₆₇	C₂²	semidirect
48. #462×C₆₇	C₂²	semidirect
49. #461×C₆₇	C₂²	semidirect
50. #461×C₆₇	C₂²	semidirect
52. #460×C₆₇	C₂²	semidirect
54. #446×C₆₇	C₂²	semidirect
55. C₈×Q₈×C₆₇	C₂²	semidirect
56. #440×C₆₇	C₂²	semidirect
57. C₄×Q₁₆×C₆₇	C₂²	semidirect
58. C₄×SD₁₆×C₆₇	C₂²	semidirect
59. C₄×SD₁₆×C₆₇	C₂²	semidirect
60. C₄×SD₁₆×C₆₇	C₂²	semidirect
61. C₄×D₈×C₆₇	C₂²	semidirect
206. C₂×C₈×C₆₇	#41	semidirect
309. #61410	C₆₇	direct (N × H)
316. #3509	C₂×C₆₇	semidirect
317. #3508	C₂×C₆₇	semidirect
318. #3400	C₂×C₆₇	semidirect
319. #3397	C₂×C₆₇	semidirect
320. #3394	C₂×C₆₇	semidirect
321. #3392	C₂×C₆₇	semidirect
322. #3154	C₂×C₆₇	semidirect
323. C₂×#492	C₂×C₆₇	semidirect
324. #3043	C₂×C₆₇	semidirect
325. #3042	C₂×C₆₇	semidirect
326. #3039	C₂×C₆₇	semidirect
327. #3038	C₂×C₆₇	semidirect
328. #2960	C₂×C₆₇	semidirect
329. #2949	C₂×C₆₇	semidirect
330. C₄×#135	C₂×C₆₇	semidirect
331. C₆₇	#61410	direct (N × H)
338. #557	C₂²×C₆₇	semidirect
339. #556	C₂²×C₆₇	semidirect
356. #497	C₂²×C₆₇	semidirect
357. #496	C₂²×C₆₇	semidirect
358. #495	C₂²×C₆₇	semidirect
361. #492	C₂²×C₆₇	semidirect
362. #464	C₂²×C₆₇	semidirect
363. #463	C₂²×C₆₇	semidirect
364. #462	C₂²×C₆₇	semidirect
365. #461	C₂²×C₆₇	semidirect
366. #461	C₂²×C₆₇	semidirect
368. #460	C₂²×C₆₇	semidirect
370. #446	C₂²×C₆₇	semidirect
371. C₈×Q₈	C₂²×C₆₇	semidirect
372. #440	C₂²×C₆₇	semidirect
373. C₄×Q₁₆	C₂²×C₆₇	semidirect
374. C₄×SD₁₆	C₂²×C₆₇	semidirect
375. C₄×SD₁₆	C₂²×C₆₇	semidirect
376. C₄×SD₁₆	C₂²×C₆₇	semidirect
377. C₄×D₈	C₂²×C₆₇	semidirect
522. C₂×C₈	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3509	2
C₆₇×#3508	2
C₆₇×#3400	2
C₆₇×#3397	2
C₆₇×#3394	2
C₆₇×#3392	2
C₆₇×#3154	2
C₂×#492×C₆₇	2
C₆₇×#3043	2
C₆₇×#3042	2
C₆₇×#3039	2
C₆₇×#3038	2
C₆₇×#2960	2
C₆₇×#2949	2
C₄×#135×C₆₇	2
#61410	67

Subgroups — C67×#61410 #30740661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61410 #30740661