Subgroups — C₆₇×#61365 #30740616

Order 17152.

1766 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3390	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3389	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3388	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3387	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3386	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3385	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3384	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3162	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3076	C₂	1	2	yes	maximal, complemented
12	C₆₇×#3076	C₂	1	4	yes	maximal, complemented
13	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#2943	C₂	1	2	yes	maximal
15	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
16	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
17	Q₈²×C₆₇	C₂²	1	2	yes	—
18	#557×C₆₇	C₂²	1	2	yes	—
19	#557×C₆₇	C₂²	1	4	yes	—
20	#554×C₆₇	—	2	4	no	—
21	#554×C₆₇	—	2	2	no	—
22	#554×C₆₇	C₂²	1	2	yes	—
23	#554×C₆₇	C₂²	1	4	yes	—
24	D₄×Q₈×C₆₇	—	2	2	no	—
25	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	#533×C₆₇	—	2	2	no	—
27	#533×C₆₇	—	2	4	no	—
28	#533×C₆₇	C₂²	1	2	yes	—
29	#533×C₆₇	C₂²	1	4	yes	—
30	C₄×#41×C₆₇	C₂²	1	2	yes	—
31	C₄×#41×C₆₇	C₂²	1	4	yes	—
32	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
33	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
34	#501×C₆₇	C₂²	1	4	yes	—
35	#501×C₆₇	C₂²	1	4	yes	—
36	#501×C₆₇	C₂²	1	4	yes	—
37	#501×C₆₇	C₂²	1	4	yes	—
38	#500×C₆₇	C₂²	1	1	yes	characteristic
39	#500×C₆₇	C₂²	1	1	yes	characteristic
40	#499×C₆₇	C₂²	1	2	yes	—
41	#499×C₆₇	C₂²	1	2	yes	—
42	#498×C₆₇	C₂²	1	2	yes	—
43	#479×C₆₇	C₂²	1	4	yes	—
44	#479×C₆₇	C₂²	1	4	yes	—
45	#479×C₆₇	C₂²	1	4	yes	—
46	#478×C₆₇	C₂²	1	4	yes	—
47	#478×C₆₇	C₂²	1	4	yes	—
48	#478×C₆₇	C₂²	1	4	yes	—
49	#477×C₆₇	C₂²	1	2	yes	—
50	#477×C₆₇	C₂²	1	4	yes	—
51	#476×C₆₇	C₂²	1	2	yes	—
52	#476×C₆₇	C₂²	1	4	yes	—
53	#475×C₆₇	C₂²	1	2	yes	—
54	#475×C₆₇	C₂²	1	4	yes	—
55	#474×C₆₇	C₂²	1	2	yes	—
56	#474×C₆₇	C₂²	1	4	yes	—
57	#446×C₆₇	C₂²	1	2	yes	—
58	#446×C₆₇	C₂²	1	2	yes	—
59	#446×C₆₇	C₂²	1	2	yes	—
60	#446×C₆₇	C₂²	1	2	yes	—
61	#442×C₆₇	C₂²	1	4	yes	—
62	#441×C₆₇	C₂²	1	4	yes	—
63	#440×C₆₇	C₂²	1	4	yes	—
64	#440×C₆₇	C₂²	1	4	yes	—
65	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
66	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
67	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
68	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
69	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
70	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
71	#427×C₆₇	C₂²	1	2	yes	—
72	#427×C₆₇	C₂²	1	4	yes	—
73	#427×C₆₇	C₂²	1	2	yes	—
74	#427×C₆₇	C₂²	1	1	yes	characteristic
75	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
76	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
77	#416×C₆₇	C₂²	1	2	yes	—
78	#416×C₆₇	C₂²	1	4	yes	—
79	C₂×#97×C₆₇	C₂²	1	2	yes	—
80	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
81	C₂×#41×C₆₇	C₂³	1	2	yes	—
82	C₂²×Q₈×C₆₇	—	2	2	no	—
83	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
84	#135×C₆₇	Q₈	1	8	yes	complemented
85	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
87	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
90	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
91	C₄⋊Q₈×C₆₇	—	2	2	no	—
92	C₄⋊Q₈×C₆₇	—	2	4	no	—
93	C₄⋊Q₈×C₆₇	—	2	2	no	—
94	C₄⋊Q₈×C₆₇	—	2	4	no	—
95	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
96	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
97	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
98	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
99	#125×C₆₇	—	2	4	no	—
100	#125×C₆₇	—	2	4	no	—
101	#125×C₆₇	—	2	4	no	—
102	#125×C₆₇	C₂³	1	4	yes	—
103	#125×C₆₇	C₂³	1	4	yes	—
104	#125×C₆₇	C₂³	1	4	yes	—
105	C₂²⋊Q₈×C₆₇	—	2	2	no	—
106	C₂²⋊Q₈×C₆₇	—	2	4	no	—
107	C₂²⋊Q₈×C₆₇	—	2	2	no	—
108	C₂²⋊Q₈×C₆₇	—	2	4	no	—
109	C₂²⋊Q₈×C₆₇	—	2	2	no	—
110	C₂²⋊Q₈×C₆₇	—	2	4	no	—
111	C₂²⋊Q₈×C₆₇	—	2	2	no	—
112	C₂²⋊Q₈×C₆₇	—	2	4	no	—
113	C₂²⋊Q₈×C₆₇	—	2	4	no	—
114	C₂²⋊Q₈×C₆₇	—	2	4	no	—
115	C₂²⋊Q₈×C₆₇	—	2	4	no	—
116	C₄×Q₈×C₆₇	—	2	2	no	—
117	C₄×Q₈×C₆₇	—	2	4	no	—
118	C₄×Q₈×C₆₇	—	2	2	no	—
119	C₄×Q₈×C₆₇	—	2	2	no	—
120	C₄×Q₈×C₆₇	—	2	4	no	—
121	C₄×Q₈×C₆₇	—	2	4	no	—
122	C₄×Q₈×C₆₇	—	2	2	no	—
123	C₄×Q₈×C₆₇	D₄	1	2	yes	—
124	C₄×Q₈×C₆₇	D₄	1	2	yes	—
125	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
126	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
127	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
128	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
129	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
130	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
131	C₄×D₄×C₆₇	—	2	2	no	—
132	C₄×D₄×C₆₇	—	2	4	no	—
133	C₄×D₄×C₆₇	—	2	2	no	—
134	C₄×D₄×C₆₇	—	2	4	no	—
135	C₄×D₄×C₆₇	C₂³	1	2	yes	—
136	C₄×D₄×C₆₇	C₂³	1	4	yes	—
137	C₄×D₄×C₆₇	C₂³	1	4	yes	—
138	C₄×D₄×C₆₇	C₂³	1	2	yes	—
139	#117×C₆₇	—	2	4	no	—
140	#117×C₆₇	—	2	2	no	—
141	#117×C₆₇	C₂³	1	2	yes	—
142	#117×C₆₇	C₂³	1	4	yes	—
143	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
144	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
145	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
146	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
147	C₂×C₄²×C₆₇	—	2	2	no	—
148	C₂×C₄²×C₆₇	—	2	4	no	—
149	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
150	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
151	#107×C₆₇	C₂³	1	2	yes	—
152	#107×C₆₇	C₂³	1	4	yes	—
153	#107×C₆₇	C₂³	1	4	yes	—
154	#107×C₆₇	C₂³	1	2	yes	—
155	#107×C₆₇	C₂³	1	2	yes	—
156	#107×C₆₇	C₂³	1	2	yes	—
157	#107×C₆₇	C₂³	1	1	yes	characteristic
158	#107×C₆₇	C₂³	1	1	yes	characteristic
159	#106×C₆₇	C₂³	1	4	yes	—
160	#106×C₆₇	C₂³	1	4	yes	—
161	#106×C₆₇	C₂³	1	4	yes	—
162	#106×C₆₇	C₂³	1	4	yes	—
163	#106×C₆₇	C₂³	1	2	yes	—
164	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
165	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
166	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
167	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
168	#103×C₆₇	C₂³	1	2	yes	—
169	#103×C₆₇	C₂³	1	4	yes	—
170	#103×C₆₇	C₂³	1	4	yes	—
171	#103×C₆₇	C₂³	1	2	yes	—
172	#102×C₆₇	C₂³	1	2	yes	—
173	#102×C₆₇	C₂³	1	4	yes	—
174	#102×C₆₇	C₂³	1	4	yes	—
175	#102×C₆₇	C₂³	1	2	yes	—
176	#97×C₆₇	C₂³	1	4	yes	—
177	#97×C₆₇	C₂³	1	4	yes	—
178	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
179	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
180	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
181	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
182	#41×C₆₇	—	2	4	no	—
183	#41×C₆₇	C₂×Q₈	1	8	yes	complemented
184	C₂×Q₈×C₆₇	—	2	4	no	—
185	C₂×Q₈×C₆₇	—	2	2	no	—
186	C₂×Q₈×C₆₇	—	2	4	no	—
187	C₂×Q₈×C₆₇	—	2	2	no	—
188	C₂×Q₈×C₆₇	—	4	2	no	—
189	C₂×Q₈×C₆₇	—	4	8	no	—
190	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
191	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
192	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
193	C₂×D₄×C₆₇	—	2	2	no	—
194	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
195	C₂²×C₄×C₆₇	—	2	2	no	—
196	C₂²×C₄×C₆₇	—	2	4	no	—
197	C₂²×C₄×C₆₇	—	2	2	no	—
198	C₂²×C₄×C₆₇	—	2	2	no	—
199	C₂²×C₄×C₆₇	—	2	4	no	—
200	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
201	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
202	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
203	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
204	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
205	Q₁₆×C₆₇	C₂×Q₈	1	4	yes	complemented
206	SD₁₆×C₆₇	C₂×Q₈	1	8	yes	complemented
207	D₈×C₆₇	C₂×Q₈	1	4	yes	complemented
208	C₂×C₈×C₆₇	—	2	2	no	—
209	C₂×C₈×C₆₇	—	2	4	no	—
210	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	complemented
211	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
212	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
213	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
214	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
215	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
216	C₄⋊C₄×C₆₇	—	2	4	no	—
217	C₄⋊C₄×C₆₇	—	2	4	no	—
218	C₄⋊C₄×C₆₇	—	2	4	no	—
219	C₄⋊C₄×C₆₇	—	2	4	no	—
220	C₄⋊C₄×C₆₇	—	2	2	no	—
221	C₄⋊C₄×C₆₇	—	2	4	no	—
222	C₄⋊C₄×C₆₇	—	2	4	no	—
223	C₄⋊C₄×C₆₇	—	2	2	no	—
224	C₄⋊C₄×C₆₇	—	2	2	no	—
225	C₄⋊C₄×C₆₇	—	2	4	no	—
226	C₄⋊C₄×C₆₇	—	2	2	no	—
227	C₄⋊C₄×C₆₇	—	2	4	no	—
228	C₄⋊C₄×C₆₇	—	2	4	no	—
229	C₄⋊C₄×C₆₇	—	2	4	no	—
230	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
231	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
232	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
233	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
234	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
235	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
236	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
237	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
238	C₂²⋊C₄×C₆₇	—	2	2	no	—
239	C₂²⋊C₄×C₆₇	—	2	4	no	—
240	C₂²⋊C₄×C₆₇	—	2	4	no	—
241	C₂²⋊C₄×C₆₇	—	2	4	no	—
242	C₂²⋊C₄×C₆₇	—	2	2	no	—
243	C₂²⋊C₄×C₆₇	—	2	2	no	—
244	C₄²×C₆₇	—	2	2	no	—
245	C₄²×C₆₇	—	2	4	no	—
246	C₄²×C₆₇	—	2	4	no	—
247	C₄²×C₆₇	—	2	2	no	—
248	C₄²×C₆₇	—	2	2	no	—
249	C₄²×C₆₇	—	2	4	no	—
250	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
251	C₄²×C₆₇	C₂×D₄	1	4	yes	—
252	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
253	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
254	C₄²×C₆₇	C₂⁴	1	4	yes	—
255	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
256	C₂³×C₆₇	—	2	2	no	—
257	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
258	Q₈×C₆₇	—	2	4	no	—
259	Q₈×C₆₇	—	2	2	no	—
260	Q₈×C₆₇	—	4	2	no	—
261	Q₈×C₆₇	—	4	8	no	—
262	Q₈×C₆₇	—	4	4	no	—
263	Q₈×C₆₇	—	8	4	no	—
264	Q₈×C₆₇	—	8	2	no	—
265	Q₈×C₆₇	C₂²×Q₈	1	4	yes	—
266	D₄×C₆₇	—	2	8	no	—
267	D₄×C₆₇	—	2	2	no	—
268	D₄×C₆₇	C₂²×Q₈	1	4	yes	—
269	C₂×C₄×C₆₇	—	2	2	no	—
270	C₂×C₄×C₆₇	—	2	8	no	—
271	C₂×C₄×C₆₇	—	2	2	no	—
272	C₂×C₄×C₆₇	—	2	4	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	2	4	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	4	1	no	—
277	C₂×C₄×C₆₇	—	4	2	no	—
278	C₂×C₄×C₆₇	—	4	2	no	—
279	C₂×C₄×C₆₇	—	4	8	no	—
280	C₂×C₄×C₆₇	—	4	4	no	—
281	C₂×C₄×C₆₇	—	8	2	no	—
282	C₂×C₄×C₆₇	—	8	4	no	—
283	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
285	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
286	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
289	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
290	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
291	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
292	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
293	C₈×C₆₇	—	2	1	no	—
294	C₈×C₆₇	—	2	4	no	—
295	C₈×C₆₇	—	2	1	no	—
296	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
297	C₈×C₆₇	C₂²×Q₈	1	2	yes	—
298	C₂²×C₆₇	—	2	1	no	—
299	C₂²×C₆₇	—	2	1	no	—
300	C₂²×C₆₇	—	2	4	no	—
301	C₂²×C₆₇	—	4	2	no	—
302	C₂²×C₆₇	—	4	2	no	—
303	C₂²×C₆₇	D₄×Q₈	1	2	yes	—
304	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
305	C₄×C₆₇	—	2	1	no	—
306	C₄×C₆₇	—	2	4	no	—
307	C₄×C₆₇	—	2	1	no	—
308	C₄×C₆₇	—	2	4	no	—
309	C₄×C₆₇	—	2	1	no	—
310	C₄×C₆₇	—	2	4	no	—
311	C₄×C₆₇	—	4	2	no	—
312	C₄×C₆₇	—	4	4	no	—
313	C₄×C₆₇	—	4	4	no	—
314	C₄×C₆₇	—	4	2	no	—
315	C₄×C₆₇	D₄×Q₈	1	2	yes	—
316	C₄×C₆₇	#575	1	2	yes	—
317	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
318	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
319	#61365	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
320	C₂×C₆₇	—	2	2	no	—
321	C₂×C₆₇	—	4	4	no	—
322	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
323	C₂×C₆₇	C₂×#575	1	1	yes	characteristic
324	C₂×C₆₇	#3584	1	1	yes	characteristic
325	Q₈×#41	C₂×C₆₇	1	2	yes	complemented
326	#3390	C₂×C₆₇	1	2	yes	complemented
327	#3389	C₂×C₆₇	1	2	yes	complemented
328	#3388	C₂×C₆₇	1	4	yes	complemented
329	#3387	C₂×C₆₇	1	2	yes	complemented
330	#3386	C₂×C₆₇	1	2	yes	complemented
331	#3385	C₂×C₆₇	1	2	yes	complemented
332	#3384	C₂×C₆₇	1	2	yes	complemented
333	#3162	C₂×C₆₇	1	2	yes	complemented
334	#3160	C₂×C₆₇	1	1	yes	characteristic, complemented
335	#3076	C₂×C₆₇	1	2	yes	complemented
336	#3076	C₂×C₆₇	1	4	yes	complemented
337	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
338	#2943	C₂×C₆₇	1	2	yes	—
339	C₄×#135	C₂×C₆₇	1	1	yes	characteristic
340	C₆₇	#61365	1	1	yes	characteristic, direct product factor, Sylow
341	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
342	Q₈²	C₂²×C₆₇	1	2	yes	—
343	#557	C₂²×C₆₇	1	2	yes	—
344	#557	C₂²×C₆₇	1	4	yes	—
345	#554	—	2	4	no	—
346	#554	—	2	2	no	—
347	#554	C₂²×C₆₇	1	2	yes	—
348	#554	C₂²×C₆₇	1	4	yes	—
349	D₄×Q₈	—	2	2	no	—
350	D₄×Q₈	C₂²×C₆₇	1	2	yes	—
351	#533	—	2	2	no	—
352	#533	—	2	4	no	—
353	#533	C₂²×C₆₇	1	2	yes	—
354	#533	C₂²×C₆₇	1	4	yes	—
355	C₄×#41	C₂²×C₆₇	1	2	yes	—
356	C₄×#41	C₂²×C₆₇	1	4	yes	—
357	C₂×C₄×Q₈	—	2	2	no	—
358	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
359	#501	C₂²×C₆₇	1	4	yes	—
360	#501	C₂²×C₆₇	1	4	yes	—
361	#501	C₂²×C₆₇	1	4	yes	—
362	#501	C₂²×C₆₇	1	4	yes	—
363	#500	C₂²×C₆₇	1	1	yes	characteristic
364	#500	C₂²×C₆₇	1	1	yes	characteristic
365	#499	C₂²×C₆₇	1	2	yes	—
366	#499	C₂²×C₆₇	1	2	yes	—
367	#498	C₂²×C₆₇	1	2	yes	—
368	#479	C₂²×C₆₇	1	4	yes	—
369	#479	C₂²×C₆₇	1	4	yes	—
370	#479	C₂²×C₆₇	1	4	yes	—
371	#478	C₂²×C₆₇	1	4	yes	—
372	#478	C₂²×C₆₇	1	4	yes	—
373	#478	C₂²×C₆₇	1	4	yes	—
374	#477	C₂²×C₆₇	1	2	yes	—
375	#477	C₂²×C₆₇	1	4	yes	—
376	#476	C₂²×C₆₇	1	2	yes	—
377	#476	C₂²×C₆₇	1	4	yes	—
378	#475	C₂²×C₆₇	1	2	yes	—
379	#475	C₂²×C₆₇	1	4	yes	—
380	#474	C₂²×C₆₇	1	2	yes	—
381	#474	C₂²×C₆₇	1	4	yes	—
382	#446	C₂²×C₆₇	1	2	yes	—
383	#446	C₂²×C₆₇	1	2	yes	—
384	#446	C₂²×C₆₇	1	2	yes	—
385	#446	C₂²×C₆₇	1	2	yes	—
386	#442	C₂²×C₆₇	1	4	yes	—
387	#441	C₂²×C₆₇	1	4	yes	—
388	#440	C₂²×C₆₇	1	4	yes	—
389	#440	C₂²×C₆₇	1	4	yes	—
390	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
391	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
392	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
393	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
394	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
395	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
396	#427	C₂²×C₆₇	1	2	yes	—
397	#427	C₂²×C₆₇	1	4	yes	—
398	#427	C₂²×C₆₇	1	2	yes	—
399	#427	C₂²×C₆₇	1	1	yes	characteristic
400	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
401	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
402	#416	C₂²×C₆₇	1	2	yes	—
403	#416	C₂²×C₆₇	1	4	yes	—
404	C₂×#97	C₂²×C₆₇	1	2	yes	—
405	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
406	C₂×#41	C₂³×C₆₇	1	2	yes	—
407	C₂²×Q₈	—	2	2	no	—
408	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
409	#135	Q₈×C₆₇	1	8	yes	complemented
410	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
411	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
412	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
413	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
414	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
415	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
416	C₄⋊Q₈	—	2	2	no	—
417	C₄⋊Q₈	—	2	4	no	—
418	C₄⋊Q₈	—	2	2	no	—
419	C₄⋊Q₈	—	2	4	no	—
420	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
421	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
422	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
423	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
424	#125	—	2	4	no	—
425	#125	—	2	4	no	—
426	#125	—	2	4	no	—
427	#125	C₂³×C₆₇	1	4	yes	—
428	#125	C₂³×C₆₇	1	4	yes	—
429	#125	C₂³×C₆₇	1	4	yes	—
430	C₂²⋊Q₈	—	2	2	no	—
431	C₂²⋊Q₈	—	2	4	no	—
432	C₂²⋊Q₈	—	2	2	no	—
433	C₂²⋊Q₈	—	2	4	no	—
434	C₂²⋊Q₈	—	2	2	no	—
435	C₂²⋊Q₈	—	2	4	no	—
436	C₂²⋊Q₈	—	2	2	no	—
437	C₂²⋊Q₈	—	2	4	no	—
438	C₂²⋊Q₈	—	2	4	no	—
439	C₂²⋊Q₈	—	2	4	no	—
440	C₂²⋊Q₈	—	2	4	no	—
441	C₄×Q₈	—	2	2	no	—
442	C₄×Q₈	—	2	4	no	—
443	C₄×Q₈	—	2	2	no	—
444	C₄×Q₈	—	2	2	no	—
445	C₄×Q₈	—	2	4	no	—
446	C₄×Q₈	—	2	4	no	—
447	C₄×Q₈	—	2	2	no	—
448	C₄×Q₈	D₄×C₆₇	1	2	yes	—
449	C₄×Q₈	D₄×C₆₇	1	2	yes	—
450	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
451	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
452	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
453	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
454	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
455	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
456	C₄×D₄	—	2	2	no	—
457	C₄×D₄	—	2	4	no	—
458	C₄×D₄	—	2	2	no	—
459	C₄×D₄	—	2	4	no	—
460	C₄×D₄	C₂³×C₆₇	1	2	yes	—
461	C₄×D₄	C₂³×C₆₇	1	4	yes	—
462	C₄×D₄	C₂³×C₆₇	1	2	yes	—
463	C₄×D₄	C₂³×C₆₇	1	4	yes	—
464	#117	—	2	4	no	—
465	#117	—	2	2	no	—
466	#117	C₂³×C₆₇	1	2	yes	—
467	#117	C₂³×C₆₇	1	4	yes	—
468	C₂×C₄⋊C₄	—	2	4	no	—
469	C₂×C₄⋊C₄	—	2	2	no	—
470	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
471	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
472	C₂×C₄²	—	2	2	no	—
473	C₂×C₄²	—	2	4	no	—
474	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
475	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
476	#107	C₂³×C₆₇	1	1	yes	characteristic
477	#107	C₂³×C₆₇	1	1	yes	characteristic
478	#107	C₂³×C₆₇	1	2	yes	—
479	#107	C₂³×C₆₇	1	2	yes	—
480	#107	C₂³×C₆₇	1	2	yes	—
481	#107	C₂³×C₆₇	1	4	yes	—
482	#107	C₂³×C₆₇	1	4	yes	—
483	#107	C₂³×C₆₇	1	2	yes	—
484	#106	C₂³×C₆₇	1	2	yes	—
485	#106	C₂³×C₆₇	1	4	yes	—
486	#106	C₂³×C₆₇	1	4	yes	—
487	#106	C₂³×C₆₇	1	4	yes	—
488	#106	C₂³×C₆₇	1	4	yes	—
489	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
490	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
491	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
492	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
493	#103	C₂³×C₆₇	1	2	yes	—
494	#103	C₂³×C₆₇	1	2	yes	—
495	#103	C₂³×C₆₇	1	4	yes	—
496	#103	C₂³×C₆₇	1	4	yes	—
497	#102	C₂³×C₆₇	1	2	yes	—
498	#102	C₂³×C₆₇	1	2	yes	—
499	#102	C₂³×C₆₇	1	4	yes	—
500	#102	C₂³×C₆₇	1	4	yes	—
501	#97	C₂³×C₆₇	1	4	yes	—
502	#97	C₂³×C₆₇	1	4	yes	—
503	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
504	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
505	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
506	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
507	#41	—	2	4	no	—
508	#41	C₂×Q₈×C₆₇	1	8	yes	complemented
509	C₂×Q₈	—	2	4	no	—
510	C₂×Q₈	—	2	2	no	—
511	C₂×Q₈	—	2	4	no	—
512	C₂×Q₈	—	2	2	no	—
513	C₂×Q₈	—	4	2	no	—
514	C₂×Q₈	—	4	8	no	—
515	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
516	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
517	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
518	C₂×D₄	—	2	2	no	—
519	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
520	C₂²×C₄	—	2	2	no	—
521	C₂²×C₄	—	2	4	no	—
522	C₂²×C₄	—	2	2	no	—
523	C₂²×C₄	—	2	2	no	—
524	C₂²×C₄	—	2	4	no	—
525	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
526	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
527	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
528	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
529	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
530	Q₁₆	C₂×Q₈×C₆₇	1	4	yes	complemented
531	SD₁₆	C₂×Q₈×C₆₇	1	8	yes	complemented
532	D₈	C₂×Q₈×C₆₇	1	4	yes	complemented
533	C₂×C₈	—	2	2	no	—
534	C₂×C₈	—	2	4	no	—
535	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	complemented
536	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
537	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
538	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
539	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
540	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
541	C₄⋊C₄	—	2	4	no	—
542	C₄⋊C₄	—	2	4	no	—
543	C₄⋊C₄	—	2	4	no	—
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	2	2	no	—
546	C₄⋊C₄	—	2	4	no	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	2	no	—
549	C₄⋊C₄	—	2	2	no	—
550	C₄⋊C₄	—	2	4	no	—
551	C₄⋊C₄	—	2	2	no	—
552	C₄⋊C₄	—	2	4	no	—
553	C₄⋊C₄	—	2	4	no	—
554	C₄⋊C₄	—	2	4	no	—
555	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
556	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
557	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
558	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
559	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
560	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
561	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
562	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
563	C₂²⋊C₄	—	2	2	no	—
564	C₂²⋊C₄	—	2	4	no	—
565	C₂²⋊C₄	—	2	4	no	—
566	C₂²⋊C₄	—	2	4	no	—
567	C₂²⋊C₄	—	2	2	no	—
568	C₂²⋊C₄	—	2	2	no	—
569	C₄²	—	2	2	no	—
570	C₄²	—	2	4	no	—
571	C₄²	—	2	2	no	—
572	C₄²	—	2	4	no	—
573	C₄²	—	2	4	no	—
574	C₄²	—	2	2	no	—
575	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
576	C₄²	C₂×D₄×C₆₇	1	4	yes	—
577	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
578	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
579	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
580	C₄²	C₂⁴×C₆₇	1	4	yes	—
581	C₂³	—	2	2	no	—
582	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
583	Q₈	—	2	4	no	—
584	Q₈	—	2	2	no	—
585	Q₈	—	4	2	no	—
586	Q₈	—	4	8	no	—
587	Q₈	—	4	4	no	—
588	Q₈	—	8	4	no	—
589	Q₈	—	8	2	no	—
590	Q₈	C₂²×Q₈×C₆₇	1	4	yes	—
591	D₄	—	2	8	no	—
592	D₄	—	2	2	no	—
593	D₄	C₂²×Q₈×C₆₇	1	4	yes	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	8	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	4	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	4	2	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	2	no	—
604	C₂×C₄	—	4	8	no	—
605	C₂×C₄	—	4	4	no	—
606	C₂×C₄	—	8	2	no	—
607	C₂×C₄	—	8	4	no	—
608	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
609	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
610	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
611	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
612	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
613	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
614	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
615	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
616	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
617	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
618	C₈	—	2	1	no	—
619	C₈	—	2	1	no	—
620	C₈	—	2	4	no	—
621	C₈	C₂²×Q₈×C₆₇	1	2	yes	—
622	C₈	C₂²×Q₈×C₆₇	1	2	yes	—
623	C₂²	—	2	1	no	—
624	C₂²	—	2	1	no	—
625	C₂²	—	2	4	no	—
626	C₂²	—	4	2	no	—
627	C₂²	—	4	2	no	—
628	C₂²	D₄×Q₈×C₆₇	1	2	yes	—
629	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
630	C₄	—	2	1	no	—
631	C₄	—	2	4	no	—
632	C₄	—	2	1	no	—
633	C₄	—	2	4	no	—
634	C₄	—	2	1	no	—
635	C₄	—	2	4	no	—
636	C₄	—	4	4	no	—
637	C₄	—	4	2	no	—
638	C₄	—	4	2	no	—
639	C₄	—	4	4	no	—
640	C₄	D₄×Q₈×C₆₇	1	2	yes	—
641	C₄	#575×C₆₇	1	2	yes	—
642	C₄	C₂³×Q₈×C₆₇	1	1	yes	characteristic
643	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
644	C₂	—	2	2	no	—
645	C₂	—	4	4	no	—
646	C₂	C₂×D₄×Q₈×C₆₇	1	1	yes	characteristic
647	C₂	C₂×#575×C₆₇	1	1	yes	characteristic
648	C₂	C₆₇×#3584	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂³×D₄
prime radical	C₂×#135×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×#41×C₆₇	C₂	semidirect
2. C₆₇×#3390	C₂	semidirect
3. C₆₇×#3389	C₂	semidirect
4. C₆₇×#3388	C₂	semidirect
5. C₆₇×#3387	C₂	semidirect
6. C₆₇×#3386	C₂	semidirect
7. C₆₇×#3385	C₂	semidirect
8. C₆₇×#3384	C₂	semidirect
9. C₆₇×#3162	C₂	semidirect
10. C₆₇×#3160	C₂	semidirect
11. C₆₇×#3076	C₂	semidirect
12. C₆₇×#3076	C₂	semidirect
13. C₂×#446×C₆₇	C₂	semidirect
84. #135×C₆₇	Q₈	semidirect
183. #41×C₆₇	C₂×Q₈	semidirect
205. Q₁₆×C₆₇	C₂×Q₈	semidirect
206. SD₁₆×C₆₇	C₂×Q₈	semidirect
207. D₈×C₆₇	C₂×Q₈	semidirect
210. C₂×C₈×C₆₇	C₂×Q₈	semidirect
319. #61365	C₆₇	direct (N × H)
325. Q₈×#41	C₂×C₆₇	semidirect
326. #3390	C₂×C₆₇	semidirect
327. #3389	C₂×C₆₇	semidirect
328. #3388	C₂×C₆₇	semidirect
329. #3387	C₂×C₆₇	semidirect
330. #3386	C₂×C₆₇	semidirect
331. #3385	C₂×C₆₇	semidirect
332. #3384	C₂×C₆₇	semidirect
333. #3162	C₂×C₆₇	semidirect
334. #3160	C₂×C₆₇	semidirect
335. #3076	C₂×C₆₇	semidirect
336. #3076	C₂×C₆₇	semidirect
337. C₂×#446	C₂×C₆₇	semidirect
340. C₆₇	#61365	direct (N × H)
409. #135	Q₈×C₆₇	semidirect
508. #41	C₂×Q₈×C₆₇	semidirect
530. Q₁₆	C₂×Q₈×C₆₇	semidirect
531. SD₁₆	C₂×Q₈×C₆₇	semidirect
532. D₈	C₂×Q₈×C₆₇	semidirect
535. C₂×C₈	C₂×Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#41×C₆₇	2
C₆₇×#3390	2
C₆₇×#3389	2
C₆₇×#3388	2
C₆₇×#3387	2
C₆₇×#3386	2
C₆₇×#3385	2
C₆₇×#3384	2
C₆₇×#3162	2
C₆₇×#3160	2
C₆₇×#3076	2
C₂×#446×C₆₇	2
C₆₇×#2943	2
C₄×#135×C₆₇	2
#61365	67

Subgroups — C67×#61365 #30740616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61365 #30740616