Subgroups — C₆₇×#59445 #30738696

Order 17152.

1588 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2774	C₂	1	4	yes	maximal
2	C₆₇×#2570	C₂	1	2	yes	maximal
3	C₆₇×#2567	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2567	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2552	C₂	1	2	yes	maximal
6	C₆₇×#2511	C₂	1	2	yes	maximal
7	C₆₇×#2508	C₂	1	2	yes	maximal
8	C₆₇×#2506	C₂	1	2	yes	maximal
9	C₆₇×#2462	C₂	1	4	yes	maximal
10	C₆₇×#2446	C₂	1	2	yes	maximal
11	C₆₇×#2370	C₂	1	2	yes	maximal
12	C₆₇×#2355	C₂	1	1	yes	maximal, characteristic
13	C₆₇×#2355	C₂	1	2	yes	maximal
14	Q₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
15	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
16	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
17	C₄×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#125×C₆₇	C₂²	1	4	yes	—
21	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	#400×C₆₇	C₂²	1	4	yes	—
27	#400×C₆₇	C₂²	1	4	yes	—
28	#400×C₆₇	C₂²	1	4	yes	—
29	#398×C₆₇	C₂²	1	4	yes	—
30	#398×C₆₇	C₂²	1	4	yes	—
31	#398×C₆₇	C₂²	1	4	yes	—
32	#395×C₆₇	C₂²	1	4	yes	—
33	#395×C₆₇	C₂²	1	4	yes	—
34	#391×C₆₇	C₂²	1	1	yes	characteristic
35	#391×C₆₇	C₂²	1	2	yes	—
36	#391×C₆₇	C₂²	1	2	yes	—
37	#391×C₆₇	C₂²	1	2	yes	—
38	#391×C₆₇	C₂²	1	1	yes	characteristic
39	#391×C₆₇	C₂²	1	2	yes	—
40	#389×C₆₇	C₂²	1	1	yes	characteristic
41	#389×C₆₇	C₂²	1	2	yes	—
42	#389×C₆₇	C₂²	1	2	yes	—
43	#389×C₆₇	C₂²	1	2	yes	—
44	#389×C₆₇	C₂²	1	1	yes	characteristic
45	#389×C₆₇	C₂²	1	2	yes	—
46	#389×C₆₇	C₂²	1	2	yes	—
47	#389×C₆₇	C₂²	1	2	yes	—
48	#389×C₆₇	C₂²	1	2	yes	—
49	#389×C₆₇	C₂²	1	4	yes	—
50	#389×C₆₇	C₂²	1	4	yes	—
51	#389×C₆₇	C₂²	1	2	yes	—
52	#389×C₆₇	C₂²	1	2	yes	—
53	#387×C₆₇	C₂²	1	4	yes	—
54	#387×C₆₇	C₂²	1	4	yes	—
55	#387×C₆₇	C₂²	1	4	yes	—
56	#387×C₆₇	C₂²	1	4	yes	—
57	#387×C₆₇	C₂²	1	2	yes	—
58	#387×C₆₇	C₂²	1	4	yes	—
59	#387×C₆₇	C₂²	1	4	yes	—
60	#387×C₆₇	C₂²	1	2	yes	—
61	#387×C₆₇	C₂²	1	4	yes	—
62	#384×C₆₇	C₂²	1	2	yes	—
63	#384×C₆₇	C₂²	1	2	yes	—
64	#384×C₆₇	C₂²	1	1	yes	characteristic
65	#384×C₆₇	C₂²	1	2	yes	—
66	#384×C₆₇	C₂²	1	1	yes	characteristic
67	#382×C₆₇	C₂²	1	4	yes	—
68	#382×C₆₇	C₂²	1	2	yes	—
69	#382×C₆₇	C₂²	1	1	yes	characteristic
70	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
71	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
72	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
73	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
74	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
75	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
76	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
77	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
78	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
79	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
80	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
81	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
82	C₄³×C₆₇	C₂²	1	1	yes	characteristic
83	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
84	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
86	C₄⋊Q₈×C₆₇	—	2	4	no	—
87	C₄⋊Q₈×C₆₇	—	2	4	no	—
88	#125×C₆₇	—	4	8	no	—
89	#125×C₆₇	Q₈	1	8	yes	complemented
90	C₄×Q₈×C₆₇	—	2	8	no	—
91	C₄×Q₈×C₆₇	—	2	4	no	—
92	C₄×Q₈×C₆₇	—	2	4	no	—
93	C₄×Q₈×C₆₇	—	4	4	no	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
116	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
118	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
122	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
124	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
125	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
127	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
132	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
133	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
136	#95×C₆₇	C₂³	1	2	yes	—
137	#95×C₆₇	C₂³	1	2	yes	—
138	#95×C₆₇	C₂³	1	4	yes	—
139	#95×C₆₇	C₂³	1	2	yes	—
140	#95×C₆₇	C₂³	1	4	yes	—
141	#95×C₆₇	C₂³	1	2	yes	—
142	#95×C₆₇	C₂³	1	2	yes	—
143	#95×C₆₇	C₂³	1	2	yes	—
144	#95×C₆₇	C₂³	1	2	yes	—
145	#95×C₆₇	C₂³	1	2	yes	—
146	#95×C₆₇	C₂³	1	4	yes	—
147	#95×C₆₇	C₂³	1	2	yes	—
148	#95×C₆₇	C₂³	1	4	yes	—
149	#95×C₆₇	C₂³	1	4	yes	—
150	#95×C₆₇	C₂³	1	4	yes	—
151	#95×C₆₇	C₂³	1	2	yes	—
152	#95×C₆₇	C₂³	1	4	yes	—
153	#95×C₆₇	C₂³	1	4	yes	—
154	#95×C₆₇	C₂³	1	4	yes	—
155	#95×C₆₇	C₂³	1	4	yes	—
156	C₂×Q₈×C₆₇	—	2	2	no	—
157	C₂×Q₈×C₆₇	—	2	2	no	—
158	C₂×Q₈×C₆₇	—	2	4	no	—
159	C₂×Q₈×C₆₇	—	2	2	no	—
160	C₂×Q₈×C₆₇	—	2	2	no	—
161	C₂×Q₈×C₆₇	—	2	2	no	—
162	C₂×Q₈×C₆₇	—	4	2	no	—
163	C₂×Q₈×C₆₇	—	4	2	no	—
164	C₂×Q₈×C₆₇	#41	1	4	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
182	C₄⋊C₄×C₆₇	—	2	4	no	—
183	C₄⋊C₄×C₆₇	—	2	8	no	—
184	C₄⋊C₄×C₆₇	—	2	4	no	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	2	4	no	—
188	C₄⋊C₄×C₆₇	—	2	4	no	—
189	C₄⋊C₄×C₆₇	—	2	4	no	—
190	C₄⋊C₄×C₆₇	—	2	4	no	—
191	C₄⋊C₄×C₆₇	—	2	4	no	—
192	C₄⋊C₄×C₆₇	—	2	8	no	—
193	C₄⋊C₄×C₆₇	—	2	4	no	—
194	C₄⋊C₄×C₆₇	—	2	4	no	—
195	C₄⋊C₄×C₆₇	—	2	4	no	—
196	C₄⋊C₄×C₆₇	—	2	2	no	—
197	C₄⋊C₄×C₆₇	—	2	2	no	—
198	C₄⋊C₄×C₆₇	—	4	2	no	—
199	C₄⋊C₄×C₆₇	—	4	2	no	—
200	C₄⋊C₄×C₆₇	—	4	4	no	—
201	C₄⋊C₄×C₆₇	—	4	4	no	—
202	C₄⋊C₄×C₆₇	—	4	2	no	—
203	C₄⋊C₄×C₆₇	—	4	4	no	—
204	C₄⋊C₄×C₆₇	—	4	4	no	—
205	C₄⋊C₄×C₆₇	—	4	4	no	—
206	C₄⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
207	C₄⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
208	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
209	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
210	C₄²×C₆₇	—	2	8	no	—
211	C₄²×C₆₇	—	2	4	no	—
212	C₄²×C₆₇	—	2	4	no	—
213	C₄²×C₆₇	—	2	4	no	—
214	C₄²×C₆₇	—	2	2	no	—
215	C₄²×C₆₇	—	2	2	no	—
216	C₄²×C₆₇	—	2	2	no	—
217	C₄²×C₆₇	—	2	4	no	—
218	C₄²×C₆₇	—	2	2	no	—
219	C₄²×C₆₇	—	4	1	no	—
220	C₄²×C₆₇	—	4	4	no	—
221	C₄²×C₆₇	—	4	4	no	—
222	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
223	C₄²×C₆₇	#41	1	4	yes	—
224	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
225	Q₈×C₆₇	—	2	8	no	—
226	Q₈×C₆₇	—	4	8	no	—
227	Q₈×C₆₇	—	4	4	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	—	2	1	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	1	no	—
233	C₂×C₄×C₆₇	—	2	1	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	1	no	—
236	C₂×C₄×C₆₇	—	2	2	no	—
237	C₂×C₄×C₆₇	—	2	1	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	1	no	—
240	C₂×C₄×C₆₇	—	2	1	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	4	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	2	no	—
253	C₂×C₄×C₆₇	—	2	2	no	—
254	C₂×C₄×C₆₇	—	2	1	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	1	no	—
257	C₂×C₄×C₆₇	—	2	4	no	—
258	C₂×C₄×C₆₇	—	2	2	no	—
259	C₂×C₄×C₆₇	—	2	2	no	—
260	C₂×C₄×C₆₇	—	2	2	no	—
261	C₂×C₄×C₆₇	—	2	1	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	2	no	—
264	C₂×C₄×C₆₇	—	2	2	no	—
265	C₂×C₄×C₆₇	—	2	4	no	—
266	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
267	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
268	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
269	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
270	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
271	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
272	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
273	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
274	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
275	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
276	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
277	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
278	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
279	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
280	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
281	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
282	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
283	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
284	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
285	C₂²×C₆₇	#585	1	1	yes	characteristic
286	C₄×C₆₇	—	2	4	no	—
287	C₄×C₆₇	—	2	4	no	—
288	C₄×C₆₇	—	2	2	no	—
289	C₄×C₆₇	—	2	2	no	—
290	C₄×C₆₇	—	2	4	no	—
291	C₄×C₆₇	—	2	2	no	—
292	C₄×C₆₇	—	2	2	no	—
293	C₄×C₆₇	—	4	2	no	—
294	C₄×C₆₇	—	4	2	no	—
295	C₄×C₆₇	—	4	2	no	—
296	C₄×C₆₇	—	4	4	no	—
297	C₄×C₆₇	—	4	4	no	—
298	C₄×C₆₇	—	4	2	no	—
299	C₄×C₆₇	—	4	1	no	—
300	C₄×C₆₇	—	4	2	no	—
301	C₄×C₆₇	#552	1	4	yes	—
302	C₄×C₆₇	#557	1	4	yes	—
303	#59445	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
304	C₂×C₆₇	C₂×#552	1	1	yes	characteristic
305	C₂×C₆₇	C₂×#557	1	1	yes	characteristic
306	C₂×C₆₇	Q₈×#41	1	1	yes	characteristic
307	C₂×C₆₇	#3503	1	1	yes	characteristic
308	C₂×C₆₇	#3506	1	1	yes	characteristic
309	C₂×C₆₇	#3522	1	1	yes	characteristic
310	C₂×C₆₇	#3555	1	1	yes	characteristic
311	#2774	C₂×C₆₇	1	4	yes	—
312	#2570	C₂×C₆₇	1	2	yes	—
313	#2567	C₂×C₆₇	1	1	yes	characteristic
314	#2567	C₂×C₆₇	1	1	yes	characteristic
315	#2552	C₂×C₆₇	1	2	yes	—
316	#2511	C₂×C₆₇	1	2	yes	—
317	#2508	C₂×C₆₇	1	2	yes	—
318	#2506	C₂×C₆₇	1	2	yes	—
319	#2462	C₂×C₆₇	1	4	yes	—
320	#2446	C₂×C₆₇	1	2	yes	—
321	#2370	C₂×C₆₇	1	2	yes	—
322	#2355	C₂×C₆₇	1	1	yes	characteristic
323	#2355	C₂×C₆₇	1	2	yes	—
324	Q₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
325	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
326	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
327	C₄×#125	C₂×C₆₇	1	1	yes	characteristic
328	C₆₇	#59445	1	1	yes	characteristic, direct product factor, Sylow
329	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
330	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
331	C₂×#125	C₂²×C₆₇	1	4	yes	—
332	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
334	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
335	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
336	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
337	#400	C₂²×C₆₇	1	4	yes	—
338	#400	C₂²×C₆₇	1	4	yes	—
339	#400	C₂²×C₆₇	1	4	yes	—
340	#398	C₂²×C₆₇	1	4	yes	—
341	#398	C₂²×C₆₇	1	4	yes	—
342	#398	C₂²×C₆₇	1	4	yes	—
343	#395	C₂²×C₆₇	1	4	yes	—
344	#395	C₂²×C₆₇	1	4	yes	—
345	#391	C₂²×C₆₇	1	1	yes	characteristic
346	#391	C₂²×C₆₇	1	1	yes	characteristic
347	#391	C₂²×C₆₇	1	2	yes	—
348	#391	C₂²×C₆₇	1	2	yes	—
349	#391	C₂²×C₆₇	1	2	yes	—
350	#391	C₂²×C₆₇	1	2	yes	—
351	#389	C₂²×C₆₇	1	1	yes	characteristic
352	#389	C₂²×C₆₇	1	1	yes	characteristic
353	#389	C₂²×C₆₇	1	2	yes	—
354	#389	C₂²×C₆₇	1	2	yes	—
355	#389	C₂²×C₆₇	1	2	yes	—
356	#389	C₂²×C₆₇	1	2	yes	—
357	#389	C₂²×C₆₇	1	2	yes	—
358	#389	C₂²×C₆₇	1	2	yes	—
359	#389	C₂²×C₆₇	1	2	yes	—
360	#389	C₂²×C₆₇	1	2	yes	—
361	#389	C₂²×C₆₇	1	4	yes	—
362	#389	C₂²×C₆₇	1	2	yes	—
363	#389	C₂²×C₆₇	1	4	yes	—
364	#387	C₂²×C₆₇	1	2	yes	—
365	#387	C₂²×C₆₇	1	2	yes	—
366	#387	C₂²×C₆₇	1	4	yes	—
367	#387	C₂²×C₆₇	1	4	yes	—
368	#387	C₂²×C₆₇	1	4	yes	—
369	#387	C₂²×C₆₇	1	4	yes	—
370	#387	C₂²×C₆₇	1	4	yes	—
371	#387	C₂²×C₆₇	1	4	yes	—
372	#387	C₂²×C₆₇	1	4	yes	—
373	#384	C₂²×C₆₇	1	1	yes	characteristic
374	#384	C₂²×C₆₇	1	1	yes	characteristic
375	#384	C₂²×C₆₇	1	2	yes	—
376	#384	C₂²×C₆₇	1	2	yes	—
377	#384	C₂²×C₆₇	1	2	yes	—
378	#382	C₂²×C₆₇	1	1	yes	characteristic
379	#382	C₂²×C₆₇	1	2	yes	—
380	#382	C₂²×C₆₇	1	4	yes	—
381	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
382	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
383	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
384	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
385	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
386	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
387	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
388	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
389	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
390	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
391	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
392	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
393	C₄³	C₂²×C₆₇	1	1	yes	characteristic
394	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
395	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
396	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
397	C₄⋊Q₈	—	2	4	no	—
398	C₄⋊Q₈	—	2	4	no	—
399	#125	—	4	8	no	—
400	#125	Q₈×C₆₇	1	8	yes	complemented
401	C₄×Q₈	—	2	8	no	—
402	C₄×Q₈	—	2	4	no	—
403	C₄×Q₈	—	2	4	no	—
404	C₄×Q₈	—	4	4	no	—
405	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
406	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
411	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
412	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
413	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
415	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
416	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
418	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
423	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
424	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
425	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
426	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
427	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
428	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
429	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
430	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
433	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
434	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
435	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
436	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
437	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
438	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
440	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
441	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
442	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
443	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
444	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
445	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
447	#95	C₂³×C₆₇	1	2	yes	—
448	#95	C₂³×C₆₇	1	2	yes	—
449	#95	C₂³×C₆₇	1	4	yes	—
450	#95	C₂³×C₆₇	1	2	yes	—
451	#95	C₂³×C₆₇	1	4	yes	—
452	#95	C₂³×C₆₇	1	2	yes	—
453	#95	C₂³×C₆₇	1	2	yes	—
454	#95	C₂³×C₆₇	1	2	yes	—
455	#95	C₂³×C₆₇	1	2	yes	—
456	#95	C₂³×C₆₇	1	2	yes	—
457	#95	C₂³×C₆₇	1	4	yes	—
458	#95	C₂³×C₆₇	1	2	yes	—
459	#95	C₂³×C₆₇	1	4	yes	—
460	#95	C₂³×C₆₇	1	4	yes	—
461	#95	C₂³×C₆₇	1	4	yes	—
462	#95	C₂³×C₆₇	1	2	yes	—
463	#95	C₂³×C₆₇	1	4	yes	—
464	#95	C₂³×C₆₇	1	4	yes	—
465	#95	C₂³×C₆₇	1	4	yes	—
466	#95	C₂³×C₆₇	1	4	yes	—
467	C₂×Q₈	—	2	2	no	—
468	C₂×Q₈	—	2	2	no	—
469	C₂×Q₈	—	2	4	no	—
470	C₂×Q₈	—	2	2	no	—
471	C₂×Q₈	—	2	2	no	—
472	C₂×Q₈	—	2	2	no	—
473	C₂×Q₈	—	4	2	no	—
474	C₂×Q₈	—	4	2	no	—
475	C₂×Q₈	#41×C₆₇	1	4	yes	—
476	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
477	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
478	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
479	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
480	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
481	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
482	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
484	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
485	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
486	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
487	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
488	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
489	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
490	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
491	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
492	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	8	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	4	no	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	4	no	—
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	8	no	—
504	C₄⋊C₄	—	2	4	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	4	2	no	—
510	C₄⋊C₄	—	4	4	no	—
511	C₄⋊C₄	—	4	4	no	—
512	C₄⋊C₄	—	4	4	no	—
513	C₄⋊C₄	—	4	2	no	—
514	C₄⋊C₄	—	4	2	no	—
515	C₄⋊C₄	—	4	4	no	—
516	C₄⋊C₄	—	4	4	no	—
517	C₄⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
518	C₄⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
519	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
520	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
521	C₄²	—	2	8	no	—
522	C₄²	—	2	4	no	—
523	C₄²	—	2	4	no	—
524	C₄²	—	2	4	no	—
525	C₄²	—	2	2	no	—
526	C₄²	—	2	2	no	—
527	C₄²	—	2	2	no	—
528	C₄²	—	2	4	no	—
529	C₄²	—	2	2	no	—
530	C₄²	—	4	4	no	—
531	C₄²	—	4	4	no	—
532	C₄²	—	4	1	no	—
533	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
534	C₄²	#41×C₆₇	1	4	yes	—
535	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
536	Q₈	—	2	8	no	—
537	Q₈	—	4	8	no	—
538	Q₈	—	4	4	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
578	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
579	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
580	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
581	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
582	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
583	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
584	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
585	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
586	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
587	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
588	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
589	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
590	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
591	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
592	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
593	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
594	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
595	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
596	C₂²	#585×C₆₇	1	1	yes	characteristic
597	C₄	—	2	2	no	—
598	C₄	—	2	4	no	—
599	C₄	—	2	4	no	—
600	C₄	—	2	2	no	—
601	C₄	—	2	2	no	—
602	C₄	—	2	4	no	—
603	C₄	—	2	2	no	—
604	C₄	—	4	4	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	1	no	—
607	C₄	—	4	2	no	—
608	C₄	—	4	2	no	—
609	C₄	—	4	4	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	2	no	—
612	C₄	#552×C₆₇	1	4	yes	—
613	C₄	#557×C₆₇	1	4	yes	—
614	C₂	C₂×#552×C₆₇	1	1	yes	characteristic
615	C₂	C₂×#557×C₆₇	1	1	yes	characteristic
616	C₂	Q₈×#41×C₆₇	1	1	yes	characteristic
617	C₂	C₆₇×#3503	1	1	yes	characteristic
618	C₂	C₆₇×#3506	1	1	yes	characteristic
619	C₂	C₆₇×#3522	1	1	yes	characteristic
620	C₂	C₆₇×#3555	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
89. #125×C₆₇	Q₈	semidirect
303. #59445	C₆₇	direct (N × H)
328. C₆₇	#59445	direct (N × H)
400. #125	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2774	2
C₆₇×#2570	2
C₆₇×#2567	2
C₆₇×#2552	2
C₆₇×#2511	2
C₆₇×#2508	2
C₆₇×#2506	2
C₆₇×#2462	2
C₆₇×#2446	2
C₆₇×#2370	2
C₆₇×#2355	2
Q₈×C₄⋊C₄×C₆₇	2
C₄×C₄⋊Q₈×C₆₇	2
C₄×#125×C₆₇	2
#59445	67

Subgroups — C67×#59445 #30738696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59445 #30738696