Subgroups — C₆₇×#59861 #30739112

Order 17152.

2062 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2808	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2804	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2724	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2597	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2593	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2561	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2484	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2458	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2405	C₂	1	2	yes	maximal
10	C₆₇×#2374	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2342	C₂	1	4	yes	maximal, complemented
12	C₄×C₂²⋊Q₈×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₂×#390×C₆₇	C₂	1	1	yes	maximal, characteristic
14	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#123×C₆₇	C₂²	1	2	yes	—
16	C₂×#123×C₆₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
19	C₂×#117×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	#400×C₆₇	C₂²	1	2	yes	complemented
25	#400×C₆₇	C₂²	1	2	yes	complemented
26	#398×C₆₇	C₂²	1	2	yes	complemented
27	#398×C₆₇	C₂²	1	2	yes	complemented
28	#397×C₆₇	C₂²	1	4	yes	complemented
29	#397×C₆₇	C₂²	1	4	yes	complemented
30	#394×C₆₇	C₂²	1	4	yes	—
31	#394×C₆₇	C₂²	1	4	yes	—
32	#393×C₆₇	C₂²	1	4	yes	complemented
33	#393×C₆₇	C₂²	1	4	yes	complemented
34	#391×C₆₇	C₂²	1	2	yes	complemented
35	#390×C₆₇	C₂²	1	1	yes	characteristic
36	#390×C₆₇	C₂²	1	1	yes	characteristic
37	#390×C₆₇	C₂²	1	1	yes	characteristic
38	#390×C₆₇	C₂²	1	4	yes	—
39	#390×C₆₇	C₂²	1	1	yes	characteristic
40	#389×C₆₇	C₂²	1	2	yes	complemented
41	#388×C₆₇	C₂²	1	4	yes	complemented
42	#388×C₆₇	C₂²	1	4	yes	complemented
43	#388×C₆₇	C₂²	1	4	yes	complemented
44	#388×C₆₇	C₂²	1	4	yes	complemented
45	#388×C₆₇	C₂²	1	4	yes	complemented
46	#388×C₆₇	C₂²	1	4	yes	complemented
47	#388×C₆₇	C₂²	1	4	yes	complemented
48	#388×C₆₇	C₂²	1	4	yes	complemented
49	#387×C₆₇	C₂²	1	4	yes	complemented
50	#387×C₆₇	C₂²	1	4	yes	complemented
51	#387×C₆₇	C₂²	1	4	yes	complemented
52	#386×C₆₇	C₂²	1	2	yes	—
53	#386×C₆₇	C₂²	1	2	yes	—
54	#385×C₆₇	C₂²	1	4	yes	—
55	#385×C₆₇	C₂²	1	4	yes	complemented
56	#385×C₆₇	C₂²	1	4	yes	complemented
57	#381×C₆₇	C₂²	1	2	yes	—
58	#381×C₆₇	C₂²	1	4	yes	—
59	#380×C₆₇	C₂²	1	2	yes	—
60	#379×C₆₇	C₂²	1	2	yes	—
61	#379×C₆₇	C₂²	1	4	yes	—
62	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
63	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
64	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
65	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
66	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
67	C₂⁵×C₆₇	C₂³	1	2	yes	—
68	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
69	C₂²×D₄×C₆₇	—	2	4	no	—
70	C₂²×D₄×C₆₇	—	4	2	no	—
71	C₂²×D₄×C₆₇	—	4	2	no	—
72	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
73	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
77	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
78	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
79	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
80	#123×C₆₇	—	2	8	no	—
81	#123×C₆₇	—	4	4	no	—
82	C₂²⋊Q₈×C₆₇	—	2	8	no	—
83	C₄×Q₈×C₆₇	—	4	4	no	—
84	#117×C₆₇	—	4	4	no	—
85	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
98	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
99	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
108	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
113	C₂×C₄²×C₆₇	—	4	2	no	—
114	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
115	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
118	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
119	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
120	#95×C₆₇	C₂³	1	4	yes	—
121	#95×C₆₇	C₂³	1	4	yes	—
122	#95×C₆₇	C₂³	1	4	yes	—
123	#95×C₆₇	C₂³	1	4	yes	—
124	#95×C₆₇	C₂³	1	4	yes	—
125	#95×C₆₇	C₂³	1	4	yes	—
126	#95×C₆₇	C₂³	1	4	yes	—
127	#95×C₆₇	C₂³	1	4	yes	—
128	#95×C₆₇	C₂³	1	4	yes	—
129	#95×C₆₇	C₂³	1	4	yes	—
130	#95×C₆₇	C₂³	1	2	yes	—
131	#95×C₆₇	C₂³	1	2	yes	—
132	C₂⁴×C₆₇	—	2	4	no	—
133	C₂⁴×C₆₇	—	2	8	no	—
134	C₂⁴×C₆₇	—	2	4	no	—
135	C₂⁴×C₆₇	—	4	2	no	—
136	C₂⁴×C₆₇	—	4	4	no	—
137	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
138	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
139	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
140	C₂×Q₈×C₆₇	—	2	4	no	—
141	C₂×Q₈×C₆₇	—	4	2	no	—
142	C₂×Q₈×C₆₇	—	4	2	no	—
143	C₂×D₄×C₆₇	—	2	4	no	—
144	C₂×D₄×C₆₇	—	2	2	no	—
145	C₂×D₄×C₆₇	—	2	4	no	—
146	C₂×D₄×C₆₇	—	4	1	no	—
147	C₂×D₄×C₆₇	—	4	1	no	—
148	C₂×D₄×C₆₇	—	4	1	no	—
149	C₂×D₄×C₆₇	—	4	2	no	—
150	C₂×D₄×C₆₇	—	4	1	no	—
151	C₂×D₄×C₆₇	—	4	16	no	—
152	C₂×D₄×C₆₇	#41	1	4	yes	—
153	C₂²×C₄×C₆₇	—	2	2	no	—
154	C₂²×C₄×C₆₇	—	2	2	no	—
155	C₂²×C₄×C₆₇	—	2	2	no	—
156	C₂²×C₄×C₆₇	—	2	8	no	—
157	C₂²×C₄×C₆₇	—	2	4	no	—
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	—	4	2	no	—
160	C₂²×C₄×C₆₇	—	4	2	no	—
161	C₂²×C₄×C₆₇	—	4	2	no	—
162	C₂²×C₄×C₆₇	—	4	2	no	—
163	C₂²×C₄×C₆₇	—	4	2	no	—
164	C₂²×C₄×C₆₇	—	4	4	no	—
165	C₂²×C₄×C₆₇	#41	1	8	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	4	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	2	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
176	C₄⋊C₄×C₆₇	—	2	4	no	—
177	C₄⋊C₄×C₆₇	—	2	8	no	—
178	C₄⋊C₄×C₆₇	—	2	8	no	—
179	C₄⋊C₄×C₆₇	—	2	8	no	—
180	C₄⋊C₄×C₆₇	—	4	1	no	—
181	C₄⋊C₄×C₆₇	—	4	2	no	—
182	C₄⋊C₄×C₆₇	—	4	1	no	—
183	C₄⋊C₄×C₆₇	—	4	2	no	—
184	C₄⋊C₄×C₆₇	—	4	2	no	—
185	C₄⋊C₄×C₆₇	—	4	2	no	—
186	C₄⋊C₄×C₆₇	—	4	2	no	—
187	C₂²⋊C₄×C₆₇	—	2	8	no	—
188	C₂²⋊C₄×C₆₇	—	2	4	no	—
189	C₂²⋊C₄×C₆₇	—	2	8	no	—
190	C₂²⋊C₄×C₆₇	—	2	8	no	—
191	C₂²⋊C₄×C₆₇	—	2	8	no	—
192	C₂²⋊C₄×C₆₇	—	2	8	no	—
193	C₂²⋊C₄×C₆₇	—	2	8	no	—
194	C₂²⋊C₄×C₆₇	—	2	8	no	—
195	C₂²⋊C₄×C₆₇	—	4	4	no	—
196	C₂²⋊C₄×C₆₇	—	4	4	no	—
197	C₂²⋊C₄×C₆₇	—	4	4	no	—
198	C₂²⋊C₄×C₆₇	—	4	4	no	—
199	C₂²⋊C₄×C₆₇	—	4	2	no	—
200	C₂²⋊C₄×C₆₇	—	4	4	no	—
201	C₄²×C₆₇	—	2	4	no	—
202	C₄²×C₆₇	—	4	1	no	—
203	C₄²×C₆₇	—	4	1	no	—
204	C₄²×C₆₇	—	4	4	no	—
205	C₄²×C₆₇	—	4	4	no	—
206	C₄²×C₆₇	—	4	4	no	—
207	C₂³×C₆₇	—	2	1	no	—
208	C₂³×C₆₇	—	2	1	no	—
209	C₂³×C₆₇	—	2	2	no	—
210	C₂³×C₆₇	—	2	4	no	—
211	C₂³×C₆₇	—	2	2	no	—
212	C₂³×C₆₇	—	2	2	no	—
213	C₂³×C₆₇	—	2	4	no	—
214	C₂³×C₆₇	—	2	4	no	—
215	C₂³×C₆₇	—	2	2	no	—
216	C₂³×C₆₇	—	2	2	no	—
217	C₂³×C₆₇	—	2	2	no	—
218	C₂³×C₆₇	—	2	4	no	—
219	C₂³×C₆₇	—	4	8	no	—
220	C₂³×C₆₇	—	4	16	no	—
221	C₂³×C₆₇	—	4	8	no	—
222	C₂³×C₆₇	—	8	8	no	—
223	C₂³×C₆₇	—	8	4	no	—
224	C₂³×C₆₇	C₂×#41	1	4	yes	—
225	C₂³×C₆₇	C₂×#41	1	4	yes	—
226	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
227	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
228	Q₈×C₆₇	—	4	8	no	—
229	D₄×C₆₇	—	4	4	no	—
230	D₄×C₆₇	—	4	8	no	—
231	D₄×C₆₇	—	4	4	no	—
232	C₂×C₄×C₆₇	—	2	8	no	—
233	C₂×C₄×C₆₇	—	2	1	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	2	no	—
236	C₂×C₄×C₆₇	—	2	1	no	—
237	C₂×C₄×C₆₇	—	2	1	no	—
238	C₂×C₄×C₆₇	—	2	1	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	4	no	—
243	C₂×C₄×C₆₇	—	2	4	no	—
244	C₂×C₄×C₆₇	—	2	4	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	4	no	—
249	C₂×C₄×C₆₇	—	2	4	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	4	no	—
254	C₂×C₄×C₆₇	—	2	4	no	—
255	C₂×C₄×C₆₇	—	2	4	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	4	8	no	—
258	C₂×C₄×C₆₇	—	4	8	no	—
259	C₂×C₄×C₆₇	—	8	4	no	—
260	C₂×C₄×C₆₇	—	8	4	no	—
261	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
262	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
263	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
264	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
265	C₂²×C₆₇	—	2	4	no	—
266	C₂²×C₆₇	—	2	2	no	—
267	C₂²×C₆₇	—	2	4	no	—
268	C₂²×C₆₇	—	2	2	no	—
269	C₂²×C₆₇	—	2	8	no	—
270	C₂²×C₆₇	—	2	4	no	—
271	C₂²×C₆₇	—	4	2	no	—
272	C₂²×C₆₇	—	4	4	no	—
273	C₂²×C₆₇	—	4	2	no	—
274	C₂²×C₆₇	—	4	4	no	—
275	C₂²×C₆₇	—	4	2	no	—
276	C₂²×C₆₇	—	4	2	no	—
277	C₂²×C₆₇	—	4	4	no	—
278	C₂²×C₆₇	—	4	2	no	—
279	C₂²×C₆₇	—	8	16	no	—
280	C₂²×C₆₇	#559	1	4	yes	—
281	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
282	C₂²×C₆₇	C₂²×#41	1	2	yes	—
283	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
284	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
285	C₂²×C₆₇	#585	1	2	yes	—
286	C₄×C₆₇	—	2	2	no	—
287	C₄×C₆₇	—	2	2	no	—
288	C₄×C₆₇	—	2	4	no	—
289	C₄×C₆₇	—	2	4	no	—
290	C₄×C₆₇	—	4	4	no	—
291	C₄×C₆₇	—	4	2	no	—
292	C₄×C₆₇	—	4	2	no	—
293	C₄×C₆₇	—	4	4	no	—
294	C₄×C₆₇	—	4	4	no	—
295	C₄×C₆₇	—	4	4	no	—
296	#59861	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
297	C₂×C₆₇	—	2	4	no	—
298	C₂×C₆₇	—	4	4	no	—
299	C₂×C₆₇	—	8	2	no	—
300	C₂×C₆₇	#3475	1	2	yes	—
301	C₂×C₆₇	C₂×#559	1	1	yes	characteristic
302	C₂×C₆₇	#3490	1	1	yes	characteristic
303	C₂×C₆₇	#3505	1	2	yes	—
304	C₂×C₆₇	#3545	1	1	yes	characteristic
305	#2808	C₂×C₆₇	1	2	yes	complemented
306	#2804	C₂×C₆₇	1	1	yes	characteristic, complemented
307	#2724	C₂×C₆₇	1	1	yes	characteristic, complemented
308	#2597	C₂×C₆₇	1	4	yes	complemented
309	#2593	C₂×C₆₇	1	2	yes	complemented
310	#2561	C₂×C₆₇	1	4	yes	complemented
311	#2484	C₂×C₆₇	1	2	yes	complemented
312	#2458	C₂×C₆₇	1	4	yes	complemented
313	#2405	C₂×C₆₇	1	2	yes	—
314	#2374	C₂×C₆₇	1	2	yes	complemented
315	#2342	C₂×C₆₇	1	4	yes	complemented
316	C₄×C₂²⋊Q₈	C₂×C₆₇	1	2	yes	complemented
317	C₂×#390	C₂×C₆₇	1	1	yes	characteristic
318	C₆₇	#59861	1	1	yes	characteristic, direct product factor, Sylow
319	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
320	C₂×#123	C₂²×C₆₇	1	2	yes	—
321	C₂×#123	C₂²×C₆₇	1	2	yes	—
322	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
323	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
324	C₂×#117	C₂²×C₆₇	1	2	yes	—
325	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
326	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
327	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
328	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
329	#400	C₂²×C₆₇	1	2	yes	complemented
330	#400	C₂²×C₆₇	1	2	yes	complemented
331	#398	C₂²×C₆₇	1	2	yes	complemented
332	#398	C₂²×C₆₇	1	2	yes	complemented
333	#397	C₂²×C₆₇	1	4	yes	complemented
334	#397	C₂²×C₆₇	1	4	yes	complemented
335	#394	C₂²×C₆₇	1	4	yes	—
336	#394	C₂²×C₆₇	1	4	yes	—
337	#393	C₂²×C₆₇	1	4	yes	complemented
338	#393	C₂²×C₆₇	1	4	yes	complemented
339	#391	C₂²×C₆₇	1	2	yes	complemented
340	#390	C₂²×C₆₇	1	1	yes	characteristic
341	#390	C₂²×C₆₇	1	4	yes	—
342	#390	C₂²×C₆₇	1	1	yes	characteristic
343	#390	C₂²×C₆₇	1	1	yes	characteristic
344	#390	C₂²×C₆₇	1	1	yes	characteristic
345	#389	C₂²×C₆₇	1	2	yes	complemented
346	#388	C₂²×C₆₇	1	4	yes	complemented
347	#388	C₂²×C₆₇	1	4	yes	complemented
348	#388	C₂²×C₆₇	1	4	yes	complemented
349	#388	C₂²×C₆₇	1	4	yes	complemented
350	#388	C₂²×C₆₇	1	4	yes	complemented
351	#388	C₂²×C₆₇	1	4	yes	complemented
352	#388	C₂²×C₆₇	1	4	yes	complemented
353	#388	C₂²×C₆₇	1	4	yes	complemented
354	#387	C₂²×C₆₇	1	4	yes	complemented
355	#387	C₂²×C₆₇	1	4	yes	complemented
356	#387	C₂²×C₆₇	1	4	yes	complemented
357	#386	C₂²×C₆₇	1	2	yes	—
358	#386	C₂²×C₆₇	1	2	yes	—
359	#385	C₂²×C₆₇	1	4	yes	—
360	#385	C₂²×C₆₇	1	4	yes	complemented
361	#385	C₂²×C₆₇	1	4	yes	complemented
362	#381	C₂²×C₆₇	1	2	yes	—
363	#381	C₂²×C₆₇	1	4	yes	—
364	#380	C₂²×C₆₇	1	2	yes	—
365	#379	C₂²×C₆₇	1	2	yes	—
366	#379	C₂²×C₆₇	1	4	yes	—
367	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
368	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
369	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
370	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
371	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
372	C₂⁵	C₂³×C₆₇	1	2	yes	—
373	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
374	C₂²×D₄	—	2	4	no	—
375	C₂²×D₄	—	4	2	no	—
376	C₂²×D₄	—	4	2	no	—
377	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
378	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
379	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
380	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
381	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
382	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
383	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
384	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
385	#123	—	2	8	no	—
386	#123	—	4	4	no	—
387	C₂²⋊Q₈	—	2	8	no	—
388	C₄×Q₈	—	4	4	no	—
389	#117	—	4	4	no	—
390	C₂×C₄⋊C₄	—	4	2	no	—
391	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
392	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
393	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
395	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
397	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
398	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
399	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
400	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
401	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
402	C₂×C₂²⋊C₄	—	2	16	no	—
403	C₂×C₂²⋊C₄	—	4	4	no	—
404	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
418	C₂×C₄²	—	4	2	no	—
419	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
420	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
421	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
422	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
423	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
424	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
425	#95	C₂³×C₆₇	1	4	yes	—
426	#95	C₂³×C₆₇	1	4	yes	—
427	#95	C₂³×C₆₇	1	4	yes	—
428	#95	C₂³×C₆₇	1	4	yes	—
429	#95	C₂³×C₆₇	1	4	yes	—
430	#95	C₂³×C₆₇	1	4	yes	—
431	#95	C₂³×C₆₇	1	4	yes	—
432	#95	C₂³×C₆₇	1	4	yes	—
433	#95	C₂³×C₆₇	1	4	yes	—
434	#95	C₂³×C₆₇	1	4	yes	—
435	#95	C₂³×C₆₇	1	2	yes	—
436	#95	C₂³×C₆₇	1	2	yes	—
437	C₂⁴	—	2	4	no	—
438	C₂⁴	—	2	8	no	—
439	C₂⁴	—	2	4	no	—
440	C₂⁴	—	4	2	no	—
441	C₂⁴	—	4	4	no	—
442	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
443	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
444	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
445	C₂×Q₈	—	2	4	no	—
446	C₂×Q₈	—	4	2	no	—
447	C₂×Q₈	—	4	2	no	—
448	C₂×D₄	—	2	4	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	4	no	—
451	C₂×D₄	—	4	1	no	—
452	C₂×D₄	—	4	1	no	—
453	C₂×D₄	—	4	1	no	—
454	C₂×D₄	—	4	2	no	—
455	C₂×D₄	—	4	1	no	—
456	C₂×D₄	—	4	16	no	—
457	C₂×D₄	#41×C₆₇	1	4	yes	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	8	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	4	2	no	—
465	C₂²×C₄	—	4	2	no	—
466	C₂²×C₄	—	4	2	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	—	4	2	no	—
469	C₂²×C₄	—	4	4	no	—
470	C₂²×C₄	#41×C₆₇	1	8	yes	—
471	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
472	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
473	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
474	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
475	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
476	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
478	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
479	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
480	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
481	C₄⋊C₄	—	2	4	no	—
482	C₄⋊C₄	—	2	8	no	—
483	C₄⋊C₄	—	2	8	no	—
484	C₄⋊C₄	—	2	8	no	—
485	C₄⋊C₄	—	4	1	no	—
486	C₄⋊C₄	—	4	2	no	—
487	C₄⋊C₄	—	4	1	no	—
488	C₄⋊C₄	—	4	2	no	—
489	C₄⋊C₄	—	4	2	no	—
490	C₄⋊C₄	—	4	2	no	—
491	C₄⋊C₄	—	4	2	no	—
492	C₂²⋊C₄	—	2	8	no	—
493	C₂²⋊C₄	—	2	4	no	—
494	C₂²⋊C₄	—	2	8	no	—
495	C₂²⋊C₄	—	2	8	no	—
496	C₂²⋊C₄	—	2	8	no	—
497	C₂²⋊C₄	—	2	8	no	—
498	C₂²⋊C₄	—	2	8	no	—
499	C₂²⋊C₄	—	2	8	no	—
500	C₂²⋊C₄	—	4	4	no	—
501	C₂²⋊C₄	—	4	2	no	—
502	C₂²⋊C₄	—	4	4	no	—
503	C₂²⋊C₄	—	4	4	no	—
504	C₂²⋊C₄	—	4	4	no	—
505	C₂²⋊C₄	—	4	4	no	—
506	C₄²	—	2	4	no	—
507	C₄²	—	4	1	no	—
508	C₄²	—	4	1	no	—
509	C₄²	—	4	4	no	—
510	C₄²	—	4	4	no	—
511	C₄²	—	4	4	no	—
512	C₂³	—	2	1	no	—
513	C₂³	—	2	1	no	—
514	C₂³	—	2	2	no	—
515	C₂³	—	2	4	no	—
516	C₂³	—	2	2	no	—
517	C₂³	—	2	2	no	—
518	C₂³	—	2	4	no	—
519	C₂³	—	2	4	no	—
520	C₂³	—	2	2	no	—
521	C₂³	—	2	2	no	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	4	no	—
524	C₂³	—	4	8	no	—
525	C₂³	—	4	16	no	—
526	C₂³	—	4	8	no	—
527	C₂³	—	8	4	no	—
528	C₂³	—	8	8	no	—
529	C₂³	C₂×#41×C₆₇	1	4	yes	—
530	C₂³	C₂×#41×C₆₇	1	4	yes	—
531	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
532	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
533	Q₈	—	4	8	no	—
534	D₄	—	4	4	no	—
535	D₄	—	4	8	no	—
536	D₄	—	4	4	no	—
537	C₂×C₄	—	2	8	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	4	8	no	—
563	C₂×C₄	—	4	8	no	—
564	C₂×C₄	—	8	4	no	—
565	C₂×C₄	—	8	4	no	—
566	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
567	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
568	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
569	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	4	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	4	no	—
574	C₂²	—	2	8	no	—
575	C₂²	—	2	4	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	4	2	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	2	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	4	4	no	—
584	C₂²	—	8	16	no	—
585	C₂²	#559×C₆₇	1	4	yes	—
586	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
587	C₂²	C₂²×#41×C₆₇	1	2	yes	—
588	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
589	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
590	C₂²	#585×C₆₇	1	2	yes	—
591	C₄	—	2	2	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	4	no	—
594	C₄	—	2	4	no	—
595	C₄	—	4	4	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	2	no	—
598	C₄	—	4	2	no	—
599	C₄	—	4	4	no	—
600	C₄	—	4	4	no	—
601	C₂	—	2	4	no	—
602	C₂	—	4	4	no	—
603	C₂	—	8	2	no	—
604	C₂	C₆₇×#3475	1	2	yes	—
605	C₂	C₂×#559×C₆₇	1	1	yes	characteristic
606	C₂	C₆₇×#3490	1	1	yes	characteristic
607	C₂	C₆₇×#3505	1	2	yes	—
608	C₂	C₆₇×#3545	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
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
1. C₆₇×#2808	C₂	semidirect
2. C₆₇×#2804	C₂	semidirect
3. C₆₇×#2724	C₂	semidirect
4. C₆₇×#2597	C₂	semidirect
5. C₆₇×#2593	C₂	semidirect
6. C₆₇×#2561	C₂	semidirect
7. C₆₇×#2484	C₂	semidirect
8. C₆₇×#2458	C₂	semidirect
10. C₆₇×#2374	C₂	semidirect
11. C₆₇×#2342	C₂	semidirect
12. C₄×C₂²⋊Q₈×C₆₇	C₂	semidirect
18. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
24. #400×C₆₇	C₂²	semidirect
25. #400×C₆₇	C₂²	semidirect
26. #398×C₆₇	C₂²	semidirect
27. #398×C₆₇	C₂²	semidirect
28. #397×C₆₇	C₂²	semidirect
29. #397×C₆₇	C₂²	semidirect
32. #393×C₆₇	C₂²	semidirect
33. #393×C₆₇	C₂²	semidirect
34. #391×C₆₇	C₂²	semidirect
40. #389×C₆₇	C₂²	semidirect
41. #388×C₆₇	C₂²	semidirect
42. #388×C₆₇	C₂²	semidirect
43. #388×C₆₇	C₂²	semidirect
44. #388×C₆₇	C₂²	semidirect
45. #388×C₆₇	C₂²	semidirect
46. #388×C₆₇	C₂²	semidirect
47. #388×C₆₇	C₂²	semidirect
48. #388×C₆₇	C₂²	semidirect
49. #387×C₆₇	C₂²	semidirect
50. #387×C₆₇	C₂²	semidirect
51. #387×C₆₇	C₂²	semidirect
55. #385×C₆₇	C₂²	semidirect
56. #385×C₆₇	C₂²	semidirect
62. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
63. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
65. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
66. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
296. #59861	C₆₇	direct (N × H)
305. #2808	C₂×C₆₇	semidirect
306. #2804	C₂×C₆₇	semidirect
307. #2724	C₂×C₆₇	semidirect
308. #2597	C₂×C₆₇	semidirect
309. #2593	C₂×C₆₇	semidirect
310. #2561	C₂×C₆₇	semidirect
311. #2484	C₂×C₆₇	semidirect
312. #2458	C₂×C₆₇	semidirect
314. #2374	C₂×C₆₇	semidirect
315. #2342	C₂×C₆₇	semidirect
316. C₄×C₂²⋊Q₈	C₂×C₆₇	semidirect
318. C₆₇	#59861	direct (N × H)
323. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
329. #400	C₂²×C₆₇	semidirect
330. #400	C₂²×C₆₇	semidirect
331. #398	C₂²×C₆₇	semidirect
332. #398	C₂²×C₆₇	semidirect
333. #397	C₂²×C₆₇	semidirect
334. #397	C₂²×C₆₇	semidirect
337. #393	C₂²×C₆₇	semidirect
338. #393	C₂²×C₆₇	semidirect
339. #391	C₂²×C₆₇	semidirect
345. #389	C₂²×C₆₇	semidirect
346. #388	C₂²×C₆₇	semidirect
347. #388	C₂²×C₆₇	semidirect
348. #388	C₂²×C₆₇	semidirect
349. #388	C₂²×C₆₇	semidirect
350. #388	C₂²×C₆₇	semidirect
351. #388	C₂²×C₆₇	semidirect
352. #388	C₂²×C₆₇	semidirect
353. #388	C₂²×C₆₇	semidirect
354. #387	C₂²×C₆₇	semidirect
355. #387	C₂²×C₆₇	semidirect
356. #387	C₂²×C₆₇	semidirect
360. #385	C₂²×C₆₇	semidirect
361. #385	C₂²×C₆₇	semidirect
367. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
368. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
370. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
371. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2808	2
C₆₇×#2804	2
C₆₇×#2724	2
C₆₇×#2597	2
C₆₇×#2593	2
C₆₇×#2561	2
C₆₇×#2484	2
C₆₇×#2458	2
C₆₇×#2405	2
C₆₇×#2374	2
C₆₇×#2342	2
C₄×C₂²⋊Q₈×C₆₇	2
C₂×#390×C₆₇	2
#59861	67

Subgroups — C67×#59861 #30739112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59861 #30739112