Subgroups — C₆₇×#59632 #30738883

Order 17152.

2044 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2752	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2709	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2602	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2597	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2541	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#2490	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2490	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2487	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2455	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2445	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2370	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2366	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2361	C₂	1	2	yes	maximal, complemented
14	C₄×#123×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×#121×C₆₇	C₂	1	2	yes	maximal, complemented
16	C₂×#127×C₆₇	C₂²	1	2	yes	—
17	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#125×C₆₇	C₂²	1	2	yes	—
19	C₂×#123×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#123×C₆₇	C₂²	1	4	yes	—
21	C₂×#121×C₆₇	C₂²	1	2	yes	—
22	C₂×#121×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
27	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	#400×C₆₇	C₂²	1	2	yes	—
30	#399×C₆₇	C₂²	1	2	yes	—
31	#397×C₆₇	C₂²	1	4	yes	—
32	#397×C₆₇	C₂²	1	4	yes	—
33	#394×C₆₇	C₂²	1	4	yes	—
34	#394×C₆₇	C₂²	1	4	yes	—
35	#394×C₆₇	C₂²	1	4	yes	—
36	#394×C₆₇	C₂²	1	4	yes	—
37	#392×C₆₇	C₂²	1	2	yes	—
38	#392×C₆₇	C₂²	1	2	yes	—
39	#390×C₆₇	C₂²	1	2	yes	—
40	#390×C₆₇	C₂²	1	2	yes	—
41	#390×C₆₇	C₂²	1	4	yes	—
42	#390×C₆₇	C₂²	1	2	yes	—
43	#390×C₆₇	C₂²	1	4	yes	—
44	#390×C₆₇	C₂²	1	2	yes	—
45	#389×C₆₇	C₂²	1	2	yes	—
46	#389×C₆₇	C₂²	1	4	yes	—
47	#389×C₆₇	C₂²	1	2	yes	—
48	#388×C₆₇	C₂²	1	4	yes	—
49	#388×C₆₇	C₂²	1	2	yes	—
50	#388×C₆₇	C₂²	1	4	yes	—
51	#388×C₆₇	C₂²	1	4	yes	—
52	#388×C₆₇	C₂²	1	4	yes	—
53	#387×C₆₇	C₂²	1	2	yes	—
54	#387×C₆₇	C₂²	1	4	yes	—
55	#387×C₆₇	C₂²	1	4	yes	—
56	#386×C₆₇	C₂²	1	4	yes	—
57	#386×C₆₇	C₂²	1	1	yes	characteristic
58	#386×C₆₇	C₂²	1	4	yes	—
59	#386×C₆₇	C₂²	1	4	yes	—
60	#386×C₆₇	C₂²	1	2	yes	—
61	#385×C₆₇	C₂²	1	1	yes	characteristic
62	#385×C₆₇	C₂²	1	4	yes	—
63	#385×C₆₇	C₂²	1	2	yes	—
64	#381×C₆₇	C₂²	1	1	yes	characteristic
65	#380×C₆₇	C₂²	1	2	yes	—
66	#380×C₆₇	C₂²	1	1	yes	characteristic
67	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
68	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
69	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
70	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
71	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
72	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
73	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
74	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
75	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
76	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
77	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
78	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
79	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
80	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
86	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
87	#127×C₆₇	—	2	8	no	—
88	#127×C₆₇	—	4	2	no	—
89	#125×C₆₇	—	2	8	no	—
90	#123×C₆₇	—	2	4	no	—
91	#123×C₆₇	—	4	8	no	—
92	#121×C₆₇	—	2	8	no	—
93	#121×C₆₇	—	2	8	no	—
94	C₄×D₄×C₆₇	—	4	4	no	—
95	C₄×D₄×C₆₇	—	4	2	no	—
96	C₄×D₄×C₆₇	—	4	8	no	—
97	C₄×D₄×C₆₇	—	4	8	no	—
98	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	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	4	yes	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
110	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
121	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
123	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
124	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
125	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
126	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
127	C₂×C₄²×C₆₇	—	4	2	no	—
128	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
129	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
132	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
133	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
134	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
135	#95×C₆₇	C₂³	1	4	yes	—
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	2	yes	—
141	#95×C₆₇	C₂³	1	2	yes	—
142	#95×C₆₇	C₂³	1	4	yes	—
143	#95×C₆₇	C₂³	1	4	yes	—
144	#95×C₆₇	C₂³	1	4	yes	—
145	#95×C₆₇	C₂³	1	2	yes	—
146	#95×C₆₇	C₂³	1	2	yes	—
147	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
148	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
149	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
150	C₂×D₄×C₆₇	—	2	8	no	—
151	C₂×D₄×C₆₇	—	2	8	no	—
152	C₂×D₄×C₆₇	—	2	8	no	—
153	C₂×D₄×C₆₇	—	2	8	no	—
154	C₂×D₄×C₆₇	—	2	4	no	—
155	C₂×D₄×C₆₇	—	2	4	no	—
156	C₂×D₄×C₆₇	—	2	4	no	—
157	C₂×D₄×C₆₇	—	2	4	no	—
158	C₂×D₄×C₆₇	—	4	1	no	—
159	C₂×D₄×C₆₇	—	4	4	no	—
160	C₂×D₄×C₆₇	—	4	4	no	—
161	C₂×D₄×C₆₇	—	4	1	no	—
162	C₂×D₄×C₆₇	—	4	4	no	—
163	C₂×D₄×C₆₇	—	4	2	no	—
164	C₂×D₄×C₆₇	—	4	2	no	—
165	C₂×D₄×C₆₇	—	4	2	no	—
166	C₂×D₄×C₆₇	#41	1	4	yes	—
167	C₂²×C₄×C₆₇	—	2	2	no	—
168	C₂²×C₄×C₆₇	—	2	4	no	—
169	C₂²×C₄×C₆₇	—	2	4	no	—
170	C₂²×C₄×C₆₇	—	2	2	no	—
171	C₂²×C₄×C₆₇	—	2	2	no	—
172	C₂²×C₄×C₆₇	—	2	4	no	—
173	C₂²×C₄×C₆₇	—	2	8	no	—
174	C₂²×C₄×C₆₇	—	4	1	no	—
175	C₂²×C₄×C₆₇	—	4	1	no	—
176	C₂²×C₄×C₆₇	—	4	1	no	—
177	C₂²×C₄×C₆₇	—	4	2	no	—
178	C₂²×C₄×C₆₇	—	4	4	no	—
179	C₂²×C₄×C₆₇	—	4	4	no	—
180	C₂²×C₄×C₆₇	—	4	4	no	—
181	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
187	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
188	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
190	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
191	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
193	C₄⋊C₄×C₆₇	—	2	4	no	—
194	C₄⋊C₄×C₆₇	—	2	8	no	—
195	C₄⋊C₄×C₆₇	—	2	8	no	—
196	C₄⋊C₄×C₆₇	—	2	4	no	—
197	C₄⋊C₄×C₆₇	—	2	4	no	—
198	C₄⋊C₄×C₆₇	—	2	8	no	—
199	C₄⋊C₄×C₆₇	—	2	4	no	—
200	C₄⋊C₄×C₆₇	—	4	2	no	—
201	C₄⋊C₄×C₆₇	—	4	1	no	—
202	C₄⋊C₄×C₆₇	—	4	4	no	—
203	C₄⋊C₄×C₆₇	#41	1	8	yes	—
204	C₂²⋊C₄×C₆₇	—	2	2	no	—
205	C₂²⋊C₄×C₆₇	—	2	4	no	—
206	C₂²⋊C₄×C₆₇	—	2	8	no	—
207	C₂²⋊C₄×C₆₇	—	2	8	no	—
208	C₂²⋊C₄×C₆₇	—	2	4	no	—
209	C₂²⋊C₄×C₆₇	—	2	8	no	—
210	C₂²⋊C₄×C₆₇	—	4	2	no	—
211	C₂²⋊C₄×C₆₇	—	4	1	no	—
212	C₂²⋊C₄×C₆₇	—	4	4	no	—
213	C₂²⋊C₄×C₆₇	—	4	4	no	—
214	C₂²⋊C₄×C₆₇	—	4	4	no	—
215	C₂²⋊C₄×C₆₇	—	4	1	no	—
216	C₂²⋊C₄×C₆₇	—	4	1	no	—
217	C₂²⋊C₄×C₆₇	—	4	4	no	—
218	C₂²⋊C₄×C₆₇	—	4	4	no	—
219	C₂²⋊C₄×C₆₇	—	4	4	no	—
220	C₂²⋊C₄×C₆₇	—	4	4	no	—
221	C₄²×C₆₇	—	2	4	no	—
222	C₄²×C₆₇	—	2	4	no	—
223	C₄²×C₆₇	—	2	4	no	—
224	C₄²×C₆₇	—	2	2	no	—
225	C₄²×C₆₇	—	4	2	no	—
226	C₄²×C₆₇	—	4	4	no	—
227	C₄²×C₆₇	—	4	4	no	—
228	C₂³×C₆₇	—	2	2	no	—
229	C₂³×C₆₇	—	2	2	no	—
230	C₂³×C₆₇	—	2	2	no	—
231	C₂³×C₆₇	—	2	2	no	—
232	C₂³×C₆₇	—	2	1	no	—
233	C₂³×C₆₇	—	2	1	no	—
234	C₂³×C₆₇	—	2	1	no	—
235	C₂³×C₆₇	—	2	1	no	—
236	C₂³×C₆₇	—	2	4	no	—
237	C₂³×C₆₇	—	2	4	no	—
238	C₂³×C₆₇	—	2	4	no	—
239	C₂³×C₆₇	—	2	4	no	—
240	C₂³×C₆₇	—	2	4	no	—
241	C₂³×C₆₇	—	2	4	no	—
242	C₂³×C₆₇	—	2	4	no	—
243	C₂³×C₆₇	C₂×#41	1	2	yes	—
244	C₂³×C₆₇	C₂×#41	1	2	yes	—
245	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
246	D₄×C₆₇	—	4	4	no	—
247	D₄×C₆₇	—	4	16	no	—
248	D₄×C₆₇	—	4	16	no	—
249	D₄×C₆₇	—	4	8	no	—
250	D₄×C₆₇	—	8	4	no	—
251	D₄×C₆₇	—	8	4	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	2	no	—
254	C₂×C₄×C₆₇	—	2	1	no	—
255	C₂×C₄×C₆₇	—	2	4	no	—
256	C₂×C₄×C₆₇	—	2	2	no	—
257	C₂×C₄×C₆₇	—	2	1	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	2	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	2	no	—
264	C₂×C₄×C₆₇	—	2	4	no	—
265	C₂×C₄×C₆₇	—	2	2	no	—
266	C₂×C₄×C₆₇	—	2	1	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	2	1	no	—
269	C₂×C₄×C₆₇	—	2	4	no	—
270	C₂×C₄×C₆₇	—	2	4	no	—
271	C₂×C₄×C₆₇	—	2	2	no	—
272	C₂×C₄×C₆₇	—	2	4	no	—
273	C₂×C₄×C₆₇	—	2	4	no	—
274	C₂×C₄×C₆₇	—	2	1	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	2	2	no	—
277	C₂×C₄×C₆₇	—	2	4	no	—
278	C₂×C₄×C₆₇	—	2	2	no	—
279	C₂×C₄×C₆₇	—	4	4	no	—
280	C₂×C₄×C₆₇	—	4	4	no	—
281	C₂×C₄×C₆₇	—	4	4	no	—
282	C₂×C₄×C₆₇	—	4	4	no	—
283	C₂×C₄×C₆₇	—	8	8	no	—
284	C₂×C₄×C₆₇	—	8	4	no	—
285	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
286	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
287	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
288	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
289	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
290	C₂²×C₆₇	—	2	4	no	—
291	C₂²×C₆₇	—	2	4	no	—
292	C₂²×C₆₇	—	2	2	no	—
293	C₂²×C₆₇	—	2	2	no	—
294	C₂²×C₆₇	—	4	1	no	—
295	C₂²×C₆₇	—	4	1	no	—
296	C₂²×C₆₇	—	4	2	no	—
297	C₂²×C₆₇	—	4	2	no	—
298	C₂²×C₆₇	—	4	1	no	—
299	C₂²×C₆₇	—	4	1	no	—
300	C₂²×C₆₇	—	4	4	no	—
301	C₂²×C₆₇	—	4	4	no	—
302	C₂²×C₆₇	—	4	4	no	—
303	C₂²×C₆₇	—	4	4	no	—
304	C₂²×C₆₇	—	4	4	no	—
305	C₂²×C₆₇	—	4	4	no	—
306	C₂²×C₆₇	—	4	4	no	—
307	C₂²×C₆₇	C₂²×#41	1	2	yes	—
308	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
309	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
310	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
311	C₂²×C₆₇	#585	1	1	yes	characteristic
312	C₄×C₆₇	—	2	8	no	—
313	C₄×C₆₇	—	2	2	no	—
314	C₄×C₆₇	—	2	4	no	—
315	C₄×C₆₇	—	4	1	no	—
316	C₄×C₆₇	—	4	1	no	—
317	C₄×C₆₇	—	4	2	no	—
318	C₄×C₆₇	—	4	2	no	—
319	C₄×C₆₇	—	4	2	no	—
320	C₄×C₆₇	—	4	4	no	—
321	C₄×C₆₇	—	4	4	no	—
322	C₄×C₆₇	—	4	1	no	—
323	C₄×C₆₇	#559	1	4	yes	—
324	#59632	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
325	C₂×C₆₇	—	4	2	no	—
326	C₂×C₆₇	—	4	2	no	—
327	C₂×C₆₇	—	8	4	no	—
328	C₂×C₆₇	C₂×#559	1	1	yes	characteristic
329	C₂×C₆₇	#3481	1	2	yes	—
330	C₂×C₆₇	#3505	1	1	yes	characteristic
331	C₂×C₆₇	#3517	1	2	yes	—
332	C₂×C₆₇	#3537	1	1	yes	characteristic
333	#2752	C₂×C₆₇	1	2	yes	complemented
334	#2709	C₂×C₆₇	1	2	yes	complemented
335	#2602	C₂×C₆₇	1	4	yes	complemented
336	#2597	C₂×C₆₇	1	4	yes	complemented
337	#2541	C₂×C₆₇	1	1	yes	characteristic
338	#2490	C₂×C₆₇	1	2	yes	complemented
339	#2490	C₂×C₆₇	1	2	yes	complemented
340	#2487	C₂×C₆₇	1	1	yes	characteristic, complemented
341	#2455	C₂×C₆₇	1	2	yes	complemented
342	#2445	C₂×C₆₇	1	2	yes	complemented
343	#2370	C₂×C₆₇	1	2	yes	complemented
344	#2366	C₂×C₆₇	1	2	yes	complemented
345	#2361	C₂×C₆₇	1	2	yes	complemented
346	C₄×#123	C₂×C₆₇	1	1	yes	characteristic, complemented
347	C₄×#121	C₂×C₆₇	1	2	yes	complemented
348	C₆₇	#59632	1	1	yes	characteristic, direct product factor, Sylow
349	C₂×#127	C₂²×C₆₇	1	2	yes	—
350	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
351	C₂×#125	C₂²×C₆₇	1	2	yes	—
352	C₂×#123	C₂²×C₆₇	1	1	yes	characteristic
353	C₂×#123	C₂²×C₆₇	1	4	yes	—
354	C₂×#121	C₂²×C₆₇	1	2	yes	—
355	C₂×#121	C₂²×C₆₇	1	2	yes	—
356	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
357	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
358	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
359	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
360	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
361	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
362	#400	C₂²×C₆₇	1	2	yes	—
363	#399	C₂²×C₆₇	1	2	yes	—
364	#397	C₂²×C₆₇	1	4	yes	—
365	#397	C₂²×C₆₇	1	4	yes	—
366	#394	C₂²×C₆₇	1	4	yes	—
367	#394	C₂²×C₆₇	1	4	yes	—
368	#394	C₂²×C₆₇	1	4	yes	—
369	#394	C₂²×C₆₇	1	4	yes	—
370	#392	C₂²×C₆₇	1	2	yes	—
371	#392	C₂²×C₆₇	1	2	yes	—
372	#390	C₂²×C₆₇	1	2	yes	—
373	#390	C₂²×C₆₇	1	4	yes	—
374	#390	C₂²×C₆₇	1	2	yes	—
375	#390	C₂²×C₆₇	1	4	yes	—
376	#390	C₂²×C₆₇	1	2	yes	—
377	#390	C₂²×C₆₇	1	2	yes	—
378	#389	C₂²×C₆₇	1	4	yes	—
379	#389	C₂²×C₆₇	1	2	yes	—
380	#389	C₂²×C₆₇	1	2	yes	—
381	#388	C₂²×C₆₇	1	4	yes	—
382	#388	C₂²×C₆₇	1	4	yes	—
383	#388	C₂²×C₆₇	1	4	yes	—
384	#388	C₂²×C₆₇	1	4	yes	—
385	#388	C₂²×C₆₇	1	2	yes	—
386	#387	C₂²×C₆₇	1	2	yes	—
387	#387	C₂²×C₆₇	1	4	yes	—
388	#387	C₂²×C₆₇	1	4	yes	—
389	#386	C₂²×C₆₇	1	4	yes	—
390	#386	C₂²×C₆₇	1	4	yes	—
391	#386	C₂²×C₆₇	1	1	yes	characteristic
392	#386	C₂²×C₆₇	1	2	yes	—
393	#386	C₂²×C₆₇	1	4	yes	—
394	#385	C₂²×C₆₇	1	4	yes	—
395	#385	C₂²×C₆₇	1	2	yes	—
396	#385	C₂²×C₆₇	1	1	yes	characteristic
397	#381	C₂²×C₆₇	1	1	yes	characteristic
398	#380	C₂²×C₆₇	1	1	yes	characteristic
399	#380	C₂²×C₆₇	1	2	yes	—
400	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
401	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
402	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
403	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
404	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
405	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
406	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
407	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
408	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
409	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
410	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
411	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
412	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
413	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
414	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
415	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
416	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
417	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
418	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
419	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
420	#127	—	2	8	no	—
421	#127	—	4	2	no	—
422	#125	—	2	8	no	—
423	#123	—	2	4	no	—
424	#123	—	4	8	no	—
425	#121	—	2	8	no	—
426	#121	—	2	8	no	—
427	C₄×D₄	—	4	4	no	—
428	C₄×D₄	—	4	2	no	—
429	C₄×D₄	—	4	8	no	—
430	C₄×D₄	—	4	8	no	—
431	C₂×C₄⋊C₄	—	4	2	no	—
432	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
433	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
434	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
435	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
436	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
437	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
438	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
439	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
440	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
441	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
442	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
443	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
444	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
445	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
446	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
447	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
448	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
449	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
450	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
451	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
452	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
453	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
454	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
455	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
456	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
457	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
458	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
459	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
460	C₂×C₄²	—	4	2	no	—
461	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
462	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
463	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
464	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
465	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
466	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
467	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
468	#95	C₂³×C₆₇	1	4	yes	—
469	#95	C₂³×C₆₇	1	2	yes	—
470	#95	C₂³×C₆₇	1	2	yes	—
471	#95	C₂³×C₆₇	1	4	yes	—
472	#95	C₂³×C₆₇	1	2	yes	—
473	#95	C₂³×C₆₇	1	2	yes	—
474	#95	C₂³×C₆₇	1	2	yes	—
475	#95	C₂³×C₆₇	1	4	yes	—
476	#95	C₂³×C₆₇	1	4	yes	—
477	#95	C₂³×C₆₇	1	4	yes	—
478	#95	C₂³×C₆₇	1	2	yes	—
479	#95	C₂³×C₆₇	1	2	yes	—
480	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
481	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
482	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
483	C₂×D₄	—	2	8	no	—
484	C₂×D₄	—	2	8	no	—
485	C₂×D₄	—	2	8	no	—
486	C₂×D₄	—	2	8	no	—
487	C₂×D₄	—	2	4	no	—
488	C₂×D₄	—	2	4	no	—
489	C₂×D₄	—	2	4	no	—
490	C₂×D₄	—	2	4	no	—
491	C₂×D₄	—	4	1	no	—
492	C₂×D₄	—	4	4	no	—
493	C₂×D₄	—	4	4	no	—
494	C₂×D₄	—	4	1	no	—
495	C₂×D₄	—	4	4	no	—
496	C₂×D₄	—	4	2	no	—
497	C₂×D₄	—	4	2	no	—
498	C₂×D₄	—	4	2	no	—
499	C₂×D₄	#41×C₆₇	1	4	yes	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	4	no	—
502	C₂²×C₄	—	2	4	no	—
503	C₂²×C₄	—	2	2	no	—
504	C₂²×C₄	—	2	2	no	—
505	C₂²×C₄	—	2	4	no	—
506	C₂²×C₄	—	2	8	no	—
507	C₂²×C₄	—	4	1	no	—
508	C₂²×C₄	—	4	1	no	—
509	C₂²×C₄	—	4	1	no	—
510	C₂²×C₄	—	4	2	no	—
511	C₂²×C₄	—	4	4	no	—
512	C₂²×C₄	—	4	4	no	—
513	C₂²×C₄	—	4	4	no	—
514	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
515	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
516	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
517	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
518	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
519	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
520	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
521	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
522	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
523	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
524	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
525	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
526	C₄⋊C₄	—	2	4	no	—
527	C₄⋊C₄	—	2	8	no	—
528	C₄⋊C₄	—	2	8	no	—
529	C₄⋊C₄	—	2	4	no	—
530	C₄⋊C₄	—	2	4	no	—
531	C₄⋊C₄	—	2	8	no	—
532	C₄⋊C₄	—	2	4	no	—
533	C₄⋊C₄	—	4	2	no	—
534	C₄⋊C₄	—	4	1	no	—
535	C₄⋊C₄	—	4	4	no	—
536	C₄⋊C₄	#41×C₆₇	1	8	yes	—
537	C₂²⋊C₄	—	2	2	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₂²⋊C₄	—	2	8	no	—
540	C₂²⋊C₄	—	2	8	no	—
541	C₂²⋊C₄	—	2	4	no	—
542	C₂²⋊C₄	—	2	8	no	—
543	C₂²⋊C₄	—	4	4	no	—
544	C₂²⋊C₄	—	4	1	no	—
545	C₂²⋊C₄	—	4	1	no	—
546	C₂²⋊C₄	—	4	4	no	—
547	C₂²⋊C₄	—	4	4	no	—
548	C₂²⋊C₄	—	4	4	no	—
549	C₂²⋊C₄	—	4	4	no	—
550	C₂²⋊C₄	—	4	2	no	—
551	C₂²⋊C₄	—	4	1	no	—
552	C₂²⋊C₄	—	4	4	no	—
553	C₂²⋊C₄	—	4	4	no	—
554	C₄²	—	2	4	no	—
555	C₄²	—	2	4	no	—
556	C₄²	—	2	4	no	—
557	C₄²	—	2	2	no	—
558	C₄²	—	4	2	no	—
559	C₄²	—	4	4	no	—
560	C₄²	—	4	4	no	—
561	C₂³	—	2	2	no	—
562	C₂³	—	2	2	no	—
563	C₂³	—	2	2	no	—
564	C₂³	—	2	2	no	—
565	C₂³	—	2	1	no	—
566	C₂³	—	2	1	no	—
567	C₂³	—	2	1	no	—
568	C₂³	—	2	1	no	—
569	C₂³	—	2	4	no	—
570	C₂³	—	2	4	no	—
571	C₂³	—	2	4	no	—
572	C₂³	—	2	4	no	—
573	C₂³	—	2	4	no	—
574	C₂³	—	2	4	no	—
575	C₂³	—	2	4	no	—
576	C₂³	C₂×#41×C₆₇	1	2	yes	—
577	C₂³	C₂×#41×C₆₇	1	2	yes	—
578	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
579	D₄	—	4	4	no	—
580	D₄	—	4	16	no	—
581	D₄	—	4	16	no	—
582	D₄	—	4	8	no	—
583	D₄	—	8	4	no	—
584	D₄	—	8	4	no	—
585	C₂×C₄	—	2	4	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	4	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	4	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	4	4	no	—
613	C₂×C₄	—	4	4	no	—
614	C₂×C₄	—	4	4	no	—
615	C₂×C₄	—	4	4	no	—
616	C₂×C₄	—	8	8	no	—
617	C₂×C₄	—	8	4	no	—
618	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
619	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
620	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
621	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
622	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
623	C₂²	—	2	4	no	—
624	C₂²	—	2	4	no	—
625	C₂²	—	2	2	no	—
626	C₂²	—	2	2	no	—
627	C₂²	—	4	1	no	—
628	C₂²	—	4	1	no	—
629	C₂²	—	4	1	no	—
630	C₂²	—	4	1	no	—
631	C₂²	—	4	2	no	—
632	C₂²	—	4	2	no	—
633	C₂²	—	4	4	no	—
634	C₂²	—	4	4	no	—
635	C₂²	—	4	4	no	—
636	C₂²	—	4	4	no	—
637	C₂²	—	4	4	no	—
638	C₂²	—	4	4	no	—
639	C₂²	—	4	4	no	—
640	C₂²	C₂²×#41×C₆₇	1	2	yes	—
641	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
642	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
643	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
644	C₂²	#585×C₆₇	1	1	yes	characteristic
645	C₄	—	2	4	no	—
646	C₄	—	2	8	no	—
647	C₄	—	2	2	no	—
648	C₄	—	4	4	no	—
649	C₄	—	4	1	no	—
650	C₄	—	4	1	no	—
651	C₄	—	4	1	no	—
652	C₄	—	4	2	no	—
653	C₄	—	4	2	no	—
654	C₄	—	4	2	no	—
655	C₄	—	4	4	no	—
656	C₄	#559×C₆₇	1	4	yes	—
657	C₂	—	4	2	no	—
658	C₂	—	4	2	no	—
659	C₂	—	8	4	no	—
660	C₂	C₂×#559×C₆₇	1	1	yes	characteristic
661	C₂	C₆₇×#3481	1	2	yes	—
662	C₂	C₆₇×#3505	1	1	yes	characteristic
663	C₂	C₆₇×#3517	1	2	yes	—
664	C₂	C₆₇×#3537	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₆₇×#2541	C₂
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×#2752	C₂	semidirect
2. C₆₇×#2709	C₂	semidirect
3. C₆₇×#2602	C₂	semidirect
4. C₆₇×#2597	C₂	semidirect
6. C₆₇×#2490	C₂	semidirect
7. C₆₇×#2490	C₂	semidirect
8. C₆₇×#2487	C₂	semidirect
9. C₆₇×#2455	C₂	semidirect
10. C₆₇×#2445	C₂	semidirect
11. C₆₇×#2370	C₂	semidirect
12. C₆₇×#2366	C₂	semidirect
13. C₆₇×#2361	C₂	semidirect
14. C₄×#123×C₆₇	C₂	semidirect
15. C₄×#121×C₆₇	C₂	semidirect
324. #59632	C₆₇	direct (N × H)
333. #2752	C₂×C₆₇	semidirect
334. #2709	C₂×C₆₇	semidirect
335. #2602	C₂×C₆₇	semidirect
336. #2597	C₂×C₆₇	semidirect
338. #2490	C₂×C₆₇	semidirect
339. #2490	C₂×C₆₇	semidirect
340. #2487	C₂×C₆₇	semidirect
341. #2455	C₂×C₆₇	semidirect
342. #2445	C₂×C₆₇	semidirect
343. #2370	C₂×C₆₇	semidirect
344. #2366	C₂×C₆₇	semidirect
345. #2361	C₂×C₆₇	semidirect
346. C₄×#123	C₂×C₆₇	semidirect
347. C₄×#121	C₂×C₆₇	semidirect
348. C₆₇	#59632	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2752	2
C₆₇×#2709	2
C₆₇×#2602	2
C₆₇×#2597	2
C₆₇×#2541	2
C₆₇×#2490	2
C₆₇×#2487	2
C₆₇×#2455	2
C₆₇×#2445	2
C₆₇×#2370	2
C₆₇×#2366	2
C₆₇×#2361	2
C₄×#123×C₆₇	2
C₄×#121×C₆₇	2
#59632	67

Subgroups — C67×#59632 #30738883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59632 #30738883