Subgroups — C₆₇×#14030 #30693281

Order 17152.

1326 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#389×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#382×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1443	C₂	1	2	yes	maximal
5	C₆₇×#1442	C₂	1	2	yes	maximal
6	C₆₇×#1436	C₂	1	2	yes	maximal
7	C₆₇×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
8	C₆₇×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
9	C₆₇×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1434	C₂	1	2	yes	maximal
11	C₄×#95×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#389×C₆₇	—	2	4	no	—
22	#382×C₆₇	C₄	1	8	yes	complemented
23	C₄×C₄⋊C₄×C₆₇	—	2	4	no	—
24	C₂×#95×C₆₇	C₂²	1	2	yes	—
25	C₂×#95×C₆₇	C₂²	1	2	yes	—
26	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#95×C₆₇	C₂²	1	2	yes	—
31	C₂×#95×C₆₇	C₂²	1	2	yes	—
32	C₂×#95×C₆₇	C₂²	1	2	yes	—
33	C₂×#95×C₆₇	C₂²	1	2	yes	—
34	C₂×#95×C₆₇	C₂²	1	2	yes	—
35	C₂×#95×C₆₇	C₂²	1	2	yes	—
36	C₂×#95×C₆₇	C₂²	1	2	yes	—
37	C₂×#95×C₆₇	C₂²	1	2	yes	—
38	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
63	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
64	C₂×C₄²×C₆₇	—	2	2	no	—
65	C₂×C₄²×C₆₇	—	2	2	no	—
66	C₂×C₄²×C₆₇	—	2	2	no	—
67	C₂×C₄²×C₆₇	—	2	2	no	—
68	C₂×C₄²×C₆₇	—	2	4	no	—
69	C₂×C₄²×C₆₇	—	2	2	no	—
70	C₂×C₄²×C₆₇	—	2	4	no	—
71	C₂×C₄²×C₆₇	—	2	2	no	—
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	—	4	2	no	—
74	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
75	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
76	#95×C₆₇	—	2	4	no	—
77	#95×C₆₇	—	2	4	no	—
78	#95×C₆₇	—	2	2	no	—
79	#95×C₆₇	—	2	2	no	—
80	#95×C₆₇	—	2	2	no	—
81	#95×C₆₇	—	2	2	no	—
82	#95×C₆₇	—	2	4	no	—
83	#95×C₆₇	—	2	4	no	—
84	#95×C₆₇	—	4	2	no	—
85	#95×C₆₇	—	4	2	no	—
86	#95×C₆₇	—	4	2	no	—
87	#95×C₆₇	—	4	2	no	—
88	#95×C₆₇	C₂×C₄	1	8	yes	—
89	#95×C₆₇	C₂×C₄	1	8	yes	—
90	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	—	2	1	no	—
92	C₂²×C₄×C₆₇	—	2	1	no	—
93	C₂²×C₄×C₆₇	—	2	1	no	—
94	C₂²×C₄×C₆₇	—	2	1	no	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	—	2	1	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	1	no	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	1	no	—
120	C₂²×C₄×C₆₇	—	2	1	no	—
121	C₂²×C₄×C₆₇	—	2	1	no	—
122	C₂²×C₄×C₆₇	—	2	1	no	—
123	C₂²×C₄×C₆₇	—	2	1	no	—
124	C₂²×C₄×C₆₇	—	2	1	no	—
125	C₂²×C₄×C₆₇	—	2	1	no	—
126	C₂²×C₄×C₆₇	—	2	1	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	1	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	—	2	1	no	—
133	C₂²×C₄×C₆₇	—	2	1	no	—
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	—	2	1	no	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	1	no	—
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	2	2	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	2	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	2	no	—
150	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
152	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
153	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
154	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
155	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
156	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
157	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
158	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
159	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
160	C₂²×C₄×C₆₇	#41	1	2	yes	—
161	C₂²×C₄×C₆₇	#41	1	2	yes	—
162	C₂²×C₄×C₆₇	#41	1	2	yes	—
163	C₂²×C₄×C₆₇	#41	1	2	yes	—
164	C₂²×C₄×C₆₇	#41	1	2	yes	—
165	C₂²×C₄×C₆₇	#41	1	2	yes	—
166	C₂²×C₄×C₆₇	#41	1	2	yes	—
167	C₂²×C₄×C₆₇	#41	1	2	yes	—
168	C₄⋊C₄×C₆₇	—	2	8	no	—
169	C₄⋊C₄×C₆₇	—	2	8	no	—
170	C₄⋊C₄×C₆₇	—	2	8	no	—
171	C₄⋊C₄×C₆₇	—	4	8	no	—
172	C₄⋊C₄×C₆₇	—	4	4	no	—
173	C₄²×C₆₇	—	2	8	no	—
174	C₄²×C₆₇	—	2	8	no	—
175	C₄²×C₆₇	—	4	8	no	—
176	C₄²×C₆₇	—	4	4	no	—
177	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
178	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
179	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
180	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
181	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
182	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
183	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
184	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
185	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
186	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
187	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
188	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
190	C₂³×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
191	C₂³×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	2	2	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	2	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	2	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	4	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	2	2	no	—
213	C₂×C₄×C₆₇	—	2	2	no	—
214	C₂×C₄×C₆₇	—	2	2	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	4	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	2	2	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	4	1	no	—
223	C₂×C₄×C₆₇	—	4	1	no	—
224	C₂×C₄×C₆₇	—	4	2	no	—
225	C₂×C₄×C₆₇	—	4	1	no	—
226	C₂×C₄×C₆₇	—	4	2	no	—
227	C₂×C₄×C₆₇	—	4	1	no	—
228	C₂×C₄×C₆₇	—	4	2	no	—
229	C₂×C₄×C₆₇	—	4	2	no	—
230	C₂×C₄×C₆₇	—	4	1	no	—
231	C₂×C₄×C₆₇	—	4	1	no	—
232	C₂×C₄×C₆₇	—	4	2	no	—
233	C₂×C₄×C₆₇	—	4	2	no	—
234	C₂×C₄×C₆₇	—	4	1	no	—
235	C₂×C₄×C₆₇	—	4	1	no	—
236	C₂×C₄×C₆₇	—	4	1	no	—
237	C₂×C₄×C₆₇	—	4	1	no	—
238	C₂×C₄×C₆₇	—	4	1	no	—
239	C₂×C₄×C₆₇	—	4	1	no	—
240	C₂×C₄×C₆₇	—	4	2	no	—
241	C₂×C₄×C₆₇	—	4	1	no	—
242	C₂×C₄×C₆₇	—	4	1	no	—
243	C₂×C₄×C₆₇	—	4	2	no	—
244	C₂×C₄×C₆₇	—	4	1	no	—
245	C₂×C₄×C₆₇	—	4	1	no	—
246	C₂×C₄×C₆₇	—	4	1	no	—
247	C₂×C₄×C₆₇	—	4	1	no	—
248	C₂×C₄×C₆₇	—	4	2	no	—
249	C₂×C₄×C₆₇	—	4	2	no	—
250	C₂×C₄×C₆₇	—	4	1	no	—
251	C₂×C₄×C₆₇	—	4	1	no	—
252	C₂×C₄×C₆₇	—	4	2	no	—
253	C₂×C₄×C₆₇	—	4	2	no	—
254	C₂×C₄×C₆₇	—	4	2	no	—
255	C₂×C₄×C₆₇	—	4	2	no	—
256	C₂×C₄×C₆₇	—	4	2	no	—
257	C₂×C₄×C₆₇	—	4	2	no	—
258	C₂×C₄×C₆₇	—	4	2	no	—
259	C₂×C₄×C₆₇	—	4	2	no	—
260	C₂×C₄×C₆₇	—	4	2	no	—
261	C₂×C₄×C₆₇	#117	1	4	yes	—
262	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
263	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
264	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
265	C₂×C₄×C₆₇	#123	1	4	yes	—
266	C₂×C₄×C₆₇	#125	1	4	yes	—
267	C₂×C₄×C₆₇	#125	1	4	yes	—
268	C₂×C₄×C₆₇	#126	1	4	yes	—
269	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
270	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
271	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
272	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
273	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
274	C₂²×C₆₇	#520	1	1	yes	characteristic
275	C₂²×C₆₇	#520	1	1	yes	characteristic
276	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
277	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
278	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
279	C₂²×C₆₇	C₂×#125	1	1	yes	characteristic
280	C₂²×C₆₇	C₂×#126	1	1	yes	characteristic
281	C₂²×C₆₇	#529	1	1	yes	characteristic
282	C₂²×C₆₇	#529	1	1	yes	characteristic
283	C₂²×C₆₇	#533	1	1	yes	characteristic
284	C₂²×C₆₇	#540	1	1	yes	characteristic
285	C₂²×C₆₇	#541	1	1	yes	characteristic
286	C₂²×C₆₇	#544	1	1	yes	characteristic
287	C₂²×C₆₇	#544	1	1	yes	characteristic
288	C₂²×C₆₇	#546	1	1	yes	characteristic
289	C₂²×C₆₇	#546	1	1	yes	characteristic
290	C₂²×C₆₇	#548	1	1	yes	characteristic
291	C₂²×C₆₇	#548	1	1	yes	characteristic
292	C₂²×C₆₇	#552	1	1	yes	characteristic
293	C₂²×C₆₇	#552	1	1	yes	characteristic
294	C₂²×C₆₇	#553	1	1	yes	characteristic
295	C₂²×C₆₇	#553	1	1	yes	characteristic
296	C₂²×C₆₇	#555	1	1	yes	characteristic
297	C₂²×C₆₇	#555	1	1	yes	characteristic
298	C₂²×C₆₇	#556	1	1	yes	characteristic
299	C₂²×C₆₇	#556	1	1	yes	characteristic
300	C₂²×C₆₇	#556	1	1	yes	characteristic
301	C₂²×C₆₇	#556	1	1	yes	characteristic
302	C₂²×C₆₇	#559	1	1	yes	characteristic
303	C₂²×C₆₇	#559	1	1	yes	characteristic
304	C₄×C₆₇	—	2	4	no	—
305	C₄×C₆₇	—	4	2	no	—
306	C₄×C₆₇	—	4	2	no	—
307	C₄×C₆₇	—	4	4	no	—
308	C₄×C₆₇	—	4	4	no	—
309	C₄×C₆₇	—	4	4	no	—
310	C₄×C₆₇	—	4	2	no	—
311	C₄×C₆₇	—	4	2	no	—
312	C₄×C₆₇	—	4	2	no	—
313	C₄×C₆₇	—	8	2	no	—
314	C₄×C₆₇	#387	1	8	yes	—
315	#14030	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
316	C₂×C₆₇	C₂×#387	1	1	yes	characteristic
317	C₂×C₆₇	C₄×#125	1	1	yes	characteristic
318	C₂×C₆₇	#2330	1	1	yes	characteristic
319	C₂×C₆₇	#2346	1	1	yes	characteristic
320	C₂×C₆₇	#2347	1	1	yes	characteristic
321	C₂×C₆₇	#2367	1	1	yes	characteristic
322	C₂×C₆₇	#2367	1	1	yes	characteristic
323	C₂×C₆₇	#2420	1	1	yes	characteristic
324	C₂×C₆₇	#2421	1	1	yes	characteristic
325	C₂×C₆₇	#2439	1	1	yes	characteristic
326	C₂×C₆₇	#2442	1	1	yes	characteristic
327	C₂×C₆₇	#2513	1	1	yes	characteristic
328	C₂×C₆₇	#2514	1	1	yes	characteristic
329	C₂×C₆₇	#2570	1	1	yes	characteristic
330	C₂×C₆₇	#2571	1	1	yes	characteristic
331	C₂×#389	C₂×C₆₇	1	1	yes	characteristic
332	C₂×#382	C₂×C₆₇	1	1	yes	characteristic
333	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
334	#1443	C₂×C₆₇	1	2	yes	—
335	#1442	C₂×C₆₇	1	2	yes	—
336	#1436	C₂×C₆₇	1	2	yes	—
337	C₄⋊#95	C₂×C₆₇	1	1	yes	characteristic
338	C₄⋊#95	C₂×C₆₇	1	1	yes	characteristic
339	C₄⋊#95	C₂×C₆₇	1	1	yes	characteristic
340	#1434	C₂×C₆₇	1	2	yes	—
341	C₄×#95	C₂×C₆₇	1	1	yes	characteristic
342	C₆₇	#14030	1	1	yes	characteristic, direct product factor, Sylow
343	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
344	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
345	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
346	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
347	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
348	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
349	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
350	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
351	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
352	#389	—	2	4	no	—
353	#382	C₄×C₆₇	1	8	yes	complemented
354	C₄×C₄⋊C₄	—	2	4	no	—
355	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
356	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
357	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
358	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
359	C₂×#95	C₂²×C₆₇	1	2	yes	—
360	C₂×#95	C₂²×C₆₇	1	2	yes	—
361	C₂×#95	C₂²×C₆₇	1	2	yes	—
362	C₂×#95	C₂²×C₆₇	1	2	yes	—
363	C₂×#95	C₂²×C₆₇	1	2	yes	—
364	C₂×#95	C₂²×C₆₇	1	2	yes	—
365	C₂×#95	C₂²×C₆₇	1	2	yes	—
366	C₂×#95	C₂²×C₆₇	1	2	yes	—
367	C₂×#95	C₂²×C₆₇	1	2	yes	—
368	C₂×#95	C₂²×C₆₇	1	2	yes	—
369	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
370	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
371	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
372	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
373	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
374	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
375	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
376	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
377	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
378	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
379	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
380	C₂×C₄⋊C₄	—	2	2	no	—
381	C₂×C₄⋊C₄	—	2	2	no	—
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	—	2	2	no	—
384	C₂×C₄⋊C₄	—	2	2	no	—
385	C₂×C₄⋊C₄	—	2	2	no	—
386	C₂×C₄⋊C₄	—	2	4	no	—
387	C₂×C₄⋊C₄	—	2	4	no	—
388	C₂×C₄⋊C₄	—	2	2	no	—
389	C₂×C₄⋊C₄	—	2	2	no	—
390	C₂×C₄⋊C₄	—	2	2	no	—
391	C₂×C₄⋊C₄	—	4	2	no	—
392	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
393	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
394	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
395	C₂×C₄²	—	2	2	no	—
396	C₂×C₄²	—	2	2	no	—
397	C₂×C₄²	—	2	2	no	—
398	C₂×C₄²	—	2	2	no	—
399	C₂×C₄²	—	2	4	no	—
400	C₂×C₄²	—	2	2	no	—
401	C₂×C₄²	—	2	4	no	—
402	C₂×C₄²	—	2	2	no	—
403	C₂×C₄²	—	2	2	no	—
404	C₂×C₄²	—	4	2	no	—
405	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
406	C₂×C₄²	Q₈×C₆₇	1	4	yes	—
407	#95	—	2	4	no	—
408	#95	—	2	4	no	—
409	#95	—	2	2	no	—
410	#95	—	2	2	no	—
411	#95	—	2	2	no	—
412	#95	—	2	2	no	—
413	#95	—	2	4	no	—
414	#95	—	2	4	no	—
415	#95	—	4	2	no	—
416	#95	—	4	2	no	—
417	#95	—	4	2	no	—
418	#95	—	4	2	no	—
419	#95	C₂×C₄×C₆₇	1	8	yes	—
420	#95	C₂×C₄×C₆₇	1	8	yes	—
421	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	2	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
482	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
483	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
484	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
485	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
486	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
487	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
488	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
489	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
490	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
491	C₂²×C₄	#41×C₆₇	1	2	yes	—
492	C₂²×C₄	#41×C₆₇	1	2	yes	—
493	C₂²×C₄	#41×C₆₇	1	2	yes	—
494	C₂²×C₄	#41×C₆₇	1	2	yes	—
495	C₂²×C₄	#41×C₆₇	1	2	yes	—
496	C₂²×C₄	#41×C₆₇	1	2	yes	—
497	C₂²×C₄	#41×C₆₇	1	2	yes	—
498	C₂²×C₄	#41×C₆₇	1	2	yes	—
499	C₄⋊C₄	—	2	8	no	—
500	C₄⋊C₄	—	2	8	no	—
501	C₄⋊C₄	—	2	8	no	—
502	C₄⋊C₄	—	4	8	no	—
503	C₄⋊C₄	—	4	4	no	—
504	C₄²	—	2	8	no	—
505	C₄²	—	2	8	no	—
506	C₄²	—	4	8	no	—
507	C₄²	—	4	4	no	—
508	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
509	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
510	C₂³	C₂²×Q₈×C₆₇	1	1	yes	characteristic
511	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
512	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
513	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
514	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
515	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
516	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
517	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
518	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
519	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
520	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
521	C₂³	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
522	C₂³	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	2	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	2	no	—
560	C₂×C₄	—	4	2	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	2	no	—
564	C₂×C₄	—	4	2	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	2	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	2	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	2	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	2	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	2	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	2	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	#117×C₆₇	1	4	yes	—
593	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
594	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
595	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
596	C₂×C₄	#123×C₆₇	1	4	yes	—
597	C₂×C₄	#125×C₆₇	1	4	yes	—
598	C₂×C₄	#125×C₆₇	1	4	yes	—
599	C₂×C₄	#126×C₆₇	1	4	yes	—
600	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
601	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
602	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
603	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
604	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
605	C₂²	#520×C₆₇	1	1	yes	characteristic
606	C₂²	#520×C₆₇	1	1	yes	characteristic
607	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
608	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
609	C₂²	C₂×#125×C₆₇	1	1	yes	characteristic
610	C₂²	C₂×#125×C₆₇	1	1	yes	characteristic
611	C₂²	C₂×#126×C₆₇	1	1	yes	characteristic
612	C₂²	#529×C₆₇	1	1	yes	characteristic
613	C₂²	#529×C₆₇	1	1	yes	characteristic
614	C₂²	#533×C₆₇	1	1	yes	characteristic
615	C₂²	#540×C₆₇	1	1	yes	characteristic
616	C₂²	#541×C₆₇	1	1	yes	characteristic
617	C₂²	#544×C₆₇	1	1	yes	characteristic
618	C₂²	#544×C₆₇	1	1	yes	characteristic
619	C₂²	#546×C₆₇	1	1	yes	characteristic
620	C₂²	#546×C₆₇	1	1	yes	characteristic
621	C₂²	#548×C₆₇	1	1	yes	characteristic
622	C₂²	#548×C₆₇	1	1	yes	characteristic
623	C₂²	#552×C₆₇	1	1	yes	characteristic
624	C₂²	#552×C₆₇	1	1	yes	characteristic
625	C₂²	#553×C₆₇	1	1	yes	characteristic
626	C₂²	#553×C₆₇	1	1	yes	characteristic
627	C₂²	#555×C₆₇	1	1	yes	characteristic
628	C₂²	#555×C₆₇	1	1	yes	characteristic
629	C₂²	#556×C₆₇	1	1	yes	characteristic
630	C₂²	#556×C₆₇	1	1	yes	characteristic
631	C₂²	#556×C₆₇	1	1	yes	characteristic
632	C₂²	#556×C₆₇	1	1	yes	characteristic
633	C₂²	#559×C₆₇	1	1	yes	characteristic
634	C₂²	#559×C₆₇	1	1	yes	characteristic
635	C₄	—	2	4	no	—
636	C₄	—	4	2	no	—
637	C₄	—	4	2	no	—
638	C₄	—	4	4	no	—
639	C₄	—	4	4	no	—
640	C₄	—	4	4	no	—
641	C₄	—	4	2	no	—
642	C₄	—	4	2	no	—
643	C₄	—	4	2	no	—
644	C₄	—	8	2	no	—
645	C₄	#387×C₆₇	1	8	yes	—
646	C₂	C₂×#387×C₆₇	1	1	yes	characteristic
647	C₂	C₄×#125×C₆₇	1	1	yes	characteristic
648	C₂	C₆₇×#2330	1	1	yes	characteristic
649	C₂	C₆₇×#2346	1	1	yes	characteristic
650	C₂	C₆₇×#2347	1	1	yes	characteristic
651	C₂	C₆₇×#2367	1	1	yes	characteristic
652	C₂	C₆₇×#2367	1	1	yes	characteristic
653	C₂	C₆₇×#2420	1	1	yes	characteristic
654	C₂	C₆₇×#2421	1	1	yes	characteristic
655	C₂	C₆₇×#2439	1	1	yes	characteristic
656	C₂	C₆₇×#2442	1	1	yes	characteristic
657	C₂	C₆₇×#2513	1	1	yes	characteristic
658	C₂	C₆₇×#2514	1	1	yes	characteristic
659	C₂	C₆₇×#2570	1	1	yes	characteristic
660	C₂	C₆₇×#2571	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂⁴	C₂⁴×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
22. #382×C₆₇	C₄	semidirect
315. #14030	C₆₇	direct (N × H)
342. C₆₇	#14030	direct (N × H)
353. #382	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#389×C₆₇	2
C₂×#382×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1443	2
C₆₇×#1442	2
C₆₇×#1436	2
C₆₇×C₄⋊#95	2
C₆₇×#1434	2
C₄×#95×C₆₇	2
#14030	67

Subgroups — C67×#14030 #30693281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#14030 #30693281