Subgroups — C₆₇×#22595 #30701846

Order 17152.

736 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1917	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#1917	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1916	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1916	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1850	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1850	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1840	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1832	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1817	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1768	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#1752	C₂	1	1	yes	maximal, characteristic
13	C₈×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
14	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#336×C₆₇	C₂	1	1	yes	maximal, characteristic
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#383×C₆₇	C₄	1	1	yes	characteristic
30	#383×C₆₇	C₄	1	1	yes	characteristic
31	#382×C₆₇	C₄	1	1	yes	characteristic
32	#382×C₆₇	C₄	1	1	yes	characteristic
33	#381×C₆₇	C₄	1	1	yes	characteristic
34	#381×C₆₇	C₄	1	1	yes	characteristic
35	#380×C₆₇	C₄	1	1	yes	characteristic
36	#380×C₆₇	C₄	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
41	#376×C₆₇	C₂²	1	1	yes	characteristic
42	#376×C₆₇	C₂²	1	1	yes	characteristic
43	#336×C₆₇	C₂²	1	1	yes	characteristic
44	#336×C₆₇	C₂²	1	1	yes	characteristic
45	#336×C₆₇	C₂²	1	1	yes	characteristic
46	#336×C₆₇	C₂²	1	1	yes	characteristic
47	#336×C₆₇	C₂²	1	1	yes	characteristic
48	#336×C₆₇	C₂²	1	1	yes	characteristic
49	#336×C₆₇	C₂²	1	1	yes	characteristic
50	#336×C₆₇	C₂²	1	1	yes	characteristic
51	#336×C₆₇	C₂²	1	1	yes	characteristic
52	#336×C₆₇	C₂²	1	1	yes	characteristic
53	#336×C₆₇	C₂²	1	1	yes	characteristic
54	#336×C₆₇	C₂²	1	1	yes	characteristic
55	#336×C₆₇	C₂²	1	1	yes	characteristic
56	#336×C₆₇	C₂²	1	1	yes	characteristic
57	#336×C₆₇	C₂²	1	1	yes	characteristic
58	#336×C₆₇	C₂²	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₆₇	—	2	2	no	—
63	C₂²×C₈×C₆₇	—	2	2	no	—
64	C₂²×C₈×C₆₇	—	2	2	no	—
65	C₂²×C₈×C₆₇	—	2	2	no	—
66	C₂²×C₈×C₆₇	—	4	1	no	—
67	C₂²×C₈×C₆₇	—	4	1	no	—
68	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	—	2	2	no	—
85	C₂×C₄²×C₆₇	—	2	2	no	—
86	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₆₇	—	2	2	no	—
95	C₄⋊C₈×C₆₇	—	2	2	no	—
96	C₄⋊C₈×C₆₇	—	4	1	no	—
97	C₄⋊C₈×C₆₇	—	4	1	no	—
98	C₂²⋊C₈×C₆₇	—	2	2	no	—
99	C₂²⋊C₈×C₆₇	—	2	2	no	—
100	#97×C₆₇	—	2	2	no	—
101	#97×C₆₇	—	4	1	no	—
102	C₄×C₈×C₆₇	—	2	2	no	—
103	C₄×C₈×C₆₇	—	4	1	no	—
104	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
105	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
106	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
107	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
108	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
109	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
110	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
111	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
112	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
113	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
114	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
115	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
116	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
117	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
118	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
119	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
120	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	2	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	4	1	no	—
131	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	#41	1	2	yes	—
145	C₂²×C₄×C₆₇	#41	1	2	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
147	C₂×C₈×C₆₇	—	2	1	no	—
148	C₂×C₈×C₆₇	—	2	1	no	—
149	C₂×C₈×C₆₇	—	2	1	no	—
150	C₂×C₈×C₆₇	—	2	1	no	—
151	C₂×C₈×C₆₇	—	2	1	no	—
152	C₂×C₈×C₆₇	—	2	1	no	—
153	C₂×C₈×C₆₇	—	2	1	no	—
154	C₂×C₈×C₆₇	—	2	1	no	—
155	C₂×C₈×C₆₇	—	2	1	no	—
156	C₂×C₈×C₆₇	—	2	1	no	—
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	1	no	—
159	C₂×C₈×C₆₇	—	2	1	no	—
160	C₂×C₈×C₆₇	—	2	1	no	—
161	C₂×C₈×C₆₇	—	2	1	no	—
162	C₂×C₈×C₆₇	—	2	1	no	—
163	C₂×C₈×C₆₇	—	2	1	no	—
164	C₂×C₈×C₆₇	—	2	1	no	—
165	C₂×C₈×C₆₇	—	2	1	no	—
166	C₂×C₈×C₆₇	—	2	1	no	—
167	C₂×C₈×C₆₇	—	2	1	no	—
168	C₂×C₈×C₆₇	—	2	1	no	—
169	C₂×C₈×C₆₇	—	4	4	no	—
170	C₂×C₈×C₆₇	—	4	4	no	—
171	C₂×C₈×C₆₇	#41	1	2	yes	—
172	C₂×C₈×C₆₇	#41	1	2	yes	—
173	C₄⋊C₄×C₆₇	—	2	2	no	—
174	C₄⋊C₄×C₆₇	—	2	2	no	—
175	C₄⋊C₄×C₆₇	—	4	1	no	—
176	C₄⋊C₄×C₆₇	—	4	1	no	—
177	C₂²⋊C₄×C₆₇	—	2	2	no	—
178	C₂²⋊C₄×C₆₇	—	2	2	no	—
179	C₂²⋊C₄×C₆₇	—	2	2	no	—
180	C₂²⋊C₄×C₆₇	—	2	2	no	—
181	C₄²×C₆₇	—	2	2	no	—
182	C₄²×C₆₇	—	2	2	no	—
183	C₄²×C₆₇	—	2	1	no	—
184	C₄²×C₆₇	—	2	1	no	—
185	C₄²×C₆₇	—	2	1	no	—
186	C₄²×C₆₇	—	2	1	no	—
187	C₄²×C₆₇	—	2	1	no	—
188	C₄²×C₆₇	—	2	1	no	—
189	C₄²×C₆₇	—	2	1	no	—
190	C₄²×C₆₇	—	2	1	no	—
191	C₄²×C₆₇	—	2	1	no	—
192	C₄²×C₆₇	—	2	1	no	—
193	C₄²×C₆₇	—	2	1	no	—
194	C₄²×C₆₇	—	2	1	no	—
195	C₂³×C₆₇	—	2	1	no	—
196	C₂³×C₆₇	—	2	1	no	—
197	C₂³×C₆₇	—	2	1	no	—
198	C₂³×C₆₇	—	2	1	no	—
199	C₂³×C₆₇	#117	1	2	yes	—
200	C₂³×C₆₇	#131	1	2	yes	—
201	C₂³×C₆₇	#131	1	2	yes	—
202	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
203	C₂×C₄×C₆₇	—	2	1	no	—
204	C₂×C₄×C₆₇	—	2	1	no	—
205	C₂×C₄×C₆₇	—	2	1	no	—
206	C₂×C₄×C₆₇	—	2	1	no	—
207	C₂×C₄×C₆₇	—	2	2	no	—
208	C₂×C₄×C₆₇	—	2	2	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	2	no	—
211	C₂×C₄×C₆₇	—	2	1	no	—
212	C₂×C₄×C₆₇	—	2	1	no	—
213	C₂×C₄×C₆₇	—	2	1	no	—
214	C₂×C₄×C₆₇	—	2	1	no	—
215	C₂×C₄×C₆₇	—	2	1	no	—
216	C₂×C₄×C₆₇	—	2	1	no	—
217	C₂×C₄×C₆₇	—	2	1	no	—
218	C₂×C₄×C₆₇	—	2	1	no	—
219	C₂×C₄×C₆₇	—	2	1	no	—
220	C₂×C₄×C₆₇	—	2	1	no	—
221	C₂×C₄×C₆₇	—	2	1	no	—
222	C₂×C₄×C₆₇	—	2	1	no	—
223	C₂×C₄×C₆₇	—	2	1	no	—
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	1	no	—
227	C₂×C₄×C₆₇	—	2	1	no	—
228	C₂×C₄×C₆₇	—	2	1	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	—	2	1	no	—
231	C₂×C₄×C₆₇	—	2	1	no	—
232	C₂×C₄×C₆₇	—	2	1	no	—
233	C₂×C₄×C₆₇	—	2	1	no	—
234	C₂×C₄×C₆₇	—	2	1	no	—
235	C₂×C₄×C₆₇	—	2	1	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	1	no	—
240	C₂×C₄×C₆₇	—	2	1	no	—
241	C₂×C₄×C₆₇	—	2	1	no	—
242	C₂×C₄×C₆₇	—	2	1	no	—
243	C₂×C₄×C₆₇	—	2	1	no	—
244	C₂×C₄×C₆₇	—	4	4	no	—
245	C₂×C₄×C₆₇	—	4	4	no	—
246	C₂×C₄×C₆₇	#117	1	2	yes	—
247	C₂×C₄×C₆₇	#131	1	2	yes	—
248	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
249	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
250	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
251	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
252	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
253	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
254	C₈×C₆₇	—	2	2	no	—
255	C₈×C₆₇	—	2	2	no	—
256	C₈×C₆₇	—	4	1	no	—
257	C₈×C₆₇	—	4	1	no	—
258	C₈×C₆₇	—	4	1	no	—
259	C₈×C₆₇	—	4	1	no	—
260	C₈×C₆₇	—	4	1	no	—
261	C₈×C₆₇	—	4	1	no	—
262	C₂²×C₆₇	—	2	2	no	—
263	C₂²×C₆₇	—	2	2	no	—
264	C₂²×C₆₇	—	2	2	no	—
265	C₂²×C₆₇	—	2	2	no	—
266	C₂²×C₆₇	—	2	2	no	—
267	C₂²×C₆₇	—	2	2	no	—
268	C₂²×C₆₇	#433	1	4	yes	—
269	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
270	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
271	C₂²×C₆₇	#520	1	1	yes	characteristic
272	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
273	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
274	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
275	C₂²×C₆₇	#568	1	1	yes	characteristic
276	C₄×C₆₇	—	2	2	no	—
277	C₄×C₆₇	—	2	2	no	—
278	C₄×C₆₇	—	4	1	no	—
279	C₄×C₆₇	—	4	1	no	—
280	C₄×C₆₇	—	4	1	no	—
281	C₄×C₆₇	—	4	1	no	—
282	C₄×C₆₇	—	4	1	no	—
283	C₄×C₆₇	—	4	1	no	—
284	C₄×C₆₇	—	4	1	no	—
285	C₄×C₆₇	—	4	1	no	—
286	C₄×C₆₇	—	4	1	no	—
287	C₄×C₆₇	—	4	1	no	—
288	C₄×C₆₇	—	4	1	no	—
289	C₄×C₆₇	#551	1	1	yes	characteristic
290	C₄×C₆₇	#551	1	1	yes	characteristic
291	C₄×C₆₇	#552	1	1	yes	characteristic
292	C₄×C₆₇	#552	1	1	yes	characteristic
293	#22595	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
294	C₂×C₆₇	—	2	4	no	—
295	C₂×C₆₇	#2343	1	1	yes	characteristic
296	C₂×C₆₇	C₂×#433	1	1	yes	characteristic
297	C₂×C₆₇	#2924	1	1	yes	characteristic
298	C₂×C₆₇	#2966	1	1	yes	characteristic
299	C₂×C₆₇	#2968	1	1	yes	characteristic
300	C₂×C₆₇	#2992	1	1	yes	characteristic
301	C₂×C₆₇	#2994	1	1	yes	characteristic
302	#2283	C₂×C₆₇	1	1	yes	characteristic
303	#1917	C₂×C₆₇	1	1	yes	characteristic, complemented
304	#1917	C₂×C₆₇	1	1	yes	characteristic, complemented
305	#1916	C₂×C₆₇	1	1	yes	characteristic, complemented
306	#1916	C₂×C₆₇	1	1	yes	characteristic, complemented
307	#1850	C₂×C₆₇	1	1	yes	characteristic
308	#1850	C₂×C₆₇	1	1	yes	characteristic
309	#1840	C₂×C₆₇	1	1	yes	characteristic, complemented
310	#1832	C₂×C₆₇	1	1	yes	characteristic, complemented
311	#1817	C₂×C₆₇	1	1	yes	characteristic
312	#1768	C₂×C₆₇	1	1	yes	characteristic, complemented
313	#1752	C₂×C₆₇	1	1	yes	characteristic
314	C₈×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic
315	#1746	C₂×C₆₇	1	1	yes	characteristic, complemented
316	C₂×#336	C₂×C₆₇	1	1	yes	characteristic
317	C₆₇	#22595	1	1	yes	characteristic, direct product factor, Sylow
318	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
319	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
320	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
321	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
322	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
323	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
324	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
326	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
327	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
328	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
329	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
330	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
331	#383	C₄×C₆₇	1	1	yes	characteristic
332	#383	C₄×C₆₇	1	1	yes	characteristic
333	#382	C₄×C₆₇	1	1	yes	characteristic
334	#382	C₄×C₆₇	1	1	yes	characteristic
335	#381	C₄×C₆₇	1	1	yes	characteristic
336	#381	C₄×C₆₇	1	1	yes	characteristic
337	#380	C₄×C₆₇	1	1	yes	characteristic
338	#380	C₄×C₆₇	1	1	yes	characteristic
339	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
340	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
341	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
342	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
343	#376	C₂²×C₆₇	1	1	yes	characteristic
344	#376	C₂²×C₆₇	1	1	yes	characteristic
345	#336	C₂²×C₆₇	1	1	yes	characteristic
346	#336	C₂²×C₆₇	1	1	yes	characteristic
347	#336	C₂²×C₆₇	1	1	yes	characteristic
348	#336	C₂²×C₆₇	1	1	yes	characteristic
349	#336	C₂²×C₆₇	1	1	yes	characteristic
350	#336	C₂²×C₆₇	1	1	yes	characteristic
351	#336	C₂²×C₆₇	1	1	yes	characteristic
352	#336	C₂²×C₆₇	1	1	yes	characteristic
353	#336	C₂²×C₆₇	1	1	yes	characteristic
354	#336	C₂²×C₆₇	1	1	yes	characteristic
355	#336	C₂²×C₆₇	1	1	yes	characteristic
356	#336	C₂²×C₆₇	1	1	yes	characteristic
357	#336	C₂²×C₆₇	1	1	yes	characteristic
358	#336	C₂²×C₆₇	1	1	yes	characteristic
359	#336	C₂²×C₆₇	1	1	yes	characteristic
360	#336	C₂²×C₆₇	1	1	yes	characteristic
361	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
362	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
363	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
364	C₂²×C₈	—	2	2	no	—
365	C₂²×C₈	—	2	2	no	—
366	C₂²×C₈	—	2	2	no	—
367	C₂²×C₈	—	2	2	no	—
368	C₂²×C₈	—	4	1	no	—
369	C₂²×C₈	—	4	1	no	—
370	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
371	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
372	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
373	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
374	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
375	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
376	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
377	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
379	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
381	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
382	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
383	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
384	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
385	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
386	C₂×C₄²	—	2	2	no	—
387	C₂×C₄²	—	2	2	no	—
388	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
389	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
391	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
394	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
396	C₄⋊C₈	—	2	2	no	—
397	C₄⋊C₈	—	2	2	no	—
398	C₄⋊C₈	—	4	1	no	—
399	C₄⋊C₈	—	4	1	no	—
400	C₂²⋊C₈	—	2	2	no	—
401	C₂²⋊C₈	—	2	2	no	—
402	#97	—	2	2	no	—
403	#97	—	4	1	no	—
404	C₄×C₈	—	2	2	no	—
405	C₄×C₈	—	4	1	no	—
406	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
407	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
408	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
409	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
410	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
411	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
412	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
413	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
414	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
415	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
416	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
417	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
418	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
419	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
420	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
421	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
422	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
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	2	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	4	1	no	—
433	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
434	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
435	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
436	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
437	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
438	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
439	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
440	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
441	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
442	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
443	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
444	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
445	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
446	C₂²×C₄	#41×C₆₇	1	2	yes	—
447	C₂²×C₄	#41×C₆₇	1	2	yes	—
448	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	4	4	no	—
472	C₂×C₈	—	4	4	no	—
473	C₂×C₈	#41×C₆₇	1	2	yes	—
474	C₂×C₈	#41×C₆₇	1	2	yes	—
475	C₄⋊C₄	—	2	2	no	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	—	4	1	no	—
478	C₄⋊C₄	—	4	1	no	—
479	C₂²⋊C₄	—	2	2	no	—
480	C₂²⋊C₄	—	2	2	no	—
481	C₂²⋊C₄	—	2	2	no	—
482	C₂²⋊C₄	—	2	2	no	—
483	C₄²	—	2	2	no	—
484	C₄²	—	2	2	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	2	1	no	—
489	C₄²	—	2	1	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	2	1	no	—
492	C₄²	—	2	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	2	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	2	1	no	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	1	no	—
499	C₂³	—	2	1	no	—
500	C₂³	—	2	1	no	—
501	C₂³	#117×C₆₇	1	2	yes	—
502	C₂³	#131×C₆₇	1	2	yes	—
503	C₂³	#131×C₆₇	1	2	yes	—
504	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	—	4	4	no	—
548	C₂×C₄	#117×C₆₇	1	2	yes	—
549	C₂×C₄	#131×C₆₇	1	2	yes	—
550	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
551	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
552	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
553	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
554	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
555	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
556	C₈	—	2	2	no	—
557	C₈	—	2	2	no	—
558	C₈	—	4	1	no	—
559	C₈	—	4	1	no	—
560	C₈	—	4	1	no	—
561	C₈	—	4	1	no	—
562	C₈	—	4	1	no	—
563	C₈	—	4	1	no	—
564	C₂²	—	2	2	no	—
565	C₂²	—	2	2	no	—
566	C₂²	—	2	2	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	2	no	—
569	C₂²	—	2	2	no	—
570	C₂²	#433×C₆₇	1	4	yes	—
571	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
572	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
573	C₂²	#520×C₆₇	1	1	yes	characteristic
574	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
575	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
576	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
577	C₂²	#568×C₆₇	1	1	yes	characteristic
578	C₄	—	2	2	no	—
579	C₄	—	2	2	no	—
580	C₄	—	4	1	no	—
581	C₄	—	4	1	no	—
582	C₄	—	4	1	no	—
583	C₄	—	4	1	no	—
584	C₄	—	4	1	no	—
585	C₄	—	4	1	no	—
586	C₄	—	4	1	no	—
587	C₄	—	4	1	no	—
588	C₄	—	4	1	no	—
589	C₄	—	4	1	no	—
590	C₄	—	4	1	no	—
591	C₄	#551×C₆₇	1	1	yes	characteristic
592	C₄	#551×C₆₇	1	1	yes	characteristic
593	C₄	#552×C₆₇	1	1	yes	characteristic
594	C₄	#552×C₆₇	1	1	yes	characteristic
595	C₂	—	2	4	no	—
596	C₂	C₆₇×#2343	1	1	yes	characteristic
597	C₂	C₂×#433×C₆₇	1	1	yes	characteristic
598	C₂	C₆₇×#2924	1	1	yes	characteristic
599	C₂	C₆₇×#2966	1	1	yes	characteristic
600	C₂	C₆₇×#2968	1	1	yes	characteristic
601	C₂	C₆₇×#2992	1	1	yes	characteristic
602	C₂	C₆₇×#2994	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#1917	C₂	semidirect
3. C₆₇×#1917	C₂	semidirect
4. C₆₇×#1916	C₂	semidirect
5. C₆₇×#1916	C₂	semidirect
8. C₆₇×#1840	C₂	semidirect
9. C₆₇×#1832	C₂	semidirect
11. C₆₇×#1768	C₂	semidirect
14. C₆₇×#1746	C₂	semidirect
293. #22595	C₆₇	direct (N × H)
303. #1917	C₂×C₆₇	semidirect
304. #1917	C₂×C₆₇	semidirect
305. #1916	C₂×C₆₇	semidirect
306. #1916	C₂×C₆₇	semidirect
309. #1840	C₂×C₆₇	semidirect
310. #1832	C₂×C₆₇	semidirect
312. #1768	C₂×C₆₇	semidirect
315. #1746	C₂×C₆₇	semidirect
317. C₆₇	#22595	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×#2283	2
C₆₇×#1917	2
C₆₇×#1916	2
C₆₇×#1850	2
C₆₇×#1840	2
C₆₇×#1832	2
C₆₇×#1817	2
C₆₇×#1768	2
C₆₇×#1752	2
C₈×C₂²⋊C₄×C₆₇	2
C₆₇×#1746	2
C₂×#336×C₆₇	2
#22595	67

Subgroups — C67×#22595 #30701846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22595 #30701846