Subgroups — C₆₇×#22594 #30701845

Order 17152.

688 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2298	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×#1917	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1916	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1851	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×#1831	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1818	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1768	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#1753	C₂	1	1	yes	maximal, characteristic
13	C₆₇×#1752	C₂	1	1	yes	maximal, characteristic
14	C₆₇×#1745	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#1729	C₂	1	1	yes	maximal, characteristic
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#117×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×#97×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#388×C₆₇	C₄	1	1	yes	characteristic
30	#388×C₆₇	C₄	1	1	yes	characteristic
31	#388×C₆₇	C₄	1	1	yes	characteristic
32	#388×C₆₇	C₄	1	1	yes	characteristic
33	#387×C₆₇	C₄	1	1	yes	characteristic
34	#387×C₆₇	C₄	1	1	yes	characteristic
35	#387×C₆₇	C₄	1	1	yes	characteristic
36	#387×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	C₄³×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	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₆₇	—	2	2	no	—
61	C₂×M₁₆×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	—	4	1	no	—
63	C₂²×C₈×C₆₇	—	2	2	no	—
64	C₂²×C₈×C₆₇	—	2	2	no	—
65	C₂²×C₈×C₆₇	—	4	1	no	—
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
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	#117×C₆₇	—	2	2	no	—
75	#117×C₆₇	—	2	2	no	—
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×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄⋊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×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₆₇	—	2	2	no	—
99	C₄⋊C₈×C₆₇	—	2	2	no	—
100	C₄⋊C₈×C₆₇	—	4	1	no	—
101	C₄⋊C₈×C₆₇	—	4	1	no	—
102	C₂²⋊C₈×C₆₇	—	2	2	no	—
103	C₂²⋊C₈×C₆₇	—	2	2	no	—
104	#97×C₆₇	—	2	2	no	—
105	#97×C₆₇	—	2	2	no	—
106	#97×C₆₇	—	4	1	no	—
107	C₄×C₈×C₆₇	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	#41	1	2	yes	—
139	C₂²×C₄×C₆₇	#41	1	2	yes	—
140	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
141	M₁₆×C₆₇	—	4	4	no	—
142	C₂×C₈×C₆₇	—	2	1	no	—
143	C₂×C₈×C₆₇	—	2	1	no	—
144	C₂×C₈×C₆₇	—	2	1	no	—
145	C₂×C₈×C₆₇	—	2	1	no	—
146	C₂×C₈×C₆₇	—	2	1	no	—
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	—	4	4	no	—
165	C₂×C₈×C₆₇	#41	1	2	yes	—
166	C₂×C₈×C₆₇	#41	1	2	yes	—
167	C₄⋊C₄×C₆₇	—	2	2	no	—
168	C₄⋊C₄×C₆₇	—	2	2	no	—
169	C₄⋊C₄×C₆₇	—	2	2	no	—
170	C₄⋊C₄×C₆₇	—	4	1	no	—
171	C₄⋊C₄×C₆₇	—	4	1	no	—
172	C₄⋊C₄×C₆₇	—	4	1	no	—
173	C₂²⋊C₄×C₆₇	—	2	2	no	—
174	C₂²⋊C₄×C₆₇	—	2	2	no	—
175	C₂²⋊C₄×C₆₇	—	2	2	no	—
176	C₂²⋊C₄×C₆₇	—	4	1	no	—
177	C₂²⋊C₄×C₆₇	—	4	1	no	—
178	C₂²⋊C₄×C₆₇	—	4	1	no	—
179	C₄²×C₆₇	—	2	2	no	—
180	C₄²×C₆₇	—	2	1	no	—
181	C₄²×C₆₇	—	2	1	no	—
182	C₄²×C₆₇	—	2	1	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	—	4	1	no	—
193	C₄²×C₆₇	—	4	1	no	—
194	C₄²×C₆₇	—	4	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	—	2	1	no	—
200	C₂³×C₆₇	—	2	1	no	—
201	C₂³×C₆₇	#117	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	#117	1	2	yes	—
245	C₂×C₄×C₆₇	#131	1	2	yes	—
246	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
247	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
248	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
249	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
250	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
251	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
252	C₈×C₆₇	—	2	2	no	—
253	C₈×C₆₇	—	2	2	no	—
254	C₈×C₆₇	—	4	1	no	—
255	C₈×C₆₇	—	4	1	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	C₂²×C₆₇	—	2	2	no	—
261	C₂²×C₆₇	—	2	2	no	—
262	C₂²×C₆₇	—	2	2	no	—
263	C₂²×C₆₇	—	4	1	no	—
264	C₂²×C₆₇	—	4	1	no	—
265	C₂²×C₆₇	—	4	1	no	—
266	C₂²×C₆₇	—	4	1	no	—
267	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
268	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
269	C₂²×C₆₇	#520	1	1	yes	characteristic
270	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
271	C₂²×C₆₇	#568	1	1	yes	characteristic
272	C₂²×C₆₇	#568	1	1	yes	characteristic
273	C₂²×C₆₇	#568	1	1	yes	characteristic
274	C₄×C₆₇	—	2	2	no	—
275	C₄×C₆₇	—	2	2	no	—
276	C₄×C₆₇	—	4	1	no	—
277	C₄×C₆₇	—	4	1	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	#551	1	1	yes	characteristic
288	C₄×C₆₇	#551	1	1	yes	characteristic
289	C₄×C₆₇	#552	1	1	yes	characteristic
290	C₄×C₆₇	#552	1	1	yes	characteristic
291	#22594	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
292	C₂×C₆₇	—	4	2	no	—
293	C₂×C₆₇	#2343	1	1	yes	characteristic
294	C₂×C₆₇	#2923	1	1	yes	characteristic
295	C₂×C₆₇	#2925	1	1	yes	characteristic
296	C₂×C₆₇	#2967	1	1	yes	characteristic
297	C₂×C₆₇	#2969	1	1	yes	characteristic
298	C₂×C₆₇	#2992	1	1	yes	characteristic
299	C₂×C₆₇	#2995	1	1	yes	characteristic
300	#2298	C₂×C₆₇	1	1	yes	characteristic
301	#1917	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#1917	C₂×C₆₇	1	1	yes	characteristic, complemented
303	#1917	C₂×C₆₇	1	1	yes	characteristic, complemented
304	#1916	C₂×C₆₇	1	1	yes	characteristic, complemented
305	#1851	C₂×C₆₇	1	1	yes	characteristic
306	#1850	C₂×C₆₇	1	1	yes	characteristic
307	#1840	C₂×C₆₇	1	1	yes	characteristic, complemented
308	#1831	C₂×C₆₇	1	1	yes	characteristic, complemented
309	#1818	C₂×C₆₇	1	1	yes	characteristic
310	#1768	C₂×C₆₇	1	1	yes	characteristic, complemented
311	#1753	C₂×C₆₇	1	1	yes	characteristic
312	#1752	C₂×C₆₇	1	1	yes	characteristic
313	#1745	C₂×C₆₇	1	1	yes	characteristic, complemented
314	#1729	C₂×C₆₇	1	1	yes	characteristic
315	C₆₇	#22594	1	1	yes	characteristic, direct product factor, Sylow
316	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
317	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
318	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
319	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
320	C₂×C₄⋊C₈	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×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
324	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
326	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
327	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
328	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
329	#388	C₄×C₆₇	1	1	yes	characteristic
330	#388	C₄×C₆₇	1	1	yes	characteristic
331	#388	C₄×C₆₇	1	1	yes	characteristic
332	#388	C₄×C₆₇	1	1	yes	characteristic
333	#387	C₄×C₆₇	1	1	yes	characteristic
334	#387	C₄×C₆₇	1	1	yes	characteristic
335	#387	C₄×C₆₇	1	1	yes	characteristic
336	#387	C₄×C₆₇	1	1	yes	characteristic
337	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
338	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
339	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
340	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
341	#376	C₂²×C₆₇	1	1	yes	characteristic
342	C₄³	C₂²×C₆₇	1	1	yes	characteristic
343	#336	C₂²×C₆₇	1	1	yes	characteristic
344	#336	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	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
360	C₂×M₁₆	—	2	2	no	—
361	C₂×M₁₆	—	2	2	no	—
362	C₂×M₁₆	—	4	1	no	—
363	C₂²×C₈	—	2	2	no	—
364	C₂²×C₈	—	2	2	no	—
365	C₂²×C₈	—	4	1	no	—
366	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
367	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
368	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
369	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
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	#117	—	2	2	no	—
375	#117	—	2	2	no	—
376	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
377	C₂×C₄⋊C₄	C₂×C₄×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×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
387	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
388	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
389	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
390	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
391	C₂×C₄²	C₂×C₄×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	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
397	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
398	C₄⋊C₈	—	2	2	no	—
399	C₄⋊C₈	—	2	2	no	—
400	C₄⋊C₈	—	4	1	no	—
401	C₄⋊C₈	—	4	1	no	—
402	C₂²⋊C₈	—	2	2	no	—
403	C₂²⋊C₈	—	2	2	no	—
404	#97	—	2	2	no	—
405	#97	—	2	2	no	—
406	#97	—	4	1	no	—
407	C₄×C₈	—	4	1	no	—
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	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
426	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
427	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
428	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
429	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
430	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
431	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
432	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
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	#41×C₆₇	1	2	yes	—
439	C₂²×C₄	#41×C₆₇	1	2	yes	—
440	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
441	M₁₆	—	4	4	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	1	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	1	no	—
447	C₂×C₈	—	2	1	no	—
448	C₂×C₈	—	2	1	no	—
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	—	4	4	no	—
465	C₂×C₈	#41×C₆₇	1	2	yes	—
466	C₂×C₈	#41×C₆₇	1	2	yes	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	—	4	1	no	—
471	C₄⋊C₄	—	4	1	no	—
472	C₄⋊C₄	—	4	1	no	—
473	C₂²⋊C₄	—	2	2	no	—
474	C₂²⋊C₄	—	2	2	no	—
475	C₂²⋊C₄	—	2	2	no	—
476	C₂²⋊C₄	—	4	1	no	—
477	C₂²⋊C₄	—	4	1	no	—
478	C₂²⋊C₄	—	4	1	no	—
479	C₄²	—	2	2	no	—
480	C₄²	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	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	—	4	1	no	—
493	C₄²	—	4	1	no	—
494	C₄²	—	4	1	no	—
495	C₂³	—	2	1	no	—
496	C₂³	—	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	C₂³×C₄×C₆₇	1	1	yes	characteristic
503	C₂×C₄	—	2	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	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	#117×C₆₇	1	2	yes	—
545	C₂×C₄	#131×C₆₇	1	2	yes	—
546	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
547	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
548	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
549	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
550	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
551	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
552	C₈	—	2	2	no	—
553	C₈	—	2	2	no	—
554	C₈	—	4	1	no	—
555	C₈	—	4	1	no	—
556	C₈	—	4	1	no	—
557	C₈	—	4	1	no	—
558	C₈	—	4	1	no	—
559	C₈	—	4	1	no	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	4	1	no	—
564	C₂²	—	4	1	no	—
565	C₂²	—	4	1	no	—
566	C₂²	—	4	1	no	—
567	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
568	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
569	C₂²	#520×C₆₇	1	1	yes	characteristic
570	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
571	C₂²	#568×C₆₇	1	1	yes	characteristic
572	C₂²	#568×C₆₇	1	1	yes	characteristic
573	C₂²	#568×C₆₇	1	1	yes	characteristic
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	4	1	no	—
577	C₄	—	4	1	no	—
578	C₄	—	4	1	no	—
579	C₄	—	4	1	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	#551×C₆₇	1	1	yes	characteristic
588	C₄	#551×C₆₇	1	1	yes	characteristic
589	C₄	#552×C₆₇	1	1	yes	characteristic
590	C₄	#552×C₆₇	1	1	yes	characteristic
591	C₂	—	4	2	no	—
592	C₂	C₆₇×#2343	1	1	yes	characteristic
593	C₂	C₆₇×#2923	1	1	yes	characteristic
594	C₂	C₆₇×#2925	1	1	yes	characteristic
595	C₂	C₆₇×#2967	1	1	yes	characteristic
596	C₂	C₆₇×#2969	1	1	yes	characteristic
597	C₂	C₆₇×#2992	1	1	yes	characteristic
598	C₂	C₆₇×#2995	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×#1917	C₂	semidirect
5. C₆₇×#1916	C₂	semidirect
8. C₆₇×#1840	C₂	semidirect
9. C₆₇×#1831	C₂	semidirect
11. C₆₇×#1768	C₂	semidirect
14. C₆₇×#1745	C₂	semidirect
291. #22594	C₆₇	direct (N × H)
301. #1917	C₂×C₆₇	semidirect
302. #1917	C₂×C₆₇	semidirect
303. #1917	C₂×C₆₇	semidirect
304. #1916	C₂×C₆₇	semidirect
307. #1840	C₂×C₆₇	semidirect
308. #1831	C₂×C₆₇	semidirect
310. #1768	C₂×C₆₇	semidirect
313. #1745	C₂×C₆₇	semidirect
315. C₆₇	#22594	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×#2298	2
C₆₇×#1917	2
C₆₇×#1916	2
C₆₇×#1851	2
C₆₇×#1850	2
C₆₇×#1840	2
C₆₇×#1831	2
C₆₇×#1818	2
C₆₇×#1768	2
C₆₇×#1753	2
C₆₇×#1752	2
C₆₇×#1745	2
C₆₇×#1729	2
#22594	67

Subgroups — C67×#22594 #30701845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22594 #30701845