Subgroups — C₆₇×#34268 #30713519

Order 17152.

2946 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#536×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#519×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2367	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2365	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2359	C₂	1	8	yes	maximal, complemented
7	C₆₇×#2323	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2311	C₂	1	1	yes	maximal, characteristic
9	C₄×C₂²⋊Q₈×C₆₇	C₂	1	4	yes	maximal, complemented
10	C₆₇×#2296	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#2274	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	#536×C₆₇	C₄	1	16	yes	complemented
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
17	C₂×#124×C₆₇	C₂²	1	2	yes	—
18	C₂×#123×C₆₇	C₂²	1	4	yes	—
19	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
20	#519×C₆₇	—	2	8	no	—
21	C₄×#41×C₆₇	—	2	8	no	—
22	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
27	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#117×C₆₇	C₂²	1	2	yes	—
29	C₂×#117×C₆₇	C₂²	1	4	yes	—
30	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
35	#391×C₆₇	C₂²	1	2	yes	—
36	#391×C₆₇	C₂²	1	2	yes	—
37	#391×C₆₇	C₂²	1	4	yes	—
38	#391×C₆₇	C₂²	1	2	yes	—
39	#390×C₆₇	C₂²	1	2	yes	—
40	#389×C₆₇	C₂²	1	2	yes	—
41	#389×C₆₇	C₂²	1	2	yes	—
42	#389×C₆₇	C₂²	1	8	yes	—
43	#388×C₆₇	C₂²	1	8	yes	—
44	#388×C₆₇	C₂²	1	8	yes	—
45	#387×C₆₇	C₂²	1	8	yes	—
46	#387×C₆₇	C₂²	1	8	yes	—
47	#386×C₆₇	C₂²	1	4	yes	—
48	#385×C₆₇	C₂²	1	4	yes	—
49	#385×C₆₇	C₂²	1	8	yes	—
50	#380×C₆₇	C₂²	1	4	yes	—
51	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
52	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
53	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
54	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
55	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
56	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
57	C₂×#95×C₆₇	C₂²	1	4	yes	—
58	C₂×#41×C₆₇	—	2	4	no	—
59	C₂×#41×C₆₇	—	2	4	no	—
60	C₂×#41×C₆₇	C₂×C₄	1	8	yes	—
61	C₂²×Q₈×C₆₇	—	2	4	no	—
62	C₂²×Q₈×C₆₇	—	2	4	no	—
63	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
64	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
65	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
66	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
68	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
70	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
71	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
72	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
73	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
74	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₆₇	C₂×C₄	1	16	yes	complemented
76	#124×C₆₇	C₂×C₄	1	16	yes	complemented
77	#123×C₆₇	C₂×C₄	1	32	yes	complemented
78	C₂²⋊Q₈×C₆₇	C₂×C₄	1	32	yes	complemented
79	C₄×Q₈×C₆₇	—	2	8	no	—
80	C₄×Q₈×C₆₇	—	2	16	no	—
81	C₄×Q₈×C₆₇	—	2	16	no	—
82	C₄×D₄×C₆₇	—	2	8	no	—
83	C₄×D₄×C₆₇	—	2	16	no	—
84	#117×C₆₇	—	2	4	no	—
85	#117×C₆₇	—	2	16	no	—
86	#117×C₆₇	—	2	8	no	—
87	#117×C₆₇	C₂×C₄	1	8	yes	complemented
88	#117×C₆₇	D₄	1	8	yes	—
89	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
101	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
106	C₂×C₄²×C₆₇	—	2	4	no	—
107	C₂×C₄²×C₆₇	—	2	8	no	—
108	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
111	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
112	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
113	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
114	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
115	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
116	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
117	#95×C₆₇	C₂³	1	8	yes	—
118	#95×C₆₇	C₂³	1	8	yes	—
119	#95×C₆₇	C₂³	1	4	yes	—
120	#95×C₆₇	C₂³	1	8	yes	—
121	#95×C₆₇	C₂³	1	8	yes	—
122	#95×C₆₇	C₂³	1	4	yes	—
123	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
124	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
125	#41×C₆₇	—	2	32	no	—
126	C₂×Q₈×C₆₇	—	2	32	no	—
127	C₂×Q₈×C₆₇	—	2	4	no	—
128	C₂×Q₈×C₆₇	—	2	4	no	—
129	C₂×Q₈×C₆₇	—	2	2	no	—
130	C₂×Q₈×C₆₇	—	2	4	no	—
131	C₂×Q₈×C₆₇	—	2	4	no	—
132	C₂×Q₈×C₆₇	—	2	2	no	—
133	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
134	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
135	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
136	C₂×D₄×C₆₇	—	2	4	no	—
137	C₂×D₄×C₆₇	—	2	4	no	—
138	C₂×D₄×C₆₇	—	2	2	no	—
139	C₂×D₄×C₆₇	—	2	2	no	—
140	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
141	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
142	C₂²×C₄×C₆₇	—	2	4	no	—
143	C₂²×C₄×C₆₇	—	2	4	no	—
144	C₂²×C₄×C₆₇	—	2	4	no	—
145	C₂²×C₄×C₆₇	—	2	4	no	—
146	C₂²×C₄×C₆₇	—	2	4	no	—
147	C₂²×C₄×C₆₇	—	2	4	no	—
148	C₂²×C₄×C₆₇	—	2	8	no	—
149	C₂²×C₄×C₆₇	—	2	8	no	—
150	C₂²×C₄×C₆₇	—	2	4	no	—
151	C₂²×C₄×C₆₇	—	4	4	no	—
152	C₂²×C₄×C₆₇	—	4	2	no	—
153	C₂²×C₄×C₆₇	—	4	4	no	—
154	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
155	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
156	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
157	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
158	C₂²×C₄×C₆₇	#41	1	4	yes	—
159	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
160	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
169	C₄⋊C₄×C₆₇	—	2	8	no	—
170	C₄⋊C₄×C₆₇	—	2	16	no	—
171	C₄⋊C₄×C₆₇	—	2	4	no	—
172	C₄⋊C₄×C₆₇	—	2	4	no	—
173	C₄⋊C₄×C₆₇	—	2	4	no	—
174	C₄⋊C₄×C₆₇	—	2	8	no	—
175	C₄⋊C₄×C₆₇	—	4	4	no	—
176	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
177	C₄⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
178	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
179	C₂²⋊C₄×C₆₇	—	2	4	no	—
180	C₂²⋊C₄×C₆₇	—	2	8	no	—
181	C₂²⋊C₄×C₆₇	—	4	8	no	—
182	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
183	C₂²⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
184	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
185	C₄²×C₆₇	—	2	2	no	—
186	C₄²×C₆₇	—	2	2	no	—
187	C₄²×C₆₇	—	2	8	no	—
188	C₄²×C₆₇	—	2	4	no	—
189	C₄²×C₆₇	—	2	16	no	—
190	C₄²×C₆₇	—	2	4	no	—
191	C₄²×C₆₇	—	4	8	no	—
192	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
193	C₄²×C₆₇	C₂×D₄	1	8	yes	—
194	C₂³×C₆₇	—	2	2	no	—
195	C₂³×C₆₇	—	2	2	no	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	2	2	no	—
198	C₂³×C₆₇	—	2	4	no	—
199	C₂³×C₆₇	—	2	4	no	—
200	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
201	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
202	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
203	C₂³×C₆₇	C₂×#41	1	2	yes	—
204	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
205	Q₈×C₆₇	—	2	16	no	—
206	Q₈×C₆₇	—	2	16	no	—
207	Q₈×C₆₇	—	2	8	no	—
208	D₄×C₆₇	—	2	16	no	—
209	D₄×C₆₇	—	2	8	no	—
210	C₂×C₄×C₆₇	—	2	16	no	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	2	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	2	no	—
215	C₂×C₄×C₆₇	—	2	4	no	—
216	C₂×C₄×C₆₇	—	2	4	no	—
217	C₂×C₄×C₆₇	—	2	4	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	2	4	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	16	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	8	no	—
228	C₂×C₄×C₆₇	—	2	4	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	4	16	no	—
231	C₂×C₄×C₆₇	—	4	8	no	—
232	C₂×C₄×C₆₇	—	8	4	no	—
233	C₂×C₄×C₆₇	C₄×D₄	1	16	yes	—
234	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
235	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
236	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
237	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
238	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
239	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
240	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
243	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
244	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
245	C₂²×C₆₇	—	2	4	no	—
246	C₂²×C₆₇	—	2	4	no	—
247	C₂²×C₆₇	—	2	2	no	—
248	C₂²×C₆₇	—	2	2	no	—
249	C₂²×C₆₇	—	2	4	no	—
250	C₂²×C₆₇	—	2	8	no	—
251	C₂²×C₆₇	—	4	4	no	—
252	C₂²×C₆₇	—	4	2	no	—
253	C₂²×C₆₇	—	4	2	no	—
254	C₂²×C₆₇	C₂×C₄×D₄	1	4	yes	—
255	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
256	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
257	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
258	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
259	C₂²×C₆₇	#585	1	2	yes	—
260	C₄×C₆₇	—	2	4	no	—
261	C₄×C₆₇	—	2	8	no	—
262	C₄×C₆₇	—	2	8	no	—
263	C₄×C₆₇	—	2	4	no	—
264	C₄×C₆₇	—	2	8	no	—
265	C₄×C₆₇	—	4	4	no	—
266	C₄×C₆₇	—	4	2	no	—
267	C₄×C₆₇	—	4	4	no	—
268	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
269	C₄×C₆₇	C₂×C₄×D₄	1	4	yes	—
270	#34268	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
271	C₂×C₆₇	—	2	4	no	—
272	C₂×C₆₇	—	4	4	no	—
273	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
274	C₂×C₆₇	C₄×2₋¹⁺⁴	1	2	yes	—
275	C₂×C₆₇	#3489	1	2	yes	—
276	C₂×C₆₇	#3509	1	1	yes	characteristic
277	C₂×C₆₇	#3510	1	1	yes	characteristic
278	C₂×#536	C₂×C₆₇	1	1	yes	characteristic
279	C₂×#519	C₂×C₆₇	1	1	yes	characteristic, complemented
280	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
281	#2367	C₂×C₆₇	1	4	yes	complemented
282	#2365	C₂×C₆₇	1	4	yes	complemented
283	#2359	C₂×C₆₇	1	8	yes	complemented
284	#2323	C₂×C₆₇	1	4	yes	complemented
285	#2311	C₂×C₆₇	1	1	yes	characteristic
286	C₄×C₂²⋊Q₈	C₂×C₆₇	1	4	yes	complemented
287	#2296	C₂×C₆₇	1	1	yes	characteristic, complemented
288	C₂×#391	C₂×C₆₇	1	1	yes	characteristic, complemented
289	#2274	C₂×C₆₇	1	1	yes	characteristic, complemented
290	C₆₇	#34268	1	1	yes	characteristic, direct product factor, Sylow
291	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
292	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
293	#536	C₄×C₆₇	1	16	yes	complemented
294	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
295	C₂×#124	C₂²×C₆₇	1	2	yes	—
296	C₂×#123	C₂²×C₆₇	1	4	yes	—
297	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
298	#519	—	2	8	no	—
299	C₄×#41	—	2	8	no	—
300	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
301	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
302	C₂×C₄×Q₈	C₂²×C₆₇	1	4	yes	—
303	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
304	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
305	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
306	C₂×#117	C₂²×C₆₇	1	2	yes	—
307	C₂×#117	C₂²×C₆₇	1	4	yes	—
308	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
309	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
310	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
311	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
312	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
313	#391	C₂²×C₆₇	1	2	yes	—
314	#391	C₂²×C₆₇	1	2	yes	—
315	#391	C₂²×C₆₇	1	4	yes	—
316	#391	C₂²×C₆₇	1	2	yes	—
317	#390	C₂²×C₆₇	1	2	yes	—
318	#389	C₂²×C₆₇	1	2	yes	—
319	#389	C₂²×C₆₇	1	8	yes	—
320	#389	C₂²×C₆₇	1	2	yes	—
321	#388	C₂²×C₆₇	1	8	yes	—
322	#388	C₂²×C₆₇	1	8	yes	—
323	#387	C₂²×C₆₇	1	8	yes	—
324	#387	C₂²×C₆₇	1	8	yes	—
325	#386	C₂²×C₆₇	1	4	yes	—
326	#385	C₂²×C₆₇	1	4	yes	—
327	#385	C₂²×C₆₇	1	8	yes	—
328	#380	C₂²×C₆₇	1	4	yes	—
329	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
330	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
331	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
332	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
333	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
334	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
335	C₂×#95	C₂²×C₆₇	1	4	yes	—
336	C₂×#41	—	2	4	no	—
337	C₂×#41	—	2	4	no	—
338	C₂×#41	C₂×C₄×C₆₇	1	8	yes	—
339	C₂²×Q₈	—	2	4	no	—
340	C₂²×Q₈	—	2	4	no	—
341	C₂²×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
342	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
343	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
344	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
345	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
346	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
347	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
348	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
349	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
350	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
351	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
352	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
353	C₄⋊Q₈	C₂×C₄×C₆₇	1	16	yes	complemented
354	#124	C₂×C₄×C₆₇	1	16	yes	complemented
355	#123	C₂×C₄×C₆₇	1	32	yes	complemented
356	C₂²⋊Q₈	C₂×C₄×C₆₇	1	32	yes	complemented
357	C₄×Q₈	—	2	8	no	—
358	C₄×Q₈	—	2	16	no	—
359	C₄×Q₈	—	2	16	no	—
360	C₄×D₄	—	2	8	no	—
361	C₄×D₄	—	2	16	no	—
362	#117	—	2	4	no	—
363	#117	—	2	16	no	—
364	#117	—	2	8	no	—
365	#117	C₂×C₄×C₆₇	1	8	yes	complemented
366	#117	D₄×C₆₇	1	8	yes	—
367	C₂×C₄⋊C₄	—	2	4	no	—
368	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
369	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
371	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
373	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
375	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
377	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
384	C₂×C₄²	—	2	4	no	—
385	C₂×C₄²	—	2	8	no	—
386	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
388	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
389	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
390	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
391	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
393	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
394	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
395	#95	C₂³×C₆₇	1	8	yes	—
396	#95	C₂³×C₆₇	1	8	yes	—
397	#95	C₂³×C₆₇	1	4	yes	—
398	#95	C₂³×C₆₇	1	8	yes	—
399	#95	C₂³×C₆₇	1	8	yes	—
400	#95	C₂³×C₆₇	1	4	yes	—
401	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
402	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
403	#41	—	2	32	no	—
404	C₂×Q₈	—	2	32	no	—
405	C₂×Q₈	—	2	4	no	—
406	C₂×Q₈	—	2	4	no	—
407	C₂×Q₈	—	2	2	no	—
408	C₂×Q₈	—	2	4	no	—
409	C₂×Q₈	—	2	4	no	—
410	C₂×Q₈	—	2	2	no	—
411	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
412	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
413	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	4	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
419	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	8	no	—
427	C₂²×C₄	—	2	8	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	4	4	no	—
430	C₂²×C₄	—	4	2	no	—
431	C₂²×C₄	—	4	4	no	—
432	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
433	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
434	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
435	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
436	C₂²×C₄	#41×C₆₇	1	4	yes	—
437	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
438	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
441	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
442	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
444	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
445	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
446	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
447	C₄⋊C₄	—	2	8	no	—
448	C₄⋊C₄	—	2	16	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	4	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	4	4	no	—
454	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
455	C₄⋊C₄	C₂²×C₄×C₆₇	1	32	yes	—
456	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
457	C₂²⋊C₄	—	2	4	no	—
458	C₂²⋊C₄	—	2	8	no	—
459	C₂²⋊C₄	—	4	8	no	—
460	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
461	C₂²⋊C₄	C₂²×C₄×C₆₇	1	32	yes	—
462	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
463	C₄²	—	2	2	no	—
464	C₄²	—	2	2	no	—
465	C₄²	—	2	8	no	—
466	C₄²	—	2	4	no	—
467	C₄²	—	2	16	no	—
468	C₄²	—	2	4	no	—
469	C₄²	—	4	8	no	—
470	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
471	C₄²	C₂×D₄×C₆₇	1	8	yes	—
472	C₂³	—	2	2	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	4	no	—
477	C₂³	—	2	4	no	—
478	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
479	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
480	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
481	C₂³	C₂×#41×C₆₇	1	2	yes	—
482	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
483	Q₈	—	2	16	no	—
484	Q₈	—	2	16	no	—
485	Q₈	—	2	8	no	—
486	D₄	—	2	16	no	—
487	D₄	—	2	8	no	—
488	C₂×C₄	—	2	16	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	16	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	8	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	4	16	no	—
509	C₂×C₄	—	4	8	no	—
510	C₂×C₄	—	8	4	no	—
511	C₂×C₄	C₄×D₄×C₆₇	1	16	yes	—
512	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
513	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
514	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
515	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
516	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
517	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
518	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
519	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
520	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
521	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
522	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
523	C₂²	—	2	4	no	—
524	C₂²	—	2	4	no	—
525	C₂²	—	2	2	no	—
526	C₂²	—	2	2	no	—
527	C₂²	—	2	4	no	—
528	C₂²	—	2	8	no	—
529	C₂²	—	4	4	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	C₂×C₄×D₄×C₆₇	1	4	yes	—
533	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
534	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
535	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
536	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
537	C₂²	#585×C₆₇	1	2	yes	—
538	C₄	—	2	4	no	—
539	C₄	—	2	8	no	—
540	C₄	—	2	8	no	—
541	C₄	—	2	4	no	—
542	C₄	—	2	8	no	—
543	C₄	—	4	2	no	—
544	C₄	—	4	4	no	—
545	C₄	—	4	4	no	—
546	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
547	C₄	C₂×C₄×D₄×C₆₇	1	4	yes	—
548	C₂	—	2	4	no	—
549	C₂	—	4	4	no	—
550	C₂	C₂²×C₄×D₄×C₆₇	1	1	yes	characteristic
551	C₂	C₄×2₋¹⁺⁴×C₆₇	1	2	yes	—
552	C₂	C₆₇×#3489	1	2	yes	—
553	C₂	C₆₇×#3509	1	1	yes	characteristic
554	C₂	C₆₇×#3510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#519×C₆₇	C₂	semidirect
4. C₆₇×#2367	C₂	semidirect
5. C₆₇×#2365	C₂	semidirect
6. C₆₇×#2359	C₂	semidirect
7. C₆₇×#2323	C₂	semidirect
9. C₄×C₂²⋊Q₈×C₆₇	C₂	semidirect
10. C₆₇×#2296	C₂	semidirect
11. C₂×#391×C₆₇	C₂	semidirect
12. C₆₇×#2274	C₂	semidirect
15. #536×C₆₇	C₄	semidirect
63. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
75. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
76. #124×C₆₇	C₂×C₄	semidirect
77. #123×C₆₇	C₂×C₄	semidirect
78. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
87. #117×C₆₇	C₂×C₄	semidirect
270. #34268	C₆₇	direct (N × H)
279. C₂×#519	C₂×C₆₇	semidirect
281. #2367	C₂×C₆₇	semidirect
282. #2365	C₂×C₆₇	semidirect
283. #2359	C₂×C₆₇	semidirect
284. #2323	C₂×C₆₇	semidirect
286. C₄×C₂²⋊Q₈	C₂×C₆₇	semidirect
287. #2296	C₂×C₆₇	semidirect
288. C₂×#391	C₂×C₆₇	semidirect
289. #2274	C₂×C₆₇	semidirect
290. C₆₇	#34268	direct (N × H)
293. #536	C₄×C₆₇	semidirect
341. C₂²×Q₈	C₂×C₄×C₆₇	semidirect
353. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
354. #124	C₂×C₄×C₆₇	semidirect
355. #123	C₂×C₄×C₆₇	semidirect
356. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
365. #117	C₂×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×#536×C₆₇	2
C₂×#519×C₆₇	2
C₂×C₄×#41×C₆₇	2
C₆₇×#2367	2
C₆₇×#2365	2
C₆₇×#2359	2
C₆₇×#2323	2
C₆₇×#2311	2
C₄×C₂²⋊Q₈×C₆₇	2
C₆₇×#2296	2
C₂×#391×C₆₇	2
C₆₇×#2274	2
#34268	67

Subgroups — C67×#34268 #30713519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34268 #30713519