Subgroups — C₆₇×#34946 #30714197

Order 17152.

2834 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2563	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2562	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2561	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2547	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2503	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2435	C₂	1	2	yes	maximal
9	C₆₇×#2398	C₂	1	4	yes	maximal, complemented
10	C₄×#124×C₆₇	C₂	1	4	yes	maximal, complemented
11	C₆₇×#2280	C₂	1	4	yes	maximal, complemented
12	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
15	#533×C₆₇	—	2	8	no	—
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
18	C₂×#125×C₆₇	C₂²	1	2	yes	complemented
19	C₂×#124×C₆₇	C₂²	1	4	yes	—
20	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
23	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
24	C₂×C₄×D₄×C₆₇	—	2	8	no	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
29	C₂×#117×C₆₇	C₂²	1	2	yes	—
30	C₂×#117×C₆₇	C₂²	1	2	yes	—
31	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
33	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	#400×C₆₇	C₂²	1	8	yes	complemented
35	#397×C₆₇	C₂²	1	8	yes	complemented
36	#394×C₆₇	C₂²	1	8	yes	—
37	#393×C₆₇	C₂²	1	8	yes	complemented
38	#391×C₆₇	C₂²	1	4	yes	complemented
39	#390×C₆₇	C₂²	1	4	yes	—
40	#388×C₆₇	C₂²	1	16	yes	complemented
41	#382×C₆₇	C₂²	1	4	yes	complemented
42	#381×C₆₇	C₂²	1	8	yes	—
43	#380×C₆₇	C₂²	1	8	yes	complemented
44	#380×C₆₇	C₂²	1	8	yes	complemented
45	#379×C₆₇	C₂²	1	8	yes	—
46	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
47	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
48	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
49	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
50	C₄×C₂²⋊C₄×C₆₇	C₂²	1	16	yes	complemented
51	#376×C₆₇	C₂²	1	2	yes	complemented
52	C₄³×C₆₇	C₂²	1	2	yes	complemented
53	C₂⁵×C₆₇	C₂³	1	2	yes	—
54	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
56	C₂²×D₄×C₆₇	—	2	8	no	—
57	C₂²×D₄×C₆₇	—	2	4	no	—
58	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
59	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
60	C₂³×C₄×C₆₇	—	2	8	no	—
61	C₂³×C₄×C₆₇	—	2	16	no	—
62	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
64	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₆₇	—	2	4	no	—
69	C₄⋊Q₈×C₆₇	—	2	4	no	—
70	#125×C₆₇	—	2	8	no	—
71	#124×C₆₇	—	2	16	no	—
72	C₂²⋊Q₈×C₆₇	—	2	8	no	—
73	C₂²⋊Q₈×C₆₇	—	2	8	no	—
74	C₄×Q₈×C₆₇	—	2	8	no	—
75	C₄×Q₈×C₆₇	—	2	8	no	—
76	C₄×D₄×C₆₇	—	2	8	no	—
77	C₄×D₄×C₆₇	—	2	8	no	—
78	C₄×D₄×C₆₇	—	2	8	no	—
79	C₄×D₄×C₆₇	—	2	8	no	—
80	#117×C₆₇	—	2	8	no	—
81	#117×C₆₇	—	2	8	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
93	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
98	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
99	C₂×C₄²×C₆₇	D₄	1	8	yes	—
100	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
101	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
102	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
103	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
108	#95×C₆₇	C₂³	1	4	yes	—
109	#95×C₆₇	C₂³	1	4	yes	—
110	#95×C₆₇	C₂³	1	16	yes	—
111	#95×C₆₇	C₂³	1	16	yes	—
112	C₂⁴×C₆₇	—	2	4	no	—
113	C₂⁴×C₆₇	—	2	8	no	—
114	C₂⁴×C₆₇	—	2	8	no	—
115	C₂⁴×C₆₇	—	4	4	no	—
116	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
117	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
118	C₂×Q₈×C₆₇	—	2	4	no	—
119	C₂×Q₈×C₆₇	—	2	4	no	—
120	C₂×Q₈×C₆₇	—	2	8	no	—
121	C₂×Q₈×C₆₇	—	4	4	no	—
122	C₂×D₄×C₆₇	—	2	32	no	—
123	C₂×D₄×C₆₇	—	2	8	no	—
124	C₂×D₄×C₆₇	—	2	4	no	—
125	C₂×D₄×C₆₇	—	2	4	no	—
126	C₂×D₄×C₆₇	—	2	4	no	—
127	C₂×D₄×C₆₇	#41	1	8	yes	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	8	no	—
131	C₂²×C₄×C₆₇	—	2	8	no	—
132	C₂²×C₄×C₆₇	—	2	8	no	—
133	C₂²×C₄×C₆₇	—	2	16	no	—
134	C₂²×C₄×C₆₇	—	2	16	no	—
135	C₂²×C₄×C₆₇	—	4	2	no	—
136	C₂²×C₄×C₆₇	—	4	32	no	—
137	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
138	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₆₇	#41	1	8	yes	—
141	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
142	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
151	C₄⋊C₄×C₆₇	—	2	4	no	—
152	C₄⋊C₄×C₆₇	—	2	4	no	—
153	C₄⋊C₄×C₆₇	—	2	4	no	—
154	C₄⋊C₄×C₆₇	—	2	4	no	—
155	C₄⋊C₄×C₆₇	—	2	8	no	—
156	C₄⋊C₄×C₆₇	—	2	8	no	—
157	C₄⋊C₄×C₆₇	—	2	4	no	—
158	C₄⋊C₄×C₆₇	—	2	4	no	—
159	C₄⋊C₄×C₆₇	—	2	8	no	—
160	C₂²⋊C₄×C₆₇	—	2	8	no	—
161	C₂²⋊C₄×C₆₇	—	2	8	no	—
162	C₂²⋊C₄×C₆₇	—	2	32	no	—
163	C₂²⋊C₄×C₆₇	—	2	8	no	—
164	C₂²⋊C₄×C₆₇	—	2	8	no	—
165	C₂²⋊C₄×C₆₇	—	4	16	no	—
166	C₄²×C₆₇	—	2	8	no	—
167	C₄²×C₆₇	—	2	8	no	—
168	C₄²×C₆₇	—	2	8	no	—
169	C₄²×C₆₇	—	4	4	no	—
170	C₄²×C₆₇	C₂×D₄	1	4	yes	—
171	C₄²×C₆₇	C₂×D₄	1	4	yes	—
172	C₄²×C₆₇	C₂×D₄	1	4	yes	—
173	C₄²×C₆₇	C₂×D₄	1	4	yes	—
174	C₂³×C₆₇	—	2	2	no	—
175	C₂³×C₆₇	—	2	2	no	—
176	C₂³×C₆₇	—	2	16	no	—
177	C₂³×C₆₇	—	2	4	no	—
178	C₂³×C₆₇	—	2	8	no	—
179	C₂³×C₆₇	—	2	4	no	—
180	C₂³×C₆₇	—	2	8	no	—
181	C₂³×C₆₇	—	4	32	no	—
182	C₂³×C₆₇	—	4	8	no	—
183	C₂³×C₆₇	—	4	8	no	—
184	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
185	C₂³×C₆₇	C₂×#41	1	8	yes	—
186	C₂³×C₆₇	C₂×#41	1	4	yes	—
187	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
188	Q₈×C₆₇	—	4	8	no	—
189	Q₈×C₆₇	—	4	8	no	—
190	D₄×C₆₇	—	2	16	no	—
191	D₄×C₆₇	—	2	16	no	—
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	8	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	8	no	—
200	C₂×C₄×C₆₇	—	2	8	no	—
201	C₂×C₄×C₆₇	—	2	8	no	—
202	C₂×C₄×C₆₇	—	2	8	no	—
203	C₂×C₄×C₆₇	—	2	8	no	—
204	C₂×C₄×C₆₇	—	2	8	no	—
205	C₂×C₄×C₆₇	—	4	4	no	—
206	C₂×C₄×C₆₇	—	4	8	no	—
207	C₂×C₄×C₆₇	—	4	16	no	—
208	C₂×C₄×C₆₇	—	4	16	no	—
209	C₂×C₄×C₆₇	#124	1	16	yes	complemented
210	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
211	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
214	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
216	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
217	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
218	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
219	C₂²×C₆₇	—	2	2	no	—
220	C₂²×C₆₇	—	2	2	no	—
221	C₂²×C₆₇	—	2	8	no	—
222	C₂²×C₆₇	—	2	8	no	—
223	C₂²×C₆₇	—	2	16	no	—
224	C₂²×C₆₇	—	4	32	no	—
225	C₂²×C₆₇	—	4	4	no	—
226	C₂²×C₆₇	—	4	4	no	—
227	C₂²×C₆₇	—	4	8	no	—
228	C₂²×C₆₇	C₂×#124	1	4	yes	—
229	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
230	C₂²×C₆₇	C₂²×#41	1	2	yes	—
231	C₂²×C₆₇	C₂²×#41	1	2	yes	—
232	C₂²×C₆₇	#585	1	2	yes	—
233	C₄×C₆₇	—	2	2	no	—
234	C₄×C₆₇	—	2	2	no	—
235	C₄×C₆₇	—	2	2	no	—
236	C₄×C₆₇	—	4	4	no	—
237	C₄×C₆₇	—	4	4	no	—
238	C₄×C₆₇	—	4	4	no	—
239	C₄×C₆₇	—	4	8	no	—
240	C₄×C₆₇	#525	1	4	yes	—
241	C₄×C₆₇	C₂×#124	1	4	yes	—
242	C₄×C₆₇	C₂×#124	1	4	yes	—
243	#34946	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
244	C₂×C₆₇	—	2	4	no	—
245	C₂×C₆₇	—	4	8	no	—
246	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
247	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
248	C₂×C₆₇	#3449	1	1	yes	characteristic
249	C₂×C₆₇	#3475	1	4	yes	—
250	C₂×#533	C₂×C₆₇	1	1	yes	characteristic, complemented
251	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
252	#2563	C₂×C₆₇	1	2	yes	complemented
253	#2562	C₂×C₆₇	1	4	yes	complemented
254	#2561	C₂×C₆₇	1	4	yes	complemented
255	#2547	C₂×C₆₇	1	2	yes	complemented
256	#2503	C₂×C₆₇	1	2	yes	complemented
257	#2435	C₂×C₆₇	1	2	yes	—
258	#2398	C₂×C₆₇	1	4	yes	complemented
259	C₄×#124	C₂×C₆₇	1	4	yes	complemented
260	#2280	C₂×C₆₇	1	4	yes	complemented
261	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
262	C₆₇	#34946	1	1	yes	characteristic, direct product factor, Sylow
263	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
264	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
265	#533	—	2	8	no	—
266	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
267	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
268	C₂×#125	C₂²×C₆₇	1	2	yes	complemented
269	C₂×#124	C₂²×C₆₇	1	4	yes	—
270	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
271	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
272	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
273	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
274	C₂×C₄×D₄	—	2	8	no	—
275	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
276	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
277	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
278	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
279	C₂×#117	C₂²×C₆₇	1	2	yes	—
280	C₂×#117	C₂²×C₆₇	1	2	yes	—
281	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
282	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
283	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
284	#400	C₂²×C₆₇	1	8	yes	complemented
285	#397	C₂²×C₆₇	1	8	yes	complemented
286	#394	C₂²×C₆₇	1	8	yes	—
287	#393	C₂²×C₆₇	1	8	yes	complemented
288	#391	C₂²×C₆₇	1	4	yes	complemented
289	#390	C₂²×C₆₇	1	4	yes	—
290	#388	C₂²×C₆₇	1	16	yes	complemented
291	#382	C₂²×C₆₇	1	4	yes	complemented
292	#381	C₂²×C₆₇	1	8	yes	—
293	#380	C₂²×C₆₇	1	8	yes	complemented
294	#380	C₂²×C₆₇	1	8	yes	complemented
295	#379	C₂²×C₆₇	1	8	yes	—
296	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
297	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
298	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
299	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
300	C₄×C₂²⋊C₄	C₂²×C₆₇	1	16	yes	complemented
301	#376	C₂²×C₆₇	1	2	yes	complemented
302	C₄³	C₂²×C₆₇	1	2	yes	complemented
303	C₂⁵	C₂³×C₆₇	1	2	yes	—
304	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
305	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
306	C₂²×D₄	—	2	8	no	—
307	C₂²×D₄	—	2	4	no	—
308	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
309	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
310	C₂³×C₄	—	2	8	no	—
311	C₂³×C₄	—	2	16	no	—
312	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
313	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
314	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
315	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
316	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
317	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
318	C₄⋊Q₈	—	2	4	no	—
319	C₄⋊Q₈	—	2	4	no	—
320	#125	—	2	8	no	—
321	#124	—	2	16	no	—
322	C₂²⋊Q₈	—	2	8	no	—
323	C₂²⋊Q₈	—	2	8	no	—
324	C₄×Q₈	—	2	8	no	—
325	C₄×Q₈	—	2	8	no	—
326	C₄×D₄	—	2	8	no	—
327	C₄×D₄	—	2	8	no	—
328	C₄×D₄	—	2	8	no	—
329	C₄×D₄	—	2	8	no	—
330	#117	—	2	8	no	—
331	#117	—	2	8	no	—
332	C₂×C₄⋊C₄	—	2	4	no	—
333	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
334	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
335	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
336	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
337	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
338	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
339	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
340	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
341	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
342	C₂×C₂²⋊C₄	—	2	8	no	—
343	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
344	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
345	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
346	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
347	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
348	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
349	C₂×C₄²	D₄×C₆₇	1	8	yes	—
350	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
351	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
352	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
353	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
354	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
355	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
356	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
357	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
358	#95	C₂³×C₆₇	1	4	yes	—
359	#95	C₂³×C₆₇	1	4	yes	—
360	#95	C₂³×C₆₇	1	16	yes	—
361	#95	C₂³×C₆₇	1	16	yes	—
362	C₂⁴	—	2	4	no	—
363	C₂⁴	—	2	8	no	—
364	C₂⁴	—	2	8	no	—
365	C₂⁴	—	4	4	no	—
366	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
367	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
368	C₂×Q₈	—	2	4	no	—
369	C₂×Q₈	—	2	4	no	—
370	C₂×Q₈	—	2	8	no	—
371	C₂×Q₈	—	4	4	no	—
372	C₂×D₄	—	2	32	no	—
373	C₂×D₄	—	2	8	no	—
374	C₂×D₄	—	2	4	no	—
375	C₂×D₄	—	2	4	no	—
376	C₂×D₄	—	2	4	no	—
377	C₂×D₄	#41×C₆₇	1	8	yes	—
378	C₂²×C₄	—	2	4	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	8	no	—
381	C₂²×C₄	—	2	8	no	—
382	C₂²×C₄	—	2	8	no	—
383	C₂²×C₄	—	2	16	no	—
384	C₂²×C₄	—	2	16	no	—
385	C₂²×C₄	—	4	2	no	—
386	C₂²×C₄	—	4	32	no	—
387	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
388	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
389	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
390	C₂²×C₄	#41×C₆₇	1	8	yes	—
391	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
392	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
393	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
394	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
395	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
396	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
397	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
398	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
399	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
400	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	8	no	—
406	C₄⋊C₄	—	2	8	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	8	no	—
410	C₂²⋊C₄	—	2	8	no	—
411	C₂²⋊C₄	—	2	8	no	—
412	C₂²⋊C₄	—	2	32	no	—
413	C₂²⋊C₄	—	2	8	no	—
414	C₂²⋊C₄	—	2	8	no	—
415	C₂²⋊C₄	—	4	16	no	—
416	C₄²	—	2	8	no	—
417	C₄²	—	2	8	no	—
418	C₄²	—	2	8	no	—
419	C₄²	—	4	4	no	—
420	C₄²	C₂×D₄×C₆₇	1	4	yes	—
421	C₄²	C₂×D₄×C₆₇	1	4	yes	—
422	C₄²	C₂×D₄×C₆₇	1	4	yes	—
423	C₄²	C₂×D₄×C₆₇	1	4	yes	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	2	no	—
426	C₂³	—	2	16	no	—
427	C₂³	—	2	4	no	—
428	C₂³	—	2	8	no	—
429	C₂³	—	2	4	no	—
430	C₂³	—	2	8	no	—
431	C₂³	—	4	32	no	—
432	C₂³	—	4	8	no	—
433	C₂³	—	4	8	no	—
434	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
435	C₂³	C₂×#41×C₆₇	1	8	yes	—
436	C₂³	C₂×#41×C₆₇	1	4	yes	—
437	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
438	Q₈	—	4	8	no	—
439	Q₈	—	4	8	no	—
440	D₄	—	2	16	no	—
441	D₄	—	2	16	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	8	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	2	8	no	—
451	C₂×C₄	—	2	8	no	—
452	C₂×C₄	—	2	8	no	—
453	C₂×C₄	—	2	8	no	—
454	C₂×C₄	—	2	8	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	4	8	no	—
457	C₂×C₄	—	4	16	no	—
458	C₂×C₄	—	4	16	no	—
459	C₂×C₄	#124×C₆₇	1	16	yes	complemented
460	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
461	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
462	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
464	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
465	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
466	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
467	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
468	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	8	no	—
472	C₂²	—	2	8	no	—
473	C₂²	—	2	16	no	—
474	C₂²	—	4	32	no	—
475	C₂²	—	4	4	no	—
476	C₂²	—	4	4	no	—
477	C₂²	—	4	8	no	—
478	C₂²	C₂×#124×C₆₇	1	4	yes	—
479	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
480	C₂²	C₂²×#41×C₆₇	1	2	yes	—
481	C₂²	C₂²×#41×C₆₇	1	2	yes	—
482	C₂²	#585×C₆₇	1	2	yes	—
483	C₄	—	2	2	no	—
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	4	4	no	—
487	C₄	—	4	4	no	—
488	C₄	—	4	4	no	—
489	C₄	—	4	8	no	—
490	C₄	#525×C₆₇	1	4	yes	—
491	C₄	C₂×#124×C₆₇	1	4	yes	—
492	C₄	C₂×#124×C₆₇	1	4	yes	—
493	C₂	—	2	4	no	—
494	C₂	—	4	8	no	—
495	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
496	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
497	C₂	C₆₇×#3449	1	1	yes	characteristic
498	C₂	C₆₇×#3475	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂³×D₄×C₆₇	C₂²
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
1. C₂×#533×C₆₇	C₂	semidirect
3. C₆₇×#2563	C₂	semidirect
4. C₆₇×#2562	C₂	semidirect
5. C₆₇×#2561	C₂	semidirect
6. C₆₇×#2547	C₂	semidirect
7. C₆₇×#2503	C₂	semidirect
9. C₆₇×#2398	C₂	semidirect
10. C₄×#124×C₆₇	C₂	semidirect
11. C₆₇×#2280	C₂	semidirect
12. C₄×#117×C₆₇	C₂	semidirect
16. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
17. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
18. C₂×#125×C₆₇	C₂²	semidirect
22. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
23. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
34. #400×C₆₇	C₂²	semidirect
35. #397×C₆₇	C₂²	semidirect
37. #393×C₆₇	C₂²	semidirect
38. #391×C₆₇	C₂²	semidirect
40. #388×C₆₇	C₂²	semidirect
41. #382×C₆₇	C₂²	semidirect
43. #380×C₆₇	C₂²	semidirect
44. #380×C₆₇	C₂²	semidirect
46. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
47. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
50. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
51. #376×C₆₇	C₂²	semidirect
52. C₄³×C₆₇	C₂²	semidirect
209. C₂×C₄×C₆₇	#124	semidirect
243. #34946	C₆₇	direct (N × H)
250. C₂×#533	C₂×C₆₇	semidirect
252. #2563	C₂×C₆₇	semidirect
253. #2562	C₂×C₆₇	semidirect
254. #2561	C₂×C₆₇	semidirect
255. #2547	C₂×C₆₇	semidirect
256. #2503	C₂×C₆₇	semidirect
258. #2398	C₂×C₆₇	semidirect
259. C₄×#124	C₂×C₆₇	semidirect
260. #2280	C₂×C₆₇	semidirect
261. C₄×#117	C₂×C₆₇	semidirect
262. C₆₇	#34946	direct (N × H)
266. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
267. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
268. C₂×#125	C₂²×C₆₇	semidirect
272. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
273. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
284. #400	C₂²×C₆₇	semidirect
285. #397	C₂²×C₆₇	semidirect
287. #393	C₂²×C₆₇	semidirect
288. #391	C₂²×C₆₇	semidirect
290. #388	C₂²×C₆₇	semidirect
291. #382	C₂²×C₆₇	semidirect
293. #380	C₂²×C₆₇	semidirect
294. #380	C₂²×C₆₇	semidirect
296. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
297. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
300. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
301. #376	C₂²×C₆₇	semidirect
302. C₄³	C₂²×C₆₇	semidirect
459. C₂×C₄	#124×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2563	2
C₆₇×#2562	2
C₆₇×#2561	2
C₆₇×#2547	2
C₆₇×#2503	2
C₆₇×#2435	2
C₆₇×#2398	2
C₄×#124×C₆₇	2
C₆₇×#2280	2
C₄×#117×C₆₇	2
#34946	67

Subgroups — C67×#34946 #30714197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34946 #30714197