Subgroups — C₆₇×#17910 #30697161

Order 17152.

720 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3105	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2966	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1586	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1585	C₂	1	2	yes	maximal, complemented
6	C₈×SD₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₈×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1573	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1490	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1483	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₆₇	—	2	1	no	—
13	#529×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	#484×C₆₇	C₄	1	1	yes	characteristic
16	#481×C₆₇	C₄	1	1	yes	characteristic
17	#478×C₆₇	C₄	1	2	yes	—
18	#463×C₆₇	C₄	1	2	yes	—
19	#450×C₆₇	C₄	1	1	yes	characteristic
20	#447×C₆₇	C₄	1	1	yes	characteristic
21	#446×C₆₇	C₂²	1	1	yes	characteristic, complemented
22	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
23	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
24	C₄×D₈×C₆₇	C₂²	1	2	yes	—
25	#436×C₆₇	C₂²	1	1	yes	characteristic, complemented
26	#435×C₆₇	—	2	1	no	—
27	C₈×D₄×C₆₇	—	2	1	no	—
28	C₈×D₄×C₆₇	—	2	1	no	—
29	C₈×D₄×C₆₇	—	2	1	no	—
30	C₈×D₄×C₆₇	C₂²	1	2	yes	—
31	C₈×D₄×C₆₇	C₂²	1	2	yes	—
32	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
33	#433×C₆₇	—	2	1	no	—
34	#432×C₆₇	C₂²	1	1	yes	characteristic
35	#431×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
38	#405×C₆₇	—	2	1	no	—
39	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	#335×C₆₇	C₂²	1	2	yes	complemented
42	#334×C₆₇	C₂²	1	2	yes	complemented
43	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	complemented
44	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
45	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
46	D₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
47	#322×C₆₇	C₂²	1	2	yes	complemented
48	C₈²×C₆₇	C₂²	1	2	yes	complemented
49	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₆₇	—	2	1	no	—
51	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
53	C₂×M₁₆×C₆₇	—	2	1	no	—
54	C₂×M₁₆×C₆₇	—	2	1	no	—
55	C₂²×C₈×C₆₇	—	2	1	no	—
56	C₂²×C₈×C₆₇	—	2	2	no	—
57	C₂²×C₈×C₆₇	—	2	2	no	—
58	C₂²×C₈×C₆₇	—	2	1	no	—
59	C₂²×C₈×C₆₇	—	4	4	no	—
60	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
62	#126×C₆₇	—	2	1	no	—
63	#125×C₆₇	C₂×C₄	1	1	yes	characteristic
64	#124×C₆₇	C₂×C₄	1	1	yes	characteristic
65	#123×C₆₇	—	2	1	no	—
66	C₂²⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
67	#121×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₆₇	—	2	1	no	—
70	C₄×D₄×C₆₇	—	2	1	no	—
71	C₄×D₄×C₆₇	—	2	1	no	—
72	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
73	C₄×D₄×C₆₇	C₂³	1	2	yes	—
74	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
75	#117×C₆₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	#107×C₆₇	C₂×C₄	1	2	yes	—
80	#106×C₆₇	C₂×C₄	1	2	yes	—
81	C₄⋊C₈×C₆₇	—	2	1	no	—
82	C₄⋊C₈×C₆₇	—	2	1	no	—
83	C₄⋊C₈×C₆₇	—	2	1	no	—
84	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
85	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
88	#103×C₆₇	C₂×C₄	1	2	yes	—
89	#102×C₆₇	C₂×C₄	1	2	yes	—
90	#102×C₆₇	C₂×C₄	1	2	yes	—
91	#102×C₆₇	C₂×C₄	1	2	yes	—
92	C₂²⋊C₈×C₆₇	—	2	1	no	—
93	C₂²⋊C₈×C₆₇	—	2	2	no	—
94	C₂²⋊C₈×C₆₇	—	2	2	no	—
95	C₂²⋊C₈×C₆₇	—	2	1	no	—
96	C₂²⋊C₈×C₆₇	—	2	1	no	—
97	C₂²⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
98	C₂²⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
99	#97×C₆₇	—	2	1	no	—
100	#97×C₆₇	—	2	1	no	—
101	#97×C₆₇	—	2	1	no	—
102	C₄×C₈×C₆₇	—	2	1	no	—
103	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
104	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
105	C₄×C₈×C₆₇	C₂³	1	2	yes	—
106	C₄×C₈×C₆₇	C₂³	1	2	yes	—
107	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
108	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
109	C₂⁴×C₆₇	—	2	1	no	—
110	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×D₄×C₆₇	—	2	4	no	—
112	C₂×D₄×C₆₇	—	2	1	no	—
113	C₂×D₄×C₆₇	—	2	1	no	—
114	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	1	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	4	2	no	—
121	C₂²×C₄×C₆₇	—	4	2	no	—
122	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
125	SD₁₆×C₆₇	—	2	4	no	—
126	D₈×C₆₇	—	2	4	no	—
127	M₁₆×C₆₇	—	2	4	no	—
128	M₁₆×C₆₇	—	4	2	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₂×C₈×C₆₇	—	2	1	no	—
133	C₂×C₈×C₆₇	—	2	1	no	—
134	C₂×C₈×C₆₇	—	4	1	no	—
135	C₂×C₈×C₆₇	—	4	4	no	—
136	C₂×C₈×C₆₇	—	4	2	no	—
137	C₂×C₈×C₆₇	—	4	2	no	—
138	C₂×C₈×C₆₇	—	4	4	no	—
139	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
141	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₄⋊C₄×C₆₇	—	2	1	no	—
146	C₄⋊C₄×C₆₇	—	2	1	no	—
147	C₄⋊C₄×C₆₇	—	2	1	no	—
148	C₄⋊C₄×C₆₇	—	2	1	no	—
149	C₄⋊C₄×C₆₇	—	2	1	no	—
150	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
153	C₂²⋊C₄×C₆₇	—	2	1	no	—
154	C₂²⋊C₄×C₆₇	—	2	2	no	—
155	C₂²⋊C₄×C₆₇	—	2	1	no	—
156	C₂²⋊C₄×C₆₇	—	2	1	no	—
157	C₂²⋊C₄×C₆₇	—	2	1	no	—
158	C₂²⋊C₄×C₆₇	—	2	2	no	—
159	C₂²⋊C₄×C₆₇	—	2	1	no	—
160	C₄²×C₆₇	—	2	1	no	—
161	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₄²×C₆₇	#41	1	1	yes	characteristic
164	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
165	C₂³×C₆₇	—	2	4	no	—
166	C₂³×C₆₇	—	2	2	no	—
167	C₂³×C₆₇	—	2	1	no	—
168	C₂³×C₆₇	—	4	2	no	—
169	C₂³×C₆₇	—	4	2	no	—
170	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
171	Q₈×C₆₇	—	2	2	no	—
172	D₄×C₆₇	—	2	4	no	—
173	D₄×C₆₇	—	2	2	no	—
174	D₄×C₆₇	—	2	2	no	—
175	D₄×C₆₇	—	4	2	no	—
176	D₄×C₆₇	#131	1	4	yes	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	4	4	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	1	no	—
189	C₂×C₄×C₆₇	—	4	1	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
193	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
197	C₈×C₆₇	—	2	4	no	—
198	C₈×C₆₇	—	2	2	no	—
199	C₈×C₆₇	—	2	2	no	—
200	C₈×C₆₇	—	2	1	no	—
201	C₈×C₆₇	—	2	1	no	—
202	C₈×C₆₇	—	2	1	no	—
203	C₈×C₆₇	—	4	1	no	—
204	C₈×C₆₇	—	4	2	no	—
205	C₈×C₆₇	—	4	1	no	—
206	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
207	C₂²×C₆₇	—	2	1	no	—
208	C₂²×C₆₇	—	2	1	no	—
209	C₂²×C₆₇	—	2	4	no	—
210	C₂²×C₆₇	—	4	8	no	—
211	C₂²×C₆₇	—	4	2	no	—
212	C₂²×C₆₇	—	4	2	no	—
213	C₂²×C₆₇	—	4	1	no	—
214	C₂²×C₆₇	—	4	1	no	—
215	C₂²×C₆₇	—	4	1	no	—
216	C₂²×C₆₇	#443	1	2	yes	—
217	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
218	C₄×C₆₇	—	2	1	no	—
219	C₄×C₆₇	—	2	1	no	—
220	C₄×C₆₇	—	2	1	no	—
221	C₄×C₆₇	—	2	1	no	—
222	C₄×C₆₇	—	4	1	no	—
223	C₄×C₆₇	—	4	1	no	—
224	C₄×C₆₇	—	4	2	no	—
225	C₄×C₆₇	—	4	1	no	—
226	C₄×C₆₇	C₂×#131	1	2	yes	—
227	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
228	C₄×C₆₇	#575	1	1	yes	characteristic
229	#17910	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₂×C₆₇	—	2	2	no	—
231	C₂×C₆₇	—	4	4	no	—
232	C₂×C₆₇	—	8	1	no	—
233	C₂×C₆₇	#2929	1	1	yes	characteristic
234	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
236	#3105	C₂×C₆₇	1	1	yes	characteristic
237	#2966	C₂×C₆₇	1	1	yes	characteristic, complemented
238	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
239	#1586	C₂×C₆₇	1	2	yes	complemented
240	#1585	C₂×C₆₇	1	2	yes	complemented
241	C₈×SD₁₆	C₂×C₆₇	1	2	yes	complemented
242	C₈×D₈	C₂×C₆₇	1	2	yes	complemented
243	#1573	C₂×C₆₇	1	1	yes	characteristic, complemented
244	#1490	C₂×C₆₇	1	1	yes	characteristic, complemented
245	#1483	C₂×C₆₇	1	1	yes	characteristic, complemented
246	C₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
247	C₆₇	#17910	1	1	yes	characteristic, direct product factor, Sylow
248	C₂³×C₈	—	2	1	no	—
249	#529	C₂²×C₆₇	1	1	yes	characteristic
250	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
251	#484	C₄×C₆₇	1	1	yes	characteristic
252	#481	C₄×C₆₇	1	1	yes	characteristic
253	#478	C₄×C₆₇	1	2	yes	—
254	#463	C₄×C₆₇	1	2	yes	—
255	#450	C₄×C₆₇	1	1	yes	characteristic
256	#447	C₄×C₆₇	1	1	yes	characteristic
257	#446	C₂²×C₆₇	1	1	yes	characteristic, complemented
258	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
259	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
260	C₄×D₈	C₂²×C₆₇	1	2	yes	—
261	#436	C₂²×C₆₇	1	1	yes	characteristic, complemented
262	#435	—	2	1	no	—
263	C₈×D₄	—	2	1	no	—
264	C₈×D₄	—	2	1	no	—
265	C₈×D₄	—	2	1	no	—
266	C₈×D₄	C₂²×C₆₇	1	2	yes	—
267	C₈×D₄	C₂²×C₆₇	1	2	yes	—
268	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
269	#433	—	2	1	no	—
270	#432	C₂²×C₆₇	1	1	yes	characteristic
271	#431	C₂²×C₆₇	1	1	yes	characteristic
272	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
273	C₂×C₂²⋊C₈	—	2	1	no	—
274	#405	—	2	1	no	—
275	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
277	#335	C₂²×C₆₇	1	2	yes	complemented
278	#334	C₂²×C₆₇	1	2	yes	complemented
279	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	complemented
280	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
281	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
282	D₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
283	#322	C₂²×C₆₇	1	2	yes	complemented
284	C₈²	C₂²×C₆₇	1	2	yes	complemented
285	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
286	C₂³×C₄	—	2	1	no	—
287	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
288	C₂×D₈	C₂×C₄×C₆₇	1	2	yes	—
289	C₂×M₁₆	—	2	1	no	—
290	C₂×M₁₆	—	2	1	no	—
291	C₂²×C₈	—	2	1	no	—
292	C₂²×C₈	—	2	2	no	—
293	C₂²×C₈	—	2	2	no	—
294	C₂²×C₈	—	2	1	no	—
295	C₂²×C₈	—	4	4	no	—
296	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
297	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
298	#126	—	2	1	no	—
299	#125	C₂×C₄×C₆₇	1	1	yes	characteristic
300	#124	C₂×C₄×C₆₇	1	1	yes	characteristic
301	#123	—	2	1	no	—
302	C₂²⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
303	#121	C₂×C₄×C₆₇	1	1	yes	characteristic
304	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
305	C₄×D₄	—	2	1	no	—
306	C₄×D₄	—	2	1	no	—
307	C₄×D₄	—	2	1	no	—
308	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
309	C₄×D₄	C₂³×C₆₇	1	2	yes	—
310	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
311	#117	—	2	1	no	—
312	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
313	C₂×C₂²⋊C₄	—	2	1	no	—
314	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
315	#107	C₂×C₄×C₆₇	1	2	yes	—
316	#106	C₂×C₄×C₆₇	1	2	yes	—
317	C₄⋊C₈	—	2	1	no	—
318	C₄⋊C₈	—	2	1	no	—
319	C₄⋊C₈	—	2	1	no	—
320	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
321	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
322	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
323	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
324	#103	C₂×C₄×C₆₇	1	2	yes	—
325	#102	C₂×C₄×C₆₇	1	2	yes	—
326	#102	C₂×C₄×C₆₇	1	2	yes	—
327	#102	C₂×C₄×C₆₇	1	2	yes	—
328	C₂²⋊C₈	—	2	1	no	—
329	C₂²⋊C₈	—	2	2	no	—
330	C₂²⋊C₈	—	2	2	no	—
331	C₂²⋊C₈	—	2	1	no	—
332	C₂²⋊C₈	—	2	1	no	—
333	C₂²⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
334	C₂²⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
335	#97	—	2	1	no	—
336	#97	—	2	1	no	—
337	#97	—	2	1	no	—
338	C₄×C₈	—	2	1	no	—
339	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
340	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
341	C₄×C₈	C₂³×C₆₇	1	2	yes	—
342	C₄×C₈	C₂³×C₆₇	1	2	yes	—
343	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
344	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
345	C₂⁴	—	2	1	no	—
346	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
347	C₂×D₄	—	2	4	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
351	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	4	2	no	—
357	C₂²×C₄	—	4	2	no	—
358	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
360	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
361	SD₁₆	—	2	4	no	—
362	D₈	—	2	4	no	—
363	M₁₆	—	2	4	no	—
364	M₁₆	—	4	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	4	1	no	—
371	C₂×C₈	—	4	4	no	—
372	C₂×C₈	—	4	2	no	—
373	C₂×C₈	—	4	4	no	—
374	C₂×C₈	—	4	2	no	—
375	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
376	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
377	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
378	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
379	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
380	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
387	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
388	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
389	C₂²⋊C₄	—	2	1	no	—
390	C₂²⋊C₄	—	2	2	no	—
391	C₂²⋊C₄	—	2	1	no	—
392	C₂²⋊C₄	—	2	1	no	—
393	C₂²⋊C₄	—	2	1	no	—
394	C₂²⋊C₄	—	2	2	no	—
395	C₂²⋊C₄	—	2	1	no	—
396	C₄²	—	2	1	no	—
397	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
398	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
399	C₄²	#41×C₆₇	1	1	yes	characteristic
400	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
401	C₂³	—	2	4	no	—
402	C₂³	—	2	2	no	—
403	C₂³	—	2	1	no	—
404	C₂³	—	4	2	no	—
405	C₂³	—	4	2	no	—
406	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
407	Q₈	—	2	2	no	—
408	D₄	—	2	4	no	—
409	D₄	—	2	2	no	—
410	D₄	—	2	2	no	—
411	D₄	—	4	2	no	—
412	D₄	#131×C₆₇	1	4	yes	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	4	4	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
428	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
429	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
430	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
431	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
433	C₈	—	2	4	no	—
434	C₈	—	2	2	no	—
435	C₈	—	2	2	no	—
436	C₈	—	2	1	no	—
437	C₈	—	2	1	no	—
438	C₈	—	2	1	no	—
439	C₈	—	4	2	no	—
440	C₈	—	4	1	no	—
441	C₈	—	4	1	no	—
442	C₈	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
443	C₂²	—	2	1	no	—
444	C₂²	—	2	1	no	—
445	C₂²	—	2	4	no	—
446	C₂²	—	4	8	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	4	1	no	—
450	C₂²	—	4	1	no	—
451	C₂²	—	4	1	no	—
452	C₂²	#443×C₆₇	1	2	yes	—
453	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
454	C₄	—	2	1	no	—
455	C₄	—	2	1	no	—
456	C₄	—	2	1	no	—
457	C₄	—	2	1	no	—
458	C₄	—	4	1	no	—
459	C₄	—	4	2	no	—
460	C₄	—	4	1	no	—
461	C₄	—	4	1	no	—
462	C₄	C₂×#131×C₆₇	1	2	yes	—
463	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
464	C₄	#575×C₆₇	1	1	yes	characteristic
465	C₂	—	2	2	no	—
466	C₂	—	4	4	no	—
467	C₂	—	8	1	no	—
468	C₂	C₆₇×#2929	1	1	yes	characteristic
469	C₂	C₄×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
470	C₂	C₂×#443×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#447×C₆₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2966	C₂	semidirect
3. C₂×C₈×D₄×C₆₇	C₂	semidirect
4. C₆₇×#1586	C₂	semidirect
5. C₆₇×#1585	C₂	semidirect
6. C₈×SD₁₆×C₆₇	C₂	semidirect
7. C₈×D₈×C₆₇	C₂	semidirect
8. C₆₇×#1573	C₂	semidirect
9. C₆₇×#1490	C₂	semidirect
10. C₆₇×#1483	C₂	semidirect
11. C₈×M₁₆×C₆₇	C₂	semidirect
21. #446×C₆₇	C₂²	semidirect
22. C₈×Q₈×C₆₇	C₂²	semidirect
25. #436×C₆₇	C₂²	semidirect
32. C₈×D₄×C₆₇	C₂²	semidirect
41. #335×C₆₇	C₂²	semidirect
42. #334×C₆₇	C₂²	semidirect
43. Q₈⋊C₈×C₆₇	C₂²	semidirect
46. D₄⋊C₈×C₆₇	C₂²	semidirect
47. #322×C₆₇	C₂²	semidirect
48. C₈²×C₆₇	C₂²	semidirect
229. #17910	C₆₇	direct (N × H)
237. #2966	C₂×C₆₇	semidirect
238. C₂×C₈×D₄	C₂×C₆₇	semidirect
239. #1586	C₂×C₆₇	semidirect
240. #1585	C₂×C₆₇	semidirect
241. C₈×SD₁₆	C₂×C₆₇	semidirect
242. C₈×D₈	C₂×C₆₇	semidirect
243. #1573	C₂×C₆₇	semidirect
244. #1490	C₂×C₆₇	semidirect
245. #1483	C₂×C₆₇	semidirect
246. C₈×M₁₆	C₂×C₆₇	semidirect
247. C₆₇	#17910	direct (N × H)
257. #446	C₂²×C₆₇	semidirect
258. C₈×Q₈	C₂²×C₆₇	semidirect
261. #436	C₂²×C₆₇	semidirect
268. C₈×D₄	C₂²×C₆₇	semidirect
277. #335	C₂²×C₆₇	semidirect
278. #334	C₂²×C₆₇	semidirect
279. Q₈⋊C₈	C₂²×C₆₇	semidirect
282. D₄⋊C₈	C₂²×C₆₇	semidirect
283. #322	C₂²×C₆₇	semidirect
284. C₈²	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3105	2
C₆₇×#2966	2
C₂×C₈×D₄×C₆₇	2
C₆₇×#1586	2
C₆₇×#1585	2
C₈×SD₁₆×C₆₇	2
C₈×D₈×C₆₇	2
C₆₇×#1573	2
C₆₇×#1490	2
C₆₇×#1483	2
C₈×M₁₆×C₆₇	2
#17910	67

Subgroups — C67×#17910 #30697161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17910 #30697161