Subgroups — C₆₇×#17915 #30697166

Order 17152.

624 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3108	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2966	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1584	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1583	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1579	C₂	1	2	yes	maximal, complemented
7	C₆₇×SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1573	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1491	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1483	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	#529×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	#483×C₆₇	C₄	1	1	yes	characteristic
15	#480×C₆₇	C₄	1	1	yes	characteristic
16	#479×C₆₇	C₄	1	2	yes	—
17	#464×C₆₇	C₄	1	2	yes	—
18	#451×C₆₇	C₄	1	1	yes	characteristic
19	#448×C₆₇	C₄	1	1	yes	characteristic
20	#446×C₆₇	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₆₇	—	2	1	no	—
22	C₈×Q₈×C₆₇	—	2	1	no	—
23	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
24	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
27	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
28	#436×C₆₇	C₂²	1	1	yes	characteristic
29	#435×C₆₇	—	2	1	no	—
30	C₈×D₄×C₆₇	—	2	1	no	—
31	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
32	#433×C₆₇	—	2	1	no	—
33	#432×C₆₇	C₂²	1	1	yes	characteristic
34	#431×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
36	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
37	#405×C₆₇	—	2	1	no	—
38	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₆₇	—	2	1	no	—
40	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	#335×C₆₇	C₂²	1	2	yes	—
42	#334×C₆₇	C₂²	1	2	yes	—
43	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
44	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
45	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
46	D₄⋊C₈×C₆₇	C₂²	1	2	yes	—
47	#322×C₆₇	C₂²	1	2	yes	—
48	#322×C₆₇	C₂²	1	2	yes	—
49	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₆₇	—	2	1	no	—
53	C₂×M₁₆×C₆₇	—	2	1	no	—
54	C₂²×C₈×C₆₇	—	2	1	no	—
55	C₂²×C₈×C₆₇	—	2	1	no	—
56	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
58	#126×C₆₇	—	2	1	no	—
59	#125×C₆₇	C₂×C₄	1	1	yes	characteristic
60	#124×C₆₇	C₂×C₄	1	1	yes	characteristic
61	#123×C₆₇	—	2	1	no	—
62	C₂²⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
63	#121×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Q₈×C₆₇	—	2	1	no	—
65	C₄×Q₈×C₆₇	—	2	1	no	—
66	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
67	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
68	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₆₇	—	2	1	no	—
70	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
71	#117×C₆₇	—	2	1	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₆₇	—	2	1	no	—
75	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
76	#107×C₆₇	C₂×C₄	1	2	yes	—
77	#106×C₆₇	C₂×C₄	1	2	yes	—
78	C₄⋊C₈×C₆₇	—	2	2	no	—
79	C₄⋊C₈×C₆₇	—	2	2	no	—
80	C₄⋊C₈×C₆₇	—	2	1	no	—
81	C₄⋊C₈×C₆₇	—	2	1	no	—
82	C₄⋊C₈×C₆₇	—	2	1	no	—
83	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
84	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
85	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
87	#103×C₆₇	C₂×C₄	1	2	yes	—
88	#103×C₆₇	C₂×C₄	1	2	yes	—
89	#103×C₆₇	C₂×C₄	1	2	yes	—
90	#102×C₆₇	C₂×C₄	1	2	yes	—
91	C₂²⋊C₈×C₆₇	—	2	1	no	—
92	C₂²⋊C₈×C₆₇	—	2	1	no	—
93	C₂²⋊C₈×C₆₇	—	2	1	no	—
94	C₂²⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₂²⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
96	#97×C₆₇	—	2	1	no	—
97	#97×C₆₇	—	2	1	no	—
98	#97×C₆₇	—	2	1	no	—
99	C₄×C₈×C₆₇	—	2	2	no	—
100	C₄×C₈×C₆₇	—	2	2	no	—
101	C₄×C₈×C₆₇	—	2	1	no	—
102	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
103	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
104	C₄×C₈×C₆₇	C₂³	1	2	yes	—
105	C₄×C₈×C₆₇	C₂³	1	2	yes	—
106	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
107	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₆₇	—	2	4	no	—
109	C₂×Q₈×C₆₇	—	2	1	no	—
110	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂×D₄×C₆₇	—	2	1	no	—
113	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	1	no	—
116	C₂²×C₄×C₆₇	—	2	1	no	—
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
120	Q₁₆×C₆₇	—	2	4	no	—
121	SD₁₆×C₆₇	—	2	4	no	—
122	M₁₆×C₆₇	—	4	2	no	—
123	M₁₆×C₆₇	—	4	2	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	2	1	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	4	1	no	—
130	C₂×C₈×C₆₇	—	4	2	no	—
131	C₂×C₈×C₆₇	—	4	2	no	—
132	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
133	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
134	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₄⋊C₄×C₆₇	—	2	1	no	—
140	C₄⋊C₄×C₆₇	—	2	1	no	—
141	C₄⋊C₄×C₆₇	—	2	1	no	—
142	C₄⋊C₄×C₆₇	—	2	1	no	—
143	C₄⋊C₄×C₆₇	—	2	2	no	—
144	C₄⋊C₄×C₆₇	—	2	1	no	—
145	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
146	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
147	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
148	C₂²⋊C₄×C₆₇	—	2	1	no	—
149	C₂²⋊C₄×C₆₇	—	2	1	no	—
150	C₂²⋊C₄×C₆₇	—	2	1	no	—
151	C₂²⋊C₄×C₆₇	—	2	1	no	—
152	C₂²⋊C₄×C₆₇	—	2	1	no	—
153	C₄²×C₆₇	—	2	2	no	—
154	C₄²×C₆₇	—	2	2	no	—
155	C₄²×C₆₇	—	2	1	no	—
156	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₆₇	#41	1	1	yes	characteristic
159	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
160	C₂³×C₆₇	—	2	1	no	—
161	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
162	Q₈×C₆₇	—	2	4	no	—
163	Q₈×C₆₇	—	2	2	no	—
164	Q₈×C₆₇	—	2	2	no	—
165	Q₈×C₆₇	#131	1	4	yes	—
166	D₄×C₆₇	—	2	2	no	—
167	D₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	1	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	—	4	1	no	—
180	C₂×C₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
187	C₈×C₆₇	—	2	2	no	—
188	C₈×C₆₇	—	2	2	no	—
189	C₈×C₆₇	—	2	1	no	—
190	C₈×C₆₇	—	2	1	no	—
191	C₈×C₆₇	—	2	1	no	—
192	C₈×C₆₇	—	4	2	no	—
193	C₈×C₆₇	—	4	1	no	—
194	C₈×C₆₇	—	4	2	no	—
195	C₈×C₆₇	—	4	1	no	—
196	C₈×C₆₇	#137	1	2	yes	—
197	C₂²×C₆₇	—	2	1	no	—
198	C₂²×C₆₇	—	2	1	no	—
199	C₂²×C₆₇	—	4	1	no	—
200	C₂²×C₆₇	—	4	1	no	—
201	C₂²×C₆₇	—	4	1	no	—
202	C₂²×C₆₇	#444	1	2	yes	—
203	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
204	C₄×C₆₇	—	2	4	no	—
205	C₄×C₆₇	—	2	1	no	—
206	C₄×C₆₇	—	2	1	no	—
207	C₄×C₆₇	—	2	1	no	—
208	C₄×C₆₇	—	2	1	no	—
209	C₄×C₆₇	—	4	1	no	—
210	C₄×C₆₇	—	4	1	no	—
211	C₄×C₆₇	—	4	2	no	—
212	C₄×C₆₇	—	4	1	no	—
213	C₄×C₆₇	C₂×#131	1	2	yes	—
214	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
215	C₄×C₆₇	#575	1	1	yes	characteristic
216	#17915	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂×C₆₇	—	2	2	no	—
218	C₂×C₆₇	—	8	1	no	—
219	C₂×C₆₇	#2929	1	1	yes	characteristic
220	C₂×C₆₇	C₄×#137	1	1	yes	characteristic
221	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
222	#3108	C₂×C₆₇	1	1	yes	characteristic
223	#2966	C₂×C₆₇	1	1	yes	characteristic
224	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
225	#1584	C₂×C₆₇	1	2	yes	complemented
226	#1583	C₂×C₆₇	1	2	yes	complemented
227	#1579	C₂×C₆₇	1	2	yes	complemented
228	SD₁₆⋊C₈	C₂×C₆₇	1	2	yes	complemented
229	#1573	C₂×C₆₇	1	1	yes	characteristic, complemented
230	#1491	C₂×C₆₇	1	1	yes	characteristic, complemented
231	#1483	C₂×C₆₇	1	1	yes	characteristic
232	#1451	C₂×C₆₇	1	1	yes	characteristic, complemented
233	C₆₇	#17915	1	1	yes	characteristic, direct product factor, Sylow
234	#529	C₂²×C₆₇	1	1	yes	characteristic
235	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
236	#483	C₄×C₆₇	1	1	yes	characteristic
237	#480	C₄×C₆₇	1	1	yes	characteristic
238	#479	C₄×C₆₇	1	2	yes	—
239	#464	C₄×C₆₇	1	2	yes	—
240	#451	C₄×C₆₇	1	1	yes	characteristic
241	#448	C₄×C₆₇	1	1	yes	characteristic
242	#446	C₂²×C₆₇	1	1	yes	characteristic
243	C₈×Q₈	—	2	1	no	—
244	C₈×Q₈	—	2	1	no	—
245	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
246	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
247	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
248	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
249	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	—
250	#436	C₂²×C₆₇	1	1	yes	characteristic
251	#435	—	2	1	no	—
252	C₈×D₄	—	2	1	no	—
253	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
254	#433	—	2	1	no	—
255	#432	C₂²×C₆₇	1	1	yes	characteristic
256	#431	C₂²×C₆₇	1	1	yes	characteristic
257	C₂×C₄⋊C₈	—	2	1	no	—
258	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
259	#405	—	2	1	no	—
260	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×C₄×C₈	—	2	1	no	—
262	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
263	#335	C₂²×C₆₇	1	2	yes	—
264	#334	C₂²×C₆₇	1	2	yes	—
265	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
266	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
267	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
268	D₄⋊C₈	C₂²×C₆₇	1	2	yes	—
269	#322	C₂²×C₆₇	1	2	yes	—
270	#322	C₂²×C₆₇	1	2	yes	—
271	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
272	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
273	C₂×SD₁₆	C₂×C₄×C₆₇	1	2	yes	—
274	C₂×M₁₆	—	2	1	no	—
275	C₂×M₁₆	—	2	1	no	—
276	C₂²×C₈	—	2	1	no	—
277	C₂²×C₈	—	2	1	no	—
278	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
279	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
280	#126	—	2	1	no	—
281	#125	C₂×C₄×C₆₇	1	1	yes	characteristic
282	#124	C₂×C₄×C₆₇	1	1	yes	characteristic
283	#123	—	2	1	no	—
284	C₂²⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
285	#121	C₂×C₄×C₆₇	1	1	yes	characteristic
286	C₄×Q₈	—	2	1	no	—
287	C₄×Q₈	—	2	1	no	—
288	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
289	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
290	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
291	C₄×D₄	—	2	1	no	—
292	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
293	#117	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
296	C₂×C₄²	—	2	1	no	—
297	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
298	#107	C₂×C₄×C₆₇	1	2	yes	—
299	#106	C₂×C₄×C₆₇	1	2	yes	—
300	C₄⋊C₈	—	2	2	no	—
301	C₄⋊C₈	—	2	2	no	—
302	C₄⋊C₈	—	2	1	no	—
303	C₄⋊C₈	—	2	1	no	—
304	C₄⋊C₈	—	2	1	no	—
305	C₄⋊C₈	C₂×C₄×C₆₇	1	2	yes	—
306	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
307	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
308	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
309	#103	C₂×C₄×C₆₇	1	2	yes	—
310	#103	C₂×C₄×C₆₇	1	2	yes	—
311	#103	C₂×C₄×C₆₇	1	2	yes	—
312	#102	C₂×C₄×C₆₇	1	2	yes	—
313	C₂²⋊C₈	—	2	1	no	—
314	C₂²⋊C₈	—	2	1	no	—
315	C₂²⋊C₈	—	2	1	no	—
316	C₂²⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
317	C₂²⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
318	#97	—	2	1	no	—
319	#97	—	2	1	no	—
320	#97	—	2	1	no	—
321	C₄×C₈	—	2	2	no	—
322	C₄×C₈	—	2	2	no	—
323	C₄×C₈	—	2	1	no	—
324	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
325	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
326	C₄×C₈	C₂³×C₆₇	1	2	yes	—
327	C₄×C₈	C₂³×C₆₇	1	2	yes	—
328	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
329	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
330	C₂×Q₈	—	2	4	no	—
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
333	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
334	C₂×D₄	—	2	1	no	—
335	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
340	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
341	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
342	Q₁₆	—	2	4	no	—
343	SD₁₆	—	2	4	no	—
344	M₁₆	—	4	2	no	—
345	M₁₆	—	4	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	4	1	no	—
352	C₂×C₈	—	4	2	no	—
353	C₂×C₈	—	4	2	no	—
354	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
355	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
356	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
357	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
358	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
359	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
368	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
369	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
370	C₂²⋊C₄	—	2	1	no	—
371	C₂²⋊C₄	—	2	1	no	—
372	C₂²⋊C₄	—	2	1	no	—
373	C₂²⋊C₄	—	2	1	no	—
374	C₂²⋊C₄	—	2	1	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	1	no	—
378	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
379	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
380	C₄²	#41×C₆₇	1	1	yes	characteristic
381	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
382	C₂³	—	2	1	no	—
383	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
384	Q₈	—	2	4	no	—
385	Q₈	—	2	2	no	—
386	Q₈	—	2	2	no	—
387	Q₈	#131×C₆₇	1	4	yes	—
388	D₄	—	2	2	no	—
389	D₄	—	4	2	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
404	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
405	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
406	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
407	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
409	C₈	—	2	2	no	—
410	C₈	—	2	2	no	—
411	C₈	—	2	1	no	—
412	C₈	—	2	1	no	—
413	C₈	—	2	1	no	—
414	C₈	—	4	2	no	—
415	C₈	—	4	1	no	—
416	C₈	—	4	2	no	—
417	C₈	—	4	1	no	—
418	C₈	#137×C₆₇	1	2	yes	—
419	C₂²	—	2	1	no	—
420	C₂²	—	2	1	no	—
421	C₂²	—	4	1	no	—
422	C₂²	—	4	1	no	—
423	C₂²	—	4	1	no	—
424	C₂²	#444×C₆₇	1	2	yes	—
425	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
426	C₄	—	2	4	no	—
427	C₄	—	2	1	no	—
428	C₄	—	2	1	no	—
429	C₄	—	2	1	no	—
430	C₄	—	2	1	no	—
431	C₄	—	4	1	no	—
432	C₄	—	4	1	no	—
433	C₄	—	4	2	no	—
434	C₄	—	4	1	no	—
435	C₄	C₂×#131×C₆₇	1	2	yes	—
436	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
437	C₄	#575×C₆₇	1	1	yes	characteristic
438	C₂	—	2	2	no	—
439	C₂	—	8	1	no	—
440	C₂	C₆₇×#2929	1	1	yes	characteristic
441	C₂	C₄×#137×C₆₇	1	1	yes	characteristic
442	C₂	C₂×#444×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	#121×C₆₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₈×Q₈×C₆₇	C₂	semidirect
4. C₆₇×#1584	C₂	semidirect
5. C₆₇×#1583	C₂	semidirect
6. C₆₇×#1579	C₂	semidirect
7. C₆₇×SD₁₆⋊C₈	C₂	semidirect
8. C₆₇×#1573	C₂	semidirect
9. C₆₇×#1491	C₂	semidirect
11. C₆₇×#1451	C₂	semidirect
216. #17915	C₆₇	direct (N × H)
224. C₂×C₈×Q₈	C₂×C₆₇	semidirect
225. #1584	C₂×C₆₇	semidirect
226. #1583	C₂×C₆₇	semidirect
227. #1579	C₂×C₆₇	semidirect
228. SD₁₆⋊C₈	C₂×C₆₇	semidirect
229. #1573	C₂×C₆₇	semidirect
230. #1491	C₂×C₆₇	semidirect
232. #1451	C₂×C₆₇	semidirect
233. C₆₇	#17915	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×#3108	2
C₆₇×#2966	2
C₂×C₈×Q₈×C₆₇	2
C₆₇×#1584	2
C₆₇×#1583	2
C₆₇×#1579	2
C₆₇×SD₁₆⋊C₈	2
C₆₇×#1573	2
C₆₇×#1491	2
C₆₇×#1483	2
C₆₇×#1451	2
#17915	67

Subgroups — C67×#17915 #30697166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17915 #30697166