Subgroups — #30761629

Order 17152.

1014 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523954	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863151	C₂	1	4	yes	maximal
4	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
6	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	4	yes	complemented
10	C₂×#14523954	—	2	2	no	—
11	C₂×#14523954	—	2	2	no	—
12	C₂×#14523954	C₄	1	4	yes	complemented
13	#16863151	—	2	8	no	—
14	C₂²×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
18	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
19	C₂×#95×C₆₇	C₄	1	2	yes	—
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₂²×C₆₇⋊C₈	—	2	2	no	—
22	C₂²×C₆₇⋊C₈	—	2	4	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	C₂²×C₆₇⋊C₈	—	2	4	no	—
28	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₆₇⋊C₈	D₄	1	4	yes	—
31	C₂²×C₆₇⋊C₈	Q₈	1	4	yes	—
32	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
33	#14523954	—	2	8	no	—
34	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
35	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
40	C₂×C₄²×C₆₇	—	2	4	no	—
41	C₂×C₄²×C₆₇	—	2	2	no	—
42	C₂×C₄²×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	—	2	4	no	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	D₄	1	2	yes	—
47	C₂×C₄²×C₆₇	D₄	1	2	yes	—
48	C₂×C₄²×C₆₇	D₄	1	2	yes	—
49	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
50	#95×C₆₇	—	2	4	no	—
51	C₂×C₆₇⋊C₈	—	2	4	no	—
52	C₂×C₆₇⋊C₈	—	2	4	no	—
53	C₂×C₆₇⋊C₈	—	2	4	no	—
54	C₂×C₆₇⋊C₈	—	2	4	no	—
55	C₂×C₆₇⋊C₈	—	2	4	no	—
56	C₂×C₆₇⋊C₈	—	2	4	no	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	2	8	no	—
59	C₂×C₆₇⋊C₈	—	4	4	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	C₄⋊C₄	1	8	yes	complemented
63	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	—	2	4	no	—
66	C₂²×C₄×C₆₇	—	2	4	no	—
67	C₂²×C₄×C₆₇	—	2	8	no	—
68	C₂²×C₄×C₆₇	—	2	4	no	—
69	C₂²×C₄×C₆₇	—	2	8	no	—
70	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
71	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	M₁₆	1	8	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	#41	1	2	yes	—
86	C₂²×C₄×C₆₇	#41	1	2	yes	—
87	C₄⋊C₄×C₆₇	—	2	16	no	—
88	C₄²×C₆₇	—	4	4	no	—
89	C₄²×C₆₇	—	4	2	no	—
90	C₄²×C₆₇	—	4	2	no	—
91	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
92	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
93	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
94	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
95	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
96	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₆₇	C₄×D₄	1	2	yes	—
99	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
100	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
101	C₂³×C₆₇	#131	1	4	yes	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	—	2	4	no	—
105	C₂×C₄×C₆₇	—	2	4	no	—
106	C₂×C₄×C₆₇	—	2	4	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	8	no	—
109	C₂×C₄×C₆₇	—	2	16	no	—
110	C₂×C₄×C₆₇	—	4	4	no	—
111	C₂×C₄×C₆₇	—	4	4	no	—
112	C₂×C₄×C₆₇	—	4	8	no	—
113	C₂×C₄×C₆₇	#95	1	4	yes	—
114	C₂×C₄×C₆₇	#95	1	4	yes	—
115	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	#117	1	2	yes	—
121	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₆₇	#121	1	2	yes	—
124	C₂×C₄×C₆₇	#121	1	2	yes	—
125	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
128	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
129	C₂×C₄×C₆₇	#125	1	2	yes	—
130	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
133	C₆₇⋊C₈	—	2	8	no	—
134	C₆₇⋊C₈	—	4	4	no	—
135	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
136	C₂²×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
137	C₂²×C₆₇	#389	1	2	yes	—
138	C₂²×C₆₇	#390	1	1	yes	characteristic
139	C₂²×C₆₇	#391	1	1	yes	characteristic
140	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
141	C₂²×C₆₇	#405	1	2	yes	—
142	C₂²×C₆₇	#407	1	2	yes	—
143	C₂²×C₆₇	#408	1	2	yes	—
144	C₂²×C₆₇	#423	1	2	yes	—
145	C₂²×C₆₇	#424	1	2	yes	—
146	C₂²×C₆₇	#435	1	8	yes	—
147	C₂²×C₆₇	#446	1	8	yes	—
148	C₄×C₆₇	—	2	4	no	—
149	C₄×C₆₇	—	2	4	no	—
150	C₄×C₆₇	—	4	8	no	—
151	C₄×C₆₇	#380	1	2	yes	—
152	C₄×C₆₇	#382	1	1	yes	characteristic
153	C₄×C₆₇	#384	1	1	yes	characteristic
154	C₄×C₆₇	#389	1	2	yes	—
155	C₄×C₆₇	#390	1	1	yes	characteristic
156	C₄×C₆₇	#391	1	1	yes	characteristic
157	#10835	—	67	1	no	maximal, Sylow
158	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
159	C₂×C₆₇	#1728	1	1	yes	characteristic
160	C₂×C₆₇	#1729	1	1	yes	characteristic
161	C₂×C₆₇	#1771	1	4	yes	—
162	C₂×C₆₇	#1940	1	4	yes	—
163	C₂×C₆₇	#1955	1	2	yes	—
164	C₂×C₆₇	#1985	1	2	yes	—
165	C₂²×C₄⋊C₈	—	67	1	no	—
166	C₂²×#97	—	67	1	no	—
167	C₂×C₄×C₄⋊C₄	D₆₇	1	1	yes	characteristic
168	C₂×#336	—	67	4	no	—
169	C₆₇	#10835	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₈	—	67	2	no	—
171	C₂³×C₈	—	67	2	no	—
172	C₂²×C₄⋊C₄	Dic₆₇	1	2	yes	—
173	C₂²×C₄²	C₂×D₆₇	1	2	yes	—
174	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
175	C₂×C₄⋊C₈	—	67	4	no	—
176	C₂×C₄⋊C₈	—	134	2	no	—
177	C₂×C₄⋊C₈	—	134	2	no	—
178	C₂×#97	—	67	4	no	—
179	C₂×#97	—	134	2	no	—
180	C₂×#97	—	134	2	no	—
181	C₄×C₄⋊C₄	—	2	2	no	—
182	C₄×C₄⋊C₄	—	2	2	no	—
183	C₂×#95	Dic₆₇	1	2	yes	—
184	#336	—	134	8	no	—
185	C₂³×C₄	C₂×Dic₆₇	1	4	yes	—
186	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
187	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
188	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
189	C₂²×C₈	—	67	4	no	—
190	C₂²×C₈	—	67	4	no	—
191	C₂²×C₈	—	67	4	no	—
192	C₂²×C₈	—	67	4	no	—
193	C₂²×C₈	—	134	4	no	—
194	C₂²×C₈	—	134	2	no	—
195	C₂²×C₈	—	134	2	no	—
196	C₂²×C₈	—	134	4	no	—
197	C₂²×C₈	—	134	2	no	—
198	C₂²×C₈	—	134	2	no	—
199	C₂²×C₈	—	134	2	no	—
200	C₂²×C₈	—	134	2	no	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	8	no	—
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	4	no	—
207	C₂×C₄²	Dic₁₃₄	1	2	yes	—
208	C₂×C₄²	C₄×D₆₇	1	2	yes	—
209	C₂×C₄²	C₄×D₆₇	1	2	yes	—
210	C₂×C₄²	D₂₆₈	1	2	yes	—
211	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
212	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
213	C₄⋊C₈	—	134	8	no	—
214	#97	—	134	8	no	—
215	#95	—	2	4	no	—
216	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
224	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
225	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
226	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
227	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
228	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
229	C₂²×C₄	#11761612	1	8	yes	—
230	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₆₇	1	2	yes	—
233	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
236	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
237	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
238	C₂²×C₄	#11761627	1	2	yes	—
239	C₂²×C₄	#11761629	1	2	yes	—
240	C₂×C₈	—	268	4	no	—
241	C₂×C₈	—	268	2	no	—
242	C₂×C₈	—	268	2	no	—
243	C₂×C₈	—	67	8	no	—
244	C₂×C₈	—	134	4	no	—
245	C₂×C₈	—	134	8	no	—
246	C₂×C₈	—	134	4	no	—
247	C₂×C₈	—	134	4	no	—
248	C₂×C₈	—	134	4	no	—
249	C₂×C₈	—	134	4	no	—
250	C₂×C₈	—	134	4	no	—
251	C₂×C₈	—	134	4	no	—
252	C₄⋊C₄	—	2	16	no	—
253	C₄²	—	4	4	no	—
254	C₄²	—	4	2	no	—
255	C₄²	—	4	2	no	—
256	C₄²	(C₂×D₆₇)⋊C₄	1	4	yes	—
257	C₄²	(C₂×D₆₇)⋊C₄	1	4	yes	—
258	C₄²	Dic₆₇⋊C₄	1	4	yes	—
259	C₄²	Dic₆₇⋊C₄	1	4	yes	—
260	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
261	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
262	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
263	C₂³	D₄×Dic₆₇	1	2	yes	—
264	C₂³	Q₈×Dic₆₇	1	2	yes	—
265	C₂³	C₂×#11761612	1	4	yes	—
266	C₂³	#14524050	1	4	yes	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	16	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	C₆₇⋊#95	1	4	yes	—
279	C₂×C₄	C₆₇⋊#95	1	4	yes	—
280	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
282	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
283	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
284	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
285	C₂×C₄	#14524001	1	2	yes	—
286	C₂×C₄	#14524005	1	2	yes	—
287	C₂×C₄	#14524008	1	2	yes	—
288	C₂×C₄	#14524014	1	2	yes	—
289	C₂×C₄	#14524018	1	2	yes	—
290	C₂×C₄	#14524019	1	2	yes	—
291	C₂×C₄	#14524023	1	2	yes	—
292	C₂×C₄	#14524030	1	1	yes	characteristic
293	C₂×C₄	#14524032	1	1	yes	characteristic
294	C₂×C₄	#14524034	1	2	yes	—
295	C₂×C₄	#14524039	1	1	yes	characteristic
296	C₂×C₄	#14524041	1	1	yes	characteristic
297	C₂×C₄	#14524042	1	2	yes	—
298	C₈	—	268	4	no	—
299	C₈	—	134	8	no	—
300	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
301	C₂²	C₄⋊C₄×Dic₆₇	1	2	yes	—
302	C₂²	#16863295	1	2	yes	—
303	C₂²	#16863299	1	1	yes	characteristic
304	C₂²	#16863303	1	1	yes	characteristic
305	C₂²	C₄×#11761612	1	2	yes	—
306	C₂²	#16863352	1	2	yes	—
307	C₂²	#16863360	1	2	yes	—
308	C₂²	#16863363	1	2	yes	—
309	C₂²	#16863420	1	2	yes	—
310	C₂²	#16863423	1	2	yes	—
311	C₂²	#16863468	1	8	yes	—
312	C₂²	#16863527	1	8	yes	—
313	C₄	—	2	4	no	—
314	C₄	—	2	4	no	—
315	C₄	—	4	8	no	—
316	C₄	#16863253	1	2	yes	—
317	C₄	#16863261	1	1	yes	characteristic
318	C₄	#16863265	1	1	yes	characteristic
319	C₄	#16863292	1	2	yes	—
320	C₄	#16863297	1	1	yes	characteristic
321	C₄	#16863301	1	1	yes	characteristic
322	C₂	#22558438	1	1	yes	characteristic
323	C₂	#22559903	1	1	yes	characteristic
324	C₂	#22559906	1	1	yes	characteristic
325	C₂	#22560084	1	4	yes	—
326	C₂	#22560933	1	4	yes	—
327	C₂	#22561008	1	2	yes	—
328	C₂	#22561150	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₆₇
commutator	C₂³×C₆₇	C₂×C₄²
Fitting	C₂×C₄×C₄⋊C₄×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₆₇
hypocenter	C₆₇	#10835
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊(C₆₇⋊C₈)	C₄	semidirect
12. C₂×#14523954	C₄	semidirect
62. C₂×C₆₇⋊C₈	C₄⋊C₄	semidirect
169. C₆₇	#10835	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₆₇⋊C₈)	2
C₂²×#14523954	2
C₂×#16863151	2
C₂×C₄×C₄⋊C₄×C₆₇	2
#10835	67