Subgroups — #30754763

Order 17152.

446 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558957	C₂	1	1	yes	maximal, characteristic
2	#22558604	C₂	1	2	yes	maximal
3	#22558533	C₂	1	2	yes	maximal
4	#22558492	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	#16863449	C₂²	1	1	yes	characteristic
8	#16863449	C₂²	1	1	yes	characteristic
9	#16863149	—	2	1	no	—
10	#16863147	—	2	1	no	—
11	Q₈⋊(C₆₇⋊C₈)	—	2	4	no	—
12	#16863122	—	2	4	no	—
13	#16863117	—	2	1	no	—
14	C₈×C₆₇⋊C₈	—	2	1	no	—
15	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
17	C₄×Q₁₆×C₆₇	—	2	1	no	—
18	C₄×Q₁₆×C₆₇	—	2	1	no	—
19	C₄×Q₁₆×C₆₇	—	2	1	no	—
20	C₄×Q₁₆×C₆₇	—	2	1	no	—
21	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
22	C₂×#103×C₆₇	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
26	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
27	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
29	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
30	C₄×C₆₇⋊C₈	—	2	1	no	—
31	C₄×C₆₇⋊C₈	—	2	1	no	—
32	C₄×C₆₇⋊C₈	—	2	2	no	—
33	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆×C₆₇	—	2	1	no	—
35	C₂×Q₁₆×C₆₇	—	2	1	no	—
36	C₂×Q₁₆×C₆₇	—	2	2	no	—
37	C₂×Q₁₆×C₆₇	C₈	1	4	yes	complemented
38	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₄×Q₈×C₆₇	—	2	2	no	—
41	C₄×Q₈×C₆₇	—	2	2	no	—
42	C₄×Q₈×C₆₇	—	2	2	no	—
43	C₄×Q₈×C₆₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	—	2	2	no	—
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	#107×C₆₇	—	2	1	no	—
49	#107×C₆₇	—	2	1	no	—
50	#103×C₆₇	—	2	2	no	—
51	#103×C₆₇	—	2	2	no	—
52	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
56	C₂×C₆₇⋊C₈	—	2	2	no	—
57	C₂×C₆₇⋊C₈	—	2	2	no	—
58	C₂×C₆₇⋊C₈	—	4	2	no	—
59	C₂×Q₈×C₆₇	—	2	2	no	—
60	C₂×Q₈×C₆₇	—	2	2	no	—
61	C₂×Q₈×C₆₇	—	2	2	no	—
62	C₂×Q₈×C₆₇	—	2	2	no	—
63	C₂×Q₈×C₆₇	—	2	2	no	—
64	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
70	Q₁₆×C₆₇	—	2	4	no	—
71	Q₁₆×C₆₇	—	2	4	no	—
72	C₂×C₈×C₆₇	—	2	1	no	—
73	C₂×C₈×C₆₇	—	2	1	no	—
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
80	C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₄⋊C₄×C₆₇	—	2	2	no	—
82	C₄⋊C₄×C₆₇	—	4	2	no	—
83	C₄⋊C₄×C₆₇	—	4	2	no	—
84	C₄²×C₆₇	—	4	2	no	—
85	C₄²×C₆₇	—	4	2	no	—
86	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₇	#41	1	1	yes	characteristic
90	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
91	Q₈×C₆₇	—	2	4	no	—
92	Q₈×C₆₇	—	2	4	no	—
93	Q₈×C₆₇	—	4	2	no	—
94	Q₈×C₆₇	—	4	2	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	4	1	no	—
98	C₂×C₄×C₆₇	—	4	2	no	—
99	C₂×C₄×C₆₇	—	4	2	no	—
100	C₂×C₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	—	8	2	no	—
102	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
106	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
107	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
112	C₈×C₆₇	—	2	1	no	—
113	C₈×C₆₇	—	2	1	no	—
114	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
115	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
116	C₆₇⋊C₈	—	4	2	no	—
117	C₆₇⋊C₈	—	4	2	no	—
118	C₆₇⋊C₈	—	8	2	no	—
119	C₂²×C₆₇	—	2	1	no	—
120	C₂²×C₆₇	—	2	1	no	—
121	C₂²×C₆₇	#390	1	1	yes	characteristic
122	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
123	C₂²×C₆₇	#436	1	1	yes	characteristic
124	C₄×C₆₇	—	2	1	no	—
125	C₄×C₆₇	—	2	1	no	—
126	C₄×C₆₇	—	2	1	no	—
127	C₄×C₆₇	—	4	4	no	—
128	C₄×C₆₇	—	8	2	no	—
129	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₄×C₆₇	#408	1	1	yes	characteristic
131	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
132	C₄×C₆₇	#436	1	1	yes	characteristic
133	C₄×C₆₇	#473	1	2	yes	—
134	#9669	—	67	1	no	maximal, Sylow
135	C₂×C₆₇	—	2	1	no	—
136	C₂×C₆₇	—	2	1	no	—
137	C₂×C₆₇	#1954	1	1	yes	characteristic
138	C₂×C₆₇	#1976	1	1	yes	characteristic
139	C₂×C₆₇	#1979	1	1	yes	characteristic
140	C₂×C₄×Q₁₆	D₆₇	1	1	yes	characteristic
141	#1564	—	67	1	no	—
142	#1479	—	67	2	no	—
143	#1461	—	67	2	no	—
144	#1450	—	67	1	no	—
145	C₆₇	#9669	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₁₆	Dic₆₇	1	1	yes	characteristic
147	C₂×C₄×Q₈	C₂×D₆₇	1	2	yes	—
148	C₄×Q₁₆	—	2	1	no	—
149	C₄×Q₁₆	—	2	1	no	—
150	C₄×Q₁₆	—	2	1	no	—
151	C₄×Q₁₆	—	2	1	no	—
152	#431	—	67	1	no	—
153	#431	—	67	1	no	—
154	#431	—	67	2	no	—
155	C₂×#107	Dic₆₇	1	1	yes	characteristic
156	C₂×#103	Dic₆₇	1	2	yes	—
157	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
158	#335	—	134	1	no	—
159	#334	—	134	1	no	—
160	Q₈⋊C₈	—	134	4	no	—
161	#324	—	134	4	no	—
162	#322	—	134	1	no	—
163	C₈²	—	134	1	no	—
164	C₂²×Q₈	C₂×Dic₆₇	1	2	yes	—
165	C₂×Q₁₆	—	2	1	no	—
166	C₂×Q₁₆	—	2	1	no	—
167	C₂×Q₁₆	—	2	2	no	—
168	C₂×Q₁₆	C₆₇⋊C₈	1	4	yes	complemented
169	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
170	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
171	C₄×Q₈	—	2	2	no	—
172	C₄×Q₈	—	2	2	no	—
173	C₄×Q₈	—	2	2	no	—
174	C₄×Q₈	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
179	#107	—	2	1	no	—
180	#107	—	2	1	no	—
181	C₄⋊C₈	—	67	2	no	—
182	C₄⋊C₈	—	134	2	no	—
183	C₄⋊C₈	—	134	1	no	—
184	#103	—	2	2	no	—
185	#103	—	2	2	no	—
186	C₂²⋊C₈	—	67	2	no	—
187	C₂²⋊C₈	—	134	2	no	—
188	C₂²⋊C₈	—	134	1	no	—
189	C₄×C₈	—	134	2	no	—
190	C₄×C₈	—	134	1	no	—
191	C₄×C₈	—	134	1	no	—
192	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
193	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	C₂×C₆₇⋊C₈	1	4	yes	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
205	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
206	C₂²×C₄	#11761627	1	1	yes	characteristic
207	Q₁₆	—	2	4	no	—
208	Q₁₆	—	2	4	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	268	2	no	—
212	C₂×C₈	—	134	2	no	—
213	C₂×C₈	—	134	2	no	—
214	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
215	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
216	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
218	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
219	C₂×C₈	#11761612	1	2	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	—	4	2	no	—
226	C₄²	D₄×D₆₇	1	1	yes	characteristic
227	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
228	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
229	C₄²	#11761627	1	1	yes	characteristic
230	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
231	Q₈	—	2	4	no	—
232	Q₈	—	2	4	no	—
233	Q₈	—	4	2	no	—
234	Q₈	—	4	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
243	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
244	C₂×C₄	#14524014	1	1	yes	characteristic
245	C₂×C₄	#14524014	1	1	yes	characteristic
246	C₂×C₄	#14524030	1	1	yes	characteristic
247	C₂×C₄	#14524039	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
249	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
250	C₂×C₄	#14524050	1	1	yes	characteristic
251	C₂×C₄	#14524050	1	1	yes	characteristic
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	268	2	no	—
255	C₈	—	268	2	no	—
256	C₈	—	536	2	no	—
257	C₈	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
258	C₈	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	#16863299	1	1	yes	characteristic
262	C₂²	D₄×C₆₇⋊C₈	1	1	yes	characteristic
263	C₂²	#16863474	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	4	no	—
268	C₄	—	8	2	no	—
269	C₄	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
270	C₄	#16863363	1	1	yes	characteristic
271	C₄	D₄×C₆₇⋊C₈	1	1	yes	characteristic
272	C₄	#16863474	1	1	yes	characteristic
273	C₄	#16863668	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	#22561004	1	1	yes	characteristic
277	C₂	#22561108	1	1	yes	characteristic
278	C₂	#22561125	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524014
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₄×Q₁₆×C₆₇	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₄×Q₁₆	D₆₇
hypocenter	C₆₇	#9669
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
37. C₂×Q₁₆×C₆₇	C₈	semidirect
145. C₆₇	#9669	semidirect
168. C₂×Q₁₆	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22558957	2
#22558604	2
#22558533	2
#22558492	2
C₂×C₄×Q₁₆×C₆₇	2
#9669	67