Subgroups — #32150672

Order 18176.

682 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986415	C₂	1	2	yes	maximal, complemented
2	#22597053	C₂	1	1	yes	maximal, characteristic
3	#22590171	C₂	1	2	yes	maximal, complemented
4	#22590164	C₂	1	4	yes	maximal, complemented
5	#22590159	C₂	1	2	yes	maximal, complemented
6	#22589953	C₂	1	2	yes	maximal, complemented
7	C₄×#14556068	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556134	C₂²	1	2	yes	—
10	#17986549	C₄	1	1	yes	characteristic
11	#17986415	—	2	8	no	—
12	C₂×#14556068	C₂²	1	1	yes	characteristic
13	C₂×#14556068	C₂²	1	1	yes	characteristic
14	C₂×#14556067	C₂²	1	4	yes	—
15	C₂×C₁₆×D₇₁	C₂²	1	4	yes	—
16	#17986078	C₄	1	2	yes	—
17	#17986078	C₄	1	2	yes	—
18	C₈×D₂₈₄	C₂²	1	2	yes	—
19	C₈×D₂₈₄	C₂²	1	2	yes	—
20	#17986023	C₄	1	1	yes	characteristic
21	#17985965	C₄	1	2	yes	—
22	(C₂×D₇₁)⋊C₁₆	C₂²	1	4	yes	—
23	(C₂×D₇₁)⋊C₁₆	C₂²	1	4	yes	—
24	#17985775	C₂²	1	2	yes	—
25	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
26	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
28	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
29	C₄⋊C₁₆×C₇₁	C₂²	1	2	yes	—
30	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
31	C₂×#11765865	C₂×C₄	1	2	yes	—
32	#14556134	—	2	4	no	—
33	#14556134	—	2	4	no	—
34	C₂×#11765849	C₂×C₄	1	2	yes	—
35	C₂×#11765848	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₇₁	C₂³	1	4	yes	—
37	#14556128	C₈	1	1	yes	characteristic
38	#14556125	C₈	1	1	yes	characteristic
39	#14556116	C₈	1	2	yes	—
40	#14556102	C₈	1	2	yes	—
41	#14556102	C₈	1	2	yes	—
42	C₄×D₂₈₄	C₂×C₄	1	2	yes	—
43	C₄×D₂₈₄	C₂×C₄	1	2	yes	—
44	#14556068	—	2	2	no	—
45	#14556068	D₄	1	4	yes	complemented
46	#14556067	—	2	8	no	—
47	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
48	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
49	C₁₆×D₇₁	—	2	8	no	—
50	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
51	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
52	(C₂×D₇₁)⋊C₈	C₂×C₄	1	4	yes	—
53	(C₂×D₇₁)⋊C₈	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₁₆×C₇₁	—	2	4	no	—
58	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
59	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
60	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
64	#11765865	—	2	4	no	—
65	#11765865	—	2	4	no	—
66	C₂×Dic₁₄₂	C₂×C₈	1	2	yes	—
67	C₂×D₂₈₄	C₂×C₈	1	2	yes	—
68	C₂×D₂₈₄	C₂×C₈	1	2	yes	—
69	C₂×D₂₈₄	C₂×C₈	1	2	yes	—
70	C₂×C₂²⋊D₇₁	C₂×C₈	1	4	yes	—
71	C₂×C₄×D₇₁	C₂²×C₄	1	4	yes	—
72	#11765849	—	2	4	no	—
73	#11765848	—	2	8	no	—
74	C₂×C₇₁⋊C₈	C₂²×C₄	1	4	yes	—
75	C₈×D₇₁	—	2	4	no	—
76	C₈×D₇₁	—	2	4	no	—
77	C₄⋊Dic₇₁	C₂×C₈	1	2	yes	—
78	C₄⋊Dic₇₁	C₂×C₈	1	2	yes	—
79	(C₂×D₇₁)⋊C₄	C₂×C₈	1	4	yes	—
80	(C₂×D₇₁)⋊C₄	C₂×C₈	1	4	yes	—
81	C₇₁⋊C₁₆	—	2	2	no	—
82	C₇₁⋊C₁₆	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₇₁	—	2	2	no	—
87	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₇₁	#41	1	2	yes	—
92	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
93	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
95	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
97	C₁₆×C₇₁	—	2	2	no	—
98	C₁₆×C₇₁	—	2	2	no	—
99	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
100	C₂²×D₇₁	C₂²×C₈	1	4	yes	—
101	C₂×C₄×C₇₁	—	2	2	no	—
102	C₂×C₄×C₇₁	—	4	1	no	—
103	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₇₁	#131	1	2	yes	—
110	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
111	C₂²⋊D₇₁	—	2	8	no	—
112	C₂×Dic₇₁	C₂²×C₈	1	4	yes	—
113	D₂₈₄	—	2	4	no	—
114	D₂₈₄	—	2	4	no	—
115	D₂₈₄	—	4	2	no	—
116	C₄×D₇₁	—	2	4	no	—
117	C₄×D₇₁	—	2	4	no	—
118	Dic₁₄₂	—	2	4	no	—
119	C₈×C₇₁	—	2	1	no	—
120	C₈×C₇₁	—	2	1	no	—
121	C₈×C₇₁	C₄×D₄	1	2	yes	—
122	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
124	C₇₁⋊C₈	—	2	4	no	—
125	C₂²×C₇₁	—	2	1	no	—
126	C₂²×C₇₁	—	2	1	no	—
127	C₂²×C₇₁	C₈×D₄	1	2	yes	—
128	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
129	C₂×D₇₁	—	2	4	no	—
130	C₂×D₇₁	—	2	4	no	—
131	C₂×D₇₁	—	2	4	no	—
132	C₄×C₇₁	—	2	1	no	—
133	C₄×C₇₁	—	2	1	no	—
134	C₄×C₇₁	—	2	1	no	—
135	C₄×C₇₁	C₈×D₄	1	2	yes	—
136	C₄×C₇₁	#504	1	2	yes	complemented
137	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
138	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
139	Dic₇₁	—	2	4	no	—
140	#33438	—	71	1	no	maximal, Sylow
141	C₂×C₇₁	—	2	2	no	—
142	C₂×C₇₁	C₂×C₈×D₄	1	1	yes	characteristic
143	C₂×C₇₁	C₂×#504	1	1	yes	characteristic
144	C₂×C₇₁	#3407	1	1	yes	characteristic
145	D₇₁	—	4	4	no	—
146	C₂×#504	—	71	2	no	—
147	#2931	—	71	1	no	—
148	#2169	—	71	2	no	—
149	#2168	—	71	4	no	—
150	D₄×C₁₆	—	71	2	no	—
151	C₂×C₄⋊C₁₆	D₇₁	1	1	yes	characteristic, complemented
152	#2113	—	71	2	no	—
153	C₄×M₃₂	—	71	1	no	—
154	C₇₁	#33438	1	1	yes	characteristic, complemented, Sylow
155	C₂×#131	—	71	2	no	—
156	#532	—	71	1	no	—
157	#504	—	142	8	no	—
158	C₂×M₃₂	—	71	1	no	—
159	C₂×M₃₂	—	71	1	no	—
160	C₂×M₃₂	—	71	4	no	—
161	C₂²×C₁₆	—	71	4	no	—
162	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
163	#436	—	71	2	no	—
164	#436	—	71	2	no	—
165	C₈×D₄	—	71	2	no	—
166	C₈×D₄	—	71	2	no	—
167	#423	—	71	1	no	—
168	#408	—	71	2	no	—
169	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
170	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
171	C₄⋊C₁₆	C₂×D₇₁	1	2	yes	—
172	C₄⋊C₁₆	C₂×D₇₁	1	1	yes	characteristic
173	C₂²⋊C₁₆	—	71	4	no	—
174	C₂²⋊C₁₆	—	71	4	no	—
175	#346	—	71	2	no	—
176	C₄×C₁₆	—	71	2	no	—
177	C₂×#41	—	71	2	no	—
178	#131	—	142	4	no	—
179	#131	—	142	4	no	—
180	C₂×M₁₆	—	71	2	no	—
181	C₂×M₁₆	—	71	4	no	—
182	C₂²×C₈	—	71	4	no	—
183	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
185	C₄⋊Q₈	—	71	1	no	—
186	#127	—	71	1	no	—
187	#124	—	71	2	no	—
188	#121	—	71	2	no	—
189	#121	—	71	2	no	—
190	C₄×D₄	—	71	2	no	—
191	C₄×D₄	—	71	2	no	—
192	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
193	M₃₂	—	71	4	no	—
194	M₃₂	—	142	8	no	—
195	M₃₂	—	142	2	no	—
196	C₂×C₁₆	—	2	4	no	—
197	C₂×C₁₆	—	71	2	no	—
198	C₂×C₁₆	—	71	2	no	—
199	C₂×C₁₆	—	142	8	no	—
200	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
201	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
202	C₄⋊C₈	—	71	2	no	—
203	C₄⋊C₈	—	71	2	no	—
204	C₂²⋊C₈	—	71	4	no	—
205	C₂²⋊C₈	—	71	4	no	—
206	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
207	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
210	#41	—	142	4	no	—
211	#41	—	142	4	no	—
212	C₂×Q₈	—	71	2	no	—
213	C₂×D₄	—	71	4	no	—
214	C₂×D₄	—	71	2	no	—
215	C₂×D₄	—	71	2	no	—
216	C₂×D₄	—	71	2	no	—
217	C₂²×C₄	—	71	4	no	—
218	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
219	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
221	M₁₆	—	142	4	no	—
222	M₁₆	—	142	8	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	71	4	no	—
225	C₂×C₈	—	142	4	no	—
226	C₂×C₈	—	142	4	no	—
227	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
230	C₂×C₈	D₄×D₇₁	1	2	yes	—
231	C₂×C₈	#11765866	1	2	yes	—
232	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
233	C₄⋊C₄	—	71	2	no	—
234	C₄⋊C₄	—	71	2	no	—
235	C₂²⋊C₄	—	71	4	no	—
236	C₂²⋊C₄	—	71	4	no	—
237	C₄²	C₈×D₇₁	1	1	yes	characteristic
238	C₄²	C₈×D₇₁	1	1	yes	characteristic
239	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
240	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
241	C₁₆	—	2	2	no	—
242	C₁₆	—	2	2	no	—
243	C₁₆	—	71	4	no	—
244	C₁₆	—	142	2	no	—
245	C₂³	—	71	4	no	—
246	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
247	Q₈	—	142	4	no	—
248	D₄	—	284	2	no	—
249	D₄	—	142	8	no	—
250	D₄	—	142	4	no	—
251	D₄	—	142	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	71	4	no	—
255	C₂×C₄	—	142	4	no	—
256	C₂×C₄	—	142	4	no	—
257	C₂×C₄	#14556090	1	2	yes	—
258	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
259	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
260	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
261	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
262	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
263	C₂×C₄	#14556136	1	2	yes	—
264	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	142	4	no	—
268	C₈	#14556090	1	2	yes	—
269	C₈	C₂×D₄×D₇₁	1	1	yes	characteristic
270	C₈	C₂×#11765866	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	142	4	no	—
274	C₂²	—	142	4	no	—
275	C₂²	—	142	4	no	—
276	C₂²	#17986064	1	2	yes	—
277	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	142	4	no	—
282	C₄	#17986064	1	2	yes	—
283	C₄	#17986415	1	2	yes	complemented
284	C₄	C₂×#14556090	1	1	yes	characteristic
285	C₄	C₂×#14556136	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	284	4	no	—
288	C₂	C₂×#17986064	1	1	yes	characteristic
289	C₂	C₂×#17986415	1	1	yes	characteristic
290	C₂	#22601442	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Fitting	C₂×C₄⋊C₁₆×C₇₁	C₂
prime radical	#17986549	C₄
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₇₁
hypocenter	C₇₁	#33438
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#17986415	C₂	semidirect
3. #22590171	C₂	semidirect
4. #22590164	C₂	semidirect
5. #22590159	C₂	semidirect
6. #22589953	C₂	semidirect
7. C₄×#14556068	C₂	semidirect
8. C₂×C₄⋊C₁₆×C₇₁	C₂	semidirect
45. #14556068	D₄	semidirect
136. C₄×C₇₁	#504	semidirect
151. C₂×C₄⋊C₁₆	D₇₁	semidirect
154. C₇₁	#33438	semidirect
283. C₄	#17986415	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986415	2
#22597053	2
#22590171	2
#22590164	2
#22590159	2
#22589953	2
C₄×#14556068	2
C₂×C₄⋊C₁₆×C₇₁	2
#33438	71