Subgroups — #32150641

Order 18176.

690 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986416	C₂	1	2	yes	maximal, complemented
2	#22590172	C₂	1	4	yes	maximal, complemented
3	D₄×C₇₁⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	#22590096	C₂	1	1	yes	maximal, characteristic, complemented
5	#22589954	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#2931	C₂	1	1	yes	maximal, characteristic
8	#17986416	—	2	8	no	—
9	C₂×#14556068	C₂²	1	2	yes	—
10	C₂×#14556068	C₂²	1	4	yes	—
11	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₇₁⋊C₁₆	C₂²	1	4	yes	—
13	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	4	yes	—
15	C₂²⋊(C₇₁⋊C₁₆)	C₂²	1	8	yes	—
16	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
17	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
18	C₂×#131×C₇₁	C₂²	1	2	yes	—
19	#532×C₇₁	C₄	1	1	yes	characteristic
20	#436×C₇₁	C₄	1	4	yes	—
21	C₈×D₄×C₇₁	C₂²	1	4	yes	—
22	#423×C₇₁	C₄	1	1	yes	characteristic
23	#408×C₇₁	C₄	1	2	yes	—
24	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
25	#14556068	—	2	8	no	—
26	#14556068	—	2	4	no	—
27	C₂×C₇₁⋊C₁₆	—	2	8	no	—
28	C₂×C₇₁⋊C₁₆	—	2	2	no	—
29	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	complemented
30	C₂×C₇₁⋊C₁₆	C₂³	1	4	yes	—
31	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
32	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
33	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
34	C₂×#41×C₇₁	C₂×C₄	1	2	yes	—
35	#131×C₇₁	—	2	4	no	—
36	#131×C₇₁	—	2	4	no	—
37	C₂×M₁₆×C₇₁	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₇₁	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₁	C₂³	1	4	yes	—
41	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₇₁	C₈	1	1	yes	characteristic
43	#127×C₇₁	C₈	1	1	yes	characteristic
44	#124×C₇₁	C₈	1	2	yes	—
45	#121×C₇₁	C₈	1	4	yes	—
46	C₄×D₄×C₇₁	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₈×C₇₁	C₂×C₄	1	4	yes	—
49	C₂²⋊C₈×C₇₁	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₇₁	C₂³	1	2	yes	—
52	C₇₁⋊C₁₆	—	2	4	no	—
53	C₇₁⋊C₁₆	—	2	2	no	—
54	C₇₁⋊C₁₆	C₂×D₄	1	2	yes	—
55	C₇₁⋊C₁₆	C₂×D₄	1	2	yes	—
56	#41×C₇₁	—	2	4	no	—
57	#41×C₇₁	—	2	4	no	—
58	C₂×Q₈×C₇₁	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₇₁	C₂×C₈	1	4	yes	—
60	C₂×D₄×C₇₁	C₂×C₈	1	4	yes	—
61	C₂×D₄×C₇₁	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₇₁	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
65	M₁₆×C₇₁	—	2	4	no	—
66	M₁₆×C₇₁	—	2	8	no	—
67	C₂×C₈×C₇₁	—	2	2	no	—
68	C₂×C₈×C₇₁	—	2	4	no	—
69	C₂×C₈×C₇₁	—	2	4	no	—
70	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₇₁	#41	1	2	yes	—
75	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
76	C₄⋊C₄×C₇₁	C₂×C₈	1	4	yes	—
77	C₂²⋊C₄×C₇₁	C₂×C₈	1	8	yes	—
78	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
81	C₂³×C₇₁	C₂²×C₈	1	4	yes	—
82	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
83	Q₈×C₇₁	—	2	4	no	—
84	D₄×C₇₁	—	2	8	no	—
85	D₄×C₇₁	—	2	4	no	—
86	D₄×C₇₁	—	2	8	no	—
87	C₂×C₄×C₇₁	—	2	4	no	—
88	C₂×C₄×C₇₁	—	2	4	no	—
89	C₂×C₄×C₇₁	—	2	4	no	—
90	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₇₁	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₇₁	#131	1	2	yes	—
96	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
97	C₈×C₇₁	—	2	4	no	—
98	C₈×C₇₁	—	2	2	no	—
99	C₈×C₇₁	C₄×D₄	1	2	yes	—
100	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
102	C₂²×C₇₁	—	2	2	no	—
103	C₂²×C₇₁	—	2	8	no	—
104	C₂²×C₇₁	—	2	4	no	—
105	C₂²×C₇₁	C₈×D₄	1	2	yes	—
106	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
107	C₄×C₇₁	—	2	2	no	—
108	C₄×C₇₁	—	2	4	no	—
109	C₄×C₇₁	C₈×D₄	1	2	yes	—
110	C₄×C₇₁	#504	1	4	yes	complemented
111	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
112	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
113	#33434	—	71	1	no	maximal, Sylow
114	C₂×C₇₁	—	2	2	no	—
115	C₂×C₇₁	—	4	4	no	—
116	C₂×C₇₁	C₂×C₈×D₄	1	1	yes	characteristic
117	C₂×C₇₁	C₂×#504	1	2	yes	—
118	C₂×#504	—	71	2	no	—
119	#2931	D₇₁	1	1	yes	characteristic
120	#2169	—	71	4	no	—
121	D₄×C₁₆	—	71	4	no	—
122	#2150	—	71	1	no	—
123	#2113	—	71	2	no	—
124	C₂×C₄×C₁₆	—	71	1	no	—
125	C₇₁	#33434	1	1	yes	characteristic, complemented, Sylow
126	C₂×#131	C₂×D₇₁	1	2	yes	—
127	#532	Dic₇₁	1	1	yes	characteristic
128	#504	—	142	8	no	—
129	C₂×M₃₂	—	71	2	no	—
130	C₂×M₃₂	—	71	4	no	—
131	C₂²×C₁₆	—	71	1	no	—
132	C₂²×C₁₆	—	71	1	no	—
133	C₂²×C₁₆	—	71	4	no	—
134	#436	Dic₇₁	1	4	yes	—
135	C₈×D₄	C₂×D₇₁	1	4	yes	—
136	#423	Dic₇₁	1	1	yes	characteristic
137	#408	Dic₇₁	1	2	yes	—
138	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
139	C₄⋊C₁₆	—	71	4	no	—
140	C₂²⋊C₁₆	—	71	8	no	—
141	C₄×C₁₆	—	71	2	no	—
142	C₄×C₁₆	—	71	2	no	—
143	C₂×#41	C₂×Dic₇₁	1	2	yes	—
144	#131	—	2	4	no	—
145	#131	—	2	4	no	—
146	C₂×M₁₆	C₂×Dic₇₁	1	2	yes	—
147	C₂×M₁₆	C₂×Dic₇₁	1	4	yes	—
148	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
149	C₂²×C₈	C₂²×D₇₁	1	4	yes	—
150	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
151	C₄⋊Q₈	C₇₁⋊C₈	1	1	yes	characteristic
152	#127	C₇₁⋊C₈	1	1	yes	characteristic
153	#124	C₇₁⋊C₈	1	2	yes	—
154	#121	C₇₁⋊C₈	1	4	yes	—
155	C₄×D₄	C₂×Dic₇₁	1	4	yes	—
156	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
157	M₃₂	—	142	8	no	—
158	M₃₂	—	142	4	no	—
159	C₂×C₁₆	—	71	4	no	—
160	C₂×C₁₆	—	71	1	no	—
161	C₂×C₁₆	—	71	1	no	—
162	C₂×C₁₆	—	71	2	no	—
163	C₂×C₁₆	—	71	4	no	—
164	C₂×C₁₆	—	142	8	no	—
165	C₂×C₁₆	—	142	2	no	—
166	C₄⋊C₈	C₂×Dic₇₁	1	4	yes	—
167	C₂²⋊C₈	C₂×Dic₇₁	1	8	yes	—
168	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
169	C₄×C₈	C₂²×D₇₁	1	2	yes	—
170	#41	—	2	4	no	—
171	#41	—	2	4	no	—
172	C₂×Q₈	C₂×C₇₁⋊C₈	1	2	yes	—
173	C₂×D₄	C₂×C₇₁⋊C₈	1	4	yes	—
174	C₂×D₄	C₂×C₇₁⋊C₈	1	4	yes	—
175	C₂×D₄	C₂×C₇₁⋊C₈	1	2	yes	—
176	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
177	C₂²×C₄	C₂²×Dic₇₁	1	4	yes	—
178	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
179	M₁₆	—	2	4	no	—
180	M₁₆	—	2	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
185	C₂×C₈	C₂²×Dic₇₁	1	4	yes	—
186	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
187	C₂×C₈	D₄×D₇₁	1	2	yes	—
188	C₂×C₈	#11765864	1	2	yes	—
189	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
190	C₄⋊C₄	C₂×C₇₁⋊C₈	1	4	yes	—
191	C₂²⋊C₄	C₂×C₇₁⋊C₈	1	8	yes	—
192	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
193	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
194	C₄²	C₂²×Dic₇₁	1	2	yes	—
195	C₁₆	—	71	2	no	—
196	C₁₆	—	71	2	no	—
197	C₁₆	—	142	4	no	—
198	C₁₆	—	142	2	no	—
199	C₂³	C₂²×C₇₁⋊C₈	1	4	yes	—
200	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	D₄	—	2	8	no	—
203	D₄	—	2	4	no	—
204	D₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	D₄×Dic₇₁	1	2	yes	—
209	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
210	C₂×C₄	C₂²×C₇₁⋊C₈	1	4	yes	—
211	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
213	C₂×C₄	#14556135	1	2	yes	—
214	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	—	2	2	no	—
217	C₈	D₄×Dic₇₁	1	2	yes	—
218	C₈	C₂×D₄×D₇₁	1	1	yes	characteristic
219	C₈	C₂×#11765864	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	8	no	—
223	C₂²	D₄×C₇₁⋊C₈	1	2	yes	—
224	C₂²	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	2	2	no	—
227	C₄	D₄×C₇₁⋊C₈	1	2	yes	—
228	C₄	#17986416	1	4	yes	complemented
229	C₄	C₂×D₄×Dic₇₁	1	1	yes	characteristic
230	C₄	C₂×#14556135	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	4	4	no	—
233	C₂	C₂×D₄×C₇₁⋊C₈	1	1	yes	characteristic
234	C₂	C₂×#17986416	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Fitting	C₇₁×#2931	C₂
prime radical	#532×C₇₁	C₄
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2931	D₇₁
hypocenter	C₇₁	#33434
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. C₂×#17986416	C₂	semidirect
2. #22590172	C₂	semidirect
3. D₄×C₇₁⋊C₁₆	C₂	semidirect
4. #22590096	C₂	semidirect
5. #22589954	C₂	semidirect
6. C₂×C₄×C₇₁⋊C₁₆	C₂	semidirect
29. C₂×C₇₁⋊C₁₆	D₄	semidirect
110. C₄×C₇₁	#504	semidirect
125. C₇₁	#33434	semidirect
228. C₄	#17986416	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986416	2
#22590172	2
D₄×C₇₁⋊C₁₆	2
#22590096	2
#22589954	2
C₂×C₄×C₇₁⋊C₁₆	2
C₇₁×#2931	2
#33434	71