Subgroups — #32150949

Order 18176.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597274	C₂	1	1	yes	maximal, characteristic
2	#22590247	C₂	1	8	yes	maximal, complemented
3	#22590100	C₂	1	2	yes	maximal
4	#22589939	C₂	1	2	yes	maximal
5	C₂×#17985775	C₂	1	1	yes	maximal, characteristic
6	C₇₁×#2150	C₂	1	1	yes	maximal, characteristic
7	#17986556	C₄	1	1	yes	characteristic
8	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
10	#17986129	C₄	1	4	yes	—
11	C₈×Dic₁₄₂	C₂²	1	4	yes	—
12	#17986025	C₄	1	2	yes	—
13	#17986023	C₄	1	1	yes	characteristic
14	#17985954	C₂²	1	2	yes	—
15	Dic₇₁⋊C₁₆	C₂²	1	8	yes	—
16	#17985775	C₂²	1	4	yes	—
17	#17985774	C₂²	1	4	yes	—
18	C₁₆×Dic₇₁	C₂²	1	4	yes	—
19	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₁₆×C₇₁	C₂²	1	4	yes	—
22	C₂×#11765849	C₂×C₄	1	2	yes	—
23	#14556128	C₈	1	2	yes	—
24	#14556120	C₈	1	2	yes	—
25	#14556107	C₈	1	4	yes	—
26	C₄×Dic₁₄₂	C₂×C₄	1	4	yes	—
27	#14556087	C₂×C₄	1	2	yes	—
28	C₂×C₇₁⋊C₁₆	—	2	2	no	—
29	C₂×C₇₁⋊C₁₆	Q₈	1	4	yes	complemented
30	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
31	C₂×C₇₁⋊C₁₆	C₂³	1	2	yes	—
32	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
33	Dic₇₁⋊C₈	C₂×C₄	1	8	yes	—
34	#14556038	C₂×C₄	1	4	yes	—
35	C₈×Dic₇₁	C₂³	1	4	yes	—
36	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
39	M₃₂×C₇₁	—	2	4	no	—
40	C₂×C₁₆×C₇₁	C₂³	1	4	yes	—
41	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₇₁	C₂³	1	2	yes	—
43	C₂×Dic₁₄₂	C₂×C₈	1	4	yes	—
44	#11765849	—	2	4	no	—
45	C₂×C₇₁⋊C₈	C₂²×C₄	1	4	yes	—
46	C₄⋊Dic₇₁	C₂×C₈	1	4	yes	—
47	C₄⋊Dic₇₁	C₂×C₈	1	4	yes	—
48	Dic₇₁⋊C₄	C₂×C₈	1	8	yes	—
49	C₂²⋊Dic₇₁	C₂×C₈	1	4	yes	—
50	C₄×Dic₇₁	C₂²×C₄	1	4	yes	—
51	C₇₁⋊C₁₆	—	2	2	no	—
52	C₇₁⋊C₁₆	C₂×Q₈	1	4	yes	—
53	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₁	—	2	2	no	—
56	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₇₁	#41	1	2	yes	—
60	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
61	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
64	C₁₆×C₇₁	—	2	4	no	—
65	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₇₁	—	4	2	no	—
67	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
69	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	#131	1	2	yes	—
72	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
73	C₂×Dic₇₁	C₂²×C₈	1	8	yes	—
74	Dic₁₄₂	—	4	4	no	—
75	C₈×C₇₁	—	2	2	no	—
76	C₈×C₇₁	C₄×Q₈	1	2	yes	—
77	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
79	C₇₁⋊C₈	—	2	4	no	—
80	C₂²×C₇₁	—	2	2	no	—
81	C₂²×C₇₁	C₈×Q₈	1	2	yes	—
82	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
83	C₄×C₇₁	—	2	2	no	—
84	C₄×C₇₁	—	2	2	no	—
85	C₄×C₇₁	C₈×Q₈	1	2	yes	—
86	C₄×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
87	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
88	Dic₇₁	—	2	8	no	—
89	#33475	—	71	1	no	maximal, Sylow
90	C₂×C₇₁	—	2	2	no	—
91	C₂×C₇₁	C₂×C₈×Q₈	1	1	yes	characteristic
92	C₂×C₇₁	#3407	1	2	yes	—
93	#2960	—	71	1	no	—
94	#2183	—	71	8	no	—
95	#2151	—	71	2	no	—
96	#2150	D₇₁	1	1	yes	characteristic
97	#2108	—	71	2	no	—
98	C₂×#346	—	71	1	no	—
99	C₇₁	#33475	1	1	yes	characteristic, complemented, Sylow
100	#533	—	71	1	no	—
101	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
102	C₂²×C₁₆	—	71	1	no	—
103	C₂²×C₁₆	—	71	1	no	—
104	#446	—	71	4	no	—
105	C₈×Q₈	—	71	4	no	—
106	#424	—	71	2	no	—
107	#423	—	71	1	no	—
108	#405	—	71	2	no	—
109	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
110	C₄⋊C₁₆	—	71	8	no	—
111	C₄⋊C₁₆	C₂×D₇₁	1	4	yes	—
112	#346	—	71	4	no	—
113	#346	—	71	4	no	—
114	C₄×C₁₆	—	71	4	no	—
115	C₂×M₁₆	—	71	2	no	—
116	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
118	C₄⋊Q₈	—	71	2	no	—
119	#125	—	71	2	no	—
120	C₂²⋊Q₈	—	71	4	no	—
121	C₄×Q₈	—	71	4	no	—
122	#117	—	71	2	no	—
123	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
124	M₃₂	—	2	4	no	—
125	C₂×C₁₆	—	71	4	no	—
126	C₂×C₁₆	—	71	2	no	—
127	C₂×C₁₆	—	71	2	no	—
128	C₂×C₁₆	—	142	2	no	—
129	C₂×C₁₆	C₂²×D₇₁	1	4	yes	—
130	C₄⋊C₈	—	71	8	no	—
131	C₄⋊C₈	—	71	4	no	—
132	#97	—	71	4	no	—
133	C₄×C₈	—	71	4	no	—
134	C₄×C₈	C₄×D₇₁	1	2	yes	—
135	C₄×C₈	C₂²×D₇₁	1	2	yes	—
136	C₂×Q₈	—	71	4	no	—
137	C₂²×C₄	C₈×D₇₁	1	2	yes	—
138	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
139	M₁₆	—	142	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	71	4	no	—
142	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
143	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₇₁	1	2	yes	—
145	C₂×C₈	#11765864	1	2	yes	—
146	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
147	C₄⋊C₄	—	71	8	no	—
148	C₄⋊C₄	—	71	4	no	—
149	C₄⋊C₄	—	71	4	no	—
150	C₂²⋊C₄	—	71	4	no	—
151	C₄²	—	71	4	no	—
152	C₄²	C₈×D₇₁	1	1	yes	characteristic
153	C₄²	C₈×D₇₁	1	1	yes	characteristic
154	C₄²	C₂×C₄×D₇₁	1	2	yes	—
155	C₁₆	—	2	4	no	—
156	C₁₆	—	71	4	no	—
157	C₁₆	—	142	2	no	—
158	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
159	Q₈	—	284	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	71	8	no	—
162	C₂×C₄	#14556093	1	2	yes	—
163	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
164	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
165	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
166	C₂×C₄	#14556136	1	2	yes	—
167	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	142	4	no	—
170	C₈	#14556093	1	2	yes	—
171	C₈	C₂×Q₈×D₇₁	1	1	yes	characteristic
172	C₈	C₂×#11765864	1	1	yes	characteristic
173	C₂²	—	2	2	no	—
174	C₂²	#17986123	1	2	yes	—
175	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	142	8	no	—
179	C₄	#17986123	1	2	yes	—
180	C₄	C₂×#14556093	1	1	yes	characteristic
181	C₄	C₂×#14556136	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×#17986123	1	1	yes	characteristic
184	C₂	#22601442	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₈
Fitting	C₇₁×#2150	C₂
prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2150	D₇₁
hypocenter	C₇₁	#33475
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #22590247	C₂	semidirect
29. C₂×C₇₁⋊C₁₆	Q₈	semidirect
99. C₇₁	#33475	semidirect

Maximal subgroups

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

subgroup	index
#22597274	2
#22590247	2
#22590100	2
#22589939	2
C₂×#17985775	2
C₇₁×#2150	2
#33475	71