Subgroups — #31887877

Order 18176.

478 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588086	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₇₁⋊#1273	C₂	1	4	yes	maximal
4	C₄×#97×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₇₁⋊C₈	C₄	1	4	yes	—
6	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
8	#17985754	C₄	1	4	yes	—
9	C₂×#97×C₇₁	C₄	1	2	yes	—
10	C₂×#97×C₇₁	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
12	C₄³×C₇₁	C₂²	1	1	yes	characteristic
13	C₂²×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
14	C₂²×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
15	C₄×C₇₁⋊C₈	—	2	8	no	—
16	C₄×C₇₁⋊C₈	—	2	4	no	—
17	C₄×C₇₁⋊C₈	—	2	4	no	—
18	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
24	#97×C₇₁	—	2	4	no	—
25	#97×C₇₁	C₈	1	8	yes	complemented
26	C₄×C₈×C₇₁	—	2	4	no	—
27	C₂×C₇₁⋊C₈	—	2	4	no	—
28	C₂×C₇₁⋊C₈	—	2	4	no	—
29	C₂×C₇₁⋊C₈	—	2	4	no	—
30	C₂×C₇₁⋊C₈	—	2	8	no	—
31	C₂×C₇₁⋊C₈	M₁₆	1	8	yes	—
32	C₂²×C₄×C₇₁	C₄²	1	2	yes	—
33	C₂²×C₄×C₇₁	C₄²	1	2	yes	—
34	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₁	—	2	4	no	—
38	C₂×C₈×C₇₁	—	2	4	no	—
39	C₂×C₈×C₇₁	C₂×C₈	1	8	yes	—
40	C₄²×C₇₁	—	2	4	no	—
41	C₄²×C₇₁	—	2	4	no	—
42	C₄²×C₇₁	C₂×C₈	1	4	yes	—
43	C₄²×C₇₁	M₁₆	1	4	yes	—
44	C₄²×C₇₁	#41	1	4	yes	—
45	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₇₁	—	2	2	no	—
47	C₂×C₄×C₇₁	—	2	2	no	—
48	C₂×C₄×C₇₁	—	2	2	no	—
49	C₂×C₄×C₇₁	—	2	2	no	—
50	C₂×C₄×C₇₁	C₄×C₈	1	4	yes	—
51	C₂×C₄×C₇₁	#97	1	4	yes	—
52	C₂×C₄×C₇₁	#117	1	2	yes	—
53	C₂×C₄×C₇₁	#117	1	2	yes	—
54	C₂×C₄×C₇₁	#117	1	2	yes	—
55	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
56	C₂×C₄×C₇₁	C₂×M₁₆	1	4	yes	—
57	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
58	C₂×C₄×C₇₁	#131	1	4	yes	—
59	C₈×C₇₁	—	2	8	no	—
60	C₇₁⋊C₈	—	2	8	no	—
61	C₇₁⋊C₈	—	4	4	no	—
62	C₂²×C₇₁	#376	1	1	yes	characteristic
63	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
64	C₂²×C₇₁	C₂×#97	1	1	yes	characteristic
65	C₂²×C₇₁	C₄×M₁₆	1	2	yes	—
66	C₂²×C₇₁	#405	1	2	yes	—
67	C₄×C₇₁	—	2	4	no	—
68	C₄×C₇₁	—	2	4	no	—
69	C₄×C₇₁	#431	1	4	yes	—
70	C₄×C₇₁	#432	1	4	yes	—
71	C₄×C₇₁	#433	1	4	yes	—
72	#8132	—	71	1	no	maximal, Sylow
73	C₂×C₇₁	C₈×M₁₆	1	2	yes	—
74	C₂×C₇₁	#1451	1	2	yes	—
75	C₂×C₇₁	#1744	1	1	yes	characteristic
76	C₂×C₇₁	#1745	1	1	yes	characteristic
77	C₂×C₇₁	#1746	1	1	yes	characteristic
78	#1744	—	71	1	no	—
79	C₄×#97	D₇₁	1	1	yes	characteristic
80	C₄²×C₈	—	71	1	no	—
81	#1273	—	71	4	no	—
82	C₇₁	#8132	1	1	yes	characteristic, complemented, Sylow
83	C₂×#97	Dic₇₁	1	2	yes	—
84	C₂×#97	Dic₇₁	1	2	yes	—
85	C₂×C₄×C₈	—	71	4	no	—
86	C₂×C₄×C₈	—	71	2	no	—
87	C₂×C₄×C₈	—	71	2	no	—
88	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
89	C₄³	C₂×D₇₁	1	1	yes	characteristic
90	#336	—	71	4	no	—
91	C₂²×C₈	—	71	4	no	—
92	C₂²×C₈	—	71	4	no	—
93	C₂²×C₈	C₂×Dic₇₁	1	4	yes	—
94	C₂×C₄²	C₄×D₇₁	1	2	yes	—
95	C₂×C₄²	C₄×D₇₁	1	2	yes	—
96	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
97	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
98	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
99	#97	—	2	4	no	—
100	#97	C₇₁⋊C₈	1	8	yes	complemented
101	C₄×C₈	—	2	4	no	—
102	C₄×C₈	—	142	8	no	—
103	C₄×C₈	—	142	4	no	—
104	C₄×C₈	—	142	4	no	—
105	C₂²×C₄	C₄×Dic₇₁	1	2	yes	—
106	C₂²×C₄	C₄×Dic₇₁	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
109	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	71	8	no	—
113	C₂×C₈	—	142	4	no	—
114	C₂×C₈	—	142	8	no	—
115	C₂×C₈	—	142	4	no	—
116	C₂×C₈	—	142	4	no	—
117	C₂×C₈	C₂×C₇₁⋊C₈	1	8	yes	—
118	C₄²	—	2	4	no	—
119	C₄²	—	2	4	no	—
120	C₄²	C₂×C₇₁⋊C₈	1	4	yes	—
121	C₄²	#11765849	1	4	yes	—
122	C₄²	#11765865	1	4	yes	—
123	C₂³	C₂×C₄×Dic₇₁	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₄×C₇₁⋊C₈	1	4	yes	—
129	C₂×C₄	#14556039	1	4	yes	—
130	C₂×C₄	#14556085	1	2	yes	—
131	C₂×C₄	#14556085	1	2	yes	—
132	C₂×C₄	#14556087	1	2	yes	—
133	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
134	C₂×C₄	M₁₆×D₇₁	1	4	yes	—
135	C₂×C₄	C₂×#11765849	1	2	yes	—
136	C₂×C₄	#14556136	1	4	yes	—
137	C₈	—	2	8	no	—
138	C₈	—	284	4	no	—
139	C₈	—	142	8	no	—
140	C₂²	#17985845	1	1	yes	characteristic
141	C₂²	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
142	C₂²	C₂×#14556039	1	1	yes	characteristic
143	C₂²	M₁₆×Dic₇₁	1	2	yes	—
144	C₂²	#17985957	1	2	yes	—
145	C₄	—	2	4	no	—
146	C₄	—	2	4	no	—
147	C₄	#17986052	1	4	yes	—
148	C₄	#17986054	1	4	yes	—
149	C₄	#17986061	1	4	yes	—
150	C₂	M₁₆×C₇₁⋊C₈	1	2	yes	—
151	C₂	#22586627	1	2	yes	—
152	C₂	#22588086	1	1	yes	characteristic
153	C₂	#22588088	1	1	yes	characteristic
154	C₂	#22588093	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄×C₈
Fitting	C₄×#97×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄²
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×#97	D₇₁
hypocenter	C₇₁	#8132
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
25. #97×C₇₁	C₈	semidirect
82. C₇₁	#8132	semidirect
100. #97	C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22588086	2
C₄²×C₇₁⋊C₈	2
C₇₁⋊#1273	2
C₄×#97×C₇₁	2
#8132	71