Subgroups — #31884288

Order 18176.

246 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556068	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985775	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1450	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556068	C₄	1	2	yes	—
5	C₂²×C₇₁⋊C₁₆	C₄	1	2	yes	—
6	#17985775	—	2	1	no	—
7	#17985775	—	2	1	no	—
8	#17985775	—	2	1	no	—
9	C₄×C₇₁⋊C₁₆	—	2	1	no	—
10	#431×C₇₁	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	#322×C₇₁	—	2	1	no	—
13	C₈²×C₇₁	—	2	1	no	—
14	#14556068	—	2	2	no	—
15	#14556068	C₈	1	4	yes	complemented
16	C₂×C₇₁⋊C₁₆	—	2	2	no	—
17	C₂×C₇₁⋊C₁₆	—	2	2	no	—
18	C₂×C₇₁⋊C₁₆	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	C₈	1	4	yes	complemented
20	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₈×C₇₁	—	2	2	no	—
24	C₂²⋊C₈×C₇₁	C₈	1	4	yes	—
25	C₄×C₈×C₇₁	—	2	2	no	—
26	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
30	C₇₁⋊C₁₆	—	2	4	no	—
31	C₇₁⋊C₁₆	—	2	4	no	—
32	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₇₁	—	2	4	no	—
35	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
42	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
43	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₇₁	M₁₆	1	2	yes	—
46	C₂³×C₇₁	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₁	—	4	2	no	—
48	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
52	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
53	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
56	C₈×C₇₁	—	4	4	no	—
57	C₈×C₇₁	C₂²⋊C₈	1	2	yes	—
58	C₈×C₇₁	C₂²⋊C₈	1	2	yes	—
59	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
60	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
61	C₂²×C₇₁	—	2	2	no	—
62	C₂²×C₇₁	C₈²	1	1	yes	characteristic
63	C₂²×C₇₁	#322	1	1	yes	characteristic
64	C₂²×C₇₁	#336	1	1	yes	characteristic
65	C₄×C₇₁	—	2	2	no	—
66	C₄×C₇₁	—	2	2	no	—
67	C₄×C₇₁	#322	1	1	yes	characteristic
68	C₄×C₇₁	#322	1	1	yes	characteristic
69	C₄×C₇₁	#336	1	1	yes	characteristic
70	C₄×C₇₁	#336	1	1	yes	characteristic
71	#7140	—	71	1	no	maximal, Sylow
72	C₂×C₇₁	—	2	1	no	—
73	C₂×C₇₁	—	2	1	no	—
74	C₂×C₇₁	#1273	1	1	yes	characteristic
75	C₂×C₇₁	#1378	1	2	yes	—
76	C₄×M₃₂	—	71	1	no	—
77	C₂×#346	—	71	1	no	—
78	#1450	D₇₁	1	1	yes	characteristic
79	C₇₁	#7140	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	—	71	2	no	—
81	C₂²×C₁₆	—	71	2	no	—
82	#431	Dic₇₁	1	2	yes	—
83	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
84	#346	—	142	1	no	—
85	#346	—	142	1	no	—
86	#346	—	142	1	no	—
87	C₄×C₁₆	—	142	1	no	—
88	#322	—	2	1	no	—
89	C₈²	—	2	1	no	—
90	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
91	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
92	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
93	M₃₂	—	71	4	no	—
94	M₃₂	—	142	2	no	—
95	C₂×C₁₆	—	71	4	no	—
96	C₂×C₁₆	—	142	2	no	—
97	C₂×C₁₆	—	142	2	no	—
98	C₂×C₁₆	—	142	2	no	—
99	C₄⋊C₈	—	2	2	no	—
100	C₂²⋊C₈	C₇₁⋊C₈	1	4	yes	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
103	C₄×C₈	D₂₈₄	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
106	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
107	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
108	C₂×C₈	—	2	4	no	—
109	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
110	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
111	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
112	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
113	C₂×C₈	C₈×D₇₁	1	2	yes	—
114	C₂×C₈	C₈×D₇₁	1	2	yes	—
115	C₂×C₈	#11765848	1	2	yes	—
116	C₂×C₈	#11765848	1	2	yes	—
117	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
118	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
119	C₄²	#11765849	1	2	yes	—
120	C₁₆	—	142	4	no	—
121	C₁₆	—	142	4	no	—
122	C₂³	C₄×C₇₁⋊C₈	1	1	yes	characteristic
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
125	C₂×C₄	C₈×Dic₇₁	1	1	yes	characteristic
126	C₂×C₄	C₈×Dic₇₁	1	1	yes	characteristic
127	C₂×C₄	#14556038	1	1	yes	characteristic
128	C₂×C₄	#14556038	1	1	yes	characteristic
129	C₂×C₄	#14556039	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
131	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
132	C₈	—	4	4	no	—
133	C₈	(C₂×D₇₁)⋊C₈	1	2	yes	—
134	C₈	(C₂×D₇₁)⋊C₈	1	2	yes	—
135	C₈	Dic₇₁⋊C₈	1	2	yes	—
136	C₈	Dic₇₁⋊C₈	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	C₈×C₇₁⋊C₈	1	1	yes	characteristic
139	C₂²	#17985720	1	1	yes	characteristic
140	C₂²	#17985754	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	2	no	—
143	C₄	#17985721	1	1	yes	characteristic
144	C₄	#17985721	1	1	yes	characteristic
145	C₄	#17985753	1	1	yes	characteristic
146	C₄	#17985753	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	C₇₁⋊#1273	1	1	yes	characteristic
150	C₂	#22586426	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	(C₂×D₇₁)⋊C₄
commutator	C₂²×C₇₁	C₈²
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₇₁×#1450	C₂
socle	C₂²×C₇₁	#336
prime radical	C₂³×C₇₁	C₄×C₈
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	#1450	D₇₁
hypocenter	C₇₁	#7140
LCS	G ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
15. #14556068	C₈	semidirect
19. C₂×C₇₁⋊C₁₆	C₈	semidirect
79. C₇₁	#7140	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556068	2
C₂×#17985775	2
C₇₁×#1450	2
#7140	71