Group #36350098 — subgroups

Subgroups — #2313×C₁₇₃ #36350098

Order 22144.

512 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 280 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic
2	#387×C₁₇₃	C₂	1	8	yes	maximal, complemented
3	#381×C₁₇₃	C₂	1	2	yes	maximal
4	#380×C₁₇₃	C₂	1	2	yes	maximal
5	C₄×C₂²⋊C₄×C₁₇₃	C₂	1	2	yes	maximal
6	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₁₇₃	C₄	1	8	yes	complemented
10	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₄×C₁₇₃	C₂²	1	4	yes	—
13	C₂×C₄²×C₁₇₃	C₂²	1	4	yes	—
14	#95×C₁₇₃	C₂²	1	4	yes	—
15	#95×C₁₇₃	C₂²	1	4	yes	—
16	#95×C₁₇₃	C₂²	1	8	yes	—
17	C₂⁴×C₁₇₃	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₁₇₃	—	2	4	no	—
19	C₂²×C₄×C₁₇₃	—	4	2	no	—
20	C₂²×C₄×C₁₇₃	—	4	2	no	—
21	C₂²×C₄×C₁₇₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₇₃	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₁₇₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₇₃	C₂³	1	8	yes	—
25	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
26	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₇₃	C₂³	1	2	yes	—
28	C₄⋊C₄×C₁₇₃	—	4	4	no	—
29	C₄⋊C₄×C₁₇₃	C₂×C₄	1	16	yes	—
30	C₂²⋊C₄×C₁₇₃	—	2	8	no	—
31	C₄²×C₁₇₃	—	4	4	no	—
32	C₂³×C₁₇₃	—	2	2	no	—
33	C₂³×C₁₇₃	—	2	2	no	—
34	C₂³×C₁₇₃	C₂²×C₄	1	2	yes	—
35	C₂³×C₁₇₃	#41	1	4	yes	—
36	C₂³×C₁₇₃	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₁₇₃	—	2	8	no	—
38	C₂×C₄×C₁₇₃	—	2	4	no	—
39	C₂×C₄×C₁₇₃	—	2	4	no	—
40	C₂×C₄×C₁₇₃	—	2	4	no	—
41	C₂×C₄×C₁₇₃	—	2	8	no	—
42	C₂×C₄×C₁₇₃	—	2	8	no	—
43	C₂×C₄×C₁₇₃	—	2	8	no	—
44	C₂×C₄×C₁₇₃	—	4	4	no	—
45	C₂×C₄×C₁₇₃	—	4	4	no	—
46	C₂×C₄×C₁₇₃	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₁₇₃	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₇₃	C₂²×C₄	1	4	yes	—
49	C₂²×C₁₇₃	—	2	2	no	—
50	C₂²×C₁₇₃	—	2	2	no	—
51	C₂²×C₁₇₃	—	2	8	no	—
52	C₂²×C₁₇₃	#117	1	4	yes	—
53	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
54	C₂²×C₁₇₃	C₂×#41	1	2	yes	—
55	C₂²×C₁₇₃	H(5,𝔽₂)	1	2	yes	—
56	C₂²×C₁₇₃	2₋¹⁺⁴	1	2	yes	—
57	C₄×C₁₇₃	—	2	4	no	—
58	C₄×C₁₇₃	—	2	4	no	—
59	C₄×C₁₇₃	—	2	4	no	—
60	C₄×C₁₇₃	—	4	8	no	—
61	C₂×C₁₇₃	—	2	4	no	—
62	C₂×C₁₇₃	C₂×#117	1	1	yes	characteristic
63	C₂×C₁₇₃	#518	1	1	yes	characteristic
64	C₂×C₁₇₃	#519	1	1	yes	characteristic
65	C₂×C₁₇₃	#539	1	4	yes	—
66	C₁₇₃	#2313	1	1	yes	characteristic, direct product factor, Sylow
67	#2313	C₁₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₄⋊C₄	C₂×C₁₇₃	1	1	yes	characteristic
69	#387	C₂×C₁₇₃	1	8	yes	complemented
70	#381	C₂×C₁₇₃	1	2	yes	—
71	#380	C₂×C₁₇₃	1	2	yes	—
72	C₄×C₂²⋊C₄	C₂×C₁₇₃	1	2	yes	—
73	C₂³×C₄	C₂²×C₁₇₃	1	1	yes	characteristic
74	C₂³×C₄	C₂²×C₁₇₃	1	1	yes	characteristic
75	C₂³×C₄	C₂²×C₁₇₃	1	1	yes	characteristic
76	C₂×C₄⋊C₄	C₄×C₁₇₃	1	8	yes	complemented
77	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	4	yes	—
78	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	4	yes	—
79	C₂×C₂²⋊C₄	C₂²×C₁₇₃	1	4	yes	—
80	C₂×C₄²	C₂²×C₁₇₃	1	4	yes	—
81	#95	C₂²×C₁₇₃	1	4	yes	—
82	#95	C₂²×C₁₇₃	1	4	yes	—
83	#95	C₂²×C₁₇₃	1	8	yes	—
84	C₂⁴	C₂³×C₁₇₃	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	C₂×C₄×C₁₇₃	1	4	yes	—
89	C₂²×C₄	C₂×C₄×C₁₇₃	1	4	yes	—
90	C₂²×C₄	C₂×C₄×C₁₇₃	1	4	yes	—
91	C₂²×C₄	C₂³×C₁₇₃	1	8	yes	—
92	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
93	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
94	C₂²×C₄	C₂³×C₁₇₃	1	2	yes	—
95	C₄⋊C₄	—	4	4	no	—
96	C₄⋊C₄	C₂×C₄×C₁₇₃	1	16	yes	—
97	C₂²⋊C₄	—	2	8	no	—
98	C₄²	—	4	4	no	—
99	C₂³	—	2	2	no	—
100	C₂³	—	2	2	no	—
101	C₂³	C₂²×C₄×C₁₇₃	1	2	yes	—
102	C₂³	#41×C₁₇₃	1	4	yes	—
103	C₂³	C₂⁴×C₁₇₃	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	8	no	—
108	C₂×C₄	—	2	8	no	—
109	C₂×C₄	—	2	8	no	—
110	C₂×C₄	—	2	8	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	C₂²×C₄×C₁₇₃	1	4	yes	—
114	C₂×C₄	C₂²×C₄×C₁₇₃	1	4	yes	—
115	C₂×C₄	C₂²×C₄×C₁₇₃	1	4	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	8	no	—
119	C₂²	#117×C₁₇₃	1	4	yes	—
120	C₂²	C₂³×C₄×C₁₇₃	1	1	yes	characteristic
121	C₂²	C₂×#41×C₁₇₃	1	2	yes	—
122	C₂²	H(5,𝔽₂)×C₁₇₃	1	2	yes	—
123	C₂²	2₋¹⁺⁴×C₁₇₃	1	2	yes	—
124	C₄	—	2	4	no	—
125	C₄	—	2	4	no	—
126	C₄	—	2	4	no	—
127	C₄	—	4	8	no	—
128	C₂	—	2	4	no	—
129	C₂	C₂×#117×C₁₇₃	1	1	yes	characteristic
130	C₂	#518×C₁₇₃	1	1	yes	characteristic
131	C₂	#519×C₁₇₃	1	1	yes	characteristic
132	C₂	#539×C₁₇₃	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊C₄×C₁₇₃

C₂

yes

maximal, characteristic

#387×C₁₇₃

C₂

yes

maximal, complemented

#381×C₁₇₃

C₂

yes

maximal

#380×C₁₇₃

C₂

yes

maximal

C₄×C₂²⋊C₄×C₁₇₃

C₂

yes

maximal

C₂³×C₄×C₁₇₃

C₂²

yes

characteristic

C₂³×C₄×C₁₇₃

C₂²

yes

characteristic

C₂³×C₄×C₁₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₇₃

C₂²

yes

—

C₂×C₄⋊C₄×C₁₇₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₇₃

C₂²

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

—

#95×C₁₇₃

C₂²

yes

—

#95×C₁₇₃

C₂²

yes

—

#95×C₁₇₃

C₂²

yes

—

C₂⁴×C₁₇₃

C₂³

yes

characteristic

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

C₂×C₄

yes

—

C₂²×C₄×C₁₇₃

C₂×C₄

yes

—

C₂²×C₄×C₁₇₃

C₂×C₄

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

C₂×C₄

yes

—

C₂²⋊C₄×C₁₇₃

—

C₄²×C₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

C₂²×C₄

yes

—

C₂³×C₁₇₃

#41

yes

—

C₂³×C₁₇₃

C₂⁴

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

#117

yes

—

C₂²×C₁₇₃

C₂³×C₄

yes

characteristic

C₂²×C₁₇₃

C₂×#41

yes

—

C₂²×C₁₇₃

H(5,𝔽₂)

yes

—

C₂²×C₁₇₃

2₋¹⁺⁴

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

—

C₁₇₃

#2313

yes

characteristic, direct product factor, Sylow

#2313

C₁₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₄⋊C₄

C₂×C₁₇₃

yes

characteristic

#387

C₂×C₁₇₃

yes

complemented

#381

C₂×C₁₇₃

yes

—

#380

C₂×C₁₇₃

yes

—

C₄×C₂²⋊C₄

C₂×C₁₇₃

yes

—

C₂³×C₄

C₂²×C₁₇₃

yes

characteristic

C₂³×C₄

C₂²×C₁₇₃

yes

characteristic

C₂³×C₄

C₂²×C₁₇₃

yes

characteristic

C₂×C₄⋊C₄

C₄×C₁₇₃

yes

complemented

C₂×C₄⋊C₄

C₂²×C₁₇₃

yes

—

C₂×C₄⋊C₄

C₂²×C₁₇₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₁₇₃

yes

—

C₂×C₄²

C₂²×C₁₇₃

yes

—

#95

C₂²×C₁₇₃

yes

—

#95

C₂²×C₁₇₃

yes

—

#95

C₂²×C₁₇₃

yes

—

C₂⁴

C₂³×C₁₇₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₁₇₃

yes

—

C₂²×C₄

C₂×C₄×C₁₇₃

yes

—

C₂²×C₄

C₂×C₄×C₁₇₃

yes

—

C₂²×C₄

C₂³×C₁₇₃

yes

—

C₂²×C₄

C₂³×C₁₇₃

yes

—

C₂²×C₄

C₂³×C₁₇₃

yes

—

C₂²×C₄

C₂³×C₁₇₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₁₇₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

100

C₂³

—

101

C₂³

C₂²×C₄×C₁₇₃

yes

—

102

C₂³

#41×C₁₇₃

yes

—

103

C₂³

C₂⁴×C₁₇₃

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×C₄×C₁₇₃

yes

—

114

C₂×C₄

C₂²×C₄×C₁₇₃

yes

—

115

C₂×C₄

C₂²×C₄×C₁₇₃

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#117×C₁₇₃

yes

—

120

C₂²

C₂³×C₄×C₁₇₃

yes

characteristic

121

C₂²

C₂×#41×C₁₇₃

yes

—

122

C₂²

H(5,𝔽₂)×C₁₇₃

yes

—

123

C₂²

2₋¹⁺⁴×C₁₇₃

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₂

—

129

C₂

C₂×#117×C₁₇₃

yes

characteristic

130

C₂

#518×C₁₇₃

yes

characteristic

131

C₂

#519×C₁₇₃

yes

characteristic

132

C₂

#539×C₁₇₃

yes

—

	subgroup	G/H
center, socle	C₂³×C₁₇₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₇₃
Frattini, cosocle	C₂³	C₂⁴×C₁₇₃
prime radical	C₂⁴×C₁₇₃	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₇₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₇₃

C₂⁴

commutator

C₂²

C₂³×C₄×C₁₇₃

Frattini, cosocle

C₂³

C₂⁴×C₁₇₃

prime radical

C₂⁴×C₁₇₃

C₂³

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₇₃ ◁ G

N	H ≅ G/N	type
2. #387×C₁₇₃	C₂	semidirect
9. C₂×C₄⋊C₄×C₁₇₃	C₄	semidirect
66. C₁₇₃	#2313	direct (N × H)
67. #2313	C₁₇₃	direct (N × H)
69. #387	C₂×C₁₇₃	semidirect
76. C₂×C₄⋊C₄	C₄×C₁₇₃	semidirect

H ≅ G/N

type

2. #387×C₁₇₃

C₂

semidirect

9. C₂×C₄⋊C₄×C₁₇₃

C₄

semidirect

66. C₁₇₃

#2313

direct (N × H)

67. #2313

C₁₇₃

direct (N × H)

69. #387

C₂×C₁₇₃

semidirect

76. C₂×C₄⋊C₄

C₄×C₁₇₃

semidirect

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

subgroup	index
C₂²×C₄⋊C₄×C₁₇₃	2
#387×C₁₇₃	2
#381×C₁₇₃	2
#380×C₁₇₃	2
C₄×C₂²⋊C₄×C₁₇₃	2
#2313	173

subgroup

index

C₂²×C₄⋊C₄×C₁₇₃

#387×C₁₇₃

#381×C₁₇₃

#380×C₁₇₃

C₄×C₂²⋊C₄×C₁₇₃

#2313

173

Subgroups — #2313×C₁₇₃ #36350098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2313×C173 #36350098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2313×C₁₇₃ #36350098