Group #36349846 — subgroups

Subgroups — #2061×C₁₇₃ #36349846

Order 22144.

230 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 92 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	#544×C₁₇₃	C₂	1	1	yes	maximal, characteristic
2	#428×C₁₇₃	C₂	1	2	yes	maximal
3	#382×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	#343×C₁₇₃	C₂	1	1	yes	maximal, characteristic
5	#339×C₁₇₃	C₂	1	2	yes	maximal
6	C₂×M₁₆×C₁₇₃	C₂²	1	2	yes	—
7	C₄⋊Q₈×C₁₇₃	—	2	2	no	—
8	#125×C₁₇₃	—	2	2	no	—
9	C₂²⋊Q₈×C₁₇₃	—	2	2	no	—
10	#117×C₁₇₃	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
14	#107×C₁₇₃	—	2	2	no	—
15	#106×C₁₇₃	—	2	2	no	—
16	#95×C₁₇₃	—	2	2	no	—
17	C₂×Q₈×C₁₇₃	—	2	2	no	—
18	C₂²×C₄×C₁₇₃	—	2	1	no	—
19	C₂²×C₄×C₁₇₃	—	2	1	no	—
20	C₂²×C₄×C₁₇₃	D₄	1	2	yes	—
21	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₇₃	—	2	2	no	—
23	M₁₆×C₁₇₃	Q₈	1	4	yes	complemented
24	C₂×C₈×C₁₇₃	—	2	2	no	—
25	C₄⋊C₄×C₁₇₃	—	2	4	no	—
26	C₄⋊C₄×C₁₇₃	—	2	2	no	—
27	C₄⋊C₄×C₁₇₃	—	2	2	no	—
28	C₄⋊C₄×C₁₇₃	—	2	2	no	—
29	C₄⋊C₄×C₁₇₃	D₄	1	2	yes	—
30	C₄⋊C₄×C₁₇₃	D₄	1	2	yes	—
31	C₂²⋊C₄×C₁₇₃	—	2	2	no	—
32	C₄²×C₁₇₃	—	2	2	no	—
33	C₄²×C₁₇₃	—	2	1	no	—
34	C₄²×C₁₇₃	Q₈	1	2	yes	complemented
35	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₁₇₃	—	8	2	no	—
37	C₂×C₄×C₁₇₃	—	2	2	no	—
38	C₂×C₄×C₁₇₃	—	2	2	no	—
39	C₂×C₄×C₁₇₃	—	2	2	no	—
40	C₂×C₄×C₁₇₃	—	2	2	no	—
41	C₂×C₄×C₁₇₃	—	4	1	no	—
42	C₂×C₄×C₁₇₃	—	4	1	no	—
43	C₂×C₄×C₁₇₃	—	4	2	no	—
44	C₂×C₄×C₁₇₃	—	4	2	no	—
45	C₂×C₄×C₁₇₃	—	4	1	no	—
46	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₇₃	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₇₃	#41	1	1	yes	characteristic
50	C₈×C₁₇₃	—	2	4	no	—
51	C₂²×C₁₇₃	—	2	1	no	—
52	C₂²×C₁₇₃	—	2	1	no	—
53	C₂²×C₁₇₃	#120	1	1	yes	characteristic
54	C₂²×C₁₇₃	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂²×C₁₇₃	C₄⋊Q₈	1	1	yes	characteristic
56	C₄×C₁₇₃	—	4	2	no	—
57	C₄×C₁₇₃	—	4	2	no	—
58	C₄×C₁₇₃	—	4	2	no	—
59	C₄×C₁₇₃	—	4	2	no	—
60	C₄×C₁₇₃	—	8	1	no	—
61	C₄×C₁₇₃	C₂²⋊Q₈	1	2	yes	—
62	C₄×C₁₇₃	C₄⋊Q₈	1	2	yes	—
63	C₂×C₁₇₃	—	2	1	no	—
64	C₂×C₁₇₃	—	2	1	no	—
65	C₂×C₁₇₃	#395	1	1	yes	characteristic
66	C₂×C₁₇₃	#454	1	1	yes	characteristic
67	C₂×C₁₇₃	#455	1	1	yes	characteristic
68	C₁₇₃	#2061	1	1	yes	characteristic, direct product factor, Sylow
69	#2061	C₁₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	#544	C₂×C₁₇₃	1	1	yes	characteristic
71	#428	C₂×C₁₇₃	1	2	yes	—
72	#382	C₂×C₁₇₃	1	1	yes	characteristic
73	#343	C₂×C₁₇₃	1	1	yes	characteristic
74	#339	C₂×C₁₇₃	1	2	yes	—
75	C₂×M₁₆	C₂²×C₁₇₃	1	2	yes	—
76	C₄⋊Q₈	—	2	2	no	—
77	#125	—	2	2	no	—
78	C₂²⋊Q₈	—	2	2	no	—
79	#117	C₂²×C₁₇₃	1	2	yes	—
80	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	1	yes	characteristic
81	C₂×C₄⋊C₄	C₂²×C₁₇₃	1	1	yes	characteristic
82	C₂×C₄²	C₂²×C₁₇₃	1	1	yes	characteristic
83	#107	—	2	2	no	—
84	#106	—	2	2	no	—
85	#95	—	2	2	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	D₄×C₁₇₃	1	2	yes	—
90	C₂²×C₄	C₂³×C₁₇₃	1	1	yes	characteristic
91	M₁₆	—	2	2	no	—
92	M₁₆	Q₈×C₁₇₃	1	4	yes	complemented
93	C₂×C₈	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	D₄×C₁₇₃	1	2	yes	—
99	C₄⋊C₄	D₄×C₁₇₃	1	2	yes	—
100	C₂²⋊C₄	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	—	2	1	no	—
103	C₄²	Q₈×C₁₇₃	1	2	yes	complemented
104	C₂³	C₂×D₄×C₁₇₃	1	1	yes	characteristic
105	Q₈	—	8	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	C₂×D₄×C₁₇₃	1	2	yes	—
116	C₂×C₄	C₂×Q₈×C₁₇₃	1	1	yes	characteristic
117	C₂×C₄	C₂×Q₈×C₁₇₃	1	2	yes	—
118	C₂×C₄	#41×C₁₇₃	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#120×C₁₇₃	1	1	yes	characteristic
123	C₂²	C₂²⋊Q₈×C₁₇₃	1	1	yes	characteristic
124	C₂²	C₄⋊Q₈×C₁₇₃	1	1	yes	characteristic
125	C₄	—	4	2	no	—
126	C₄	—	4	2	no	—
127	C₄	—	4	2	no	—
128	C₄	—	4	2	no	—
129	C₄	—	8	1	no	—
130	C₄	C₂²⋊Q₈×C₁₇₃	1	2	yes	—
131	C₄	C₄⋊Q₈×C₁₇₃	1	2	yes	—
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#395×C₁₇₃	1	1	yes	characteristic
135	C₂	#454×C₁₇₃	1	1	yes	characteristic
136	C₂	#455×C₁₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#544×C₁₇₃

C₂

yes

maximal, characteristic

#428×C₁₇₃

C₂

yes

maximal

#382×C₁₇₃

C₂

yes

maximal, characteristic

#343×C₁₇₃

C₂

yes

maximal, characteristic

#339×C₁₇₃

C₂

yes

maximal

C₂×M₁₆×C₁₇₃

C₂²

yes

—

C₄⋊Q₈×C₁₇₃

—

#125×C₁₇₃

—

C₂²⋊Q₈×C₁₇₃

—

#117×C₁₇₃

C₂²

yes

—

C₂×C₄⋊C₄×C₁₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇₃

C₂²

yes

characteristic

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

#107×C₁₇₃

—

#106×C₁₇₃

—

#95×C₁₇₃

—

C₂×Q₈×C₁₇₃

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

D₄

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

M₁₆×C₁₇₃

—

M₁₆×C₁₇₃

Q₈

yes

complemented

C₂×C₈×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

D₄

yes

—

C₄⋊C₄×C₁₇₃

D₄

yes

—

C₂²⋊C₄×C₁₇₃

—

C₄²×C₁₇₃

—

C₄²×C₁₇₃

—

C₄²×C₁₇₃

Q₈

yes

complemented

C₂³×C₁₇₃

C₂×D₄

yes

characteristic

Q₈×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

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×Q₈

yes

—

C₂×C₄×C₁₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇₃

#41

yes

characteristic

C₈×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

#120

yes

characteristic

C₂²×C₁₇₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₇₃

C₄⋊Q₈

yes

characteristic

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₂²⋊Q₈

yes

—

C₄×C₁₇₃

C₄⋊Q₈

yes

—

C₂×C₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₁₇₃

yes

characteristic, direct product factor, Sylow

#2061

C₁₇₃

yes

maximal, characteristic, direct product factor, Sylow

#544

C₂×C₁₇₃

yes

characteristic

#428

C₂×C₁₇₃

yes

—

#382

C₂×C₁₇₃

yes

characteristic

#343

C₂×C₁₇₃

yes

characteristic

#339

C₂×C₁₇₃

yes

—

C₂×M₁₆

C₂²×C₁₇₃

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂²×C₁₇₃

yes

—

C₂×C₄⋊C₄

C₂²×C₁₇₃

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₇₃

yes

characteristic

C₂×C₄²

C₂²×C₁₇₃

yes

—

—

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₇₃

yes

—

C₂²×C₄

C₂³×C₁₇₃

yes

characteristic

M₁₆

—

M₁₆

Q₈×C₁₇₃

yes

complemented

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₁₇₃

yes

—

C₄⋊C₄

D₄×C₁₇₃

yes

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

Q₈×C₁₇₃

yes

complemented

104

C₂³

C₂×D₄×C₁₇₃

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂×D₄×C₁₇₃

yes

—

116

C₂×C₄

C₂×Q₈×C₁₇₃

yes

characteristic

117

C₂×C₄

C₂×Q₈×C₁₇₃

yes

—

118

C₂×C₄

#41×C₁₇₃

yes

characteristic

119

C₈

—

120

C₂²

—

121

C₂²

—

122

C₂²

#120×C₁₇₃

yes

characteristic

123

C₂²

C₂²⋊Q₈×C₁₇₃

yes

characteristic

124

C₂²

C₄⋊Q₈×C₁₇₃

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

C₂²⋊Q₈×C₁₇₃

yes

—

131

C₄

C₄⋊Q₈×C₁₇₃

yes

—

132

C₂

—

133

C₂

—

134

C₂

#395×C₁₇₃

yes

characteristic

135

C₂

#454×C₁₇₃

yes

characteristic

136

C₂

#455×C₁₇₃

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇₃	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₇₃
prime radical	C₂³×C₁₇₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇₃ ◁ C₂²×C₄×C₁₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₇₃

#120

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₇₃

prime radical

C₂³×C₁₇₃

C₂×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₇₃ ◁ C₂²×C₄×C₁₇₃ ◁ G

N	H ≅ G/N	type
23. M₁₆×C₁₇₃	Q₈	semidirect
34. C₄²×C₁₇₃	Q₈	semidirect
68. C₁₇₃	#2061	direct (N × H)
69. #2061	C₁₇₃	direct (N × H)
92. M₁₆	Q₈×C₁₇₃	semidirect
103. C₄²	Q₈×C₁₇₃	semidirect

H ≅ G/N

type

23. M₁₆×C₁₇₃

Q₈

semidirect

34. C₄²×C₁₇₃

Q₈

semidirect

68. C₁₇₃

#2061

direct (N × H)

69. #2061

C₁₇₃

direct (N × H)

92. M₁₆

Q₈×C₁₇₃

semidirect

103. C₄²

Q₈×C₁₇₃

semidirect

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

subgroup	index
#544×C₁₇₃	2
#428×C₁₇₃	2
#382×C₁₇₃	2
#343×C₁₇₃	2
#339×C₁₇₃	2
#2061	173

subgroup

index

#544×C₁₇₃

#428×C₁₇₃

#382×C₁₇₃

#343×C₁₇₃

#339×C₁₇₃

#2061

173

Subgroups — #2061×C₁₇₃ #36349846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2061×C173 #36349846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2061×C₁₇₃ #36349846