Group #34011924 — subgroups

Subgroups — #34011924

Order 18688.

212 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 102 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	Q₈×#11924031	C₂	1	2	yes	maximal
2	#22624953	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11924031	C₂	1	2	yes	maximal
4	#22624844	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×#10596831	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11924031	—	2	2	no	—
7	C₄⋊#11924031	C₂²	1	2	yes	—
8	C₄×#11924031	—	2	2	no	—
9	C₄×#11924031	C₂²	1	1	yes	characteristic
10	C₄×#11924031	C₂²	1	1	yes	characteristic
11	Q₈×#10596831	C₂²	1	2	yes	—
12	#17988340	C₄	1	1	yes	characteristic
13	Q₈⋊#10596831	C₄	1	2	yes	—
14	C₈×#10596831	C₂²	1	1	yes	characteristic
15	C₄×Q₁₆×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924031	—	2	2	no	—
17	C₂×#11924031	—	2	1	no	—
18	C₂×#11924031	D₄	1	2	yes	—
19	C₄⋊#10596831	—	2	2	no	—
20	C₄⋊#10596831	C₂×C₄	1	2	yes	—
21	C₄×#10596831	—	2	2	no	—
22	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
23	C₄×#10596831	C₂³	1	1	yes	characteristic
24	C₂×Q₁₆×C₇₃	C₈	1	1	yes	characteristic
25	C₄×Q₈×C₇₃	C₂×C₄	1	2	yes	—
26	#107×C₇₃	C₈	1	1	yes	characteristic
27	#103×C₇₃	C₈	1	2	yes	—
28	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	#11924031	—	2	2	no	—
30	#11924031	—	4	1	no	—
31	#11924031	—	4	2	no	—
32	C₂×#10596831	—	2	2	no	—
33	C₂×#10596831	—	2	1	no	—
34	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
35	C₂×#10596831	#41	1	1	yes	characteristic
36	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
37	Q₁₆×C₇₃	C₁₆	1	4	yes	complemented
38	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
40	C₄⋊C₄×C₇₃	—	2	2	no	—
41	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
42	C₄²×C₇₃	—	2	2	no	—
43	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
44	Q₈×C₇₃	—	2	2	no	—
45	Q₈×C₇₃	C₂×C₁₆	1	4	yes	—
46	C₂×C₄×C₇₃	—	2	2	no	—
47	C₂×C₄×C₇₃	—	2	2	no	—
48	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
51	C₈×C₇₃	—	2	1	no	—
52	C₈×C₇₃	C₂×C₁₆	1	2	yes	—
53	#10596831	—	2	1	no	—
54	#10596831	—	4	1	no	—
55	#10596831	—	4	2	no	—
56	#10596831	#137	1	2	yes	—
57	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
58	C₄×C₇₃	—	2	4	no	—
59	C₄×C₇₃	—	2	1	no	—
60	C₄×C₇₃	—	4	2	no	—
61	C₄×C₇₃	#441	1	1	yes	characteristic
62	C₄×C₇₃	#444	1	1	yes	characteristic
63	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
64	C₄×C₇₃	#504	1	1	yes	characteristic
65	#12059	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	#1579	1	1	yes	characteristic
67	C₂×C₇₃	D₄×C₁₆	1	1	yes	characteristic
68	C₂×C₇₃	#2171	1	1	yes	characteristic
69	Q₈×C₁₆	—	73	2	no	—
70	C₈×Q₁₆	—	73	1	no	—
71	#1367	—	73	1	no	—
72	Q₈⋊C₁₆	—	73	2	no	—
73	#1312	—	73	1	no	—
74	C₇₃	#12059	1	1	yes	characteristic, complemented, Sylow
75	C₈×Q₈	—	73	2	no	—
76	C₄×Q₁₆	#64408	1	1	yes	characteristic
77	C₄⋊C₁₆	—	73	2	no	—
78	C₄⋊C₁₆	—	146	2	no	—
79	C₄×C₁₆	—	73	1	no	—
80	C₄×C₁₆	—	73	1	no	—
81	C₄×C₁₆	—	146	2	no	—
82	#335	—	73	1	no	—
83	Q₈⋊C₈	—	73	2	no	—
84	C₈²	—	73	1	no	—
85	C₂×Q₁₆	#10596833	1	1	yes	characteristic
86	C₄×Q₈	C₂×#64408	1	2	yes	—
87	C₂×C₁₆	—	73	2	no	—
88	C₂×C₁₆	—	146	1	no	—
89	C₂×C₁₆	—	146	2	no	—
90	#107	#10596833	1	1	yes	characteristic
91	C₄⋊C₈	—	73	2	no	—
92	C₄⋊C₈	—	146	2	no	—
93	#103	#10596833	1	2	yes	—
94	C₄×C₈	—	73	1	no	—
95	C₄×C₈	—	73	1	no	—
96	C₄×C₈	—	146	2	no	—
97	C₄×C₈	C₂×#64408	1	1	yes	characteristic
98	C₂×Q₈	C₂×#10596833	1	2	yes	—
99	Q₁₆	#11924031	1	4	yes	complemented
100	C₂×C₈	—	73	1	no	—
101	C₂×C₈	—	73	1	no	—
102	C₂×C₈	—	146	1	no	—
103	C₂×C₈	—	146	2	no	—
104	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
105	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	C₂×#10596833	1	2	yes	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂²×#64408	1	1	yes	characteristic
110	C₁₆	—	292	2	no	—
111	C₁₆	—	292	1	no	—
112	C₁₆	—	146	2	no	—
113	Q₈	—	2	2	no	—
114	Q₈	C₂×#11924031	1	4	yes	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	D₄×#64408	1	1	yes	characteristic
118	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
119	C₂×C₄	#14556678	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	292	1	no	—
122	C₈	—	292	2	no	—
123	C₈	—	73	2	no	—
124	C₈	—	146	1	no	—
125	C₈	C₂×#11924031	1	2	yes	—
126	C₂²	D₄×#10596833	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	2	no	—
130	C₄	#17989627	1	1	yes	characteristic
131	C₄	#17989633	1	1	yes	characteristic
132	C₄	C₂²×#11924031	1	1	yes	characteristic
133	C₄	#17989645	1	1	yes	characteristic
134	C₂	#22625524	1	1	yes	characteristic
135	C₂	D₄×#11924031	1	1	yes	characteristic
136	C₂	#22626321	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11924031

C₂

yes

maximal

#22624953

C₂

yes

maximal, characteristic

Q₈⋊#11924031

C₂

yes

maximal

#22624844

C₂

yes

maximal, characteristic

Q₁₆×#10596831

C₂

yes

maximal, characteristic

C₄⋊#11924031

—

C₄⋊#11924031

C₂²

yes

—

C₄×#11924031

—

C₄×#11924031

C₂²

yes

characteristic

C₄×#11924031

C₂²

yes

characteristic

Q₈×#10596831

C₂²

yes

—

#17988340

C₄

yes

characteristic

Q₈⋊#10596831

C₄

yes

—

C₈×#10596831

C₂²

yes

characteristic

C₄×Q₁₆×C₇₃

C₄

yes

—

—

D₄

yes

—

C₄⋊#10596831

—

C₄⋊#10596831

C₂×C₄

yes

—

C₄×#10596831

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂³

yes

characteristic

C₂×Q₁₆×C₇₃

C₈

yes

characteristic

C₄×Q₈×C₇₃

C₂×C₄

yes

—

#107×C₇₃

C₈

yes

characteristic

#103×C₇₃

C₈

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

—

—

C₂×D₄

yes

yes

C₂×Q₈×C₇₃

C₂×C₈

yes

—

Q₁₆×C₇₃

C₁₆

yes

complemented

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

#131

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

C₂×C₁₆

yes

—

—

—

—

yes

—

C₂²×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂²×C₁₆

yes

C₄×C₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

D₄×C₁₆

yes

characteristic

C₂×C₇₃

#2171

yes

characteristic

Q₈×C₁₆

—

C₈×Q₁₆

—

#1367

—

Q₈⋊C₁₆

—

#1312

—

C₇₃

#12059

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×Q₁₆

#64408

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

146

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

146

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₂×Q₁₆

#10596833

yes

characteristic

C₄×Q₈

C₂×#64408

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

#107

#10596833

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

146

—

#103

#10596833

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

146

—

C₄×C₈

C₂×#64408

yes

characteristic

C₂×Q₈

C₂×#10596833

yes

—

Q₁₆

#11924031

yes

complemented

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

146

—

103

C₂×C₈

—

146

—

104

C₂×C₈

C₂×#10596833

yes

characteristic

105

C₂×C₈

C₂×#10596833

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×#10596833

yes

—

108

C₄²

—

109

C₄²

C₂²×#64408

yes

characteristic

110

C₁₆

—

292

—

111

C₁₆

—

292

—

112

C₁₆

—

146

—

113

Q₈

—

114

Q₈

C₂×#11924031

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

D₄×#64408

yes

characteristic

118

C₂×C₄

C₂²×#10596833

yes

characteristic

119

C₂×C₄

#14556678

yes

characteristic

120

C₈

—

121

C₈

—

292

—

122

C₈

—

292

—

123

C₈

—

124

C₈

—

146

—

125

C₈

C₂×#11924031

yes

—

126

C₂²

D₄×#10596833

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#17989627

yes

characteristic

131

C₄

#17989633

yes

characteristic

132

C₄

C₂²×#11924031

yes

characteristic

133

C₄

#17989645

yes

characteristic

134

C₂

#22625524

yes

characteristic

135

C₂

D₄×#11924031

yes

characteristic

136

C₂

#22626321

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#64408
Fitting	C₄×Q₁₆×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	C₈×D₄
cosocle	C₄×#10596831	C₂³
hypercenter	C₄×Q₁₆	#64408
hypocenter	C₇₃	#12059
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×#64408

commutator

C₄×C₇₃

C₂²×C₁₆

Frattini

C₄²

C₂²×#64408

Fitting

C₄×Q₁₆×C₇₃

C₄

socle, prime radical

C₂²×C₇₃

C₈×D₄

C₂³

C₄×Q₁₆

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
37. Q₁₆×C₇₃	C₁₆	semidirect
74. C₇₃	#12059	semidirect
99. Q₁₆	#11924031	semidirect

H ≅ G/N

type

37. Q₁₆×C₇₃

C₁₆

semidirect

74. C₇₃

#12059

semidirect

99. Q₁₆

#11924031

semidirect

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

subgroup	index
Q₈×#11924031	2
#22624953	2
Q₈⋊#11924031	2
#22624844	2
Q₁₆×#10596831	2
#12059	73

subgroup

index

Q₁₆×#10596831

Subgroups — #34011924

Conjugacy classes of subgroups

Complements

Maximal subgroups