Group #34000014 — subgroups

Subgroups — #34000014

Order 18688.

146 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 58 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	#22624960	C₂	1	1	yes	maximal, characteristic
2	C₈×#11924031	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×#10596831	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11924031	—	2	1	no	—
5	C₄×#11924031	—	2	1	no	—
6	Q₈×#10596831	—	2	1	no	—
7	#17988340	C₄	1	1	yes	characteristic
8	Q₈⋊#10596831	—	2	1	no	—
9	C₈×#10596831	C₂²	1	1	yes	characteristic
10	C₄×Q₁₆×C₇₃	C₄	1	1	yes	characteristic
11	C₂×#11924031	—	2	2	no	—
12	C₂×#11924031	—	4	1	no	—
13	C₄⋊#10596831	—	2	1	no	—
14	C₄⋊#10596831	—	4	1	no	—
15	C₄×#10596831	—	4	1	no	—
16	C₄×#10596831	D₄	1	1	yes	characteristic
17	C₄×#10596831	D₄	1	1	yes	characteristic
18	C₂×Q₁₆×C₇₃	C₈	1	1	yes	characteristic
19	C₄×Q₈×C₇₃	—	2	1	no	—
20	#107×C₇₃	C₈	1	1	yes	characteristic
21	#103×C₇₃	—	2	1	no	—
22	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
23	#11924031	—	2	4	no	—
24	#11924031	—	8	1	no	—
25	C₂×#10596831	—	2	1	no	—
26	C₂×#10596831	—	4	1	no	—
27	C₂×#10596831	D₈	1	1	yes	characteristic
28	C₂×#10596831	SD₁₆	1	1	yes	characteristic
29	C₂×Q₈×C₇₃	—	2	1	no	—
30	Q₁₆×C₇₃	—	2	1	no	—
31	Q₁₆×C₇₃	C₁₆	1	2	yes	complemented
32	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
34	C₄⋊C₄×C₇₃	—	2	1	no	—
35	C₄⋊C₄×C₇₃	—	4	1	no	—
36	C₄²×C₇₃	—	4	1	no	—
37	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
38	Q₈×C₇₃	—	2	2	no	—
39	Q₈×C₇₃	—	4	1	no	—
40	C₂×C₄×C₇₃	—	4	1	no	—
41	C₂×C₄×C₇₃	—	4	1	no	—
42	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
43	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
44	C₂×C₄×C₇₃	#104	1	1	yes	characteristic
45	C₈×C₇₃	—	2	1	no	—
46	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
47	C₈×C₇₃	M₃₂	1	1	yes	characteristic
48	#10596831	—	2	2	no	—
49	#10596831	—	2	1	no	—
50	#10596831	—	8	1	no	—
51	#10596831	SD₃₂	1	1	yes	characteristic
52	#10596831	Q₃₂	1	1	yes	characteristic
53	C₂²×C₇₃	D₄⋊C₈	1	1	yes	characteristic
54	C₄×C₇₃	—	2	1	no	—
55	C₄×C₇₃	—	4	2	no	—
56	C₄×C₇₃	—	8	1	no	—
57	C₄×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
58	C₄×C₇₃	#350	1	1	yes	characteristic
59	C₄×C₇₃	#358	1	1	yes	characteristic
60	C₄×C₇₃	#359	1	1	yes	characteristic
61	#7069	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	D₄⋊C₁₆	1	1	yes	characteristic
63	C₂×C₇₃	#1332	1	1	yes	characteristic
64	C₂×C₇₃	#1335	1	1	yes	characteristic
65	C₈×Q₁₆	—	73	1	no	—
66	#1370	—	73	1	no	—
67	C₈×C₁₆	—	73	1	no	—
68	C₇₃	#7069	1	1	yes	characteristic, complemented, Sylow
69	C₈×Q₈	—	146	1	no	—
70	C₄×Q₁₆	#64408	1	1	yes	characteristic
71	C₄⋊C₁₆	—	146	1	no	—
72	C₄×C₁₆	—	146	1	no	—
73	#335	—	73	1	no	—
74	Q₈⋊C₈	—	146	1	no	—
75	C₈²	—	73	1	no	—
76	C₂×Q₁₆	#10596833	1	1	yes	characteristic
77	C₄×Q₈	—	2	1	no	—
78	C₂×C₁₆	—	292	1	no	—
79	C₂×C₁₆	—	146	2	no	—
80	#107	#10596833	1	1	yes	characteristic
81	C₄⋊C₈	—	292	1	no	—
82	C₄⋊C₈	—	146	1	no	—
83	#103	—	2	1	no	—
84	C₄×C₈	—	292	1	no	—
85	C₄×C₈	—	73	1	no	—
86	C₄×C₈	—	73	1	no	—
87	C₄×C₈	C₂×#64408	1	1	yes	characteristic
88	C₂×Q₈	—	2	1	no	—
89	Q₁₆	—	2	1	no	—
90	Q₁₆	#11924031	1	2	yes	complemented
91	C₂×C₈	—	292	1	no	—
92	C₂×C₈	—	73	1	no	—
93	C₂×C₈	—	73	1	no	—
94	C₂×C₈	—	146	1	no	—
95	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
96	C₂×C₈	#11924037	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄²	—	4	1	no	—
100	C₄²	C₂²⋊#64408	1	1	yes	characteristic
101	C₁₆	—	584	1	no	—
102	C₁₆	—	146	4	no	—
103	Q₈	—	2	2	no	—
104	Q₈	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
108	C₂×C₄	#14556653	1	1	yes	characteristic
109	C₂×C₄	#14556655	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	584	1	no	—
112	C₈	—	73	1	no	—
113	C₈	—	73	1	no	—
114	C₈	—	146	2	no	—
115	C₈	—	146	1	no	—
116	C₈	C₂×#11924031	1	1	yes	characteristic
117	C₈	#14556665	1	1	yes	characteristic
118	C₂²	D₄⋊#10596833	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	4	2	no	—
121	C₄	—	8	1	no	—
122	C₄	C₂²⋊#11924031	1	1	yes	characteristic
123	C₄	#17989485	1	1	yes	characteristic
124	C₄	#17989500	1	1	yes	characteristic
125	C₄	#17989502	1	1	yes	characteristic
126	C₂	D₄⋊#11924031	1	1	yes	characteristic
127	C₂	#22624887	1	1	yes	characteristic
128	C₂	#22624893	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22624960

C₂

yes

maximal, characteristic

C₈×#11924031

C₂

yes

maximal, characteristic

Q₁₆×#10596831

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

characteristic

Q₈⋊#10596831

—

C₈×#10596831

C₂²

yes

characteristic

C₄×Q₁₆×C₇₃

C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#10596831

D₄

yes

characteristic

C₂×Q₁₆×C₇₃

C₈

yes

characteristic

C₄×Q₈×C₇₃

—

#107×C₇₃

C₈

yes

characteristic

#103×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

—

D₈

yes

characteristic

C₂×#10596831

SD₁₆

yes

characteristic

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

C₁₆

yes

complemented

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

M₁₆

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₈×C₇₃

—

C₈×C₇₃

C₂×C₁₆

yes

characteristic

C₈×C₇₃

M₃₂

yes

—

—

—

SD₃₂

yes

characteristic

#10596831

Q₃₂

yes

characteristic

C₂²×C₇₃

D₄⋊C₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊C₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

D₄⋊C₁₆

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₈×Q₁₆

—

#1370

—

C₈×C₁₆

—

C₇₃

#7069

yes

characteristic, complemented, Sylow

C₈×Q₈

—

146

—

C₄×Q₁₆

#64408

yes

characteristic

C₄⋊C₁₆

—

146

—

C₄×C₁₆

—

146

—

#335

—

Q₈⋊C₈

—

146

—

C₈²

—

C₂×Q₁₆

#10596833

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

146

—

#107

#10596833

yes

characteristic

C₄⋊C₈

—

292

—

C₄⋊C₈

—

146

—

#103

—

C₄×C₈

—

292

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

#11924031

yes

complemented

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

100

C₄²

C₂²⋊#64408

yes

characteristic

101

C₁₆

—

584

—

102

C₁₆

—

146

—

103

Q₈

—

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂²⋊#10596833

yes

characteristic

108

C₂×C₄

#14556653

yes

characteristic

109

C₂×C₄

#14556655

yes

characteristic

110

C₈

—

111

C₈

—

584

—

112

C₈

—

113

C₈

—

114

C₈

—

146

—

115

C₈

—

146

—

116

C₈

C₂×#11924031

yes

characteristic

117

C₈

#14556665

yes

characteristic

118

C₂²

D₄⋊#10596833

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

C₂²⋊#11924031

yes

characteristic

123

C₄

#17989485

yes

characteristic

124

C₄

#17989500

yes

characteristic

125

C₄

#17989502

yes

characteristic

126

C₂

D₄⋊#11924031

yes

characteristic

127

C₂

#22624887

yes

characteristic

128

C₂

#22624893

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556653
commutator	C₈×C₇₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×#64408
Fitting	C₄×Q₁₆×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	D₄⋊C₈
cosocle	C₈×#10596831	C₂²
hypercenter	C₄×Q₁₆	#64408
hypocenter	C₇₃	#7069
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

#14556653

commutator

C₈×C₇₃

C₂×C₁₆

Frattini

C₄×C₈

C₂×#64408

Fitting

C₄×Q₁₆×C₇₃

C₄

socle, prime radical

C₂²×C₇₃

D₄⋊C₈

C₂²

C₄×Q₁₆

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

N	H ≅ G/N	type
31. Q₁₆×C₇₃	C₁₆	semidirect
68. C₇₃	#7069	semidirect
90. Q₁₆	#11924031	semidirect

H ≅ G/N

type

31. Q₁₆×C₇₃

C₁₆

semidirect

68. C₇₃

#7069

semidirect

90. Q₁₆

#11924031

semidirect

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

subgroup	index
#22624960	2
C₈×#11924031	2
Q₁₆×#10596831	2
#7069	73

subgroup

index

#22624960

C₈×#11924031

Q₁₆×#10596831

#7069

Subgroups — #34000014

Conjugacy classes of subgroups

Complements

Maximal subgroups