Group #34034425 — subgroups

Subgroups — #34034425

Order 18688.

218 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×#17989695	C₂	1	2	yes	maximal
2	C₄⋊#14556681	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×#11924031	C₂	1	1	yes	maximal, characteristic
4	#17989695	—	2	4	no	—
5	C₂×#14556681	C₂²	1	4	yes	—
6	C₂²×#11924031	C₄	1	1	yes	characteristic
7	C₂²×#11924031	C₂²	1	1	yes	characteristic
8	C₄×#11924031	C₄	1	2	yes	—
9	C₄×#11924031	C₂²	1	2	yes	—
10	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
11	#14556681	—	2	4	no	—
12	C₂×#11924031	—	2	2	no	—
13	C₂×#11924031	C₂×C₄	1	2	yes	—
14	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
15	C₂×#11924031	D₄	1	2	yes	—
16	C₂×#11924031	Q₈	1	2	yes	—
17	C₂×#11924031	C₂³	1	1	yes	characteristic
18	C₂²×#10596831	C₈	1	1	yes	characteristic
19	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
20	C₄×#10596831	C₈	1	2	yes	—
21	C₄×#10596831	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₇₃	C₈	1	1	yes	characteristic
23	#11924031	—	2	2	no	—
24	#11924031	C₄⋊C₄	1	2	yes	—
25	#11924031	C₂×D₄	1	1	yes	characteristic
26	#11924031	C₂×Q₈	1	1	yes	characteristic
27	C₂×#10596831	—	2	2	no	—
28	C₂×#10596831	C₄⋊C₄	1	2	yes	—
29	C₂×#10596831	C₂×C₈	1	2	yes	—
30	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
31	C₂×#10596831	M₁₆	1	2	yes	—
32	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	C₁₆	1	2	yes	—
34	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₇₃	C₁₆	1	1	yes	characteristic
36	C₄²×C₇₃	C₁₆	1	1	yes	characteristic
37	C₄²×C₇₃	C₂×C₈	1	2	yes	—
38	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₇₃	—	2	4	no	—
40	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
41	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
42	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
44	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
45	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
46	#10596831	—	2	2	no	—
47	#10596831	C₄⋊C₈	1	2	yes	—
48	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
49	#10596831	C₂×M₁₆	1	1	yes	characteristic
50	C₂²×C₇₃	—	2	2	no	—
51	C₂²×C₇₃	C₄⋊C₁₆	1	2	yes	—
52	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
53	C₄×C₇₃	—	2	2	no	—
54	C₄×C₇₃	C₄⋊C₁₆	1	2	yes	—
55	C₄×C₇₃	M₆₄	1	4	yes	complemented
56	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
57	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
58	#13627	—	73	1	no	maximal, Sylow
59	C₂×C₇₃	—	2	2	no	—
60	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
61	C₂×C₇₃	C₂×M₆₄	1	2	yes	—
62	C₂×M₆₄	—	73	2	no	—
63	C₂×C₄×C₁₆	—	73	1	no	—
64	C₄⋊C₃₂	—	73	4	no	—
65	C₇₃	#13627	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₁₆	—	73	1	no	—
67	C₂²×C₁₆	—	73	1	no	—
68	C₂×C₄×C₈	—	73	1	no	—
69	M₆₄	—	146	4	no	—
70	C₂×C₃₂	—	73	4	no	—
71	C₄×C₁₆	—	73	2	no	—
72	C₄×C₁₆	—	73	2	no	—
73	C₂²×C₈	—	73	1	no	—
74	C₂²×C₈	—	73	1	no	—
75	C₂×C₄²	#10596842	1	1	yes	characteristic
76	C₂×C₁₆	—	73	1	no	—
77	C₂×C₁₆	—	73	1	no	—
78	C₂×C₁₆	—	73	2	no	—
79	C₂×C₁₆	—	73	2	no	—
80	C₂×C₁₆	—	73	2	no	—
81	C₂×C₁₆	—	146	2	no	—
82	C₄×C₈	—	73	2	no	—
83	C₄×C₈	—	73	2	no	—
84	C₃₂	—	146	4	no	—
85	C₂²×C₄	#11924040	1	2	yes	—
86	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
87	C₂×C₈	—	73	2	no	—
88	C₂×C₈	—	73	2	no	—
89	C₂×C₈	—	73	1	no	—
90	C₂×C₈	—	73	1	no	—
91	C₂×C₈	—	73	2	no	—
92	C₂×C₈	—	146	2	no	—
93	C₄²	#11924040	1	1	yes	characteristic
94	C₄²	#11924040	1	1	yes	characteristic
95	C₄²	C₂×#10596842	1	2	yes	—
96	C₁₆	—	73	2	no	—
97	C₁₆	—	73	1	no	—
98	C₁₆	—	73	1	no	—
99	C₁₆	—	146	2	no	—
100	C₂³	C₂×#11924040	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	C₄⋊#10596842	1	2	yes	—
103	C₂×C₄	C₂×#11924040	1	2	yes	—
104	C₂×C₄	C₂×#11924040	1	1	yes	characteristic
105	C₂×C₄	C₂×#11924040	1	2	yes	—
106	C₂×C₄	#14556688	1	2	yes	—
107	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
108	C₈	—	73	1	no	—
109	C₈	—	73	1	no	—
110	C₈	—	73	2	no	—
111	C₈	—	146	2	no	—
112	C₂²	—	2	2	no	—
113	C₂²	C₄⋊#11924040	1	2	yes	—
114	C₂²	C₂²×#11924040	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	C₄⋊#11924040	1	2	yes	—
117	C₄	#17989695	1	4	yes	complemented
118	C₄	C₂×C₄⋊#10596842	1	1	yes	characteristic
119	C₄	C₂×#14556688	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₄⋊#11924040	1	1	yes	characteristic
122	C₂	C₂×#17989695	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989695

C₂

yes

maximal

C₄⋊#14556681

C₂

yes

maximal, complemented

C₂×C₄×#11924031

C₂

yes

maximal, characteristic

#17989695

—

C₂×#14556681

C₂²

yes

—

C₂²×#11924031

C₄

yes

characteristic

C₂²×#11924031

C₂²

yes

characteristic

C₄×#11924031

C₄

yes

—

C₄×#11924031

C₂²

yes

—

C₂×C₄×#10596831

C₄

yes

—

—

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

D₄

yes

—

C₂×#11924031

Q₈

yes

—

C₂×#11924031

C₂³

yes

characteristic

C₂²×#10596831

C₈

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₈

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₈

yes

characteristic

#11924031

—

#11924031

C₄⋊C₄

yes

—

#11924031

C₂×D₄

yes

characteristic

#11924031

C₂×Q₈

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

C₂×C₈

yes

characteristic

C₂×#10596831

M₁₆

yes

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₁₆

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

C₁₆

yes

characteristic

C₄²×C₇₃

C₁₆

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

—

C₂³×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

M₃₂

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

#10596831

—

#10596831

C₄⋊C₈

yes

—

#10596831

C₂×C₄⋊C₄

yes

characteristic

#10596831

C₂×M₁₆

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

C₄⋊C₁₆

yes

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₄⋊C₁₆

yes

—

C₄×C₇₃

M₆₄

yes

complemented

C₄×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

#13627

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₇₃

C₂×M₆₄

yes

—

C₂×M₆₄

—

C₂×C₄×C₁₆

—

C₄⋊C₃₂

—

C₇₃

#13627

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

M₆₄

—

146

—

C₂×C₃₂

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

#10596842

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

C₃₂

—

146

—

C₂²×C₄

#11924040

yes

—

C₂²×C₄

C₂×#10596842

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₄²

yes

C₄²

yes

C₄²

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

146

—

100

C₂³

C₂×#11924040

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

C₄⋊#10596842

yes

—

103

C₂×C₄

C₂×#11924040

yes

—

104

C₂×C₄

C₂×#11924040

yes

characteristic

105

C₂×C₄

C₂×#11924040

yes

—

106

C₂×C₄

#14556688

yes

—

107

C₂×C₄

C₂²×#10596842

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

146

—

112

C₂²

—

113

C₂²

C₄⋊#11924040

yes

—

114

C₂²

C₂²×#11924040

yes

characteristic

115

C₄

—

116

C₄

C₄⋊#11924040

yes

—

117

C₄

#17989695

yes

complemented

118

C₄

C₂×C₄⋊#10596842

yes

characteristic

119

C₄

C₂×#14556688

yes

characteristic

120

C₂

—

121

C₂

C₂×C₄⋊#11924040

yes

characteristic

122

C₂

C₂×#17989695

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂×C₄²×C₇₃	C₈
prime radical	C₂³×C₇₃	C₂×C₁₆
cosocle	C₂×#11924031	C₂³
hypercenter	C₂×C₄²	#10596842
hypocenter	C₇₃	#13627
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10596842

commutator, socle

C₂²×C₇₃

C₂²×C₁₆

Fitting

C₂×C₄²×C₇₃

C₈

prime radical

C₂³×C₇₃

C₂×C₁₆

C₂³

C₂×C₄²

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
2. C₄⋊#14556681	C₂	semidirect
55. C₄×C₇₃	M₆₄	semidirect
65. C₇₃	#13627	semidirect
117. C₄	#17989695	semidirect

H ≅ G/N

type

2. C₄⋊#14556681

C₂

semidirect

55. C₄×C₇₃

M₆₄

semidirect

65. C₇₃

#13627

semidirect

117. C₄

#17989695

semidirect

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

subgroup	index
C₂×#17989695	2
C₄⋊#14556681	2
C₂×C₄×#11924031	2
#13627	73

subgroup

index

Subgroups — #34034425

Conjugacy classes of subgroups

Complements

Maximal subgroups