Group #34034428 — subgroups

Subgroups — #34034428

Order 18688.

206 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 109 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	1	yes	maximal, characteristic
2	C₂²×#14556681	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#14556681	C₂	1	4	yes	maximal, complemented
4	#22626140	C₂	1	1	yes	maximal, characteristic
5	#17989695	—	2	2	no	—
6	C₂×#14556681	—	2	2	no	—
7	C₂×#14556681	C₂²	1	2	yes	—
8	C₂×#14556681	C₂²	1	2	yes	—
9	C₂×#14556665	C₄	1	1	yes	characteristic
10	C₂²×#11924031	C₂²	1	1	yes	characteristic
11	#17989475	C₄	1	2	yes	—
12	C₄×#11924031	C₂²	1	2	yes	—
13	#17988543	C₄	1	1	yes	characteristic
14	#14556681	—	2	2	no	—
15	#14556681	—	2	2	no	—
16	#14556665	—	2	2	no	—
17	C₂×#11924031	C₂×C₄	1	2	yes	—
18	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
19	C₂×#11924031	D₄	1	2	yes	—
20	C₂×#11924031	Q₈	1	2	yes	—
21	C₂×#11924031	C₂³	1	1	yes	characteristic
22	C₂×#11924012	C₈	1	1	yes	characteristic
23	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
24	#14556502	C₈	1	2	yes	—
25	C₄×#10596831	C₂×C₄	1	2	yes	—
26	#117×C₇₃	C₈	1	1	yes	characteristic
27	#11924031	—	2	2	no	—
28	#11924031	C₄⋊C₄	1	2	yes	—
29	#11924031	C₂×D₄	1	1	yes	characteristic
30	#11924031	C₂×Q₈	1	1	yes	characteristic
31	#11924012	—	2	2	no	—
32	C₂×#10596831	C₄⋊C₄	1	2	yes	—
33	C₂×#10596831	C₂×C₈	1	2	yes	—
34	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
35	C₂×#10596831	M₁₆	1	2	yes	—
36	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	C₁₆	1	2	yes	—
39	C₂²⋊C₄×C₇₃	C₁₆	1	2	yes	—
40	C₄²×C₇₃	C₂×C₈	1	2	yes	—
41	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₇₃	C₂×C₁₆	1	4	yes	—
44	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
46	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
47	#10596831	—	2	2	no	—
48	#10596831	C₄⋊C₈	1	2	yes	—
49	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
50	#10596831	C₂×M₁₆	1	1	yes	characteristic
51	C₂²×C₇₃	—	2	2	no	—
52	C₂²×C₇₃	C₄⋊C₁₆	1	2	yes	—
53	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
54	C₄×C₇₃	—	2	4	no	—
55	C₄×C₇₃	C₄⋊C₁₆	1	2	yes	—
56	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
57	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
58	#13628	—	73	1	no	maximal, Sylow
59	C₂×C₇₃	—	2	2	no	—
60	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
61	C₂×C₇₃	#2258	1	2	yes	—
62	C₂×M₆₄	—	73	1	no	—
63	C₂²×C₃₂	—	73	1	no	—
64	#2108	—	73	1	no	—
65	C₄⋊C₃₂	—	73	4	no	—
66	C₇₃	#13628	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₃₂	—	73	1	no	—
68	C₂²×C₁₆	—	73	1	no	—
69	#405	—	73	1	no	—
70	M₆₄	—	146	2	no	—
71	C₂×C₃₂	—	73	2	no	—
72	C₂×C₃₂	—	73	2	no	—
73	C₂×C₃₂	—	146	2	no	—
74	#346	—	73	2	no	—
75	C₄×C₁₆	—	73	2	no	—
76	C₂×M₁₆	—	73	1	no	—
77	C₂²×C₈	—	73	1	no	—
78	#117	#10596842	1	1	yes	characteristic
79	M₃₂	—	146	2	no	—
80	C₂×C₁₆	—	73	1	no	—
81	C₂×C₁₆	—	73	1	no	—
82	C₂×C₁₆	—	73	2	no	—
83	C₂×C₁₆	—	73	2	no	—
84	C₂×C₁₆	—	73	2	no	—
85	#97	—	73	2	no	—
86	C₄×C₈	—	73	2	no	—
87	C₃₂	—	146	2	no	—
88	C₃₂	—	146	2	no	—
89	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
90	M₁₆	—	146	2	no	—
91	C₂×C₈	—	73	2	no	—
92	C₂×C₈	—	73	2	no	—
93	C₂×C₈	—	73	1	no	—
94	C₂×C₈	—	73	1	no	—
95	C₂×C₈	—	73	2	no	—
96	C₄⋊C₄	#11924040	1	2	yes	—
97	C₂²⋊C₄	#11924040	1	2	yes	—
98	C₄²	C₂×#10596842	1	2	yes	—
99	C₁₆	—	73	2	no	—
100	C₁₆	—	73	1	no	—
101	C₁₆	—	73	1	no	—
102	C₁₆	—	146	2	no	—
103	C₂³	C₂×#11924040	1	1	yes	characteristic
104	C₂×C₄	C₄⋊#10596842	1	2	yes	—
105	C₂×C₄	C₂×#11924040	1	4	yes	—
106	C₂×C₄	C₂×#11924040	1	1	yes	characteristic
107	C₂×C₄	#14556688	1	2	yes	—
108	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
109	C₈	—	73	1	no	—
110	C₈	—	73	1	no	—
111	C₈	—	73	2	no	—
112	C₈	—	146	2	no	—
113	C₂²	—	2	2	no	—
114	C₂²	C₄⋊#11924040	1	2	yes	—
115	C₂²	C₂²×#11924040	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	C₄⋊#11924040	1	2	yes	—
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	#22627164	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989695

C₂

yes

maximal, characteristic

C₂²×#14556681

C₂

yes

maximal, characteristic

C₄⋊#14556681

C₂

yes

maximal, complemented

#22626140

C₂

yes

maximal, characteristic

#17989695

—

C₂×#14556681

—

C₂×#14556681

C₂²

yes

—

C₂×#14556681

C₂²

yes

—

C₂×#14556665

C₄

yes

characteristic

C₂²×#11924031

C₂²

yes

characteristic

#17989475

C₄

yes

—

C₄×#11924031

C₂²

yes

—

#17988543

C₄

yes

—

—

—

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

D₄

yes

—

C₂×#11924031

Q₈

yes

—

C₂×#11924031

C₂³

yes

characteristic

C₂×#11924012

C₈

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

characteristic

#14556502

C₈

yes

—

C₄×#10596831

C₂×C₄

yes

—

#117×C₇₃

C₈

yes

characteristic

#11924031

—

#11924031

C₄⋊C₄

yes

—

#11924031

C₂×D₄

yes

characteristic

#11924031

C₂×Q₈

yes

characteristic

#11924012

—

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

C₂×C₈

yes

characteristic

C₄⋊C₄×C₇₃

C₁₆

yes

—

C₂²⋊C₄×C₇₃

C₁₆

yes

—

C₄²×C₇₃

C₂×C₈

yes

—

C₂³×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

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

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

#13628

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₇₃

#2258

yes

—

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

—

C₄⋊C₃₂

—

C₇₃

#13628

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

M₆₄

—

146

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

146

—

#346

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

#10596842

yes

characteristic

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₃₂

—

146

—

C₃₂

—

146

—

C₂²×C₄

C₂×#10596842

yes

characteristic

M₁₆

—

146

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

#11924040

yes

—

C₂²⋊C₄

#11924040

yes

—

C₄²

C₂×#10596842

yes

—

C₁₆

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

146

—

103

C₂³

C₂×#11924040

yes

characteristic

104

C₂×C₄

C₄⋊#10596842

yes

—

105

C₂×C₄

C₂×#11924040

yes

—

106

C₂×C₄

C₂×#11924040

yes

characteristic

107

C₂×C₄

#14556688

yes

—

108

C₂×C₄

C₂²×#10596842

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

146

—

113

C₂²

—

114

C₂²

C₄⋊#11924040

yes

—

115

C₂²

C₂²×#11924040

yes

characteristic

116

C₄

—

117

C₄

C₄⋊#11924040

yes

—

118

C₄

C₂×C₄⋊#10596842

yes

characteristic

119

C₄

C₂×#14556688

yes

characteristic

120

C₂

—

121

C₂

C₂×C₄⋊#11924040

yes

characteristic

122

C₂

#22627164

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting	#117×C₇₃	C₈
prime radical	C₂³×C₇₃	C₂×C₁₆
cosocle	C₂×#11924031	C₂³
hypercenter	#117	#10596842
hypocenter	C₇₃	#13628
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #117

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10596842

commutator, socle

C₂²×C₇₃

C₂²×C₁₆

Fitting

#117×C₇₃

C₈

prime radical

C₂³×C₇₃

C₂×C₁₆

C₂³

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ #117

N	H ≅ G/N	type
3. C₄⋊#14556681	C₂	semidirect
66. C₇₃	#13628	semidirect

H ≅ G/N

type

3. C₄⋊#14556681

C₂

semidirect

66. C₇₃

#13628

semidirect

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

subgroup	index
C₂×#17989695	2
C₂²×#14556681	2
C₄⋊#14556681	2
#22626140	2
#13628	73

subgroup

index

Subgroups — #34034428

Conjugacy classes of subgroups

Complements

Maximal subgroups