Group #34034387 — subgroups

Subgroups — #34034387

Order 18688.

238 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 141 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	#22627016	C₂	1	2	yes	maximal, complemented
4	C₄×#14556681	C₂	1	2	yes	maximal, complemented
5	#22626140	C₂	1	1	yes	maximal, characteristic
6	#17989695	C₄	1	4	yes	complemented
7	C₂×#14556681	C₄	1	4	yes	complemented
8	C₂×#14556681	C₂²	1	1	yes	characteristic
9	C₂×#14556681	C₂²	1	1	yes	characteristic
10	C₂×#14556681	C₂²	1	2	yes	—
11	C₂×#14556665	C₄	1	1	yes	characteristic
12	C₂²×#11924031	C₂²	1	1	yes	characteristic
13	#17989475	C₄	1	2	yes	—
14	C₄×#11924031	C₂²	1	2	yes	—
15	#17988543	C₄	1	1	yes	characteristic
16	#14556681	C₂×C₄	1	2	yes	—
17	#14556681	C₂×C₄	1	2	yes	—
18	#14556681	C₂×C₄	1	4	yes	—
19	#14556665	C₈	1	4	yes	—
20	C₂×#11924031	C₂×C₄	1	2	yes	—
21	C₂×#11924031	C₂×C₄	1	2	yes	—
22	C₂×#11924031	C₂×C₄	1	1	yes	characteristic
23	C₂×#11924031	C₂×C₄	1	2	yes	—
24	C₂×#11924031	C₂³	1	1	yes	characteristic
25	C₂×#11924012	C₈	1	1	yes	characteristic
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	#14556502	C₈	1	2	yes	—
28	C₄×#10596831	C₂×C₄	1	2	yes	—
29	#117×C₇₃	C₈	1	1	yes	characteristic
30	#11924031	C₄²	1	2	yes	—
31	#11924031	C₂×C₈	1	4	yes	—
32	#11924031	C₂²×C₄	1	1	yes	characteristic
33	#11924031	C₂²×C₄	1	1	yes	characteristic
34	#11924012	C₁₆	1	4	yes	—
35	C₂×#10596831	C₄²	1	2	yes	—
36	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
37	C₂×#10596831	C₂×C₈	1	2	yes	—
38	C₂×#10596831	C₂×C₈	1	2	yes	—
39	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₄⋊C₄×C₇₃	C₁₆	1	2	yes	—
42	C₂²⋊C₄×C₇₃	C₁₆	1	2	yes	—
43	C₄²×C₇₃	C₂×C₈	1	2	yes	—
44	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₇₃	C₂×C₁₆	1	4	yes	—
47	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
49	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
50	#10596831	C₄×C₈	1	2	yes	—
51	#10596831	C₂×C₁₆	1	4	yes	—
52	#10596831	C₂×C₄²	1	1	yes	characteristic
53	#10596831	C₂²×C₈	1	1	yes	characteristic
54	C₂²×C₇₃	—	2	2	no	—
55	C₂²×C₇₃	C₄×C₁₆	1	2	yes	—
56	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
57	C₄×C₇₃	—	2	4	no	—
58	C₄×C₇₃	C₄×C₁₆	1	2	yes	—
59	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
60	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
61	#13549	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	—	2	2	no	—
63	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
64	C₂×C₇₃	#2258	1	2	yes	—
65	C₂×M₆₄	—	73	1	no	—
66	C₂²×C₃₂	—	73	1	no	—
67	#2108	—	73	1	no	—
68	#1397	—	73	2	no	—
69	C₄×C₃₂	—	73	2	no	—
70	C₇₃	#13549	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₃₂	—	73	1	no	—
72	C₂²×C₁₆	—	73	1	no	—
73	#405	—	73	1	no	—
74	M₆₄	—	73	4	no	—
75	C₂×C₃₂	—	73	4	no	—
76	C₂×C₃₂	—	73	1	no	—
77	C₂×C₃₂	—	73	2	no	—
78	C₂×C₃₂	—	73	1	no	—
79	#346	—	73	2	no	—
80	C₄×C₁₆	—	73	2	no	—
81	C₂×M₁₆	—	73	1	no	—
82	C₂²×C₈	—	73	1	no	—
83	#117	#10596842	1	1	yes	characteristic
84	M₃₂	—	73	4	no	—
85	C₂×C₁₆	—	73	1	no	—
86	C₂×C₁₆	—	73	2	no	—
87	C₂×C₁₆	—	73	1	no	—
88	C₂×C₁₆	—	73	2	no	—
89	C₂×C₁₆	—	73	2	no	—
90	#97	—	73	2	no	—
91	C₄×C₈	—	73	2	no	—
92	C₃₂	—	73	2	no	—
93	C₃₂	—	73	2	no	—
94	C₃₂	—	73	4	no	—
95	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
96	M₁₆	—	73	4	no	—
97	C₂×C₈	—	73	2	no	—
98	C₂×C₈	—	73	2	no	—
99	C₂×C₈	—	73	2	no	—
100	C₂×C₈	—	73	1	no	—
101	C₂×C₈	—	73	1	no	—
102	C₄⋊C₄	#11924040	1	2	yes	—
103	C₂²⋊C₄	#11924040	1	2	yes	—
104	C₄²	C₂×#10596842	1	2	yes	—
105	C₁₆	—	73	2	no	—
106	C₁₆	—	73	4	no	—
107	C₁₆	—	73	1	no	—
108	C₁₆	—	73	1	no	—
109	C₂³	C₂×#11924040	1	1	yes	characteristic
110	C₂×C₄	C₄×#10596842	1	2	yes	—
111	C₂×C₄	C₂×#11924040	1	4	yes	—
112	C₂×C₄	C₂×#11924040	1	1	yes	characteristic
113	C₂×C₄	#14556686	1	2	yes	—
114	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
115	C₈	—	73	1	no	—
116	C₈	—	73	1	no	—
117	C₈	—	73	2	no	—
118	C₈	—	73	4	no	—
119	C₂²	—	2	2	no	—
120	C₂²	C₄×#11924040	1	2	yes	—
121	C₂²	C₂²×#11924040	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	C₄×#11924040	1	2	yes	—
124	C₄	C₂×C₄×#10596842	1	1	yes	characteristic
125	C₄	C₂×#14556686	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×C₄×#11924040	1	1	yes	characteristic
128	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

#22627016

C₂

yes

maximal, complemented

C₄×#14556681

C₂

yes

maximal, complemented

#22626140

C₂

yes

maximal, characteristic

#17989695

C₄

yes

complemented

C₂×#14556681

C₄

yes

complemented

C₂×#14556681

C₂²

yes

characteristic

C₂×#14556681

C₂²

yes

characteristic

C₂×#14556681

C₂²

yes

—

C₂×#14556665

C₄

yes

characteristic

C₂²×#11924031

C₂²

yes

characteristic

#17989475

C₄

yes

—

C₄×#11924031

C₂²

yes

—

#17988543

C₄

yes

characteristic

#14556681

C₂×C₄

yes

—

#14556681

C₂×C₄

yes

—

#14556681

C₂×C₄

yes

—

#14556665

C₈

yes

—

C₂×#11924031

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

characteristic

C₂×#11924031

C₂×C₄

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

C₄²

yes

—

#11924031

C₂×C₈

yes

—

#11924031

C₂²×C₄

yes

characteristic

#11924031

C₂²×C₄

yes

characteristic

#11924012

C₁₆

yes

—

C₂×#10596831

C₄²

yes

—

C₂×#10596831

C₂×C₈

yes

characteristic

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

C₂×C₈

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

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

#10596831

C₄×C₈

yes

—

#10596831

C₂×C₁₆

yes

—

#10596831

C₂×C₄²

yes

characteristic

#10596831

C₂²×C₈

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

C₂²×C₁₆

yes

characteristic

#13549

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂×C₄×C₁₆

yes

characteristic

C₂×C₇₃

#2258

yes

—

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

—

#1397

—

C₄×C₃₂

—

C₇₃

#13549

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

#346

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

#10596842

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₃₂

—

C₂²×C₄

C₂×#10596842

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₄⋊C₄

#11924040

yes

—

103

C₂²⋊C₄

#11924040

yes

—

104

C₄²

C₂×#10596842

yes

—

105

C₁₆

—

106

C₁₆

—

107

C₁₆

—

108

C₁₆

—

109

C₂³

C₂×#11924040

yes

characteristic

110

C₂×C₄

C₄×#10596842

yes

—

111

C₂×C₄

C₂×#11924040

yes

—

112

C₂×C₄

C₂×#11924040

yes

characteristic

113

C₂×C₄

#14556686

yes

—

114

C₂×C₄

C₂²×#10596842

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

C₄×#11924040

yes

—

121

C₂²

C₂²×#11924040

yes

characteristic

122

C₄

—

123

C₄

C₄×#11924040

yes

—

124

C₄

C₂×C₄×#10596842

yes

characteristic

125

C₄

C₂×#14556686

yes

characteristic

126

C₂

—

127

C₂

C₂×C₄×#11924040

yes

characteristic

128

C₂

#22627164

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator	C₂×C₇₃	C₂×C₄×C₁₆
Fitting	#117×C₇₃	C₈
socle	C₂²×C₇₃	C₂²×C₁₆
prime radical	C₂³×C₇₃	C₂×C₁₆
cosocle	C₂×#11924031	C₂³
hypercenter	#117	#10596842
hypocenter	C₇₃	#13549
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #117

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10596842

commutator

C₂×C₇₃

C₂×C₄×C₁₆

Fitting

#117×C₇₃

C₈

socle

C₂²×C₇₃

C₂²×C₁₆

prime radical

C₂³×C₇₃

C₂×C₁₆

C₂³

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ #117

N	H ≅ G/N	type
3. #22627016	C₂	semidirect
4. C₄×#14556681	C₂	semidirect
6. #17989695	C₄	semidirect
7. C₂×#14556681	C₄	semidirect
70. C₇₃	#13549	semidirect

H ≅ G/N

type

3. #22627016

C₂

semidirect

4. C₄×#14556681

C₂

semidirect

6. #17989695

C₄

semidirect

7. C₂×#14556681

C₄

semidirect

70. C₇₃

#13549

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
#22627016	2
C₄×#14556681	2
#22626140	2
#13549	73

subgroup

index

Subgroups — #34034387

Conjugacy classes of subgroups

Complements

Maximal subgroups