Group #34014477 — subgroups

Subgroups — #34014477

Order 18688.

206 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 128 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×#17989519	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556641	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#14556641	C₂	1	4	yes	maximal, complemented
4	#22611528	C₂	1	1	yes	maximal, characteristic
5	#17989519	—	2	2	no	—
6	C₂×#14556641	—	2	2	no	—
7	C₂×#14556641	C₂²	1	2	yes	—
8	C₂×#14556641	C₂²	1	2	yes	—
9	C₂×#14556531	C₄	1	1	yes	characteristic
10	C₂²×#11924003	C₂²	1	1	yes	characteristic
11	#17988363	C₄	1	2	yes	—
12	C₄×#11924003	C₂²	1	2	yes	—
13	#405×C₇₃	C₄	1	1	yes	characteristic
14	#14556641	—	2	2	no	—
15	#14556641	—	2	2	no	—
16	#14556531	—	2	2	no	—
17	C₂×#11924003	C₂×C₄	1	2	yes	—
18	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
19	C₂×#11924003	D₄	1	2	yes	—
20	C₂×#11924003	Q₈	1	2	yes	—
21	C₂×#11924003	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₇₃	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	#117×C₇₃	C₈	1	1	yes	characteristic
25	#97×C₇₃	C₈	1	2	yes	—
26	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
27	#11924003	—	2	2	no	—
28	#11924003	C₄⋊C₄	1	2	yes	—
29	#11924003	C₂×D₄	1	1	yes	characteristic
30	#11924003	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
32	M₁₆×C₇₃	—	2	2	no	—
33	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
37	C₂×C₈×C₇₃	C₂²×C₄	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	C₈×C₇₃	—	2	2	no	—
48	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
49	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₇₃	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	#64408	1	1	yes	characteristic
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	#10596833	1	1	yes	characteristic
77	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
78	#117	#10596833	1	1	yes	characteristic
79	M₃₂	—	146	2	no	—
80	C₂×C₁₆	—	73	2	no	—
81	C₂×C₁₆	—	73	2	no	—
82	C₂×C₁₆	—	73	1	no	—
83	C₂×C₁₆	—	73	1	no	—
84	C₂×C₁₆	—	73	2	no	—
85	#97	#10596833	1	2	yes	—
86	C₄×C₈	C₂×#64408	1	2	yes	—
87	C₃₂	—	146	2	no	—
88	C₃₂	—	146	2	no	—
89	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	C₂×C₈	C₄⋊#64408	1	2	yes	—
92	C₂×C₈	C₂×#10596833	1	2	yes	—
93	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
94	C₂×C₈	#11924038	1	2	yes	—
95	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
96	C₄⋊C₄	#11924031	1	2	yes	—
97	C₂²⋊C₄	#11924031	1	2	yes	—
98	C₄²	C₂×#10596833	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×#11924031	1	1	yes	characteristic
104	C₂×C₄	C₄⋊#10596833	1	2	yes	—
105	C₂×C₄	C₂×#11924031	1	4	yes	—
106	C₂×C₄	C₂×#11924031	1	1	yes	characteristic
107	C₂×C₄	#14556665	1	2	yes	—
108	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	C₄⋊#10596833	1	2	yes	—
111	C₈	C₂×C₄⋊#64408	1	1	yes	characteristic
112	C₈	C₂×#11924038	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	C₄⋊#11924031	1	2	yes	—
115	C₂²	C₂²×#11924031	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	C₄⋊#11924031	1	2	yes	—
118	C₄	C₂×C₄⋊#10596833	1	1	yes	characteristic
119	C₄	C₂×#14556665	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₄⋊#11924031	1	1	yes	characteristic
122	C₂	#22626354	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989519

C₂

yes

maximal, characteristic

C₂²×#14556641

C₂

yes

maximal, characteristic

C₄⋊#14556641

C₂

yes

maximal, complemented

#22611528

C₂

yes

maximal, characteristic

#17989519

—

C₂×#14556641

—

C₂×#14556641

C₂²

yes

—

C₂×#14556641

C₂²

yes

—

C₂×#14556531

C₄

yes

characteristic

C₂²×#11924003

C₂²

yes

characteristic

#17988363

C₄

yes

—

C₄×#11924003

C₂²

yes

—

#405×C₇₃

C₄

yes

—

—

—

C₂×C₄

yes

—

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

D₄

yes

—

C₂×#11924003

Q₈

yes

—

C₂×#11924003

C₂³

yes

characteristic

C₂×M₁₆×C₇₃

C₈

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

C₈

yes

characteristic

#97×C₇₃

C₈

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

#11924003

—

#11924003

C₄⋊C₄

yes

—

#11924003

C₂×D₄

yes

characteristic

#11924003

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

M₁₆

yes

—

C₂×C₈×C₇₃

C₂²×C₄

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

C₈×C₇₃

—

C₈×C₇₃

C₄⋊C₈

yes

—

C₈×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₃

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

#64408

yes

characteristic

M₆₄

—

146

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

146

—

#346

—

C₄×C₁₆

—

C₂×M₁₆

yes

C₂²×C₈

yes

yes

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

#10596833

yes

—

C₄×C₈

C₂×#64408

yes

—

C₃₂

—

146

—

C₃₂

—

146

—

C₂²×C₄

C₂×#10596833

yes

characteristic

M₁₆

—

C₂×C₈

C₄⋊#64408

yes

—

C₂×C₈

C₂×#10596833

yes

—

C₂×C₈

C₂×#10596833

yes

characteristic

C₂×C₈

#11924038

yes

—

C₂×C₈

C₂²×#64408

yes

characteristic

C₄⋊C₄

#11924031

yes

—

C₂²⋊C₄

#11924031

yes

—

C₄²

C₂×#10596833

yes

—

C₁₆

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

146

—

103

C₂³

C₂×#11924031

yes

characteristic

104

C₂×C₄

C₄⋊#10596833

yes

—

105

C₂×C₄

C₂×#11924031

yes

—

106

C₂×C₄

C₂×#11924031

yes

characteristic

107

C₂×C₄

#14556665

yes

—

108

C₂×C₄

C₂²×#10596833

yes

characteristic

109

C₈

—

110

C₈

C₄⋊#10596833

yes

—

111

C₈

C₂×C₄⋊#64408

yes

characteristic

112

C₈

C₂×#11924038

yes

characteristic

113

C₂²

—

114

C₂²

C₄⋊#11924031

yes

—

115

C₂²

C₂²×#11924031

yes

characteristic

116

C₄

—

117

C₄

C₄⋊#11924031

yes

—

118

C₄

C₂×C₄⋊#10596833

yes

characteristic

119

C₄

C₂×#14556665

yes

characteristic

120

C₂

—

121

C₂

C₂×C₄⋊#11924031

yes

characteristic

122

C₂

#22626354

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting	#405×C₇₃	C₄
prime radical	C₂³×C₇₃	C₂×C₁₆
cosocle	C₂×#11924003	C₂³
hypercenter	#405	#64408
hypocenter	C₇₃	#13628
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#64408

commutator, socle

C₂²×C₇₃

C₂²×C₁₆

Fitting

#405×C₇₃

C₄

prime radical

C₂³×C₇₃

C₂×C₁₆

C₂³

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₈ ◁ #405

N	H ≅ G/N	type
3. C₄⋊#14556641	C₂	semidirect
66. C₇₃	#13628	semidirect

H ≅ G/N

type

3. C₄⋊#14556641

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×#17989519	2
C₂²×#14556641	2
C₄⋊#14556641	2
#22611528	2
#13628	73

subgroup

index

Subgroups — #34014477

Conjugacy classes of subgroups

Complements

Maximal subgroups