Group #34000828 — subgroups

Subgroups — #34000828

Order 18688.

138 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 52 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	C₂²⋊#14556641	C₂	1	2	yes	maximal
2	C₂×#17988405	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556641	—	2	2	no	—
4	C₂²×#11924003	C₂²	1	1	yes	characteristic
5	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
6	#17988405	—	2	2	no	—
7	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
8	#14556641	—	4	2	no	—
9	C₂²×#10596831	C₈	1	1	yes	characteristic
10	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
11	C₂×#11924003	—	4	1	no	—
12	C₂×#11924003	D₄	1	2	yes	—
13	C₈×Dic₇₃	—	2	1	no	—
14	C₈×Dic₇₃	—	2	1	no	—
15	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₇₃	—	2	1	no	—
17	C₂×C₁₆×C₇₃	—	4	1	no	—
18	C₂²×Dic₇₃	C₁₆	1	1	yes	characteristic
19	C₂²×Dic₇₃	C₁₆	1	1	yes	characteristic
20	C₂×#10596831	—	2	1	no	—
21	C₂×#10596831	—	4	1	no	—
22	C₄×Dic₇₃	—	2	1	no	—
23	C₄×Dic₇₃	—	2	1	no	—
24	#11924003	—	2	2	no	—
25	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₈×C₇₃	—	2	1	no	—
27	C₂×C₈×C₇₃	—	2	1	no	—
28	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
30	C₁₆×C₇₃	—	4	1	no	—
31	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
32	C₂×C₄×C₇₃	—	2	1	no	—
33	C₂×C₄×C₇₃	—	2	1	no	—
34	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
35	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
36	C₂×Dic₇₃	—	2	1	no	—
37	C₂×Dic₇₃	—	2	1	no	—
38	C₂×Dic₇₃	—	2	1	no	—
39	C₂×Dic₇₃	—	4	1	no	—
40	C₂×Dic₇₃	C₃₂	1	2	yes	complemented
41	C₈×C₇₃	—	2	1	no	—
42	C₈×C₇₃	#99	1	1	yes	characteristic
43	C₈×C₇₃	#101	1	1	yes	characteristic
44	#10596831	—	4	1	no	—
45	C₂²×C₇₃	—	2	1	no	—
46	C₂²×C₇₃	—	2	1	no	—
47	C₂²×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
48	C₂²×C₇₃	C₂×C₃₂	1	1	yes	characteristic
49	C₂²×C₇₃	M₆₄	1	1	yes	characteristic
50	C₄×C₇₃	—	2	1	no	—
51	C₄×C₇₃	#324	1	1	yes	characteristic
52	C₄×C₇₃	#349	1	1	yes	characteristic
53	Dic₇₃	—	2	2	no	—
54	Dic₇₃	—	4	1	no	—
55	#7362	—	73	1	no	maximal, Sylow
56	C₂×C₇₃	—	2	1	no	—
57	C₂×C₇₃	—	2	1	no	—
58	C₂×C₇₃	#1322	1	1	yes	characteristic
59	C₂×C₇₃	C₂²⋊C₃₂	1	1	yes	characteristic
60	C₂×C₇₃	#1400	1	1	yes	characteristic
61	C₂×C₄⋊C₁₆	—	73	1	no	—
62	C₂²⋊C₃₂	—	73	2	no	—
63	C₇₃	#7362	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₁₆	—	73	1	no	—
65	C₂²×C₁₆	#64408	1	1	yes	characteristic
66	C₂×C₄×C₈	—	73	1	no	—
67	C₂×C₃₂	—	146	2	no	—
68	C₄⋊C₁₆	—	146	2	no	—
69	C₂²×C₈	—	73	1	no	—
70	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
71	C₂×C₄²	—	73	1	no	—
72	C₂×C₁₆	—	2	1	no	—
73	C₂×C₁₆	—	4	1	no	—
74	C₂×C₁₆	—	292	1	no	—
75	C₂×C₁₆	—	73	2	no	—
76	C₄×C₈	—	146	1	no	—
77	C₄×C₈	—	146	1	no	—
78	C₃₂	—	292	2	no	—
79	C₂²×C₄	—	73	1	no	—
80	C₂²×C₄	—	73	1	no	—
81	C₂²×C₄	#11924036	1	1	yes	characteristic
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	292	1	no	—
85	C₂×C₈	—	146	1	no	—
86	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
87	C₂×C₈	#11924038	1	1	yes	characteristic
88	C₄²	—	146	1	no	—
89	C₄²	—	146	1	no	—
90	C₁₆	—	4	1	no	—
91	C₁₆	—	146	2	no	—
92	C₂³	#14556664	1	1	yes	characteristic
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	292	1	no	—
96	C₂×C₄	—	73	2	no	—
97	C₂×C₄	—	146	1	no	—
98	C₂×C₄	—	146	1	no	—
99	C₂×C₄	—	146	1	no	—
100	C₂×C₄	#14556648	1	1	yes	characteristic
101	C₂×C₄	#14556666	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	292	1	no	—
104	C₈	#14556649	1	1	yes	characteristic
105	C₈	#14556652	1	1	yes	characteristic
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#17989482	1	1	yes	characteristic
109	C₂²	#17989518	1	1	yes	characteristic
110	C₂²	#17989520	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	292	1	no	—
113	C₄	—	146	2	no	—
114	C₄	#17989433	1	1	yes	characteristic
115	C₄	#17989484	1	1	yes	characteristic
116	C₂	—	2	1	no	—
117	C₂	—	2	1	no	—
118	C₂	#22624868	1	1	yes	characteristic
119	C₂	#22625045	1	1	yes	characteristic
120	C₂	#22625047	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²⋊#14556641

C₂

yes

maximal

C₂×#17988405

C₂

yes

maximal, characteristic

C₂×#14556641

—

C₂²×#11924003

C₂²

yes

characteristic

C₂×C₈×Dic₇₃

C₄

yes

characteristic

#17988405

—

C₂²×C₁₆×C₇₃

C₄

yes

characteristic

#14556641

—

C₂²×#10596831

C₈

yes

characteristic

C₂×C₄×Dic₇₃

C₈

yes

characteristic

C₂×#11924003

—

C₂×#11924003

D₄

yes

—

C₈×Dic₇₃

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂²×Dic₇₃

C₁₆

yes

characteristic

C₂²×Dic₇₃

C₁₆

yes

characteristic

C₂×#10596831

—

C₂×#10596831

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

M₁₆

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

M₃₂

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₃₂

yes

complemented

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²⋊C₁₆

yes

characteristic

C₂²×C₇₃

C₂×C₃₂

yes

characteristic

C₂²×C₇₃

M₆₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#7362

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1322

yes

characteristic

C₂×C₇₃

C₂²⋊C₃₂

yes

characteristic

C₂×C₇₃

#1400

yes

characteristic

C₂×C₄⋊C₁₆

—

C₂²⋊C₃₂

—

C₇₃

#7362

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

#64408

yes

characteristic

C₂×C₄×C₈

—

C₂×C₃₂

—

146

—

C₄⋊C₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

C₄×C₈

—

146

—

C₄×C₈

—

146

—

C₃₂

—

292

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#11924036

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

292

—

C₂×C₈

—

146

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

146

—

C₄²

—

146

—

C₁₆

—

C₁₆

—

146

—

C₂³

#14556664

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

—

146

—

C₂×C₄

—

146

—

100

C₂×C₄

#14556648

yes

characteristic

101

C₂×C₄

#14556666

yes

characteristic

102

C₈

—

103

C₈

—

292

—

104

C₈

#14556649

yes

characteristic

105

C₈

#14556652

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

#17989482

yes

characteristic

109

C₂²

#17989518

yes

characteristic

110

C₂²

#17989520

yes

characteristic

111

C₄

—

112

C₄

—

292

—

113

C₄

—

146

—

114

C₄

#17989433

yes

characteristic

115

C₄

#17989484

yes

characteristic

116

C₂

—

117

C₂

—

118

C₂

#22624868

yes

characteristic

119

C₂

#22625045

yes

characteristic

120

C₂

#22625047

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊#64408
commutator	C₂²×C₇₃	C₂×C₃₂
Frattini	C₂²×C₈	C₂×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂²⋊C₁₆
prime radical	C₂³×C₇₃	C₂×C₁₆
cosocle	C₂²×#11924003	C₂²
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	#7362
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₈

C₂²⋊#64408

commutator

C₂²×C₇₃

C₂×C₃₂

Frattini

C₂²×C₈

C₂×#64408

Fitting

C₂²×C₁₆×C₇₃

C₄

socle

C₂²×C₇₃

C₂²⋊C₁₆

prime radical

C₂³×C₇₃

C₂×C₁₆

C₂²

C₂²×C₁₆

C₇₃

G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
40. C₂×Dic₇₃	C₃₂	semidirect
63. C₇₃	#7362	semidirect

H ≅ G/N

type

40. C₂×Dic₇₃

C₃₂

semidirect

63. C₇₃

#7362

semidirect

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

subgroup	index
C₂²⋊#14556641	2
C₂×#17988405	2
#7362	73

subgroup

index

C₂²⋊#14556641

C₂×#17988405

#7362

Subgroups — #34000828

Conjugacy classes of subgroups

Complements

Maximal subgroups