Group #34036758 — subgroups

Subgroups — #34036758

Order 18752.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#22627623	C₂	1	2	yes	maximal, complemented
2	#22627622	C₂	1	2	yes	maximal, complemented
3	#22627614	C₂	1	2	yes	maximal, complemented
4	#22627614	C₂	1	2	yes	maximal, complemented
5	#22627603	C₂	1	1	yes	maximal, characteristic
6	#22627601	C₂	1	2	yes	maximal
7	#22627580	C₂	1	2	yes	maximal
8	C₂×#17989781	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊Q₈×C₂₉₃	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₅₈₆	C₂²	1	2	yes	—
11	C₂×Dic₅₈₆	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₉₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₉₃	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₂₉₃	C₂²	1	2	yes	—
15	C₄⋊Dic₂₉₃	C₂²	1	1	yes	characteristic
16	#17989781	C₂²	1	2	yes	—
17	#17989781	C₂²	1	2	yes	—
18	#17989781	C₂²	1	2	yes	—
19	#17989781	C₂²	1	2	yes	—
20	#17989781	C₂²	1	2	yes	—
21	#17989781	C₂²	1	2	yes	—
22	#17989781	C₂²	1	2	yes	—
23	C₂²⋊Dic₂₉₃	C₂²	1	2	yes	—
24	C₄×Dic₂₉₃	C₂²	1	2	yes	—
25	C₄×Dic₂₉₃	C₂²	1	2	yes	—
26	C₂×Q₈×C₂₉₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₂₉₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₂₉₃	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×C₂₉₃	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₂₉₃	C₂²	1	2	yes	—
31	Dic₅₈₆	—	4	2	no	—
32	Dic₅₈₆	—	4	1	no	—
33	C₂×Dic₂₉₃	—	4	1	no	—
34	C₂×Dic₂₉₃	Q₈	1	4	yes	complemented
35	C₂×Dic₂₉₃	C₂³	1	2	yes	—
36	C₂×Dic₂₉₃	C₂³	1	2	yes	—
37	C₂×Dic₂₉₃	C₂³	1	2	yes	—
38	C₂×Dic₂₉₃	C₂³	1	2	yes	—
39	C₂³×C₂₉₃	C₂³	1	1	yes	characteristic
40	Q₈×C₂₉₃	—	4	1	no	—
41	C₂×C₄×C₂₉₃	—	4	1	no	—
42	C₂×C₄×C₂₉₃	C₂³	1	2	yes	—
43	C₂×C₄×C₂₉₃	C₂³	1	2	yes	—
44	C₂×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
46	C₂²×C₂₉₃	—	2	1	no	—
47	C₂²×C₂₉₃	—	2	1	no	—
48	C₂²×C₂₉₃	C₂×Q₈	1	2	yes	—
49	C₂²×C₂₉₃	C₂⁴	1	1	yes	characteristic
50	C₄×C₂₉₃	—	2	1	no	—
51	C₄×C₂₉₃	—	2	1	no	—
52	C₄×C₂₉₃	—	2	2	no	—
53	C₄×C₂₉₃	—	2	2	no	—
54	Dic₂₉₃	—	2	2	no	—
55	Dic₂₉₃	—	2	2	no	—
56	Dic₂₉₃	—	2	2	no	—
57	Dic₂₉₃	C₂×Q₈	1	4	yes	—
58	C₂×C₂₉₃	—	2	2	no	—
59	C₂×C₂₉₃	C₂²×Q₈	1	1	yes	characteristic
60	C₂×C₂₉₃	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×C₂₉₃	2₋¹⁺⁴	1	1	yes	characteristic
62	C₂₉₃	#544	1	1	yes	characteristic, complemented, Sylow
63	#544	—	293	1	no	maximal, Sylow
64	C₄⋊Q₈	—	293	2	no	—
65	C₄⋊Q₈	—	293	2	no	—
66	#125	—	293	2	no	—
67	#125	—	293	2	no	—
68	C₂²⋊Q₈	—	293	1	no	—
69	C₂²⋊Q₈	—	293	2	no	—
70	C₂²⋊Q₈	D₂₉₃	1	1	yes	characteristic
71	#117	—	293	2	no	—
72	C₂×C₄⋊C₄	—	293	1	no	—
73	C₂×Q₈	—	293	1	no	—
74	C₂×Q₈	—	293	2	no	—
75	C₂×Q₈	C₂×D₂₉₃	1	1	yes	characteristic
76	C₂²×C₄	—	293	1	no	—
77	C₂²×C₄	—	293	1	no	—
78	C₂²×C₄	C₂×D₂₉₃	1	1	yes	characteristic
79	C₄⋊C₄	—	293	2	no	—
80	C₄⋊C₄	—	293	2	no	—
81	C₄⋊C₄	—	293	2	no	—
82	C₄⋊C₄	—	293	2	no	—
83	C₄⋊C₄	—	293	2	no	—
84	C₄⋊C₄	—	293	1	no	—
85	C₄⋊C₄	—	293	2	no	—
86	C₄⋊C₄	—	293	2	no	—
87	C₄⋊C₄	—	293	2	no	—
88	C₄⋊C₄	C₂×D₂₉₃	1	1	yes	characteristic
89	C₄⋊C₄	C₂×D₂₉₃	1	2	yes	—
90	C₂²⋊C₄	—	293	2	no	—
91	C₂²⋊C₄	C₂×D₂₉₃	1	2	yes	—
92	C₄²	—	293	2	no	—
93	C₄²	—	293	2	no	—
94	C₂³	C₂²×D₂₉₃	1	1	yes	characteristic
95	Q₈	—	4	1	no	—
96	Q₈	—	1172	1	no	—
97	Q₈	—	1172	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	293	4	no	—
100	C₂×C₄	—	293	2	no	—
101	C₂×C₄	—	293	2	no	—
102	C₂×C₄	—	293	2	no	—
103	C₂×C₄	—	293	2	no	—
104	C₂×C₄	—	1172	1	no	—
105	C₂×C₄	C₂²×D₂₉₃	1	2	yes	—
106	C₂×C₄	C₂²×D₂₉₃	1	2	yes	—
107	C₂×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	Q₈×D₂₉₃	1	2	yes	—
112	C₂²	C₂³×D₂₉₃	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	293	4	no	—
118	C₄	—	586	2	no	—
119	C₄	—	586	2	no	—
120	C₄	—	586	2	no	—
121	C₂	—	2	2	no	—
122	C₂	C₂×Q₈×D₂₉₃	1	1	yes	characteristic
123	C₂	#22627669	1	1	yes	characteristic
124	C₂	#22627671	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627623

C₂

yes

maximal, complemented

#22627622

C₂

yes

maximal, complemented

#22627614

C₂

yes

maximal, complemented

#22627614

C₂

yes

maximal, complemented

#22627603

C₂

yes

maximal, characteristic

#22627601

C₂

yes

maximal

#22627580

C₂

yes

maximal

C₂×#17989781

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₂₉₃

C₂

yes

maximal, characteristic

C₂×Dic₅₈₆

C₂²

yes

—

C₂×Dic₅₈₆

C₂²

yes

characteristic

C₂²×Dic₂₉₃

C₂²

yes

characteristic

C₂²×Dic₂₉₃

C₂²

yes

characteristic

C₄⋊Dic₂₉₃

C₂²

yes

—

C₄⋊Dic₂₉₃

C₂²

yes

characteristic

#17989781

C₂²

yes

—

#17989781

C₂²

yes

—

#17989781

C₂²

yes

—

#17989781

C₂²

yes

—

#17989781

C₂²

yes

—

#17989781

C₂²

yes

—

#17989781

C₂²

yes

—

C₂²⋊Dic₂₉₃

C₂²

yes

—

C₄×Dic₂₉₃

C₂²

yes

—

C₄×Dic₂₉₃

C₂²

yes

—

C₂×Q₈×C₂₉₃

C₂²

yes

characteristic

C₂²×C₄×C₂₉₃

C₂²

yes

characteristic

C₄⋊C₄×C₂₉₃

C₂²

yes

characteristic

C₄⋊C₄×C₂₉₃

C₂²

yes

—

C₂²⋊C₄×C₂₉₃

C₂²

yes

—

Dic₅₈₆

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

Q₈

yes

complemented

C₂×Dic₂₉₃

C₂³

yes

—

C₂×Dic₂₉₃

C₂³

yes

—

C₂×Dic₂₉₃

C₂³

yes

—

C₂×Dic₂₉₃

C₂³

yes

—

C₂³×C₂₉₃

C₂³

yes

characteristic

Q₈×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂³

yes

—

C₂×C₄×C₂₉₃

C₂³

yes

—

C₂×C₄×C₂₉₃

C₂³

yes

characteristic

C₂×C₄×C₂₉₃

C₂³

yes

characteristic

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

C₂×Q₈

yes

—

C₂²×C₂₉₃

C₂⁴

yes

characteristic

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

C₂×Q₈

yes

—

C₂×C₂₉₃

—

C₂×C₂₉₃

C₂²×Q₈

yes

characteristic

C₂×C₂₉₃

H(5,𝔽₂)

yes

characteristic

C₂×C₂₉₃

2₋¹⁺⁴

yes

characteristic

C₂₉₃

#544

yes

characteristic, complemented, Sylow

#544

—

293

maximal, Sylow

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

293

—

#125

—

293

—

#125

—

293

—

C₂²⋊Q₈

—

293

—

C₂²⋊Q₈

—

293

—

C₂²⋊Q₈

D₂₉₃

yes

characteristic

#117

—

293

—

C₂×C₄⋊C₄

—

293

—

C₂×Q₈

—

293

—

C₂×Q₈

—

293

—

C₂×Q₈

C₂×D₂₉₃

yes

characteristic

C₂²×C₄

—

293

—

C₂²×C₄

—

293

—

C₂²×C₄

C₂×D₂₉₃

yes

characteristic

C₄⋊C₄

—

293

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

293

—

C₄⋊C₄

C₂×D₂₉₃

yes

characteristic

C₄⋊C₄

C₂×D₂₉₃

yes

—

C₂²⋊C₄

—

293

—

C₂²⋊C₄

C₂×D₂₉₃

yes

—

C₄²

—

293

—

C₄²

—

293

—

C₂³

C₂²×D₂₉₃

yes

characteristic

Q₈

—

Q₈

—

1172

—

Q₈

—

1172

—

C₂×C₄

—

C₂×C₄

—

293

—

100

C₂×C₄

—

293

—

101

C₂×C₄

—

293

—

102

C₂×C₄

—

293

—

103

C₂×C₄

—

293

—

104

C₂×C₄

—

1172

—

105

C₂×C₄

C₂²×D₂₉₃

yes

—

106

C₂×C₄

C₂²×D₂₉₃

yes

—

107

C₂×C₄

C₂²×D₂₉₃

yes

characteristic

108

C₂×C₄

C₂²×D₂₉₃

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

Q₈×D₂₉₃

yes

—

112

C₂²

C₂³×D₂₉₃

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

293

—

118

C₄

—

586

—

119

C₄

—

586

—

120

C₄

—

586

—

121

C₂

—

122

C₂

C₂×Q₈×D₂₉₃

yes

characteristic

123

C₂

#22627669

yes

characteristic

124

C₂

#22627671

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₉₃
commutator, socle, cosocle	C₂²×C₂₉₃	C₂⁴
Fitting	C₂²⋊Q₈×C₂₉₃	C₂
prime radical	C₂³×C₂₉₃	C₂³
hypercenter	C₂²⋊Q₈	D₂₉₃
hypocenter	C₂₉₃	#544
LCS	G ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₉₃

commutator, socle, cosocle

C₂²×C₂₉₃

C₂⁴

Fitting

C₂²⋊Q₈×C₂₉₃

C₂

prime radical

C₂³×C₂₉₃

C₂³

hypercenter

C₂²⋊Q₈

D₂₉₃

C₂₉₃

G ▷ C₂²×C₂₉₃ ▷ C₂₉₃

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #22627623	C₂	semidirect
2. #22627622	C₂	semidirect
3. #22627614	C₂	semidirect
4. #22627614	C₂	semidirect
34. C₂×Dic₂₉₃	Q₈	semidirect
62. C₂₉₃	#544	semidirect

H ≅ G/N

type

1. #22627623

C₂

semidirect

2. #22627622

C₂

semidirect

3. #22627614

C₂

semidirect

4. #22627614

C₂

semidirect

34. C₂×Dic₂₉₃

Q₈

semidirect

62. C₂₉₃

#544

semidirect

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

subgroup	index
#22627623	2
#22627622	2
#22627614	2
#22627603	2
#22627601	2
#22627580	2
C₂×#17989781	2
C₂²⋊Q₈×C₂₉₃	2
#544	293

subgroup

index

C₂²⋊Q₈×C₂₉₃

293

Subgroups — #34036758

Conjugacy classes of subgroups

Complements

Maximal subgroups