Group #49174957 — subgroups

Subgroups — #49174957

Order 37504.

280 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 90 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×#22627691	C₂	1	1	yes	maximal, characteristic
2	C₂×#22627679	C₂	1	1	yes	maximal, characteristic
3	#34036995	C₂	1	4	yes	maximal
4	C₂×#22627589	C₂	1	1	yes	maximal, characteristic
5	C₂²×#14556880	C₂²	1	2	yes	—
6	C₂²×#14556880	C₂²	1	2	yes	—
7	#22627691	—	2	2	no	—
8	#22627679	—	2	2	no	—
9	C₂²×Dic₅₈₆	C₄	1	1	yes	characteristic
10	#22627589	—	2	2	no	—
11	#22627589	—	2	2	no	—
12	C₂×#17989781	C₄	1	2	yes	—
13	C₂×C₄×Dic₂₉₃	C₂²	1	2	yes	—
14	C₂×C₄×Dic₂₉₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₉₃	C₄	1	1	yes	characteristic
16	C₂×#14556880	—	2	2	no	—
17	C₂×#14556880	—	2	2	no	—
18	C₂×#14556880	—	2	2	no	—
19	C₂×#14556880	—	2	4	no	—
20	C₂×#14556880	Q₈	1	4	yes	complemented
21	C₂×Dic₅₈₆	—	2	4	no	—
22	C₂×Dic₅₈₆	—	2	2	no	—
23	C₂²×Dic₂₉₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₂₉₃	C₂×C₄	1	2	yes	—
25	C₂²×Dic₂₉₃	C₂³	1	1	yes	characteristic
26	#17989781	—	2	4	no	—
27	C₄×Dic₂₉₃	—	2	2	no	—
28	C₄×Dic₂₉₃	—	2	2	no	—
29	C₄×Dic₂₉₃	D₄	1	2	yes	—
30	C₄×Dic₂₉₃	D₄	1	2	yes	—
31	C₂²×C₄×C₂₉₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₂₉₃	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₂₉₃	—	2	2	no	—
34	#14556880	—	2	4	no	—
35	#14556880	—	4	2	no	—
36	Dic₅₈₆	—	2	8	no	—
37	C₂×Dic₂₉₃	—	2	2	no	—
38	C₂×Dic₂₉₃	—	2	2	no	—
39	C₂×Dic₂₉₃	—	2	2	no	—
40	C₂×Dic₂₉₃	C₂²⋊C₄	1	2	yes	—
41	C₂×Dic₂₉₃	M₁₆	1	4	yes	—
42	C₂×Dic₂₉₃	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₂₉₃	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₂₉₃	C₂×Q₈	1	2	yes	—
45	C₂×Dic₂₉₃	#41	1	2	yes	—
46	C₂³×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉₃	—	2	2	no	—
48	C₂×C₄×C₂₉₃	—	2	2	no	—
49	C₂×C₄×C₂₉₃	—	2	4	no	—
50	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₂₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₉₃	C₄×Q₈	1	2	yes	—
53	C₂²×C₂₉₃	C₂×M₁₆	1	2	yes	—
54	C₂²×C₂₉₃	#131	1	2	yes	—
55	C₄×C₂₉₃	—	2	2	no	—
56	C₄×C₂₉₃	—	4	2	no	—
57	Dic₂₉₃	—	2	4	no	—
58	Dic₂₉₃	—	2	2	no	—
59	Dic₂₉₃	C₂²⋊Q₈	1	2	yes	—
60	Dic₂₉₃	#124	1	1	yes	characteristic
61	Dic₂₉₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₂₉₃	#391	1	1	yes	characteristic
63	C₂×C₂₉₃	#407	1	1	yes	characteristic
64	C₂×C₂₉₃	#408	1	1	yes	characteristic
65	C₂×C₂₉₃	#446	1	4	yes	—
66	C₂₉₃	#1985	1	1	yes	characteristic, complemented, Sylow
67	#1985	—	293	1	no	maximal, Sylow
68	C₂×C₄×Q₈	—	293	1	no	—
69	C₂×C₄⋊C₈	—	293	1	no	—
70	C₂×#97	—	293	1	no	—
71	#336	—	293	4	no	—
72	C₂²×Q₈	—	293	1	no	—
73	C₂²×C₈	—	293	2	no	—
74	C₂²×C₈	—	293	2	no	—
75	C₄×Q₈	—	586	2	no	—
76	C₄×Q₈	—	586	2	no	—
77	C₂×C₄⋊C₄	—	293	2	no	—
78	C₂×C₄⋊C₄	#11924078	1	1	yes	characteristic
79	C₂×C₄²	—	293	1	no	—
80	C₂×C₄²	—	293	2	no	—
81	C₄⋊C₈	—	586	2	no	—
82	#97	—	586	2	no	—
83	C₂×Q₈	—	586	4	no	—
84	C₂×Q₈	—	586	2	no	—
85	C₂²×C₄	—	293	1	no	—
86	C₂²×C₄	—	293	1	no	—
87	C₂²×C₄	—	293	2	no	—
88	C₂²×C₄	C₂×#11924078	1	2	yes	—
89	C₂²×C₄	C₂×#11924078	1	1	yes	characteristic
90	C₂×C₈	—	293	4	no	—
91	C₂×C₈	—	586	2	no	—
92	C₂×C₈	—	586	2	no	—
93	C₂×C₈	—	586	2	no	—
94	C₂×C₈	—	586	4	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	586	4	no	—
97	C₄²	—	293	2	no	—
98	C₄²	—	293	2	no	—
99	C₄²	—	586	2	no	—
100	C₄²	—	586	2	no	—
101	C₂³	C₂²×#11924078	1	1	yes	characteristic
102	Q₈	—	586	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	293	1	no	—
107	C₂×C₄	—	293	1	no	—
108	C₂×C₄	—	293	2	no	—
109	C₂×C₄	—	293	2	no	—
110	C₂×C₄	—	293	2	no	—
111	C₂×C₄	—	293	4	no	—
112	C₂×C₄	—	586	2	no	—
113	C₂×C₄	—	586	2	no	—
114	C₂×C₄	—	586	2	no	—
115	C₂×C₄	C₂²⋊#11924078	1	2	yes	—
116	C₈	—	586	4	no	—
117	C₈	—	1172	2	no	—
118	C₂²	C₂×C₂²⋊#11924078	1	1	yes	characteristic
119	C₂²	Q₈×#11924078	1	2	yes	—
120	C₂²	D₂₉₃⋊M₁₆	1	2	yes	—
121	C₂²	#22627711	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	4	2	no	—
124	C₄	—	293	2	no	—
125	C₄	—	293	1	no	—
126	C₄	—	293	1	no	—
127	C₄	—	586	4	no	—
128	C₄	—	586	2	no	—
129	C₂	(C₂×#11924078)⋊Q₈	1	1	yes	characteristic
130	C₂	#34037101	1	1	yes	characteristic
131	C₂	#34037103	1	1	yes	characteristic
132	C₂	#34037181	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22627691

C₂

yes

maximal, characteristic

C₂×#22627679

C₂

yes

maximal, characteristic

#34036995

C₂

yes

maximal

C₂×#22627589

C₂

yes

maximal, characteristic

C₂²×#14556880

C₂²

yes

—

C₂²×#14556880

C₂²

yes

—

#22627691

—

#22627679

—

C₂²×Dic₅₈₆

C₄

yes

—

—

C₄

yes

—

C₂×C₄×Dic₂₉₃

C₂²

yes

—

C₂×C₄×Dic₂₉₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₉₃

C₄

yes

—

—

—

—

Q₈

yes

complemented

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂²×Dic₂₉₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₉₃

C₂×C₄

yes

—

C₂²×Dic₂₉₃

C₂³

yes

characteristic

#17989781

—

C₄×Dic₂₉₃

—

C₄×Dic₂₉₃

—

C₄×Dic₂₉₃

D₄

yes

—

C₄×Dic₂₉₃

D₄

yes

—

C₂²×C₄×C₂₉₃

C₂×C₄

yes

—

C₂²×C₄×C₂₉₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₂₉₃

—

#14556880

—

#14556880

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

C₂²⋊C₄

yes

—

C₂×Dic₂₉₃

M₁₆

yes

—

C₂×Dic₂₉₃

C₂×D₄

yes

characteristic

C₂×Dic₂₉₃

C₂×D₄

yes

characteristic

C₂×Dic₂₉₃

C₂×Q₈

yes

—

C₂×Dic₂₉₃

#41

yes

—

C₂³×C₂₉₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂²⋊C₄

yes

—

C₂²×C₂₉₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉₃

C₄×Q₈

yes

—

C₂²×C₂₉₃

C₂×M₁₆

yes

—

C₂²×C₂₉₃

#131

yes

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

C₂²⋊Q₈

yes

—

Dic₂₉₃

#124

yes

characteristic

Dic₂₉₃

C₄⋊Q₈

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

—

C₂₉₃

#1985

yes

characteristic, complemented, Sylow

#1985

—

293

maximal, Sylow

C₂×C₄×Q₈

—

293

—

C₂×C₄⋊C₈

—

293

—

C₂×#97

—

293

—

#336

—

293

—

C₂²×Q₈

—

293

—

C₂²×C₈

—

293

—

C₂²×C₈

—

293

—

C₄×Q₈

—

586

—

C₄×Q₈

—

586

—

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

#11924078

yes

characteristic

C₂×C₄²

—

293

—

C₂×C₄²

—

293

—

C₄⋊C₈

—

586

—

#97

—

586

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂²×C₄

—

293

—

C₂²×C₄

—

293

—

C₂²×C₄

—

293

—

C₂²×C₄

C₂×#11924078

yes

—

C₂²×C₄

C₂×#11924078

yes

characteristic

C₂×C₈

—

293

—

C₂×C₈

—

586

—

C₂×C₈

—

586

—

C₂×C₈

—

586

—

C₂×C₈

—

586

—

C₄⋊C₄

—

C₄⋊C₄

—

586

—

C₄²

—

293

—

C₄²

—

293

—

C₄²

—

586

—

100

C₄²

—

586

—

101

C₂³

C₂²×#11924078

yes

characteristic

102

Q₈

—

586

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

293

—

107

C₂×C₄

—

293

—

108

C₂×C₄

—

293

—

109

C₂×C₄

—

293

—

110

C₂×C₄

—

293

—

111

C₂×C₄

—

293

—

112

C₂×C₄

—

586

—

113

C₂×C₄

—

586

—

114

C₂×C₄

—

586

—

115

C₂×C₄

C₂²⋊#11924078

yes

—

116

C₈

—

586

—

117

C₈

—

1172

—

118

C₂²

C₂×C₂²⋊#11924078

yes

characteristic

119

C₂²

Q₈×#11924078

yes

—

120

C₂²

D₂₉₃⋊M₁₆

yes

—

121

C₂²

#22627711

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

293

—

125

C₄

—

293

—

126

C₄

—

293

—

127

C₄

—

586

—

128

C₄

—

586

—

129

C₂

(C₂×#11924078)⋊Q₈

yes

characteristic

130

C₂

#34037101

yes

characteristic

131

C₂

#34037103

yes

characteristic

132

C₂

#34037181

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#11924078
commutator, socle, prime radical	C₂³×C₂₉₃	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₂₉₃	C₄
cosocle	C₂²×Dic₂₉₃	C₂³
hypercenter	C₂×C₄⋊C₄	#11924078
hypocenter	C₂₉₃	#1985
LCS	G ▷ C₂³×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×#11924078

commutator, socle, prime radical

C₂³×C₂₉₃

C₂²×C₄

Fitting

C₂×C₄⋊C₄×C₂₉₃

C₄

cosocle

C₂²×Dic₂₉₃

C₂³

C₂×C₄⋊C₄

C₂₉₃

G ▷ C₂³×C₂₉₃ ▷ C₂₉₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
20. C₂×#14556880	Q₈	semidirect
66. C₂₉₃	#1985	semidirect

H ≅ G/N

type

20. C₂×#14556880

Q₈

semidirect

66. C₂₉₃

#1985

semidirect

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

subgroup	index
C₂×#22627691	2
C₂×#22627679	2
#34036995	2
C₂×#22627589	2
#1985	293

subgroup

index

293

Subgroups — #49174957

Conjugacy classes of subgroups

Complements

Maximal subgroups