Group #49174955 — subgroups

Subgroups — #49174955

Order 37504.

292 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 110 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×C₄⋊#14556880	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#14556880	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	C₄⋊#14556880	—	2	2	no	—
8	C₄×#14556880	—	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	2	no	—
22	C₂×Dic₅₈₆	—	2	2	no	—
23	C₂×Dic₅₈₆	C₈	1	4	yes	complemented
24	C₂²×Dic₂₉₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₂₉₃	C₂×C₄	1	2	yes	—
26	C₂²×Dic₂₉₃	C₂³	1	1	yes	characteristic
27	#17989781	—	2	4	no	—
28	C₄×Dic₂₉₃	—	2	2	no	—
29	C₄×Dic₂₉₃	—	2	2	no	—
30	C₄×Dic₂₉₃	D₄	1	2	yes	—
31	C₄×Dic₂₉₃	D₄	1	2	yes	—
32	C₂²×C₄×C₂₉₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₂₉₃	C₂×C₄	1	1	yes	characteristic
34	C₄⋊C₄×C₂₉₃	—	2	2	no	—
35	#14556880	—	2	4	no	—
36	#14556880	—	4	2	no	—
37	Dic₅₈₆	—	2	8	no	—
38	C₂×Dic₂₉₃	—	2	2	no	—
39	C₂×Dic₂₉₃	—	2	2	no	—
40	C₂×Dic₂₉₃	—	2	2	no	—
41	C₂×Dic₂₉₃	C₂²⋊C₄	1	2	yes	—
42	C₂×Dic₂₉₃	C₂×C₈	1	4	yes	—
43	C₂×Dic₂₉₃	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₂₉₃	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₂₉₃	C₂×Q₈	1	2	yes	—
46	C₂×Dic₂₉₃	#41	1	2	yes	—
47	C₂³×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₉₃	—	2	2	no	—
49	C₂×C₄×C₂₉₃	—	2	4	no	—
50	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₂₉₃	C₂×C₈	1	2	yes	—
52	C₂×C₄×C₂₉₃	M₁₆	1	2	yes	—
53	C₂²×C₂₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₂₉₃	C₄×Q₈	1	2	yes	—
55	C₂²×C₂₉₃	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₂₉₃	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₂₉₃	#131	1	2	yes	—
58	C₄×C₂₉₃	—	4	2	no	—
59	C₄×C₂₉₃	C₂²⋊C₈	1	4	yes	—
60	Dic₂₉₃	—	2	4	no	—
61	Dic₂₉₃	—	2	2	no	—
62	Dic₂₉₃	C₂²⋊Q₈	1	2	yes	—
63	Dic₂₉₃	#124	1	1	yes	characteristic
64	Dic₂₉₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₂₉₃	#391	1	1	yes	characteristic
66	C₂×C₂₉₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₂₉₃	#408	1	1	yes	characteristic
68	C₂×C₂₉₃	C₈×Q₈	1	2	yes	—
69	C₂×C₂₉₃	#446	1	2	yes	—
70	C₂₉₃	#1984	1	1	yes	characteristic, complemented, Sylow
71	#1984	—	293	1	no	maximal, Sylow
72	C₂×C₄×Q₈	—	293	1	no	—
73	C₂×C₄⋊C₈	—	293	1	no	—
74	C₂×C₄×C₈	—	293	1	no	—
75	#336	—	293	4	no	—
76	C₂²×Q₈	—	293	1	no	—
77	C₂²×C₈	—	293	2	no	—
78	C₂²×C₈	—	293	2	no	—
79	C₄×Q₈	—	586	2	no	—
80	C₄×Q₈	—	586	2	no	—
81	C₂×C₄⋊C₄	—	293	2	no	—
82	C₂×C₄⋊C₄	#11924078	1	1	yes	characteristic
83	C₂×C₄²	—	293	1	no	—
84	C₂×C₄²	—	293	2	no	—
85	C₄⋊C₈	—	586	2	no	—
86	C₄×C₈	—	586	2	no	—
87	C₂×Q₈	—	293	4	no	—
88	C₂×Q₈	—	586	2	no	—
89	C₂×Q₈	—	586	2	no	—
90	C₂²×C₄	—	293	1	no	—
91	C₂²×C₄	—	293	1	no	—
92	C₂²×C₄	—	293	2	no	—
93	C₂²×C₄	C₂×#11924078	1	2	yes	—
94	C₂²×C₄	C₂×#11924078	1	1	yes	characteristic
95	C₂×C₈	—	293	4	no	—
96	C₂×C₈	—	586	2	no	—
97	C₂×C₈	—	586	2	no	—
98	C₂×C₈	—	586	2	no	—
99	C₂×C₈	—	586	4	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	586	4	no	—
102	C₄²	—	293	2	no	—
103	C₄²	—	293	2	no	—
104	C₄²	—	586	2	no	—
105	C₄²	—	586	2	no	—
106	C₂³	C₂²×#11924078	1	1	yes	characteristic
107	Q₈	—	586	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	293	1	no	—
111	C₂×C₄	—	293	1	no	—
112	C₂×C₄	—	293	2	no	—
113	C₂×C₄	—	293	2	no	—
114	C₂×C₄	—	293	2	no	—
115	C₂×C₄	—	293	4	no	—
116	C₂×C₄	—	586	2	no	—
117	C₂×C₄	—	586	2	no	—
118	C₂×C₄	—	586	2	no	—
119	C₂×C₄	C₂²⋊#11924078	1	2	yes	—
120	C₂×C₄	D₂₉₃⋊C₈	1	2	yes	—
121	C₂×C₄	#17989812	1	2	yes	—
122	C₈	—	586	4	no	—
123	C₈	—	1172	2	no	—
124	C₂²	C₂×C₂²⋊#11924078	1	1	yes	characteristic
125	C₂²	Q₈×#11924078	1	2	yes	—
126	C₂²	C₂×D₂₉₃⋊C₈	1	1	yes	characteristic
127	C₂²	C₂×#17989812	1	1	yes	characteristic
128	C₂²	#22627711	1	2	yes	—
129	C₄	—	4	2	no	—
130	C₄	—	293	2	no	—
131	C₄	—	293	1	no	—
132	C₄	—	293	1	no	—
133	C₄	—	586	4	no	—
134	C₄	—	586	2	no	—
135	C₄	#22627681	1	4	yes	—
136	C₂	(C₂×#11924078)⋊Q₈	1	1	yes	characteristic
137	C₂	C₂×#22627681	1	1	yes	characteristic
138	C₂	#34037103	1	1	yes	characteristic
139	C₂	#34037178	1	2	yes	—
140	C₂	#34037179	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#14556880

C₂

yes

maximal, characteristic

C₂×C₄×#14556880

C₂

yes

maximal, characteristic

#34036995

C₂

yes

maximal

C₂×#22627589

C₂

yes

maximal, characteristic

C₂²×#14556880

C₂²

yes

—

C₂²×#14556880

C₂²

yes

—

C₄⋊#14556880

—

C₄×#14556880

—

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

—

C₂×Dic₅₈₆

C₈

yes

complemented

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

C₂×C₈

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

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉₃

C₂×C₈

yes

—

C₂×C₄×C₂₉₃

M₁₆

yes

—

C₂²×C₂₉₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉₃

C₄×Q₈

yes

—

C₂²×C₂₉₃

C₂²×C₈

yes

characteristic

C₂²×C₂₉₃

C₂×M₁₆

yes

characteristic

C₂²×C₂₉₃

#131

yes

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂²⋊C₈

yes

—

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

C₂²⋊Q₈

yes

—

Dic₂₉₃

#124

yes

characteristic

Dic₂₉₃

C₄⋊Q₈

yes

characteristic

C₂×C₂₉₃

#391

yes

characteristic

C₂×C₂₉₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₉₃

#408

yes

characteristic

C₂×C₂₉₃

C₈×Q₈

yes

—

C₂×C₂₉₃

#446

yes

—

C₂₉₃

#1984

yes

characteristic, complemented, Sylow

#1984

—

293

maximal, Sylow

C₂×C₄×Q₈

—

293

—

C₂×C₄⋊C₈

—

293

—

C₂×C₄×C₈

—

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

—

C₄×C₈

—

586

—

C₂×Q₈

—

293

—

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

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

586

—

102

C₄²

—

293

—

103

C₄²

—

293

—

104

C₄²

—

586

—

105

C₄²

—

586

—

106

C₂³

C₂²×#11924078

yes

characteristic

107

Q₈

—

586

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

293

—

111

C₂×C₄

—

293

—

112

C₂×C₄

—

293

—

113

C₂×C₄

—

293

—

114

C₂×C₄

—

293

—

115

C₂×C₄

—

293

—

116

C₂×C₄

—

586

—

117

C₂×C₄

—

586

—

118

C₂×C₄

—

586

—

119

C₂×C₄

C₂²⋊#11924078

yes

—

120

C₂×C₄

D₂₉₃⋊C₈

yes

—

121

C₂×C₄

#17989812

yes

—

122

C₈

—

586

—

123

C₈

—

1172

—

124

C₂²

C₂×C₂²⋊#11924078

yes

characteristic

125

C₂²

Q₈×#11924078

yes

—

126

C₂²

C₂×D₂₉₃⋊C₈

yes

characteristic

127

C₂²

C₂×#17989812

yes

characteristic

128

C₂²

#22627711

yes

—

129

C₄

—

130

C₄

—

293

—

131

C₄

—

293

—

132

C₄

—

293

—

133

C₄

—

586

—

134

C₄

—

586

—

135

C₄

#22627681

yes

—

136

C₂

(C₂×#11924078)⋊Q₈

yes

characteristic

137

C₂

C₂×#22627681

yes

characteristic

138

C₂

#34037103

yes

characteristic

139

C₂

#34037178

yes

—

140

C₂

#34037179

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#11924078
commutator	C₂²×C₂₉₃	C₂²×C₈
Fitting	C₂×C₄⋊C₄×C₂₉₃	C₄
socle, prime radical	C₂³×C₂₉₃	C₂²×C₄
cosocle	C₂²×Dic₂₉₃	C₂³
hypercenter	C₂×C₄⋊C₄	#11924078
hypocenter	C₂₉₃	#1984
LCS	G ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×#11924078

commutator

C₂²×C₂₉₃

C₂²×C₈

Fitting

C₂×C₄⋊C₄×C₂₉₃

C₄

socle, prime radical

C₂³×C₂₉₃

C₂²×C₄

cosocle

C₂²×Dic₂₉₃

C₂³

C₂×C₄⋊C₄

C₂₉₃

G ▷ C₂²×C₂₉₃ ▷ C₂₉₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
20. C₂×#14556880	Q₈	semidirect
23. C₂×Dic₅₈₆	C₈	semidirect
70. C₂₉₃	#1984	semidirect

H ≅ G/N

type

20. C₂×#14556880

Q₈

semidirect

23. C₂×Dic₅₈₆

C₈

semidirect

70. C₂₉₃

#1984

semidirect

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

subgroup	index
C₂×C₄⋊#14556880	2
C₂×C₄×#14556880	2
#34036995	2
C₂×#22627589	2
#1984	293

subgroup

index

293

Subgroups — #49174955

Conjugacy classes of subgroups

Complements

Maximal subgroups