Group #49174972 — subgroups

Subgroups — #49174972

Order 37504.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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	#34037133	C₂	1	1	yes	maximal, characteristic
2	#34037065	C₂	1	1	yes	maximal, characteristic
3	#34036999	C₂	1	4	yes	maximal
4	#34036685	C₂	1	1	yes	maximal, characteristic
5	C₂×#17989812	C₂²	1	2	yes	—
6	C₂×C₄×#11924078	C₂²	1	2	yes	—
7	C₄⋊#14556880	—	2	2	no	—
8	(C₂×#11924078)⋊C₄	—	2	2	no	—
9	#22627623	C₄	1	2	yes	complemented
10	#22627616	C₄	1	1	yes	characteristic, complemented
11	#22627602	—	2	2	no	—
12	#22627589	—	2	2	no	—
13	#22627582	C₂²	1	2	yes	—
14	C₄²×D₂₉₃	C₂²	1	1	yes	characteristic
15	#125×C₂₉₃	C₄	1	1	yes	characteristic, complemented
16	C₂²×#11924078	—	2	2	no	—
17	#17989812	—	2	4	no	—
18	C₂×#14556880	—	2	2	no	—
19	C₄×#11924078	—	2	2	no	—
20	C₄×#11924078	Q₈	1	4	yes	complemented
21	C₂×Dic₅₈₆	—	2	2	no	—
22	C₂×C₄×D₂₉₃	D₄	1	2	yes	—
23	C₂×C₄×D₂₉₃	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₂₉₃	C₂×C₄	1	2	yes	—
25	#17989781	—	2	2	no	—
26	#17989781	—	2	2	no	—
27	#17989779	—	2	2	no	—
28	C₄×Dic₂₉₃	—	2	2	no	—
29	C₄×Dic₂₉₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₂₉₃	D₄	1	2	yes	—
31	C₄⋊C₄×C₂₉₃	—	2	2	no	—
32	C₄⋊C₄×C₂₉₃	C₂×C₄	1	2	yes	—
33	C₄²×C₂₉₃	C₂×C₄	1	1	yes	characteristic
34	C₂×#11924078	—	2	4	no	—
35	C₂×#11924078	—	4	4	no	—
36	#14556880	—	4	2	no	—
37	C₂²×D₂₉₃	C₂×D₄	1	1	yes	characteristic
38	Dic₅₈₆	—	4	4	no	—
39	C₂×Dic₂₉₃	—	2	2	no	—
40	C₂×Dic₂₉₃	C₂²⋊C₄	1	2	yes	—
41	C₂×Dic₂₉₃	C₂×D₄	1	1	yes	characteristic
42	C₄×D₂₉₃	—	4	2	no	—
43	C₄×D₂₉₃	C₂×Q₈	1	2	yes	—
44	C₄×D₂₉₃	#41	1	2	yes	—
45	C₂×C₄×C₂₉₃	—	2	2	no	—
46	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₂₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₂₉₃	—	2	2	no	—
50	C₂×D₂₉₃	C₂²⋊Q₈	1	2	yes	—
51	#11924078	—	4	4	no	—
52	C₄×C₂₉₃	—	2	2	no	—
53	C₄×C₂₉₃	—	4	2	no	—
54	C₄×C₂₉₃	C₄×Q₈	1	2	yes	—
55	Dic₂₉₃	—	2	2	no	—
56	Dic₂₉₃	—	4	2	no	—
57	Dic₂₉₃	#124	1	1	yes	characteristic
58	Dic₂₉₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₂₉₃	#391	1	1	yes	characteristic
60	C₂×C₂₉₃	#421	1	2	yes	—
61	D₂₉₃	—	2	2	no	—
62	C₂₉₃	#1995	1	1	yes	characteristic, complemented, Sylow
63	#1995	—	293	1	no	maximal, Sylow
64	#533	—	293	1	no	—
65	#423	—	293	1	no	—
66	#376	—	293	1	no	—
67	#339	—	293	4	no	—
68	C₂×M₁₆	—	293	2	no	—
69	C₄⋊Q₈	—	293	2	no	—
70	#125	—	293	1	no	—
71	#125	#11924078	1	1	yes	characteristic, complemented
72	C₂²⋊Q₈	—	586	2	no	—
73	C₄×Q₈	—	586	2	no	—
74	#117	—	293	2	no	—
75	C₂×C₄²	—	293	1	no	—
76	C₂×C₄²	—	293	2	no	—
77	C₄⋊C₈	—	586	2	no	—
78	#95	—	586	2	no	—
79	C₂×Q₈	—	586	2	no	—
80	C₂²×C₄	—	293	1	no	—
81	C₂²×C₄	—	293	2	no	—
82	C₂²×C₄	—	586	2	no	—
83	M₁₆	—	586	4	no	—
84	C₂×C₈	—	586	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	293	2	no	—
87	C₄⋊C₄	—	586	2	no	—
88	C₄⋊C₄	—	586	2	no	—
89	C₄⋊C₄	C₂×#11924078	1	2	yes	—
90	C₂²⋊C₄	—	586	2	no	—
91	C₄²	—	293	2	no	—
92	C₄²	—	293	1	no	—
93	C₄²	—	293	4	no	—
94	C₄²	—	586	2	no	—
95	C₄²	—	586	2	no	—
96	C₄²	C₂×#11924078	1	1	yes	characteristic
97	C₂³	—	293	1	no	—
98	Q₈	—	1172	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	293	1	no	—
101	C₂×C₄	—	293	2	no	—
102	C₂×C₄	—	293	2	no	—
103	C₂×C₄	—	293	2	no	—
104	C₂×C₄	—	586	4	no	—
105	C₂×C₄	—	586	2	no	—
106	C₂×C₄	—	1172	2	no	—
107	C₂×C₄	—	1172	4	no	—
108	C₂×C₄	C₂²⋊#11924078	1	2	yes	—
109	C₂×C₄	C₂²×#11924078	1	1	yes	characteristic
110	C₈	—	1172	2	no	—
111	C₂²	—	293	2	no	—
112	C₂²	—	586	2	no	—
113	C₂²	C₂×C₂²⋊#11924078	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	2	no	—
116	C₄	—	293	1	no	—
117	C₄	—	293	1	no	—
118	C₄	—	586	2	no	—
119	C₄	—	1172	4	no	—
120	C₄	—	1172	2	no	—
121	C₄	Q₈×#11924078	1	2	yes	—
122	C₂	—	586	2	no	—
123	C₂	(C₂×#11924078)⋊Q₈	1	1	yes	characteristic
124	C₂	#34037126	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#34037133

C₂

yes

maximal, characteristic

#34037065

C₂

yes

maximal, characteristic

#34036999

C₂

yes

maximal

#34036685

C₂

yes

maximal, characteristic

C₂×#17989812

C₂²

yes

—

C₂×C₄×#11924078

C₂²

yes

—

C₄⋊#14556880

—

(C₂×#11924078)⋊C₄

—

#22627623

C₄

yes

complemented

#22627616

C₄

yes

characteristic, complemented

#22627602

—

#22627589

—

#22627582

C₂²

yes

—

C₄²×D₂₉₃

C₂²

yes

characteristic

#125×C₂₉₃

C₄

yes

characteristic, complemented

—

—

—

—

Q₈

yes

complemented

C₂×Dic₅₈₆

—

C₂×C₄×D₂₉₃

D₄

yes

—

C₂×C₄×D₂₉₃

C₂³

yes

characteristic

C₄⋊Dic₂₉₃

C₂×C₄

yes

—

#17989781

—

#17989781

—

#17989779

—

C₄×Dic₂₉₃

—

C₄×Dic₂₉₃

C₂×C₄

yes

characteristic

C₄×Dic₂₉₃

D₄

yes

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

C₂×C₄

yes

—

C₄²×C₂₉₃

C₂×C₄

yes

—

—

—

C₂²×D₂₉₃

C₂×D₄

yes

characteristic

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

C₂²⋊C₄

yes

—

C₂×Dic₂₉₃

C₂×D₄

yes

characteristic

C₄×D₂₉₃

—

C₄×D₂₉₃

C₂×Q₈

yes

—

C₄×D₂₉₃

#41

yes

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉₃

C₂²×C₄

yes

characteristic

C₂²×C₂₉₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₉₃

—

C₂×D₂₉₃

C₂²⋊Q₈

yes

—

#11924078

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₄×Q₈

yes

—

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

#124

yes

characteristic

Dic₂₉₃

C₄⋊Q₈

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

—

D₂₉₃

—

C₂₉₃

#1995

yes

characteristic, complemented, Sylow

#1995

—

293

maximal, Sylow

#533

—

293

—

#423

—

293

—

#376

—

293

—

#339

—

293

—

C₂×M₁₆

—

293

—

C₄⋊Q₈

—

293

—

#125

—

293

—

#125

#11924078

yes

characteristic, complemented

C₂²⋊Q₈

—

586

—

C₄×Q₈

—

586

—

#117

—

293

—

C₂×C₄²

—

293

—

C₂×C₄²

—

293

—

C₄⋊C₈

—

586

—

#95

—

586

—

C₂×Q₈

—

586

—

C₂²×C₄

—

293

—

C₂²×C₄

—

293

—

C₂²×C₄

—

586

—

M₁₆

—

586

—

C₂×C₈

—

586

—

C₄⋊C₄

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

586

—

C₄⋊C₄

—

586

—

C₄⋊C₄

C₂×#11924078

yes

—

C₂²⋊C₄

—

586

—

C₄²

—

293

—

C₄²

—

293

—

C₄²

—

293

—

C₄²

—

586

—

C₄²

—

586

—

C₄²

C₂×#11924078

yes

characteristic

C₂³

—

293

—

Q₈

—

1172

—

C₂×C₄

—

100

C₂×C₄

—

293

—

101

C₂×C₄

—

293

—

102

C₂×C₄

—

293

—

103

C₂×C₄

—

293

—

104

C₂×C₄

—

586

—

105

C₂×C₄

—

586

—

106

C₂×C₄

—

1172

—

107

C₂×C₄

—

1172

—

108

C₂×C₄

C₂²⋊#11924078

yes

—

109

C₂×C₄

C₂²×#11924078

yes

characteristic

110

C₈

—

1172

—

111

C₂²

—

293

—

112

C₂²

—

586

—

113

C₂²

C₂×C₂²⋊#11924078

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

293

—

117

C₄

—

293

—

118

C₄

—

586

—

119

C₄

—

1172

—

120

C₄

—

1172

—

121

C₄

Q₈×#11924078

yes

—

122

C₂

—

586

—

123

C₂

(C₂×#11924078)⋊Q₈

yes

characteristic

124

C₂

#34037126

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#11924078
commutator	C₂×C₄×C₂₉₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#11924078
Fitting	#125×C₂₉₃	C₄
socle	C₂²×C₂₉₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₂₉₃	C₂×D₄
cosocle	C₂×C₄×D₂₉₃	C₂³
hypercenter	#125	#11924078
hypocenter	C₂₉₃	#1995
LCS	G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#11924078

commutator

C₂×C₄×C₂₉₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#11924078

Fitting

#125×C₂₉₃

C₄

socle

C₂²×C₂₉₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₂₉₃

C₂×D₄

cosocle

C₂×C₄×D₂₉₃

C₂³

C₂₉₃

G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
9. #22627623	C₄	semidirect
10. #22627616	C₄	semidirect
15. #125×C₂₉₃	C₄	semidirect
20. C₄×#11924078	Q₈	semidirect
62. C₂₉₃	#1995	semidirect
71. #125	#11924078	semidirect

H ≅ G/N

type

9. #22627623

C₄

semidirect

10. #22627616

C₄

semidirect

15. #125×C₂₉₃

C₄

semidirect

20. C₄×#11924078

Q₈

semidirect

62. C₂₉₃

#1995

semidirect

71. #125

#11924078

semidirect

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

subgroup	index
#34037133	2
#34037065	2
#34036999	2
#34036685	2
#1995	293

subgroup

index

293

Subgroups — #49174972

Conjugacy classes of subgroups

Complements

Maximal subgroups