Group #49174029 — subgroups

Subgroups — #49174029

Order 37504.

248 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 84 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	#34037145	C₂	1	2	yes	maximal
2	#34036975	C₂	1	4	yes	maximal
3	C₂×#22627589	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#14556880	—	2	2	no	—
5	C₂²⋊#14556880	C₂²	1	4	yes	—
6	C₄×#14556880	—	2	2	no	—
7	C₂²×Dic₅₈₆	C₄	1	1	yes	characteristic
8	#22627589	—	2	2	no	—
9	#22627589	—	2	2	no	—
10	C₂×#17989781	C₂²	1	2	yes	—
11	C₂×C₄×Dic₂₉₃	C₄	1	2	yes	—
12	C₂×C₄×Dic₂₉₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₉₃	C₄	1	1	yes	characteristic
14	C₂×#14556880	—	2	4	no	—
15	C₂×Dic₅₈₆	—	2	2	no	—
16	C₂×Dic₅₈₆	—	2	2	no	—
17	C₂×Dic₅₈₆	C₈	1	4	yes	complemented
18	C₂²×Dic₂₉₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₂₉₃	C₂×C₄	1	2	yes	—
20	C₂²×Dic₂₉₃	C₂³	1	1	yes	characteristic
21	#17989781	—	2	4	no	—
22	C₄×Dic₂₉₃	—	2	2	no	—
23	C₄×Dic₂₉₃	—	2	2	no	—
24	C₄×Dic₂₉₃	D₄	1	2	yes	—
25	C₄×Dic₂₉₃	D₄	1	2	yes	—
26	C₂²×C₄×C₂₉₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₉₃	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₄×C₂₉₃	—	2	2	no	—
29	#14556880	—	4	4	no	—
30	Dic₅₈₆	—	2	8	no	—
31	C₂×Dic₂₉₃	—	2	2	no	—
32	C₂×Dic₂₉₃	—	2	2	no	—
33	C₂×Dic₂₉₃	—	2	2	no	—
34	C₂×Dic₂₉₃	—	4	1	no	—
35	C₂×Dic₂₉₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×Dic₂₉₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×Dic₂₉₃	C₂×C₈	1	4	yes	—
38	C₂×Dic₂₉₃	C₂×D₄	1	2	yes	—
39	C₂³×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₉₃	—	2	2	no	—
41	C₂×C₄×C₂₉₃	—	4	2	no	—
42	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₉₃	C₂×C₈	1	2	yes	—
44	C₂×C₄×C₂₉₃	M₁₆	1	2	yes	—
45	C₂²×C₂₉₃	—	2	1	no	—
46	C₂²×C₂₉₃	—	2	1	no	—
47	C₂²×C₂₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₂₉₃	C₂²×C₈	1	1	yes	characteristic
49	C₂²×C₂₉₃	C₂×M₁₆	1	1	yes	characteristic
50	C₄×C₂₉₃	—	4	2	no	—
51	C₄×C₂₉₃	C₂²⋊C₈	1	4	yes	—
52	Dic₂₉₃	—	2	4	no	—
53	Dic₂₉₃	—	2	2	no	—
54	Dic₂₉₃	—	2	1	no	—
55	Dic₂₉₃	#101	1	2	yes	—
56	C₂×C₂₉₃	—	2	1	no	—
57	C₂×C₂₉₃	—	2	1	no	—
58	C₂×C₂₉₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₂₉₃	#410	1	1	yes	characteristic
60	C₂×C₂₉₃	C₂×#101	1	1	yes	characteristic
61	C₂₉₃	#1461	1	1	yes	characteristic, complemented, Sylow
62	#1461	—	293	1	no	maximal, Sylow
63	C₂×C₄×Q₈	—	293	1	no	—
64	#431	—	293	2	no	—
65	#324	—	293	4	no	—
66	C₂²×Q₈	—	293	1	no	—
67	C₄×Q₈	—	586	2	no	—
68	C₄×Q₈	—	586	2	no	—
69	C₂×C₄⋊C₄	—	293	2	no	—
70	C₂×C₄⋊C₄	#11924078	1	1	yes	characteristic
71	C₂×C₄²	—	293	1	no	—
72	C₂×C₄²	—	293	2	no	—
73	C₄⋊C₈	—	586	2	no	—
74	C₂²⋊C₈	—	293	4	no	—
75	C₄×C₈	—	586	2	no	—
76	C₂×Q₈	—	293	4	no	—
77	C₂×Q₈	—	586	2	no	—
78	C₂×Q₈	—	586	2	no	—
79	C₂²×C₄	—	293	1	no	—
80	C₂²×C₄	—	293	2	no	—
81	C₂²×C₄	—	293	1	no	—
82	C₂²×C₄	C₂×#11924078	1	2	yes	—
83	C₂²×C₄	C₂×#11924078	1	1	yes	characteristic
84	C₂×C₈	—	586	4	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	586	4	no	—
87	C₄²	—	293	2	no	—
88	C₄²	—	293	2	no	—
89	C₄²	—	586	2	no	—
90	C₄²	—	586	2	no	—
91	C₂³	C₂²×#11924078	1	1	yes	characteristic
92	Q₈	—	586	8	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	293	1	no	—
96	C₂×C₄	—	293	1	no	—
97	C₂×C₄	—	293	2	no	—
98	C₂×C₄	—	293	4	no	—
99	C₂×C₄	—	586	2	no	—
100	C₂×C₄	—	586	2	no	—
101	C₂×C₄	—	586	2	no	—
102	C₂×C₄	—	1172	1	no	—
103	C₂×C₄	C₂²⋊#11924078	1	2	yes	—
104	C₂×C₄	D₂₉₃⋊C₈	1	2	yes	—
105	C₂×C₄	#17989812	1	2	yes	—
106	C₈	—	1172	4	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	C₂×C₂²⋊#11924078	1	1	yes	characteristic
110	C₂²	C₂×D₂₉₃⋊C₈	1	1	yes	characteristic
111	C₂²	C₂×#17989812	1	1	yes	characteristic
112	C₄	—	4	2	no	—
113	C₄	—	293	2	no	—
114	C₄	—	586	4	no	—
115	C₄	—	586	2	no	—
116	C₄	—	586	1	no	—
117	C₄	#22627681	1	4	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	C₂×#22627681	1	1	yes	characteristic
121	C₂	#34037109	1	1	yes	characteristic
122	C₂	D₂₉₃⋊#101	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#34037145

C₂

yes

maximal

#34036975

C₂

yes

maximal

C₂×#22627589

C₂

yes

maximal, characteristic

C₄⋊#14556880

—

C₂²⋊#14556880

C₂²

yes

—

C₄×#14556880

—

C₂²×Dic₅₈₆

C₄

yes

—

—

C₂²

yes

—

C₂×C₄×Dic₂₉₃

C₄

yes

—

C₂×C₄×Dic₂₉₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₉₃

C₄

yes

characteristic

C₂×#14556880

—

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

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉₃

C₂×C₈

yes

—

C₂×Dic₂₉₃

C₂×D₄

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

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉₃

C₂²×C₈

yes

characteristic

C₂²×C₂₉₃

C₂×M₁₆

yes

characteristic

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂²⋊C₈

yes

—

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

#101

yes

—

C₂×C₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

C₂×C₂²⋊C₈

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂₉₃

yes

characteristic, complemented, Sylow

#1461

—

293

maximal, Sylow

C₂×C₄×Q₈

—

293

—

#431

—

293

—

#324

—

293

—

C₂²×Q₈

—

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

—

C₂²⋊C₈

—

293

—

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

—

586

—

C₄⋊C₄

—

C₄⋊C₄

—

586

—

C₄²

—

293

—

C₄²

—

293

—

C₄²

—

586

—

C₄²

—

586

—

C₂³

C₂²×#11924078

yes

characteristic

Q₈

—

586

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

293

—

C₂×C₄

—

293

—

C₂×C₄

—

293

—

C₂×C₄

—

293

—

C₂×C₄

—

586

—

100

C₂×C₄

—

586

—

101

C₂×C₄

—

586

—

102

C₂×C₄

—

1172

—

103

C₂×C₄

C₂²⋊#11924078

yes

—

104

C₂×C₄

D₂₉₃⋊C₈

yes

—

105

C₂×C₄

#17989812

yes

—

106

C₈

—

1172

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×C₂²⋊#11924078

yes

characteristic

110

C₂²

C₂×D₂₉₃⋊C₈

yes

characteristic

111

C₂²

C₂×#17989812

yes

characteristic

112

C₄

—

113

C₄

—

293

—

114

C₄

—

586

—

115

C₄

—

586

—

116

C₄

—

586

—

117

C₄

#22627681

yes

—

118

C₂

—

119

C₂

—

120

C₂

C₂×#22627681

yes

characteristic

121

C₂

#34037109

yes

characteristic

122

C₂

D₂₉₃⋊#101

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#11924078
commutator	C₂²×C₂₉₃	C₂²×C₈
Frattini	C₂³	C₂²×#11924078
Fitting	C₂×C₄⋊C₄×C₂₉₃	C₄
socle	C₂²×C₂₉₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₉₃	C₂²×C₄
cosocle	C₂²×Dic₂₉₃	C₂³
hypercenter	C₂×C₄⋊C₄	#11924078
hypocenter	C₂₉₃	#1461
LCS	G ▷ C₂²×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊#11924078

commutator

C₂²×C₂₉₃

C₂²×C₈

Frattini

C₂³

C₂²×#11924078

Fitting

C₂×C₄⋊C₄×C₂₉₃

C₄

socle

C₂²×C₂₉₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₉₃

C₂²×C₄

cosocle

C₂²×Dic₂₉₃

C₂³

C₂×C₄⋊C₄

C₂₉₃

G ▷ C₂²×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
17. C₂×Dic₅₈₆	C₈	semidirect
61. C₂₉₃	#1461	semidirect

H ≅ G/N

type

17. C₂×Dic₅₈₆

C₈

semidirect

61. C₂₉₃

#1461

semidirect

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

subgroup	index
#34037145	2
#34036975	2
C₂×#22627589	2
#1461	293

subgroup

index

293

Subgroups — #49174029

Conjugacy classes of subgroups

Complements

Maximal subgroups