Group #49169125 — subgroups

Subgroups — C₂×#34036498 #49169125

Order 37504.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22627624	C₂	1	1	yes	maximal, characteristic
2	C₂×#22627616	C₂	1	1	yes	maximal, characteristic
3	#34036498	C₂	1	8	yes	maximal, direct product factor
4	C₂×#22627549	C₂	1	4	yes	maximal
5	C₂×#97×C₂₉₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₈₆	C₂²	1	2	yes	—
7	#22627624	—	2	2	no	—
8	#22627624	C₂²	1	4	yes	—
9	#22627616	—	2	2	no	—
10	#22627616	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₂₉₃	—	2	2	no	—
12	C₂×C₄⋊Dic₂₉₃	C₂²	1	2	yes	—
13	C₂×#17989781	—	2	2	no	—
14	#22627549	C₂²	1	16	yes	—
15	C₂²×C₈×C₂₉₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₂₉₃	C₂²	1	1	yes	characteristic
17	#97×C₂₉₃	C₂²	1	4	yes	—
18	C₂×Dic₅₈₆	—	2	2	no	—
19	C₂×Dic₅₈₆	—	2	8	no	—
20	C₂×Dic₅₈₆	C₂³	1	4	yes	—
21	C₂²×Dic₂₉₃	—	2	2	no	—
22	C₂²×Dic₂₉₃	—	2	2	no	—
23	C₄⋊Dic₂₉₃	—	2	2	no	—
24	C₄⋊Dic₂₉₃	—	2	8	no	—
25	C₄⋊Dic₂₉₃	C₂³	1	4	yes	—
26	#17989781	—	2	8	no	—
27	C₂²×C₄×C₂₉₃	D₄	1	2	yes	—
28	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₂₉₃	—	2	4	no	—
30	C₂×C₈×C₂₉₃	C₂³	1	4	yes	—
31	C₄²×C₂₉₃	D₄	1	2	yes	—
32	C₄²×C₂₉₃	C₂³	1	2	yes	—
33	Dic₅₈₆	—	2	8	no	—
34	Dic₅₈₆	—	4	4	no	—
35	C₂×Dic₂₉₃	—	2	4	no	—
36	C₂×Dic₂₉₃	—	2	4	no	—
37	C₂×Dic₂₉₃	—	4	4	no	—
38	C₂×Dic₂₉₃	—	4	4	no	—
39	C₂³×C₂₉₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₉₃	—	2	4	no	—
41	C₂×C₄×C₂₉₃	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₉₃	#41	1	4	yes	—
44	C₂×C₄×C₂₉₃	C₂⁴	1	1	yes	characteristic
45	C₈×C₂₉₃	—	2	4	no	—
46	C₂²×C₂₉₃	#124	1	2	yes	—
47	C₂²×C₂₉₃	#137	1	4	yes	—
48	C₂²×C₂₉₃	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₂₉₃	—	2	4	no	—
50	C₄×C₂₉₃	#124	1	2	yes	—
51	C₄×C₂₉₃	C₂×#41	1	2	yes	—
52	Dic₂₉₃	—	4	4	no	—
53	Dic₂₉₃	—	4	4	no	—
54	C₂×C₂₉₃	#491	1	4	yes	complemented
55	C₂×C₂₉₃	C₂×#124	1	1	yes	characteristic
56	C₂×C₂₉₃	C₂×#137	1	2	yes	—
57	C₂₉₃	C₂×#491	1	1	yes	characteristic, complemented, Sylow
58	C₂×#491	—	293	1	no	maximal, Sylow
59	C₂×C₄⋊Q₈	—	293	1	no	—
60	C₂×#125	—	293	1	no	—
61	#491	—	293	8	no	—
62	C₂×#103	—	293	4	no	—
63	C₂×#97	D₂₉₃	1	1	yes	characteristic
64	C₂²×Q₈	—	293	2	no	—
65	C₂²×C₈	C₂×D₂₉₃	1	2	yes	—
66	C₄⋊Q₈	—	293	4	no	—
67	C₄⋊Q₈	—	586	2	no	—
68	#125	—	293	4	no	—
69	#125	—	586	2	no	—
70	C₂×C₄⋊C₄	—	293	2	no	—
71	C₂×C₄⋊C₄	—	586	2	no	—
72	C₂×C₄⋊C₄	—	586	2	no	—
73	C₂×C₄²	C₂×D₂₉₃	1	1	yes	characteristic
74	#103	—	293	16	no	—
75	#97	C₂×D₂₉₃	1	4	yes	—
76	C₂×Q₈	—	293	4	no	—
77	C₂×Q₈	—	586	2	no	—
78	C₂×Q₈	—	586	8	no	—
79	C₂²×C₄	—	586	2	no	—
80	C₂²×C₄	—	586	2	no	—
81	C₂²×C₄	D₁₁₇₂	1	2	yes	—
82	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	C₂²×D₂₉₃	1	4	yes	—
85	C₄⋊C₄	—	293	4	no	—
86	C₄⋊C₄	—	586	2	no	—
87	C₄⋊C₄	—	586	8	no	—
88	C₄⋊C₄	—	586	8	no	—
89	C₄²	D₁₁₇₂	1	2	yes	—
90	C₄²	C₂²×D₂₉₃	1	2	yes	—
91	C₂³	C₂×D₁₁₇₂	1	1	yes	characteristic
92	Q₈	—	586	8	no	—
93	Q₈	—	1172	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	586	4	no	—
96	C₂×C₄	—	586	4	no	—
97	C₂×C₄	—	1172	4	no	—
98	C₂×C₄	—	1172	4	no	—
99	C₂×C₄	C₂×D₁₁₇₂	1	4	yes	—
100	C₂×C₄	C₂×D₁₁₇₂	1	1	yes	characteristic
101	C₂×C₄	#17989802	1	4	yes	—
102	C₂×C₄	C₂³×D₂₉₃	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₂²	#22627612	1	2	yes	—
105	C₂²	#22627657	1	4	yes	—
106	C₂²	C₂²×D₁₁₇₂	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	1172	4	no	—
109	C₄	—	1172	4	no	—
110	C₄	#22627612	1	2	yes	—
111	C₄	C₂×#17989802	1	2	yes	—
112	C₂	#34036498	1	4	yes	direct product factor
113	C₂	C₂×#22627612	1	1	yes	characteristic
114	C₂	C₂×#22627657	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22627624

C₂

yes

maximal, characteristic

C₂×#22627616

C₂

yes

maximal, characteristic

#34036498

C₂

yes

maximal, direct product factor

C₂×#22627549

C₂

yes

maximal

C₂×#97×C₂₉₃

C₂

yes

maximal, characteristic

C₂²×Dic₅₈₆

C₂²

yes

—

#22627624

—

#22627624

C₂²

yes

—

#22627616

—

#22627616

C₂²

yes

—

C₂×C₄⋊Dic₂₉₃

—

C₂×C₄⋊Dic₂₉₃

C₂²

yes

—

C₂×#17989781

—

#22627549

C₂²

yes

—

C₂²×C₈×C₂₉₃

C₂²

yes

—

C₂×C₄²×C₂₉₃

C₂²

yes

characteristic

#97×C₂₉₃

C₂²

yes

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

C₂³

yes

—

C₂²×Dic₂₉₃

—

C₂²×Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

C₂³

yes

—

#17989781

—

C₂²×C₄×C₂₉₃

D₄

yes

—

C₂²×C₄×C₂₉₃

C₂³

yes

characteristic

C₂×C₈×C₂₉₃

—

C₂×C₈×C₂₉₃

C₂³

yes

—

C₄²×C₂₉₃

D₄

yes

—

C₄²×C₂₉₃

C₂³

yes

—

Dic₅₈₆

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂³×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

#41

yes

—

C₂×C₄×C₂₉₃

C₂⁴

yes

characteristic

C₈×C₂₉₃

—

C₂²×C₂₉₃

#124

yes

—

C₂²×C₂₉₃

#137

yes

—

C₂²×C₂₉₃

C₂²×D₄

yes

characteristic

C₄×C₂₉₃

—

C₄×C₂₉₃

#124

yes

—

C₄×C₂₉₃

C₂×#41

yes

—

Dic₂₉₃

—

Dic₂₉₃

—

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

—

C₂₉₃

C₂×#491

yes

characteristic, complemented, Sylow

C₂×#491

—

293

maximal, Sylow

C₂×C₄⋊Q₈

—

293

—

C₂×#125

—

293

—

#491

—

293

—

C₂×#103

—

293

—

C₂×#97

D₂₉₃

yes

characteristic

C₂²×Q₈

—

293

—

C₂²×C₈

C₂×D₂₉₃

yes

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

586

—

#125

—

293

—

#125

—

586

—

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

—

586

—

C₂×C₄⋊C₄

—

586

—

C₂×C₄²

C₂×D₂₉₃

yes

characteristic

#103

—

293

—

#97

C₂×D₂₉₃

yes

—

C₂×Q₈

—

293

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂²×C₄

—

586

—

C₂²×C₄

—

586

—

C₂²×C₄

D₁₁₇₂

yes

—

C₂²×C₄

C₂²×D₂₉₃

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂²×D₂₉₃

yes

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

586

—

C₄⋊C₄

—

586

—

C₄⋊C₄

—

586

—

C₄²

D₁₁₇₂

yes

—

C₄²

C₂²×D₂₉₃

yes

—

C₂³

C₂×D₁₁₇₂

yes

characteristic

Q₈

—

586

—

Q₈

—

1172

—

C₂×C₄

—

C₂×C₄

—

586

—

C₂×C₄

—

586

—

C₂×C₄

—

1172

—

C₂×C₄

—

1172

—

C₂×C₄

C₂×D₁₁₇₂

yes

—

100

C₂×C₄

C₂×D₁₁₇₂

yes

characteristic

101

C₂×C₄

#17989802

yes

—

102

C₂×C₄

C₂³×D₂₉₃

yes

characteristic

103

C₈

—

104

C₂²

#22627612

yes

—

105

C₂²

#22627657

yes

—

106

C₂²

C₂²×D₁₁₇₂

yes

characteristic

107

C₄

—

108

C₄

—

1172

—

109

C₄

—

1172

—

110

C₄

#22627612

yes

—

111

C₄

C₂×#17989802

yes

—

112

C₂

#34036498

yes

direct product factor

113

C₂

C₂×#22627612

yes

characteristic

114

C₂

C₂×#22627657

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₁₇₂
commutator, cosocle	C₂×C₄×C₂₉₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉₃
Fitting	C₂×#97×C₂₉₃	C₂
socle, prime radical	C₂³×C₂₉₃	C₂×D₄
hypercenter	C₂×#97	D₂₉₃
hypocenter	C₂₉₃	C₂×#491
LCS	G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₁₇₂

commutator, cosocle

C₂×C₄×C₂₉₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉₃

Fitting

C₂×#97×C₂₉₃

C₂

socle, prime radical

C₂³×C₂₉₃

C₂×D₄

hypercenter

C₂×#97

D₂₉₃

hypocenter

C₂₉₃

C₂×#491

LCS

G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
3. #34036498	C₂	direct (N × H)
54. C₂×C₂₉₃	#491	semidirect
57. C₂₉₃	C₂×#491	semidirect
112. C₂	#34036498	direct (N × H)

H ≅ G/N

type

3. #34036498

C₂

direct (N × H)

54. C₂×C₂₉₃

#491

semidirect

57. C₂₉₃

C₂×#491

semidirect

112. C₂

#34036498

direct (N × H)

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

subgroup	index
C₂×#22627624	2
C₂×#22627616	2
#34036498	2
C₂×#22627549	2
C₂×#97×C₂₉₃	2
C₂×#491	293

subgroup

index

C₂×#97×C₂₉₃

293

Subgroups — C₂×#34036498 #49169125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#34036498 #49169125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#34036498 #49169125