Group #22612496 — subgroups

Subgroups — C₄×#14556588 #22612496

Order 9344.

924 conjugacy classes of subgroups; only the 277 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 277 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×#14556588	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₂₉₂	C₂	1	6	yes	maximal, complemented
3	#17988489	C₂	1	6	yes	maximal, complemented
4	#17988457	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄³×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₉₂	C₂²	1	6	yes	—
7	C₂²×C₄×D₇₃	C₂²	1	4	yes	—
8	#14556588	C₄	1	8	yes	direct product factor
9	C₄×D₂₉₂	—	2	24	no	—
10	C₂×C₄⋊Dic₇₃	C₂²	1	6	yes	—
11	C₂×#11924005	C₂²	1	12	yes	—
12	C₂×C₄²×C₇₃	C₂²	1	3	yes	—
13	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₇₃	C₂²	1	3	yes	—
15	C₂³×D₇₃	C₂³	1	4	yes	—
16	C₂×D₂₉₂	—	2	12	no	—
17	C₂×D₂₉₂	—	2	12	no	—
18	C₂×D₂₉₂	C₂×C₄	1	24	yes	complemented
19	C₂²×Dic₇₃	C₂³	1	4	yes	—
20	C₂×C₄×D₇₃	—	2	24	no	—
21	C₄⋊Dic₇₃	—	2	12	no	—
22	#11924005	—	2	24	no	—
23	C₂²×C₄×C₇₃	C₂³	1	3	yes	—
24	C₂²×C₄×C₇₃	C₂³	1	3	yes	—
25	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
26	C₄²×C₇₃	—	2	6	no	—
27	C₄²×C₇₃	C₂×C₄	1	4	yes	complemented
28	C₄²×C₇₃	D₄	1	12	yes	complemented
29	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
30	C₂²×D₇₃	—	2	12	no	—
31	C₂²×D₇₃	—	2	12	no	—
32	C₂²×D₇₃	C₂²×C₄	1	8	yes	—
33	C₂×C₄×C₇₃	—	2	6	no	—
34	C₂×C₄×C₇₃	C₂²×C₄	1	6	yes	—
35	C₂×C₄×C₇₃	C₂×D₄	1	6	yes	—
36	C₂×C₄×C₇₃	C₂×D₄	1	6	yes	—
37	C₂×C₄×C₇₃	C₂×D₄	1	6	yes	—
38	C₂×C₄×C₇₃	#41	1	6	yes	—
39	C₂×Dic₇₃	—	2	12	no	—
40	D₂₉₂	—	2	48	no	—
41	C₄×D₇₃	—	4	16	no	—
42	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
43	C₂²×C₇₃	C₂²×D₄	1	3	yes	—
44	C₂²×C₇₃	C₂×#41	1	3	yes	—
45	C₂×D₇₃	—	2	24	no	—
46	C₂×D₇₃	—	4	4	no	—
47	C₂×D₇₃	—	4	12	no	—
48	C₄×C₇₃	—	2	6	no	—
49	C₄×C₇₃	C₄×D₄	1	12	yes	complemented
50	C₄×C₇₃	#127	1	4	yes	complemented
51	Dic₇₃	—	4	4	no	—
52	C₂×C₇₃	C₂×C₄×D₄	1	3	yes	—
53	C₂×C₇₃	C₂×#127	1	1	yes	characteristic
54	C₂×C₇₃	#532	1	3	yes	—
55	D₇₃	—	4	8	no	—
56	C₄×#127	—	73	1	no	maximal, Sylow
57	C₇₃	C₄×#127	1	1	yes	characteristic, complemented, Sylow
58	C₂×#127	—	73	1	no	—
59	C₂×C₄×D₄	—	73	6	no	—
60	#390	—	73	6	no	—
61	#384	—	73	1	no	—
62	C₄³	D₇₃	1	1	yes	characteristic, complemented
63	C₂²×D₄	—	73	6	no	—
64	C₂³×C₄	—	73	4	no	—
65	#127	—	73	8	no	—
66	C₄×D₄	—	146	24	no	—
67	C₂×C₄⋊C₄	—	73	6	no	—
68	C₂×C₂²⋊C₄	—	73	12	no	—
69	C₂×C₄²	C₂×D₇₃	1	3	yes	—
70	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
71	C₂×C₄²	C₂×D₇₃	1	3	yes	—
72	C₂⁴	—	73	4	no	—
73	C₂×D₄	—	73	24	no	—
74	C₂×D₄	—	146	12	no	—
75	C₂×D₄	—	146	12	no	—
76	C₂²×C₄	—	73	4	no	—
77	C₂²×C₄	—	146	24	no	—
78	C₂²×C₄	C₂²×D₇₃	1	3	yes	—
79	C₂²×C₄	C₂²×D₇₃	1	3	yes	—
80	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
81	C₄⋊C₄	—	146	12	no	—
82	C₂²⋊C₄	—	146	24	no	—
83	C₄²	—	2	6	no	—
84	C₄²	C₄×D₇₃	1	4	yes	complemented
85	C₄²	D₂₉₂	1	12	yes	complemented
86	C₂³	—	73	8	no	—
87	C₂³	—	146	12	no	—
88	C₂³	—	146	12	no	—
89	C₂³	C₂³×D₇₃	1	1	yes	characteristic
90	D₄	—	146	48	no	—
91	C₂×C₄	—	2	6	no	—
92	C₂×C₄	—	292	16	no	—
93	C₂×C₄	—	146	12	no	—
94	C₂×C₄	C₂×C₄×D₇₃	1	6	yes	—
95	C₂×C₄	C₂×D₂₉₂	1	6	yes	—
96	C₂×C₄	C₂×D₂₉₂	1	6	yes	—
97	C₂×C₄	C₂×D₂₉₂	1	6	yes	—
98	C₂×C₄	#11924028	1	6	yes	—
99	C₂²	—	292	4	no	—
100	C₂²	—	292	12	no	—
101	C₂²	—	146	24	no	—
102	C₂²	C₂²×C₄×D₇₃	1	1	yes	characteristic
103	C₂²	C₂²×D₂₉₂	1	3	yes	—
104	C₂²	C₂×#11924028	1	3	yes	—
105	C₄	—	2	6	no	—
106	C₄	—	292	4	no	—
107	C₄	C₄×D₂₉₂	1	12	yes	complemented
108	C₄	#14556588	1	4	yes	direct product factor
109	C₂	—	292	8	no	—
110	C₂	C₂×C₄×D₂₉₂	1	3	yes	—
111	C₂	C₂×#14556588	1	1	yes	characteristic
112	C₂	#17989137	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556588

C₂

yes

maximal, characteristic

C₂×C₄×D₂₉₂

C₂

yes

maximal, complemented

#17988489

C₂

yes

maximal, complemented

#17988457

C₂

yes

maximal, characteristic, complemented

C₄³×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₉₂

C₂²

yes

—

C₂²×C₄×D₇₃

C₂²

yes

—

#14556588

C₄

yes

direct product factor

C₄×D₂₉₂

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×#11924005

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂×C₄²×C₇₃

C₂²

yes

—

C₂³×D₇₃

C₂³

yes

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂×C₄

yes

complemented

C₂²×Dic₇₃

C₂³

yes

—

C₂×C₄×D₇₃

—

C₄⋊Dic₇₃

—

#11924005

—

C₂²×C₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

complemented

C₄²×C₇₃

D₄

yes

complemented

C₂³×C₇₃

C₂⁴

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

D₂₉₂

—

C₄×D₇₃

—

C₂²×C₇₃

C₂³×C₄

yes

characteristic

C₂²×C₇₃

C₂²×D₄

yes

—

C₂²×C₇₃

C₂×#41

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₄

yes

complemented

C₄×C₇₃

#127

yes

complemented

Dic₇₃

—

C₂×C₇₃

C₂×C₄×D₄

yes

—

C₂×C₇₃

C₂×#127

yes

characteristic

C₂×C₇₃

#532

yes

—

D₇₃

—

C₄×#127

—

maximal, Sylow

C₇₃

C₄×#127

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×C₄×D₄

—

#390

—

#384

—

C₄³

D₇₃

yes

characteristic, complemented

C₂²×D₄

—

C₂³×C₄

—

#127

—

C₄×D₄

—

146

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₇₃

yes

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₂×C₄²

C₂×D₇₃

yes

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

146

—

C₂×D₄

—

146

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂²×C₄

C₂²×D₇₃

yes

—

C₂²×C₄

C₂²×D₇₃

yes

—

C₂²×C₄

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

146

—

C₄²

—

C₄²

C₄×D₇₃

yes

complemented

C₄²

D₂₉₂

yes

complemented

C₂³

—

C₂³

—

146

—

C₂³

—

146

—

C₂³

C₂³×D₇₃

yes

characteristic

D₄

—

146

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

—

146

—

C₂×C₄

C₂×C₄×D₇₃

yes

—

C₂×C₄

C₂×D₂₉₂

yes

—

C₂×C₄

C₂×D₂₉₂

yes

—

C₂×C₄

C₂×D₂₉₂

yes

—

C₂×C₄

#11924028

yes

—

C₂²

—

292

—

100

C₂²

—

292

—

101

C₂²

—

146

—

102

C₂²

C₂²×C₄×D₇₃

yes

characteristic

103

C₂²

C₂²×D₂₉₂

yes

—

104

C₂²

C₂×#11924028

yes

—

105

C₄

—

106

C₄

—

292

—

107

C₄

C₄×D₂₉₂

yes

complemented

108

C₄

#14556588

yes

direct product factor

109

C₂

—

292

—

110

C₂

C₂×C₄×D₂₉₂

yes

—

111

C₂

C₂×#14556588

yes

characteristic

112

C₂

#17989137

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂³×C₄
Frattini	C₂³	C₂³×D₇₃
Fitting	C₄³×C₇₃	C₂
socle	C₂³×C₇₃	C₂⁴
hypercenter	C₄³	D₇₃
hypocenter	C₇₃	C₄×#127
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center

C₂²×C₄

C₂²×D₇₃

commutator

C₂²×C₇₃

C₂³×C₄

Frattini

C₂³

C₂³×D₇₃

Fitting

C₄³×C₇₃

C₂

socle

C₂³×C₇₃

C₂⁴

hypercenter

C₄³

D₇₃

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
2. C₂×C₄×D₂₉₂	C₂	semidirect
3. #17988489	C₂	semidirect
4. #17988457	C₂	semidirect
5. C₄³×C₇₃	C₂	semidirect
8. #14556588	C₄	direct (N × H)
18. C₂×D₂₉₂	C₂×C₄	semidirect
27. C₄²×C₇₃	C₂×C₄	semidirect
28. C₄²×C₇₃	D₄	semidirect
49. C₄×C₇₃	C₄×D₄	semidirect
50. C₄×C₇₃	#127	semidirect
57. C₇₃	C₄×#127	semidirect
62. C₄³	D₇₃	semidirect
84. C₄²	C₄×D₇₃	semidirect
85. C₄²	D₂₉₂	semidirect
107. C₄	C₄×D₂₉₂	semidirect
108. C₄	#14556588	direct (N × H)

H ≅ G/N

type

2. C₂×C₄×D₂₉₂

C₂

semidirect

3. #17988489

C₂

semidirect

4. #17988457

C₂

semidirect

5. C₄³×C₇₃

C₂

semidirect

8. #14556588

C₄

direct (N × H)

18. C₂×D₂₉₂

C₂×C₄

semidirect

27. C₄²×C₇₃

C₂×C₄

semidirect

28. C₄²×C₇₃

D₄

semidirect

49. C₄×C₇₃

C₄×D₄

semidirect

50. C₄×C₇₃

#127

semidirect

57. C₇₃

C₄×#127

semidirect

62. C₄³

D₇₃

semidirect

84. C₄²

C₄×D₇₃

semidirect

85. C₄²

D₂₉₂

semidirect

107. C₄

C₄×D₂₉₂

semidirect

108. C₄

#14556588

direct (N × H)

Subgroups — C₄×#14556588 #22612496

Conjugacy classes of subgroups

Complements

Subgroups — C4×#14556588 #22612496

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#14556588 #22612496