Group #22623558 — subgroups

Subgroups — #22623558

Order 9344.

1202 conjugacy classes of subgroups; only the 491 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 491 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	#17989303	C₂	1	9	yes	maximal, complemented
2	#17989259	C₂	1	6	yes	maximal, complemented
3	#17989138	C₂	1	9	yes	maximal, complemented
4	C₄×#11924029	C₂	1	6	yes	maximal, complemented
5	Q₈²×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924029	C₂²	1	6	yes	—
7	C₂×#11924029	C₂²	1	6	yes	—
8	#14556588	C₂²	1	9	yes	—
9	#14556580	C₂²	1	18	yes	—
10	#14556566	C₂²	1	36	yes	—
11	Q₈×Dic₇₃	C₂²	1	2	yes	—
12	C₄×D₂₉₂	C₂²	1	18	yes	—
13	#14556553	C₂²	1	18	yes	—
14	#14556549	C₂²	1	18	yes	—
15	C₄²×D₇₃	C₂²	1	9	yes	—
16	C₄⋊Q₈×C₇₃	C₂²	1	9	yes	—
17	C₄×Q₈×C₇₃	C₂²	1	6	yes	—
18	#11924029	—	2	24	no	—
19	#11924029	—	4	12	no	—
20	C₂×D₂₉₂	C₂³	1	18	yes	—
21	C₂×D₂₉₂	C₂³	1	18	yes	—
22	C₂×C₄×D₇₃	C₂³	1	18	yes	—
23	C₂×C₄×D₇₃	C₂³	1	9	yes	—
24	C₄⋊Dic₇₃	C₂³	1	6	yes	—
25	#11924005	C₂³	1	36	yes	—
26	C₄×Dic₇₃	C₂³	1	6	yes	—
27	C₄×Dic₇₃	C₂³	1	9	yes	—
28	C₂×Q₈×C₇₃	C₂³	1	6	yes	—
29	C₂×Q₈×C₇₃	C₂³	1	2	yes	—
30	C₄⋊C₄×C₇₃	C₂³	1	18	yes	—
31	C₄²×C₇₃	C₂³	1	9	yes	—
32	C₂²×D₇₃	C₂⁴	1	9	yes	—
33	Q₈×C₇₃	—	4	6	no	—
34	Q₈×C₇₃	#41	1	8	yes	complemented
35	C₂×C₄×C₇₃	C₂⁴	1	9	yes	—
36	C₂×C₄×C₇₃	C₂⁴	1	6	yes	—
37	C₂×Dic₇₃	C₂⁴	1	6	yes	—
38	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
39	D₂₉₂	—	2	36	no	—
40	D₂₉₂	—	4	18	no	—
41	C₄×D₇₃	—	2	36	no	—
42	C₄×D₇₃	—	2	18	no	—
43	C₂²×C₇₃	C₂⁵	1	1	yes	characteristic
44	C₂×D₇₃	—	2	18	no	—
45	C₂×D₇₃	—	2	9	no	—
46	C₄×C₇₃	—	2	9	no	—
47	C₄×C₇₃	C₂×#41	1	12	yes	—
48	Dic₇₃	—	2	6	no	—
49	Dic₇₃	H(5,𝔽₂)	1	2	yes	—
50	C₂×C₇₃	C₂²×#41	1	2	yes	—
51	C₂×C₇₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
52	D₇₃	—	4	9	no	—
53	#3483	—	73	1	no	maximal, Sylow
54	C₇₃	#3483	1	1	yes	characteristic, complemented, Sylow
55	Q₈²	D₇₃	1	1	yes	characteristic, complemented
56	#555	—	73	9	no	—
57	#550	—	73	6	no	—
58	#532	—	73	9	no	—
59	C₄×#41	—	73	6	no	—
60	C₂×#41	—	73	6	no	—
61	C₂×#41	—	73	6	no	—
62	C₄⋊Q₈	C₂×D₇₃	1	9	yes	—
63	#127	—	73	9	no	—
64	#124	—	73	18	no	—
65	#121	—	73	36	no	—
66	C₄×Q₈	—	73	2	no	—
67	C₄×Q₈	C₂×D₇₃	1	6	yes	—
68	C₄×D₄	—	73	18	no	—
69	C₄×D₄	—	73	18	no	—
70	#117	—	73	18	no	—
71	C₂×C₄²	—	73	9	no	—
72	#41	—	292	12	no	—
73	#41	—	146	24	no	—
74	C₂×Q₈	C₂²×D₇₃	1	6	yes	—
75	C₂×Q₈	C₂²×D₇₃	1	2	yes	—
76	C₂×D₄	—	73	18	no	—
77	C₂×D₄	—	73	18	no	—
78	C₂²×C₄	—	73	18	no	—
79	C₂²×C₄	—	73	9	no	—
80	C₄⋊C₄	—	73	6	no	—
81	C₄⋊C₄	C₂²×D₇₃	1	18	yes	—
82	C₂²⋊C₄	—	73	36	no	—
83	C₄²	—	73	6	no	—
84	C₄²	—	73	9	no	—
85	C₄²	C₂²×D₇₃	1	9	yes	—
86	C₂³	—	73	9	no	—
87	Q₈	—	4	6	no	—
88	Q₈	#11924029	1	8	yes	complemented
89	D₄	—	292	18	no	—
90	D₄	—	146	36	no	—
91	C₂×C₄	—	73	1	no	—
92	C₂×C₄	—	73	6	no	—
93	C₂×C₄	—	146	18	no	—
94	C₂×C₄	—	146	36	no	—
95	C₂×C₄	C₂³×D₇₃	1	9	yes	—
96	C₂×C₄	C₂³×D₇₃	1	6	yes	—
97	C₂²	—	146	18	no	—
98	C₂²	—	146	9	no	—
99	C₂²	C₂⁴×D₇₃	1	1	yes	characteristic
100	C₄	—	2	9	no	—
101	C₄	—	73	2	no	—
102	C₄	—	146	6	no	—
103	C₄	C₂×#11924029	1	12	yes	—
104	C₂	—	292	9	no	—
105	C₂	C₂²×#11924029	1	2	yes	—
106	C₂	H(5,𝔽₂)×D₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989303

C₂

yes

maximal, complemented

#17989259

C₂

yes

maximal, complemented

#17989138

C₂

yes

maximal, complemented

C₄×#11924029

C₂

yes

maximal, complemented

Q₈²×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924029

C₂²

yes

—

C₂×#11924029

C₂²

yes

—

#14556588

C₂²

yes

—

#14556580

C₂²

yes

—

#14556566

C₂²

yes

—

Q₈×Dic₇₃

C₂²

yes

—

C₄×D₂₉₂

C₂²

yes

—

#14556553

C₂²

yes

—

#14556549

C₂²

yes

—

C₄²×D₇₃

C₂²

yes

—

C₄⋊Q₈×C₇₃

C₂²

yes

—

C₄×Q₈×C₇₃

C₂²

yes

—

#11924029

—

#11924029

—

C₂×D₂₉₂

C₂³

yes

—

C₂×D₂₉₂

C₂³

yes

—

C₂×C₄×D₇₃

C₂³

yes

—

C₂×C₄×D₇₃

C₂³

yes

—

C₄⋊Dic₇₃

C₂³

yes

—

#11924005

C₂³

yes

—

C₄×Dic₇₃

C₂³

yes

—

C₄×Dic₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₄²×C₇₃

C₂³

yes

—

C₂²×D₇₃

C₂⁴

yes

—

Q₈×C₇₃

—

Q₈×C₇₃

#41

yes

complemented

C₂×C₄×C₇₃

C₂⁴

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

—

C₂×Dic₇₃

C₂⁴

yes

—

C₂×Dic₇₃

C₂⁴

yes

characteristic

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₂²×C₇₃

C₂⁵

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#41

yes

—

Dic₇₃

—

Dic₇₃

H(5,𝔽₂)

yes

—

C₂×C₇₃

C₂²×#41

yes

—

C₂×C₇₃

C₂×H(5,𝔽₂)

yes

characteristic

D₇₃

—

#3483

—

maximal, Sylow

C₇₃

#3483

yes

characteristic, complemented, Sylow

Q₈²

D₇₃

yes

characteristic, complemented

#555

—

#550

—

#532

—

C₄×#41

—

C₂×#41

—

C₂×#41

—

C₄⋊Q₈

C₂×D₇₃

yes

—

#127

—

#124

—

#121

—

C₄×Q₈

—

C₄×Q₈

C₂×D₇₃

yes

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

#41

—

292

—

#41

—

146

—

C₂×Q₈

C₂²×D₇₃

yes

—

C₂×Q₈

C₂²×D₇₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₇₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₇₃

yes

—

C₂³

—

Q₈

—

Q₈

#11924029

yes

complemented

D₄

—

292

—

D₄

—

146

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

—

146

—

C₂×C₄

C₂³×D₇₃

yes

—

C₂×C₄

C₂³×D₇₃

yes

—

C₂²

—

146

—

C₂²

—

146

—

C₂²

C₂⁴×D₇₃

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

146

—

103

C₄

C₂×#11924029

yes

—

104

C₂

—

292

—

105

C₂

C₂²×#11924029

yes

—

106

C₂

H(5,𝔽₂)×D₇₃

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₇₃
commutator, socle	C₂²×C₇₃	C₂⁵
Fitting	Q₈²×C₇₃	C₂
hypercenter	Q₈²	D₇₃
hypocenter	C₇₃	#3483
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ Q₈²

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₇₃

commutator, socle

C₂²×C₇₃

C₂⁵

Fitting

Q₈²×C₇₃

C₂

hypercenter

Q₈²

D₇₃

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ Q₈²

N	H ≅ G/N	type
1. #17989303	C₂	semidirect
2. #17989259	C₂	semidirect
3. #17989138	C₂	semidirect
4. C₄×#11924029	C₂	semidirect
5. Q₈²×C₇₃	C₂	semidirect
34. Q₈×C₇₃	#41	semidirect
54. C₇₃	#3483	semidirect
55. Q₈²	D₇₃	semidirect
88. Q₈	#11924029	semidirect

H ≅ G/N

type

1. #17989303

C₂

semidirect

2. #17989259

C₂

semidirect

3. #17989138

C₂

semidirect

4. C₄×#11924029

C₂

semidirect

5. Q₈²×C₇₃

C₂

semidirect

34. Q₈×C₇₃

#41

semidirect

54. C₇₃

#3483

semidirect

55. Q₈²

D₇₃

semidirect

88. Q₈

#11924029

semidirect

Subgroups — #22623558

Conjugacy classes of subgroups

Complements