Group #38764705 — subgroups

Subgroups — #1633×C₁₉₉ #38764705

Order 25472.

206 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 76 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	Q₈²×C₁₉₉	C₂	1	1	yes	maximal, characteristic
2	#494×C₁₉₉	C₂	1	1	yes	maximal, characteristic
3	#477×C₁₉₉	C₂	1	2	yes	maximal
4	#333×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊C₈×C₁₉₉	C₂	1	2	yes	maximal
6	C₂×Q₁₆×C₁₉₉	—	2	2	no	—
7	C₄⋊Q₈×C₁₉₉	—	2	2	no	—
8	C₄⋊Q₈×C₁₉₉	—	2	2	no	—
9	C₄⋊Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₄×Q₈×C₁₉₉	—	2	2	no	—
12	C₄×Q₈×C₁₉₉	C₂²	1	2	yes	—
13	#107×C₁₉₉	—	2	2	no	—
14	C₄⋊C₈×C₁₉₉	C₂²	1	2	yes	—
15	#103×C₁₉₉	—	2	2	no	—
16	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₉₉	—	2	2	no	—
18	C₂×Q₈×C₁₉₉	—	2	1	no	—
19	C₂×Q₈×C₁₉₉	D₄	1	2	yes	—
20	C₂×Q₈×C₁₉₉	D₄	1	2	yes	—
21	Q₁₆×C₁₉₉	—	4	4	no	—
22	C₂×C₈×C₁₉₉	—	2	1	no	—
23	C₂×C₈×C₁₉₉	—	2	2	no	—
24	C₄⋊C₄×C₁₉₉	—	2	4	no	—
25	C₄⋊C₄×C₁₉₉	—	2	2	no	—
26	C₄⋊C₄×C₁₉₉	—	2	2	no	—
27	C₄⋊C₄×C₁₉₉	—	2	2	no	—
28	C₄⋊C₄×C₁₉₉	D₄	1	2	yes	—
29	C₄²×C₁₉₉	—	2	2	no	—
30	C₄²×C₁₉₉	—	2	2	no	—
31	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
32	Q₈×C₁₉₉	—	2	2	no	—
33	Q₈×C₁₉₉	—	4	2	no	—
34	Q₈×C₁₉₉	—	4	2	no	—
35	Q₈×C₁₉₉	—	8	2	no	—
36	Q₈×C₁₉₉	Q₁₆	1	4	yes	complemented
37	C₂×C₄×C₁₉₉	—	2	2	no	—
38	C₂×C₄×C₁₉₉	—	2	2	no	—
39	C₂×C₄×C₁₉₉	—	2	2	no	—
40	C₂×C₄×C₁₉₉	—	4	1	no	—
41	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
43	C₈×C₁₉₉	—	2	2	no	—
44	C₈×C₁₉₉	—	4	2	no	—
45	C₂²×C₁₉₉	#120	1	1	yes	characteristic
46	C₄×C₁₉₉	—	2	4	no	—
47	C₄×C₁₉₉	—	2	1	no	—
48	C₄×C₁₉₉	—	4	2	no	—
49	C₄×C₁₉₉	—	4	2	no	—
50	C₄×C₁₉₉	—	8	1	no	—
51	C₄×C₁₉₉	C₂×Q₁₆	1	2	yes	—
52	C₄×C₁₉₉	#137	1	2	yes	—
53	C₂×C₁₉₉	#451	1	2	yes	—
54	C₂×C₁₉₉	#453	1	1	yes	characteristic
55	C₁₉₉	#1633	1	1	yes	characteristic, direct product factor, Sylow
56	#1633	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	Q₈²	C₂×C₁₉₉	1	1	yes	characteristic
58	#494	C₂×C₁₉₉	1	1	yes	characteristic
59	#477	C₂×C₁₉₉	1	2	yes	—
60	#333	C₂×C₁₉₉	1	1	yes	characteristic
61	Q₈⋊C₈	C₂×C₁₉₉	1	2	yes	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	C₂²×C₁₉₉	1	1	yes	characteristic
66	C₄⋊Q₈	C₂²×C₁₉₉	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂²×C₁₉₉	1	2	yes	—
69	#107	—	2	2	no	—
70	C₄⋊C₈	C₂²×C₁₉₉	1	2	yes	—
71	#103	—	2	2	no	—
72	C₄×C₈	C₂²×C₁₉₉	1	1	yes	characteristic
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	2	1	no	—
75	C₂×Q₈	D₄×C₁₉₉	1	2	yes	—
76	C₂×Q₈	D₄×C₁₉₉	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	D₄×C₁₉₉	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂³×C₁₉₉	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	4	2	no	—
91	Q₈	—	8	2	no	—
92	Q₈	Q₁₆×C₁₉₉	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	C₂×D₄×C₁₉₉	1	1	yes	characteristic
98	C₂×C₄	C₂×D₄×C₁₉₉	1	2	yes	—
99	C₈	—	2	2	no	—
100	C₈	—	4	2	no	—
101	C₂²	#120×C₁₉₉	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	8	1	no	—
107	C₄	C₂×Q₁₆×C₁₉₉	1	2	yes	—
108	C₄	#137×C₁₉₉	1	2	yes	—
109	C₂	#451×C₁₉₉	1	2	yes	—
110	C₂	#453×C₁₉₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈²×C₁₉₉

C₂

yes

maximal, characteristic

#494×C₁₉₉

C₂

yes

maximal, characteristic

#477×C₁₉₉

C₂

yes

maximal

#333×C₁₉₉

C₂

yes

maximal, characteristic

Q₈⋊C₈×C₁₉₉

C₂

yes

maximal

C₂×Q₁₆×C₁₉₉

—

C₄⋊Q₈×C₁₉₉

—

C₄⋊Q₈×C₁₉₉

—

C₄⋊Q₈×C₁₉₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₉

C₂²

yes

characteristic

C₄×Q₈×C₁₉₉

—

C₄×Q₈×C₁₉₉

C₂²

yes

—

#107×C₁₉₉

—

C₄⋊C₈×C₁₉₉

C₂²

yes

—

#103×C₁₉₉

—

C₄×C₈×C₁₉₉

C₂²

yes

characteristic

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

D₄

yes

—

C₂×Q₈×C₁₉₉

D₄

yes

—

Q₁₆×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

D₄

yes

—

C₄²×C₁₉₉

—

C₄²×C₁₉₉

—

C₄²×C₁₉₉

C₂³

yes

characteristic

Q₈×C₁₉₉

—

Q₈×C₁₉₉

—

Q₈×C₁₉₉

—

Q₈×C₁₉₉

—

Q₈×C₁₉₉

Q₁₆

yes

complemented

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₈×C₁₉₉

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

#120

yes

characteristic

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×Q₁₆

yes

—

C₄×C₁₉₉

#137

yes

—

C₂×C₁₉₉

#451

yes

—

C₂×C₁₉₉

#453

yes

characteristic

C₁₉₉

#1633

yes

characteristic, direct product factor, Sylow

#1633

C₁₉₉

yes

maximal, characteristic, direct product factor, Sylow

Q₈²

C₂×C₁₉₉

yes

characteristic

#494

C₂×C₁₉₉

yes

characteristic

#477

C₂×C₁₉₉

yes

—

#333

C₂×C₁₉₉

yes

characteristic

Q₈⋊C₈

C₂×C₁₉₉

yes

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₁₉₉

yes

characteristic

C₄⋊Q₈

C₂²×C₁₉₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂²×C₁₉₉

yes

—

#107

—

C₄⋊C₈

C₂²×C₁₉₉

yes

—

#103

—

C₄×C₈

C₂²×C₁₉₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄×C₁₉₉

yes

—

C₂×Q₈

D₄×C₁₉₉

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₁₉₉

yes

—

C₄²

—

C₄²

—

C₄²

C₂³×C₁₉₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

Q₁₆×C₁₉₉

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₁₉₉

yes

characteristic

C₂×C₄

C₂×D₄×C₁₉₉

yes

—

C₈

—

100

C₈

—

101

C₂²

#120×C₁₉₉

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

C₂×Q₁₆×C₁₉₉

yes

—

108

C₄

#137×C₁₉₉

yes

—

109

C₂

#451×C₁₉₉

yes

—

110

C₂

#453×C₁₉₉

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₉₉	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₁₉₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉₉ ◁ C₄²×C₁₉₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₉₉

#120

commutator, Frattini, cosocle

C₄²

C₂³×C₁₉₉

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉₉ ◁ C₄²×C₁₉₉ ◁ G

N	H ≅ G/N	type
36. Q₈×C₁₉₉	Q₁₆	semidirect
55. C₁₉₉	#1633	direct (N × H)
56. #1633	C₁₉₉	direct (N × H)
92. Q₈	Q₁₆×C₁₉₉	semidirect

H ≅ G/N

type

36. Q₈×C₁₉₉

Q₁₆

semidirect

55. C₁₉₉

#1633

direct (N × H)

56. #1633

C₁₉₉

direct (N × H)

92. Q₈

Q₁₆×C₁₉₉

semidirect

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

subgroup	index
Q₈²×C₁₉₉	2
#494×C₁₉₉	2
#477×C₁₉₉	2
#333×C₁₉₉	2
Q₈⋊C₈×C₁₉₉	2
#1633	199

subgroup

index

Q₈²×C₁₉₉

#494×C₁₉₉

#477×C₁₉₉

#333×C₁₉₉

Q₈⋊C₈×C₁₉₉

#1633

199

Subgroups — #1633×C₁₉₉ #38764705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1633×C199 #38764705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1633×C₁₉₉ #38764705