Group #38767760 — subgroups

Subgroups — #38767760

Order 25472.

148 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 65 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	#27164148	C₂	1	2	yes	maximal
2	C₄×Dic₇₉₆	C₂	1	1	yes	maximal, characteristic
3	#27163769	C₂	1	1	yes	maximal, characteristic
4	#27163750	C₂	1	2	yes	maximal
5	#322×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₉₆	C₄	1	1	yes	characteristic
7	C₄×Dic₃₉₈	C₂²	1	2	yes	—
8	#20263889	C₄	1	1	yes	characteristic
9	C₄⋊(C₁₉₉⋊C₈)	C₂²	1	2	yes	—
10	Dic₁₉₉⋊C₈	—	2	2	no	—
11	#20263880	C₄	1	2	yes	—
12	#20263865	—	2	2	no	—
13	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₉₈	C₂×C₄	1	2	yes	—
17	Dic₇₉₆	—	2	2	no	—
18	C₂×C₁₉₉⋊C₈	—	2	2	no	—
19	C₄⋊Dic₁₉₉	C₂×C₄	1	2	yes	—
20	Dic₁₉₉⋊C₄	—	2	2	no	—
21	C₄×Dic₁₉₉	—	2	2	no	—
22	C₂×C₈×C₁₉₉	—	2	1	no	—
23	C₂×C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₉₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
26	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
27	Dic₃₉₈	—	2	2	no	—
28	Dic₃₉₈	—	2	2	no	—
29	C₂×Dic₁₉₉	—	2	2	no	—
30	C₂×Dic₁₉₉	—	2	2	no	—
31	C₁₉₉⋊C₈	—	4	2	no	—
32	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₉₉	#41	1	1	yes	characteristic
35	C₈×C₁₉₉	—	2	1	no	—
36	C₈×C₁₉₉	—	2	2	no	—
37	C₈×C₁₉₉	—	4	1	no	—
38	C₈×C₁₉₉	M₁₆	1	2	yes	—
39	C₂²×C₁₉₉	C₄×D₄	1	1	yes	characteristic
40	C₄×C₁₉₉	—	2	1	no	—
41	C₄×C₁₉₉	C₂×M₁₆	1	1	yes	characteristic
42	C₄×C₁₉₉	#131	1	1	yes	characteristic
43	C₄×C₁₉₉	#137	1	2	yes	—
44	Dic₁₉₉	—	4	2	no	—
45	Dic₁₉₉	—	4	2	no	—
46	C₂×C₁₉₉	#436	1	1	yes	characteristic
47	C₂×C₁₉₉	#441	1	1	yes	characteristic
48	C₂×C₁₉₉	#444	1	1	yes	characteristic
49	C₁₉₉	#1593	1	1	yes	characteristic, complemented, Sylow
50	#1593	—	199	1	no	maximal, Sylow
51	#446	—	199	2	no	—
52	C₄×Q₁₆	—	199	1	no	—
53	#334	—	199	1	no	—
54	Q₈⋊C₈	—	199	2	no	—
55	#322	D₁₉₉	1	1	yes	characteristic
56	C₂×Q₁₆	—	199	1	no	—
57	C₄×Q₈	—	199	2	no	—
58	#107	—	199	1	no	—
59	C₄⋊C₈	—	398	2	no	—
60	C₄⋊C₈	—	199	2	no	—
61	#103	—	199	2	no	—
62	#97	—	398	2	no	—
63	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
64	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
65	C₄×C₈	C₂×D₁₉₉	1	1	yes	characteristic
66	C₂×Q₈	—	199	2	no	—
67	Q₁₆	—	398	2	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	398	2	no	—
70	C₂×C₈	C₄×D₁₉₉	1	1	yes	characteristic
71	C₂×C₈	C₄×D₁₉₉	1	1	yes	characteristic
72	C₂×C₈	D₇₉₆	1	2	yes	—
73	C₄⋊C₄	—	398	2	no	—
74	C₄⋊C₄	—	199	2	no	—
75	C₄²	—	398	2	no	—
76	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
77	Q₈	—	398	2	no	—
78	Q₈	—	398	2	no	—
79	C₂×C₄	—	398	2	no	—
80	C₂×C₄	—	398	2	no	—
81	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
82	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
83	C₂×C₄	#15691886	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	1	no	—
87	C₈	—	796	2	no	—
88	C₈	#15691869	1	2	yes	—
89	C₂²	C₄×D₇₉₆	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	796	2	no	—
92	C₄	—	796	2	no	—
93	C₄	C₂×#15691869	1	1	yes	characteristic
94	C₄	#20263961	1	1	yes	characteristic
95	C₄	#20263988	1	2	yes	—
96	C₂	#27164097	1	1	yes	characteristic
97	C₂	#27164126	1	1	yes	characteristic
98	C₂	#27164141	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27164148

C₂

yes

maximal

C₄×Dic₇₉₆

C₂

yes

maximal, characteristic

#27163769

C₂

yes

maximal, characteristic

#27163750

C₂

yes

maximal

#322×C₁₉₉

C₂

yes

maximal, characteristic

C₂×Dic₇₉₆

C₄

yes

characteristic

C₄×Dic₃₉₈

C₂²

yes

—

#20263889

C₄

yes

characteristic

C₄⋊(C₁₉₉⋊C₈)

C₂²

yes

—

Dic₁₉₉⋊C₈

—

#20263880

C₄

yes

—

#20263865

—

C₄×C₈×C₁₉₉

C₂²

yes

characteristic

C₄×C₈×C₁₉₉

C₂²

yes

characteristic

C₄×C₈×C₁₉₉

C₂²

yes

characteristic

C₂×Dic₃₉₈

C₂×C₄

yes

—

Dic₇₉₆

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

C₂×C₄

yes

—

Dic₁₉₉⋊C₄

—

C₄×Dic₁₉₉

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₉

D₄

yes

—

C₄²×C₁₉₉

C₂³

yes

characteristic

Dic₃₉₈

—

Dic₃₉₈

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

characteristic

C₈×C₁₉₉

—

C₈×C₁₉₉

—

C₈×C₁₉₉

—

C₈×C₁₉₉

M₁₆

yes

—

C₂²×C₁₉₉

C₄×D₄

yes

characteristic

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×M₁₆

yes

C₄×C₁₉₉

yes

C₄×C₁₉₉

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₁₉₉

yes

characteristic, complemented, Sylow

#1593

—

199

maximal, Sylow

#446

—

199

—

C₄×Q₁₆

—

199

—

#334

—

199

—

Q₈⋊C₈

—

199

—

#322

D₁₉₉

yes

characteristic

C₂×Q₁₆

—

199

—

C₄×Q₈

—

199

—

#107

—

199

—

C₄⋊C₈

—

398

—

C₄⋊C₈

—

199

—

#103

—

199

—

#97

—

398

—

C₄×C₈

C₂×D₁₉₉

yes

characteristic

C₄×C₈

C₂×D₁₉₉

yes

characteristic

C₄×C₈

C₂×D₁₉₉

yes

characteristic

C₂×Q₈

—

199

—

Q₁₆

—

398

—

C₂×C₈

—

C₂×C₈

—

398

—

C₂×C₈

C₄×D₁₉₉

yes

characteristic

C₂×C₈

C₄×D₁₉₉

yes

characteristic

C₂×C₈

D₇₉₆

yes

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

199

—

C₄²

—

398

—

C₄²

C₂²×D₁₉₉

yes

characteristic

Q₈

—

398

—

Q₈

—

398

—

C₂×C₄

—

398

—

C₂×C₄

—

398

—

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

C₂×C₄

C₂×D₇₉₆

yes

characteristic

C₂×C₄

#15691886

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

796

—

C₈

#15691869

yes

—

C₂²

C₄×D₇₉₆

yes

characteristic

C₄

—

C₄

—

796

—

C₄

—

796

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	C₂×D₇₉₆
commutator	C₂×C₄×C₁₉₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₉₉
Fitting	#322×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₄×D₄
cosocle	C₄²×C₁₉₉	C₂³
hypercenter	#322	D₁₉₉
hypocenter	C₁₉₉	#1593
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₇₉₆

commutator

C₂×C₄×C₁₉₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₉₉

Fitting

#322×C₁₉₉

C₂

socle, prime radical

C₂²×C₁₉₉

C₄×D₄

cosocle

C₄²×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
49. C₁₉₉	#1593	semidirect

H ≅ G/N

type

49. C₁₉₉

#1593

semidirect

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

subgroup	index
#27164148	2
C₄×Dic₇₉₆	2
#27163769	2
#27163750	2
#322×C₁₉₉	2
#1593	199

subgroup

index

C₄×Dic₇₉₆

#322×C₁₉₉

199

Subgroups — #38767760

Conjugacy classes of subgroups

Complements

Maximal subgroups