Subgroups — #11772725

Order 1152.

103 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10593838	C₂	1	1	yes	maximal, characteristic
2	#10593259	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63725	C₂	1	1	yes	maximal, characteristic, complemented
4	#64157	C₄	1	1	yes	characteristic
5	#63773	—	2	1	no	—
6	#63773	C₄	1	1	yes	characteristic
7	C₂×#3708	—	2	1	no	—
8	C₂×#3708	C₂²	1	1	yes	characteristic
9	#63725	—	2	1	no	—
10	#63725	—	2	1	no	—
11	C₃²×#41	C₈	1	1	yes	characteristic
12	#3748	—	2	1	no	—
13	#3748	C₈	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
15	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
16	#3708	—	2	1	no	—
17	#3708	D₄	1	1	yes	characteristic
18	#3708	D₄	1	1	yes	characteristic
19	#1401	—	9	1	no	maximal, Sylow
20	#948	—	12	1	no	—
21	C₃²×Q₈	C₁₆	1	1	yes	characteristic
22	C₃²×D₄	—	2	1	no	—
23	C₃²×D₄	C₁₆	1	1	yes	characteristic
24	C₂×C₃²×C₄	—	2	1	no	—
25	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
26	C₃⋊(C₃⋊C₈)	—	2	1	no	—
27	C₃⋊(C₃⋊C₈)	C₂²⋊C₄	1	1	yes	characteristic
28	C₃⋊(C₃⋊C₈)	M₁₆	1	1	yes	characteristic
29	#504	—	9	1	no	—
30	M₆₄	—	9	1	no	—
31	C₂×C₃₂	—	9	1	no	—
32	C₃×#41	—	4	1	no	—
33	#208	—	12	1	no	—
34	#208	—	12	2	no	—
35	C₂×C₃⋊C₈	—	12	1	no	—
36	C₂×C₃⋊C₈	—	12	2	no	—
37	C₂²×C₃²	—	2	1	no	—
38	C₂²×C₃²	M₃₂	1	1	yes	characteristic
39	C₃²×C₄	—	2	1	no	—
40	C₃²×C₄	C₂²⋊C₈	1	1	yes	characteristic
41	C₃²×C₄	C₂×C₁₆	1	1	yes	characteristic
42	#131	—	9	1	no	—
43	M₃₂	—	9	1	no	—
44	M₃₂	—	18	1	no	—
45	C₂×C₁₆	—	9	1	no	—
46	C₂×C₁₆	—	18	1	no	—
47	C₃₂	—	18	1	no	—
48	C₃₂	—	18	1	no	—
49	C₃×Q₈	—	4	1	no	—
50	C₃×D₄	—	4	2	no	—
51	C₃×D₄	—	4	1	no	—
52	C₂×C₃×C₄	—	4	2	no	—
53	C₂×C₃×C₄	—	4	1	no	—
54	C₃⋊C₈	—	12	1	no	—
55	C₃⋊C₈	—	12	1	no	—
56	C₃⋊C₈	—	12	2	no	—
57	C₂×C₃²	—	2	1	no	—
58	C₂×C₃²	—	4	1	no	—
59	C₂×C₃²	C₂²⋊C₁₆	1	1	yes	characteristic
60	#41	AGL(1,𝔽₉)	1	1	yes	characteristic
61	M₁₆	—	9	1	no	—
62	M₁₆	—	18	1	no	—
63	C₂×C₈	—	9	1	no	—
64	C₂×C₈	—	18	1	no	—
65	C₁₆	—	9	1	no	—
66	C₁₆	—	9	1	no	—
67	C₁₆	—	18	1	no	—
68	C₂²×C₃	—	4	2	no	—
69	C₂²×C₃	—	4	1	no	—
70	C₃×C₄	—	4	2	no	—
71	C₃×C₄	—	4	1	no	—
72	C₃×C₄	—	4	1	no	—
73	C₃²	#1401	1	1	yes	characteristic, complemented, Sylow
74	Q₈	#3771	1	1	yes	characteristic
75	D₄	—	2	1	no	—
76	D₄	#3771	1	1	yes	characteristic
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	C₂×AGL(1,𝔽₉)	1	1	yes	characteristic
79	C₈	—	9	1	no	—
80	C₈	—	9	1	no	—
81	C₈	—	18	1	no	—
82	C₂×C₃	—	4	1	no	—
83	C₂×C₃	—	8	1	no	—
84	C₂×C₃	—	8	2	no	—
85	C₂²	—	2	1	no	—
86	C₂²	#64218	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	C₂×#3771	1	1	yes	characteristic
89	C₄	C₂²⋊AGL(1,𝔽₉)	1	1	yes	characteristic
90	C₃	—	4	1	no	—
91	C₂	—	2	1	no	—
92	C₂	—	4	1	no	—
93	C₂	C₂²⋊#3771	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊AGL(1,𝔽₉)
commutator	C₃²×C₄	C₂×C₁₆
Frattini	C₂×C₄	C₂×AGL(1,𝔽₉)
Fitting, prime radical	C₃²×#41	C₈
socle	C₂×C₃²	C₂²⋊C₁₆
cosocle	C₂×#3708	C₂²
hypercenter	#41	AGL(1,𝔽₉)
hypocenter	C₃²	#1401
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #10593259	C₂	semidirect
3. C₂×#63725	C₂	semidirect
73. C₃²	#1401	semidirect

Maximal subgroups

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

subgroup	index
#10593838	2
#10593259	2
C₂×#63725	2
#1401	9