Subgroups — #11772980

Order 1152.

234 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63626	C₂	1	1	yes	maximal, characteristic
2	C₃²×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#10589372	C₂	1	1	yes	maximal, characteristic, complemented
4	#67604	—	3	4	no	maximal
5	C₂×#3745	—	2	1	no	—
6	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
7	#63646	—	2	1	no	—
8	#63626	—	2	1	no	—
9	#63626	C₄	1	2	yes	—
10	C₃²×C₃₂	—	2	1	no	—
11	C₂×D₄₈	—	3	4	no	—
12	C₃×M₆₄	D₃	1	4	yes	complemented
13	#4506	—	3	4	no	—
14	C₂×C₄⋊(C₃⋊D₃)	—	4	1	no	—
15	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
16	#3748	—	4	1	no	—
17	#3745	—	2	2	no	—
18	#3745	—	4	1	no	—
19	C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₃²×C₁₆	D₄	1	1	yes	characteristic
21	#1419	—	9	1	no	Sylow
22	C₂×D₂₄	—	6	4	no	—
23	C₂×C₃×C₁₆	C₂×D₃	1	4	yes	—
24	#819	—	6	4	no	—
25	D₄₈	—	3	8	no	—
26	D₄₈	—	6	4	no	—
27	C₃×C₃₂	—	2	4	no	—
28	C₂²×C₃⋊D₃	—	8	1	no	—
29	C₂×C₃²×C₄	D₈	1	1	yes	characteristic
30	C₄⋊(C₃⋊D₃)	—	4	2	no	—
31	C₄⋊(C₃⋊D₃)	—	8	1	no	—
32	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
33	C₃²×C₈	SD₁₆	1	1	yes	characteristic
34	C₃⋊(C₃⋊C₈)	—	8	1	no	—
35	C₂×D₁₆	—	9	1	no	—
36	M₆₄	C₃⋊D₃	1	1	yes	characteristic, complemented
37	#368	—	9	1	no	—
38	C₂×D₁₂	—	12	4	no	—
39	C₂×C₃×C₈	D₁₂	1	4	yes	—
40	#208	—	12	4	no	—
41	D₂₄	—	6	8	no	—
42	D₂₄	—	12	4	no	—
43	C₃×C₁₆	C₄×D₃	1	4	yes	—
44	C₃×C₁₆	#67	1	4	yes	—
45	C₂²×C₃²	SD₃₂	1	1	yes	characteristic
46	C₂×C₃⋊D₃	—	8	2	no	—
47	C₂×C₃⋊D₃	—	16	2	no	—
48	C₃²×C₄	#102	1	1	yes	characteristic
49	C₃²×C₄	D₁₆	1	1	yes	characteristic
50	C₂×D₈	—	18	1	no	—
51	D₁₆	—	9	2	no	—
52	D₁₆	—	18	1	no	—
53	C₂×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
54	#108	—	18	1	no	—
55	C₃₂	—	2	1	no	—
56	C₂²×D₃	—	24	4	no	—
57	C₂×C₃×C₄	D₂₄	1	4	yes	—
58	D₁₂	—	12	8	no	—
59	D₁₂	—	24	4	no	—
60	C₃×C₈	#204	1	4	yes	—
61	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
62	C₃⋊C₈	—	24	4	no	—
63	C₂×C₃²	—	2	1	no	—
64	C₂×C₃²	#357	1	1	yes	characteristic
65	C₃⋊D₃	—	16	2	no	—
66	C₂×D₄	—	36	1	no	—
67	D₈	—	18	2	no	—
68	D₈	—	36	1	no	—
69	M₁₆	—	36	1	no	—
70	C₂×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
71	C₁₆	C₄×C₃⋊D₃	1	1	yes	characteristic
72	C₁₆	#638	1	1	yes	characteristic
73	C₂²×C₃	#800	1	4	yes	—
74	C₂×D₃	—	24	8	no	—
75	C₂×D₃	—	48	8	no	—
76	C₃×C₄	D₄₈	1	4	yes	—
77	C₃×C₄	#821	1	4	yes	—
78	C₃²	#1419	1	1	yes	characteristic, complemented, Sylow
79	C₂³	—	72	1	no	—
80	D₄	—	36	2	no	—
81	D₄	—	72	1	no	—
82	C₂×C₄	#3745	1	1	yes	characteristic
83	C₈	—	72	1	no	—
84	C₈	#3744	1	1	yes	characteristic
85	C₈	#3752	1	1	yes	characteristic
86	C₂×C₃	—	2	4	no	—
87	C₂×C₃	#4509	1	4	yes	—
88	D₃	—	48	8	no	—
89	C₂²	—	72	2	no	—
90	C₂²	—	144	2	no	—
91	C₂²	#63627	1	1	yes	characteristic
92	C₄	#63626	1	1	yes	characteristic
93	C₄	#63648	1	1	yes	characteristic
94	C₃	#67604	1	4	yes	complemented
95	C₂	—	2	1	no	—
96	C₂	—	144	2	no	—
97	C₂	#10589375	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10589375
commutator	C₃²×C₁₆	C₂×C₄
Frattini	C₂×C₁₆	C₂×C₃⋊D₃
Fitting	C₃²×M₆₄	C₂
socle	C₂×C₃²	#357
prime radical	C₂×#63626	C₂
cosocle	C₂×C₃²×C₁₆	C₂²
hypercenter	M₆₄	C₃⋊D₃
hypocenter	C₃²	#1419
LCS	G ▷ C₃²×C₁₆ ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₃²×M₆₄	C₂	semidirect
3. #10589372	C₂	semidirect
12. C₃×M₆₄	D₃	semidirect
36. M₆₄	C₃⋊D₃	semidirect
78. C₃²	#1419	semidirect
94. C₃	#67604	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63626	2
C₃²×M₆₄	2
#10589372	2
#67604	3