Subgroups — #11772702

Order 1152.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#10589365	C₂	1	2	yes	maximal, complemented
3	#67809	—	3	4	no	maximal
4	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
5	C₃²×M₃₂	—	2	2	no	—
6	C₂×C₃²×C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
8	C₃⋊(C₃⋊C₃₂)	—	2	2	no	—
9	C₂×C₃×M₃₂	D₃	1	4	yes	—
10	#4499	—	3	8	no	—
11	C₂²×C₃²×C₄	C₈	1	1	yes	characteristic
12	C₂×C₃²×C₈	—	2	1	no	—
13	C₂×C₃²×C₈	—	2	1	no	—
14	C₂×C₃²×C₈	C₈	1	1	yes	characteristic
15	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₃²×C₁₆	—	2	1	no	—
17	C₃²×C₁₆	D₄	1	2	yes	—
18	#1400	—	9	1	no	Sylow
19	C₂²×C₃×C₈	Dic₃	1	4	yes	—
20	C₃×M₃₂	—	2	8	no	—
21	C₂×C₃×C₁₆	Dic₃	1	4	yes	—
22	C₂×C₃×C₁₆	C₂×D₃	1	4	yes	—
23	C₃⋊C₃₂	—	6	8	no	—
24	C₂³×C₃²	C₁₆	1	1	yes	characteristic
25	C₂×C₃²×C₄	—	2	1	no	—
26	C₂×C₃²×C₄	—	2	1	no	—
27	C₂×C₃²×C₄	C₁₆	1	1	yes	characteristic
28	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
29	C₃²×C₈	—	2	1	no	—
30	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
31	C₃²×C₈	M₁₆	1	1	yes	characteristic
32	C₂×M₃₂	C₃⋊D₃	1	1	yes	characteristic
33	M₆₄	—	9	2	no	—
34	C₂²×C₃×C₄	C₃⋊C₈	1	4	yes	—
35	C₂×C₃×C₈	—	2	4	no	—
36	C₂×C₃×C₈	—	2	4	no	—
37	C₂×C₃×C₈	C₃⋊C₈	1	4	yes	—
38	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
39	C₃×C₁₆	—	2	4	no	—
40	C₃×C₁₆	#67	1	8	yes	—
41	C₂²×C₃²	—	2	1	no	—
42	C₂²×C₃²	—	4	1	no	—
43	C₂²×C₃²	C₂×C₁₆	1	1	yes	characteristic
44	C₃²×C₄	—	2	1	no	—
45	C₃²×C₄	C₂²⋊C₈	1	1	yes	characteristic
46	C₃²×C₄	M₃₂	1	1	yes	characteristic
47	C₂²×C₈	C₃⋊Dic₃	1	1	yes	characteristic
48	M₃₂	—	2	2	no	—
49	C₂×C₁₆	C₃⋊Dic₃	1	1	yes	characteristic
50	C₂×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
51	C₃₂	—	18	2	no	—
52	C₂³×C₃	C₃⋊C₁₆	1	4	yes	—
53	C₂×C₃×C₄	—	2	4	no	—
54	C₂×C₃×C₄	—	2	4	no	—
55	C₂×C₃×C₄	C₃⋊C₁₆	1	4	yes	—
56	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
57	C₃×C₈	—	2	4	no	—
58	C₃×C₈	#208	1	4	yes	—
59	C₃×C₈	#217	1	4	yes	—
60	C₂×C₃²	—	2	1	no	—
61	C₂×C₃²	—	4	1	no	—
62	C₂×C₃²	C₂²⋊C₁₆	1	1	yes	characteristic
63	C₂²×C₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
67	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	#638	1	2	yes	—
70	C₂²×C₃	—	2	4	no	—
71	C₂²×C₃	—	4	4	no	—
72	C₂²×C₃	C₂×C₃⋊C₁₆	1	4	yes	—
73	C₃×C₄	—	2	4	no	—
74	C₃×C₄	#812	1	4	yes	—
75	C₃×C₄	#830	1	4	yes	—
76	C₃²	#1400	1	1	yes	characteristic, complemented, Sylow
77	C₂³	C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
81	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
82	C₈	—	2	1	no	—
83	C₈	#3748	1	1	yes	characteristic
84	C₈	#3757	1	1	yes	characteristic
85	C₂×C₃	—	2	4	no	—
86	C₂×C₃	—	4	4	no	—
87	C₂×C₃	#4549	1	4	yes	—
88	C₂²	—	2	1	no	—
89	C₂²	—	4	1	no	—
90	C₂²	C₂×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	#63639	1	1	yes	characteristic
93	C₄	#63657	1	1	yes	characteristic
94	C₃	#67809	1	4	yes	complemented
95	C₂	—	2	1	no	—
96	C₂	—	4	1	no	—
97	C₂	#10589415	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#3757
commutator	C₂²×C₃²	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×C₃⋊D₃
Fitting	C₂×C₃²×M₃₂	C₂
socle	C₂×C₃²	C₂²⋊C₁₆
prime radical	C₂³×C₃²	C₁₆
cosocle	C₂×C₃²×C₁₆	C₂²
hypercenter	C₂×M₃₂	C₃⋊D₃
hypocenter	C₃²	#1400
LCS	G ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #10589365	C₂	semidirect
76. C₃²	#1400	semidirect
94. C₃	#67809	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×M₃₂	2
#10589365	2
#67809	3