Subgroups — C₂×#63700 #10593118

Order 576.

381 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#63700	C₂	1	4	yes	maximal, direct product factor
2	C₂×#41⋊C₉	C₂	1	3	yes	maximal, complemented
3	C₂×C₃×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic
4	C₂×Q₈×C₉	—	4	1	no	maximal
5	#41⋊C₉	C₂²	1	6	yes	complemented
6	C₂×Q₈⋊C₉	C₂²	1	1	yes	characteristic, complemented
7	C₃×H(5,𝔽₂)	—	3	4	no	—
8	C₃×H(5,𝔽₂)	C₂×C₃	1	4	yes	—
9	C₂×C₃×#41	—	3	1	no	—
10	C₂×C₃×#41	C₂×C₃	1	3	yes	—
11	C₂²×C₃×D₄	—	3	3	no	—
12	Q₈×C₉	—	4	4	no	—
13	C₂×C₄×C₉	—	4	3	no	—
14	Q₈⋊C₉	C₂³	1	1	yes	characteristic, complemented
15	C₂×H(5,𝔽₂)	C₉	1	1	yes	characteristic, complemented, Sylow
16	C₂⁴×C₃	—	3	2	no	—
17	C₃×#41	—	3	4	no	—
18	C₃×#41	—	3	4	no	—
19	C₃×#41	—	3	6	no	—
20	C₃×#41	C₂²×C₃	1	6	yes	—
21	C₂×C₃×Q₈	A₄	1	1	yes	characteristic
22	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
23	C₂×C₃×D₄	—	3	3	no	—
24	C₂×C₃×D₄	—	3	12	no	—
25	C₂×C₃×D₄	—	3	12	no	—
26	C₂×C₃×D₄	—	3	3	no	—
27	C₂²×C₃×C₄	—	3	3	no	—
28	C₂²×C₉	—	4	1	no	—
29	C₄×C₉	—	4	6	no	—
30	H(5,𝔽₂)	—	3	4	no	—
31	H(5,𝔽₂)	C₂×C₉	1	4	yes	complemented
32	C₂×#41	—	3	1	no	—
33	C₂×#41	C₂×C₉	1	3	yes	—
34	C₂²×D₄	—	3	3	no	—
35	C₂³×C₃	—	3	8	no	—
36	C₂³×C₃	—	3	3	no	—
37	C₂³×C₃	—	12	4	no	—
38	C₃×Q₈	—	3	1	no	—
39	C₃×Q₈	C₂×A₄	1	4	yes	—
40	C₃×Q₈	C₂³×C₃	1	1	yes	characteristic
41	C₃×D₄	—	3	6	no	—
42	C₃×D₄	—	3	6	no	—
43	C₃×D₄	—	3	12	no	—
44	C₂×C₃×C₄	—	3	1	no	—
45	C₂×C₃×C₄	—	3	6	no	—
46	C₂×C₃×C₄	—	3	6	no	—
47	C₂×C₃×C₄	C₂×A₄	1	3	yes	—
48	C₂×C₉	—	4	2	no	—
49	C₂×C₉	—	4	1	no	—
50	C₂⁴	—	3	2	no	—
51	#41	—	3	4	no	—
52	#41	—	3	4	no	—
53	#41	—	3	6	no	—
54	#41	C₂²×C₉	1	6	yes	—
55	C₂×Q₈	C₂²⋊C₉	1	1	yes	characteristic
56	C₂×Q₈	C₂²×C₉	1	1	yes	characteristic
57	C₂×D₄	—	3	3	no	—
58	C₂×D₄	—	3	12	no	—
59	C₂×D₄	—	3	12	no	—
60	C₂×D₄	—	3	3	no	—
61	C₂²×C₄	—	3	3	no	—
62	C₂²×C₃	—	3	6	no	—
63	C₂²×C₃	—	6	6	no	—
64	C₂²×C₃	—	12	8	no	—
65	C₂²×C₃	C₂²×A₄	1	1	yes	characteristic
66	C₃×C₄	—	3	1	no	—
67	C₃×C₄	—	3	1	no	—
68	C₃×C₄	C₂²×A₄	1	6	yes	—
69	C₉	—	4	1	no	Sylow
70	C₂³	—	3	8	no	—
71	C₂³	—	3	3	no	—
72	C₂³	—	12	4	no	—
73	Q₈	—	3	1	no	—
74	Q₈	C₂×C₂²⋊C₉	1	4	yes	—
75	Q₈	C₂³×C₉	1	1	yes	characteristic
76	D₄	—	3	6	no	—
77	D₄	—	3	6	no	—
78	D₄	—	3	12	no	—
79	C₂×C₄	—	3	1	no	—
80	C₂×C₄	—	3	6	no	—
81	C₂×C₄	—	3	6	no	—
82	C₂×C₄	C₂×C₂²⋊C₉	1	3	yes	—
83	C₂×C₃	—	6	6	no	—
84	C₂×C₃	#994	1	2	yes	—
85	C₂×C₃	C₂³×A₄	1	1	yes	characteristic
86	C₂²	—	3	6	no	—
87	C₂²	—	6	6	no	—
88	C₂²	—	12	8	no	—
89	C₂²	C₂²×C₂²⋊C₉	1	1	yes	characteristic
90	C₄	—	3	1	no	—
91	C₄	—	3	1	no	—
92	C₄	C₂²×C₂²⋊C₉	1	6	yes	—
93	C₃	C₂×#994	1	1	yes	characteristic
94	C₂	—	6	6	no	—
95	C₂	#63700	1	2	yes	direct product factor
96	C₂	C₂³×C₂²⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×A₄
commutator, hypocenter	Q₈	C₂³×C₉
Frattini	C₂×C₃	C₂³×A₄
Fitting, prime radical	C₂×C₃×H(5,𝔽₂)	C₃
cosocle	C₃×Q₈	C₂³×C₃
hypercenter	C₂×C₃×Q₈	A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. #63700	C₂	direct (N × H)
2. C₂×#41⋊C₉	C₂	semidirect
5. #41⋊C₉	C₂²	semidirect
6. C₂×Q₈⋊C₉	C₂²	semidirect
14. Q₈⋊C₉	C₂³	semidirect
15. C₂×H(5,𝔽₂)	C₉	semidirect
31. H(5,𝔽₂)	C₂×C₉	semidirect
95. C₂	#63700	direct (N × H)

Maximal subgroups

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

subgroup	index
#63700	2
C₂×#41⋊C₉	2
C₂×C₃×H(5,𝔽₂)	3
C₂×Q₈×C₉	4

Subgroups — C2×#63700 #10593118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#63700 #10593118