Subgroups — #42121603

Order 27712.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357216	C₂	1	1	yes	maximal, characteristic
2	#28357135	C₂	1	1	yes	maximal, characteristic, complemented
3	#99×C₄₃₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₇₃₂	C₄	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊D₄₃₃	—	2	1	no	—
6	C₂×C₂²⋊D₄₃₃	—	2	1	no	—
7	C₂²⋊Dic₄₃₃	—	2	1	no	—
8	C₄×Dic₄₃₃	C₄	1	1	yes	characteristic, complemented
9	C₂×D₄×C₄₃₃	C₂²	1	1	yes	characteristic
10	C₂²⋊C₄×C₄₃₃	—	2	1	no	—
11	C₂²×D₄₃₃	—	2	1	no	—
12	D₁₇₃₂	—	4	1	no	—
13	C₂²⋊D₄₃₃	—	4	2	no	—
14	C₂²⋊D₄₃₃	—	4	2	no	—
15	C₂×Dic₄₃₃	—	2	1	no	—
16	C₂×Dic₄₃₃	—	4	1	no	—
17	C₂³×C₄₃₃	D₄	1	1	yes	characteristic
18	C₂³×C₄₃₃	D₄	1	1	yes	characteristic
19	D₄×C₄₃₃	—	4	1	no	—
20	C₂×C₄×C₄₃₃	—	4	1	no	—
21	C₂×C₄×C₄₃₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄₃₃	—	2	1	no	—
23	C₂²×C₄₃₃	—	2	1	no	—
24	C₂²×C₄₃₃	—	4	1	no	—
25	C₂²×C₄₃₃	—	4	1	no	—
26	C₂²×C₄₃₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₄₃₃	—	4	1	no	—
28	C₂×D₄₃₃	—	4	1	no	—
29	C₂×D₄₃₃	—	4	1	no	—
30	C₄×C₄₃₃	—	2	1	no	—
31	C₄×C₄₃₃	—	8	1	no	—
32	Dic₄₃₃	—	2	2	no	—
33	Dic₄₃₃	—	8	1	no	—
34	C₂×C₄₃₃	—	2	1	no	—
35	C₂×C₄₃₃	—	4	1	no	—
36	C₂×C₄₃₃	—	4	1	no	—
37	C₂×C₄₃₃	#99	1	1	yes	characteristic
38	D₄₃₃	—	8	1	no	—
39	C₄₃₃	#353	1	1	yes	characteristic, complemented, Sylow
40	#353	—	433	1	no	maximal, Sylow
41	#127	—	433	1	no	—
42	#99	—	433	1	no	—
43	#99	D₄₃₃	1	1	yes	characteristic, complemented
44	C₂×D₄	—	866	1	no	—
45	C₂×D₄	—	866	1	no	—
46	C₂×D₄	—	433	1	no	—
47	C₂×D₄	C₂×D₄₃₃	1	1	yes	characteristic
48	C₂²⋊C₄	—	2	1	no	—
49	C₂²⋊C₄	—	866	1	no	—
50	C₄²	—	433	1	no	—
51	C₂³	—	866	1	no	—
52	C₂³	C₂²⋊D₄₃₃	1	1	yes	characteristic
53	C₂³	D₁₇₃₂	1	1	yes	characteristic
54	D₄	—	4	1	no	—
55	D₄	—	1732	1	no	—
56	D₄	—	1732	2	no	—
57	D₄	—	1732	2	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	866	1	no	—
60	C₂×C₄	—	1732	1	no	—
61	C₂×C₄	C₄×D₄₃₃	1	1	yes	characteristic
62	C₂²	—	2	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	1732	1	no	—
67	C₂²	—	1732	1	no	—
68	C₂²	—	1732	1	no	—
69	C₂²	#20294074	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	8	1	no	—
72	C₄	—	866	2	no	—
73	C₄	—	3464	1	no	—
74	C₂	—	2	1	no	—
75	C₂	—	4	1	no	—
76	C₂	—	4	1	no	—
77	C₂	—	3464	1	no	—
78	C₂	#28357134	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28357134
commutator	C₂×C₄×C₄₃₃	C₂×C₄
Frattini	C₂×D₄	C₂×D₄₃₃
Fitting	#99×C₄₃₃	C₂
socle	C₂×C₄₃₃	#99
prime radical	#28357216	C₂
cosocle	C₂×D₄×C₄₃₃	C₂²
hypercenter	#99	D₄₃₃
hypocenter	C₄₃₃	#353
LCS	G ▷ C₂×C₄×C₄₃₃ ▷ C₂²×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #99

Complements

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

N	H ≅ G/N	type
2. #28357135	C₂	semidirect
3. #99×C₄₃₃	C₂	semidirect
4. C₂×D₁₇₃₂	C₄	semidirect
8. C₄×Dic₄₃₃	C₄	semidirect
39. C₄₃₃	#353	semidirect
43. #99	D₄₃₃	semidirect

Maximal subgroups

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

subgroup	index
#28357216	2
#28357135	2
#99×C₄₃₃	2
#353	433