Subgroups — #28411757

Order 14656.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20301021	C₂	1	2	yes	maximal
2	C₂×Dic₉₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20300994	C₂	1	1	yes	maximal, characteristic
4	#20300906	C₂	1	2	yes	maximal, complemented
5	#108×C₂₂₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16814666	C₂²	1	1	yes	characteristic
7	C₂×Dic₄₅₈	C₂²	1	2	yes	—
8	Dic₉₁₆	—	2	2	no	—
9	Dic₉₁₆	—	2	1	no	—
10	Dic₉₁₆	—	2	1	no	—
11	#16814655	—	2	2	no	—
12	#16814650	C₂²	1	1	yes	characteristic
13	#16814649	—	2	1	no	—
14	C₈×D₂₂₉	—	2	1	no	—
15	M₁₆×C₂₂₉	C₂²	1	2	yes	—
16	C₂×C₈×C₂₂₉	C₂²	1	1	yes	characteristic
17	Dic₄₅₈	—	2	2	no	—
18	Dic₄₅₈	—	2	2	no	—
19	Dic₄₅₈	D₄	1	1	yes	characteristic
20	D₉₁₆	D₄	1	1	yes	characteristic
21	C₂²⋊D₂₂₉	—	2	1	no	—
22	C₂×Dic₂₂₉	—	2	2	no	—
23	C₄×D₂₂₉	—	2	1	no	—
24	#14270080	—	2	1	no	—
25	C₂×C₄×C₂₂₉	C₂³	1	1	yes	characteristic
26	C₈×C₂₂₉	—	2	2	no	—
27	C₈×C₂₂₉	D₄	1	1	yes	characteristic
28	C₈×C₂₂₉	D₄	1	1	yes	characteristic
29	C₂²×C₂₂₉	#41	1	1	yes	characteristic
30	C₂×D₂₂₉	—	2	1	no	—
31	C₄×C₂₂₉	C₂×D₄	1	1	yes	characteristic
32	C₄×C₂₂₉	C₂×D₄	1	1	yes	characteristic
33	Dic₂₂₉	—	2	1	no	—
34	Dic₂₂₉	—	4	2	no	—
35	C₂×C₂₂₉	—	2	1	no	—
36	C₂×C₂₂₉	#121	1	1	yes	characteristic
37	D₂₂₉	—	4	1	no	—
38	C₂₂₉	#473	1	1	yes	characteristic, complemented, Sylow
39	#473	—	229	1	no	maximal, Sylow
40	#137	—	229	2	no	—
41	C₂×Q₁₆	—	229	1	no	—
42	#131	—	229	1	no	—
43	#108	D₂₂₉	1	1	yes	characteristic, complemented
44	#101	—	229	2	no	—
45	#41	—	229	1	no	—
46	C₂×Q₈	—	229	2	no	—
47	Q₁₆	—	458	2	no	—
48	Q₁₆	—	458	1	no	—
49	Q₁₆	—	458	1	no	—
50	SD₁₆	—	458	2	no	—
51	M₁₆	—	458	1	no	—
52	M₁₆	—	229	1	no	—
53	M₁₆	C₂×D₂₂₉	1	2	yes	—
54	C₂×C₈	—	458	1	no	—
55	C₂×C₈	C₂×D₂₂₉	1	1	yes	characteristic
56	Q₈	—	458	2	no	—
57	Q₈	—	458	2	no	—
58	Q₈	—	229	1	no	—
59	D₄	—	458	1	no	—
60	D₄	—	229	1	no	—
61	C₂×C₄	—	458	2	no	—
62	C₂×C₄	—	458	1	no	—
63	C₂×C₄	C₂²×D₂₂₉	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₈	—	458	1	no	—
66	C₈	D₉₁₆	1	1	yes	characteristic
67	C₈	D₉₁₆	1	1	yes	characteristic
68	C₂²	—	458	1	no	—
69	C₂²	#16814667	1	1	yes	characteristic
70	C₄	—	916	2	no	—
71	C₄	—	458	1	no	—
72	C₄	D₄×D₂₂₉	1	1	yes	characteristic
73	C₄	C₂×D₉₁₆	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	916	1	no	—
76	C₂	#20300963	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20300963
commutator, cosocle	C₂×C₄×C₂₂₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₉
Fitting	#108×C₂₂₉	C₂
socle	C₂×C₂₂₉	#121
prime radical	#16814666	C₂²
hypercenter	#108	D₂₂₉
hypocenter	C₂₂₉	#473
LCS	G ▷ C₂×C₄×C₂₂₉ ▷ C₂×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₉₁₆	C₂	semidirect
4. #20300906	C₂	semidirect
5. #108×C₂₂₉	C₂	semidirect
38. C₂₂₉	#473	semidirect
43. #108	D₂₂₉	semidirect

Maximal subgroups

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

subgroup	index
#20301021	2
C₂×Dic₉₁₆	2
#20300994	2
#20300906	2
#108×C₂₂₉	2
#473	229