Subgroups — #28365319

Order 13952.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295431	C₂	1	1	yes	maximal, characteristic
2	#20294853	C₂	1	2	yes	maximal
3	#20294851	C₂	1	2	yes	maximal
4	#20294802	C₂	1	1	yes	maximal, characteristic
5	#499×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812699	—	2	1	no	—
7	C₂×#13182004	C₂²	1	2	yes	—
8	#16812633	—	2	1	no	—
9	#16812633	C₂²	1	2	yes	—
10	#125×C₁₀₉	—	2	1	no	—
11	#107×C₁₀₉	C₂²	1	1	yes	characteristic
12	#107×C₁₀₉	C₂²	1	1	yes	characteristic
13	#106×C₁₀₉	—	2	1	no	—
14	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₈	—	4	1	no	—
16	C₄⋊Dic₁₀₉	—	2	2	no	—
17	C₄⋊Dic₁₀₉	—	4	1	no	—
18	#13182004	Q₈	1	4	yes	complemented
19	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₀₉	—	2	1	no	—
22	C₄⋊C₄×C₁₀₉	—	2	1	no	—
23	C₄⋊C₄×C₁₀₉	—	4	1	no	—
24	C₄⋊C₄×C₁₀₉	—	4	1	no	—
25	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	—	4	1	no	—
27	C₂×C₄×C₁₀₉	—	4	1	no	—
28	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	—	4	2	no	—
32	Dic₂₁₈	—	8	2	no	—
33	C₈×C₁₀₉	Q₁₆	1	2	yes	—
34	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
35	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₀₉	—	2	1	no	—
37	C₄×C₁₀₉	—	2	1	no	—
38	C₄×C₁₀₉	—	8	1	no	—
39	C₄×C₁₀₉	—	8	1	no	—
40	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
41	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
42	Dic₁₀₉	—	8	2	no	—
43	C₂×C₁₀₉	#500	1	1	yes	characteristic
44	C₂×C₁₀₉	#509	1	1	yes	characteristic
45	C₂×C₁₀₉	#510	1	1	yes	characteristic
46	#2255	—	109	1	no	maximal, Sylow
47	C₁₀₉	#2255	1	1	yes	characteristic, complemented, Sylow
48	#500	—	109	1	no	—
49	#499	D₁₀₉	1	1	yes	characteristic
50	#367	—	109	2	no	—
51	#366	—	109	2	no	—
52	#346	—	109	1	no	—
53	C₄⋊Q₈	—	218	1	no	—
54	#125	—	2	1	no	—
55	C₂×C₁₆	—	109	2	no	—
56	#107	—	109	2	no	—
57	#107	—	218	1	no	—
58	#107	C₂×D₁₀₉	1	1	yes	characteristic
59	#107	C₂×D₁₀₉	1	1	yes	characteristic
60	#106	—	2	1	no	—
61	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
62	C₂×Q₈	—	436	1	no	—
63	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
65	C₄⋊C₄	—	2	1	no	—
66	C₄⋊C₄	—	2	1	no	—
67	C₄⋊C₄	—	4	1	no	—
68	C₄⋊C₄	—	4	1	no	—
69	C₄⋊C₄	—	436	1	no	—
70	C₄⋊C₄	—	218	2	no	—
71	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
72	C₁₆	—	109	4	no	—
73	Q₈	—	872	2	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	436	2	no	—
77	C₂×C₄	#13182014	1	1	yes	characteristic
78	C₂×C₄	#13182014	1	1	yes	characteristic
79	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
80	C₈	#13182019	1	2	yes	—
81	C₈	Q₈×D₁₀₉	1	2	yes	—
82	C₂²	C₂×#13182014	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	8	1	no	—
86	C₄	—	8	1	no	—
87	C₄	—	872	2	no	—
88	C₄	#16812697	1	1	yes	characteristic
89	C₄	C₂×#13182019	1	1	yes	characteristic
90	C₂	#20295429	1	1	yes	characteristic
91	C₂	#20295461	1	1	yes	characteristic
92	C₂	#20295466	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13182014
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#499×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₂×D₈
hypercenter	#499	D₁₀₉
hypocenter	C₁₀₉	#2255
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
18. #13182004	Q₈	semidirect
47. C₁₀₉	#2255	semidirect

Maximal subgroups

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

subgroup	index
#20295431	2
#20294853	2
#20294851	2
#20294802	2
#499×C₁₀₉	2
#2255	109