Subgroups — #28362634

Order 13952.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295399	C₂	1	1	yes	maximal, characteristic
2	#20295376	C₂	1	2	yes	maximal, complemented
3	#20295374	C₂	1	2	yes	maximal
4	C₈×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13182018	—	2	2	no	—
7	#16812699	C₂²	1	1	yes	characteristic
8	#16812696	C₂²	1	1	yes	characteristic
9	#16812623	—	2	4	no	—
10	#16812619	—	2	4	no	—
11	C₄×#11724784	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
13	#106×C₁₀₉	—	2	2	no	—
14	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₈	—	2	1	no	—
16	C₂×D₄₃₆	—	2	2	no	—
17	C₂×D₄₃₆	—	2	1	no	—
18	#13182018	—	4	4	no	—
19	C₂×#11724784	D₄	1	4	yes	—
20	C₄⋊Dic₁₀₉	—	2	2	no	—
21	C₂×Q₈×C₁₀₉	—	2	2	no	—
22	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₀₉	—	2	2	no	—
24	C₄⋊C₄×C₁₀₉	—	2	2	no	—
25	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂²×D₁₀₉	—	4	1	no	—
27	Q₈×C₁₀₉	—	4	4	no	—
28	C₂×C₄×C₁₀₉	—	4	2	no	—
29	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
30	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	—	4	1	no	—
32	D₄₃₆	—	4	2	no	—
33	D₄₃₆	—	4	4	no	—
34	Dic₂₁₈	—	4	2	no	—
35	C₈×C₁₀₉	SD₁₆	1	4	yes	—
36	#11724784	—	2	4	no	—
37	C₂²×C₁₀₉	#127	1	1	yes	characteristic
38	C₂×D₁₀₉	—	8	1	no	—
39	C₂×D₁₀₉	—	8	2	no	—
40	C₄×C₁₀₉	—	8	2	no	—
41	C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
42	C₄×C₁₀₉	#135	1	4	yes	—
43	Dic₁₀₉	—	8	1	no	—
44	C₂×C₁₀₉	#492	1	1	yes	characteristic
45	C₂×C₁₀₉	#495	1	2	yes	—
46	D₁₀₉	—	16	1	no	—
47	#1708	—	109	1	no	maximal, Sylow
48	C₁₀₉	#1708	1	1	yes	characteristic, complemented, Sylow
49	#498	D₁₀₉	1	1	yes	characteristic, complemented
50	#492	—	109	1	no	—
51	#487	—	109	2	no	—
52	#486	—	109	2	no	—
53	C₈²	—	109	1	no	—
54	C₂×SD₁₆	—	218	2	no	—
55	C₄⋊Q₈	—	109	1	no	—
56	C₄⋊Q₈	C₂×D₁₀₉	1	2	yes	—
57	#127	—	109	1	no	—
58	#106	—	2	2	no	—
59	#103	—	218	4	no	—
60	#102	—	218	4	no	—
61	C₄×C₈	—	109	2	no	—
62	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
63	C₂×Q₈	—	2	2	no	—
64	C₂×Q₈	—	218	1	no	—
65	C₂×D₄	—	218	2	no	—
66	C₂×D₄	—	218	1	no	—
67	SD₁₆	—	436	4	no	—
68	C₂×C₈	—	109	4	no	—
69	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	218	2	no	—
73	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
74	C₂³	—	436	1	no	—
75	Q₈	—	4	4	no	—
76	Q₈	—	436	2	no	—
77	D₄	—	436	2	no	—
78	D₄	—	436	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	—	436	1	no	—
81	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
83	C₈	—	218	4	no	—
84	C₈	#13182016	1	4	yes	—
85	C₂²	—	872	1	no	—
86	C₂²	—	872	2	no	—
87	C₂²	#16812695	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	—	872	1	no	—
90	C₄	C₂×#13182016	1	2	yes	—
91	C₄	#16812720	1	4	yes	—
92	C₂	—	1744	1	no	—
93	C₂	#20295400	1	1	yes	characteristic
94	C₂	#20295407	1	2	yes	—

	subgroup	G/H
center	C₂²	#16812695
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#498×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#127
prime radical	#16812696	C₂²
hypercenter	#498	D₁₀₉
hypocenter	C₁₀₉	#1708
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

Complements

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

N	H ≅ G/N	type
2. #20295376	C₂	semidirect
4. C₈×#11724784	C₂	semidirect
5. #498×C₁₀₉	C₂	semidirect
48. C₁₀₉	#1708	semidirect
49. #498	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295399	2
#20295376	2
#20295374	2
C₈×#11724784	2
#498×C₁₀₉	2
#1708	109