Subgroups — #28361027

Order 13952.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295284	C₂	1	1	yes	maximal, characteristic
2	#20294784	C₂	1	1	yes	maximal, characteristic
3	#349×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₃₆	C₄	1	1	yes	characteristic
5	C₂×#13182013	—	2	1	no	—
6	#16812678	—	2	1	no	—
7	#16812631	C₄	1	1	yes	characteristic
8	#16812624	—	2	1	no	—
9	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	M₃₂×C₁₀₉	—	2	1	no	—
11	C₂×Dic₂₁₈	—	2	1	no	—
12	Dic₄₃₆	—	4	1	no	—
13	#13182013	—	4	1	no	—
14	#13182013	—	4	1	no	—
15	C₂×#11724784	—	2	2	no	—
16	C₄⋊Dic₁₀₉	—	2	1	no	—
17	#13182008	—	4	1	no	—
18	C₂²⋊Dic₁₀₉	—	4	1	no	—
19	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
20	M₁₆×C₁₀₉	—	4	1	no	—
21	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
23	C₁₆×C₁₀₉	—	4	1	no	—
24	C₂³×C₁₀₉	SD₁₆	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	—	2	1	no	—
26	C₂×C₄×C₁₀₉	—	2	1	no	—
27	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
29	C₂×Dic₁₀₉	—	4	1	no	—
30	C₂×Dic₁₀₉	—	4	1	no	—
31	Dic₂₁₈	—	4	1	no	—
32	Dic₂₁₈	—	4	1	no	—
33	C₈×C₁₀₉	—	2	1	no	—
34	C₈×C₁₀₉	—	2	1	no	—
35	#11724784	—	4	2	no	—
36	C₂²×C₁₀₉	—	2	1	no	—
37	C₂²×C₁₀₉	—	4	1	no	—
38	C₂²×C₁₀₉	#102	1	1	yes	characteristic
39	C₄×C₁₀₉	—	2	1	no	—
40	C₄×C₁₀₉	#99	1	1	yes	characteristic
41	C₄×C₁₀₉	#104	1	1	yes	characteristic
42	Dic₁₀₉	—	8	1	no	—
43	Dic₁₀₉	—	8	1	no	—
44	C₂×C₁₀₉	—	2	1	no	—
45	C₂×C₁₀₉	—	4	1	no	—
46	C₂×C₁₀₉	#327	1	1	yes	characteristic
47	#1342	—	109	1	no	maximal, Sylow
48	C₁₀₉	#1342	1	1	yes	characteristic, complemented, Sylow
49	#470	—	109	1	no	—
50	#349	D₁₀₉	1	1	yes	characteristic
51	#338	—	109	1	no	—
52	C₂×Q₁₆	—	109	1	no	—
53	C₂×M₁₆	—	218	1	no	—
54	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
55	C₂²⋊Q₈	—	218	1	no	—
56	M₃₂	—	2	1	no	—
57	#106	—	109	1	no	—
58	#103	—	218	1	no	—
59	C₂×Q₈	—	218	1	no	—
60	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
61	Q₁₆	—	436	1	no	—
62	M₁₆	—	4	1	no	—
63	M₁₆	—	436	1	no	—
64	M₁₆	—	436	1	no	—
65	C₂×C₈	—	218	2	no	—
66	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
67	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
68	C₄⋊C₄	—	436	1	no	—
69	C₄⋊C₄	—	218	1	no	—
70	C₂²⋊C₄	—	436	1	no	—
71	C₁₆	—	4	1	no	—
72	C₂³	#13182018	1	1	yes	characteristic
73	Q₈	—	436	1	no	—
74	Q₈	—	436	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	436	1	no	—
78	C₂×C₄	—	436	1	no	—
79	C₂×C₄	#13182006	1	1	yes	characteristic
80	C₂×C₄	D₈₇₂	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	2	1	no	—
83	C₈	—	436	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	#16812621	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	872	1	no	—
89	C₄	—	872	1	no	—
90	C₄	#16812613	1	1	yes	characteristic
91	C₄	#16812625	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	4	1	no	—
94	C₂	#20294761	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20294761
commutator	C₂×C₈×C₁₀₉	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₁₀₉
Fitting	#349×C₁₀₉	C₂
socle	C₂×C₁₀₉	#327
prime radical	C₂³×C₁₀₉	SD₁₆
cosocle	C₂×M₁₆×C₁₀₉	C₂²
hypercenter	#349	D₁₀₉
hypocenter	C₁₀₉	#1342
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

Complements

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

N	H ≅ G/N	type
48. C₁₀₉	#1342	semidirect

Maximal subgroups

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

subgroup	index
#20295284	2
#20294784	2
#349×C₁₀₉	2
#1342	109