Subgroups — #28364792

Order 13952.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13182004	C₂	1	1	yes	maximal, characteristic
2	#20294981	C₂	1	1	yes	maximal, characteristic
3	#20294844	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	C₄	1	1	yes	characteristic
6	#16812658	C₄	1	1	yes	characteristic
7	C₂×#13182004	—	2	2	no	—
8	C₂×#13182004	C₂²	1	2	yes	—
9	#16812609	C₄	1	2	yes	—
10	C₈×Dic₁₀₉	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	M₃₂×C₁₀₉	—	2	2	no	—
13	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	—
14	#13182013	—	2	2	no	—
15	C₂×#11724784	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₁₀₉	C₈	1	2	yes	—
17	C₂²⋊Dic₁₀₉	C₈	1	2	yes	—
18	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
19	#13182004	—	2	2	no	—
20	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
23	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
24	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₁₆×C₁₀₉	—	2	2	no	—
26	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
30	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	C₂×C₈	1	4	yes	—
32	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
33	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
34	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
35	#11724784	—	2	2	no	—
36	C₂²×C₁₀₉	—	2	2	no	—
37	C₂²×C₁₀₉	C₄⋊C₈	1	2	yes	—
38	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
40	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
42	Dic₁₀₉	—	2	4	no	—
43	C₂×C₁₀₉	—	2	2	no	—
44	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₀₉	#504	1	2	yes	—
46	#2151	—	109	1	no	maximal, Sylow
47	C₁₀₉	#2151	1	1	yes	characteristic, complemented, Sylow
48	C₂×M₃₂	D₁₀₉	1	1	yes	characteristic
49	C₂²×C₁₆	—	109	1	no	—
50	#405	—	109	1	no	—
51	C₄⋊C₁₆	—	109	4	no	—
52	C₂×M₁₆	—	109	1	no	—
53	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
54	#117	—	109	1	no	—
55	M₃₂	—	2	2	no	—
56	C₂×C₁₆	—	109	2	no	—
57	C₂×C₁₆	—	218	2	no	—
58	C₂×C₁₆	C₂×D₁₀₉	1	2	yes	—
59	#97	—	109	2	no	—
60	C₄×C₈	—	109	2	no	—
61	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
62	M₁₆	—	218	2	no	—
63	C₂×C₈	—	109	2	no	—
64	C₂×C₈	Dic₂₁₈	1	2	yes	—
65	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
67	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
68	C₄⋊C₄	—	109	2	no	—
69	C₂²⋊C₄	—	109	2	no	—
70	C₄²	—	109	2	no	—
71	C₁₆	—	2	2	no	—
72	C₁₆	—	218	2	no	—
73	C₂³	C₈×D₁₀₉	1	1	yes	characteristic
74	C₂×C₄	—	109	4	no	—
75	C₂×C₄	#13182008	1	2	yes	—
76	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
77	C₂×C₄	#13182012	1	2	yes	—
78	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
79	C₈	—	218	2	no	—
80	C₈	#13182008	1	2	yes	—
81	C₈	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
82	C₈	C₂×Dic₂₁₈	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	#16812628	1	2	yes	—
85	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
86	C₄	—	218	4	no	—
87	C₄	#16812628	1	2	yes	—
88	C₄	C₂×#13182008	1	1	yes	characteristic
89	C₄	C₂×#13182012	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×#16812628	1	1	yes	characteristic
92	C₂	#20295444	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂²×C₈
Fitting	C₂×M₃₂×C₁₀₉	C₂
prime radical	C₂³×C₁₀₉	C₂×C₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	C₂×M₃₂	D₁₀₉
hypocenter	C₁₀₉	#2151
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20294844	C₂	semidirect
47. C₁₀₉	#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13182004	2
#20294981	2
#20294844	2
C₂×M₃₂×C₁₀₉	2
#2151	109