Subgroups — #28379429

Order 13952.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×#89137	C₂	1	2	yes	maximal
2	#20295953	C₂	1	1	yes	maximal, characteristic
3	#20295939	C₂	1	2	yes	maximal
4	#20295913	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×#89137	—	2	2	no	—
7	D₁₀₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16812768	C₂²	1	2	yes	—
9	#16812762	—	2	2	no	—
10	C₈×#89137	C₂²	1	1	yes	characteristic
11	Q₁₆×D₁₀₉	C₂²	1	2	yes	—
12	Dic₈₇₂	C₄	1	1	yes	characteristic, complemented
13	#16812645	C₄	1	2	yes	complemented
14	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
15	Q₃₂×C₁₀₉	C₄	1	1	yes	characteristic, complemented
16	D₁₀₉⋊C₈	D₄	1	1	yes	characteristic
17	C₄⋊#89137	—	2	2	no	—
18	C₄⋊#89137	—	4	2	no	—
19	C₄×#89137	—	4	2	no	—
20	C₄×#89137	D₄	1	1	yes	characteristic
21	#13182032	—	2	1	no	—
22	Q₈×D₁₀₉	—	2	2	no	—
23	Dic₄₃₆	C₂×C₄	1	2	yes	—
24	#13182019	—	2	2	no	—
25	C₈×D₁₀₉	C₂³	1	1	yes	characteristic
26	#13182004	C₂×C₄	1	1	yes	characteristic
27	Q₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×#89137	—	4	2	no	—
30	C₂×#89137	D₈	1	2	yes	—
31	Q₈×C₁₀₉	—	2	2	no	—
32	C₄×D₁₀₉	—	4	2	no	—
33	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
34	Dic₂₁₈	—	2	2	no	—
35	Dic₂₁₈	—	4	2	no	—
36	#11724786	—	2	1	no	—
37	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	#11724784	#41	1	1	yes	characteristic
39	C₂×D₁₀₉	C₂×D₈	1	1	yes	characteristic
40	#89137	—	2	2	no	—
41	#89137	—	8	2	no	—
42	C₄×C₁₀₉	—	4	2	no	—
43	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
44	Dic₁₀₉	—	4	2	no	—
45	Dic₁₀₉	#135	1	1	yes	characteristic
46	C₂×C₁₀₉	C₄×D₈	1	1	yes	characteristic
47	D₁₀₉	#510	1	2	yes	—
48	#2176	—	109	1	no	maximal, Sylow
49	C₁₀₉	#2176	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₃₂	—	109	1	no	—
51	C₄×Q₁₆	—	109	2	no	—
52	#367	—	109	1	no	—
53	#358	—	109	2	no	—
54	#346	—	109	1	no	—
55	C₂×Q₁₆	—	109	2	no	—
56	C₄×Q₈	—	218	2	no	—
57	Q₃₂	—	109	2	no	—
58	Q₃₂	—	109	1	no	—
59	Q₃₂	#89137	1	1	yes	characteristic, complemented
60	C₂×C₁₆	—	109	1	no	—
61	C₂×C₁₆	—	109	1	no	—
62	#107	—	109	2	no	—
63	#103	—	218	2	no	—
64	C₄×C₈	—	109	1	no	—
65	C₂×Q₈	—	218	2	no	—
66	Q₁₆	—	109	2	no	—
67	Q₁₆	—	218	2	no	—
68	Q₁₆	C₂×#89137	1	2	yes	—
69	C₂×C₈	—	109	1	no	—
70	C₂×C₈	—	109	1	no	—
71	C₄⋊C₄	—	436	2	no	—
72	C₄⋊C₄	—	218	2	no	—
73	C₄²	—	109	1	no	—
74	C₄²	—	436	2	no	—
75	C₁₆	—	109	1	no	—
76	C₁₆	—	218	1	no	—
77	C₁₆	C₂×#89137	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	Q₈	—	436	2	no	—
80	Q₈	—	218	2	no	—
81	C₂×C₄	—	109	1	no	—
82	C₂×C₄	—	109	2	no	—
83	C₂×C₄	—	436	2	no	—
84	C₂×C₄	—	436	2	no	—
85	C₈	—	109	1	no	—
86	C₈	—	218	1	no	—
87	C₈	C₂²×#89137	1	1	yes	characteristic
88	C₂²	—	109	1	no	—
89	C₄	—	4	2	no	—
90	C₄	—	872	2	no	—
91	C₄	—	109	1	no	—
92	C₄	—	436	2	no	—
93	C₄	—	218	2	no	—
94	C₄	D₄×#89137	1	1	yes	characteristic
95	C₂	—	109	2	no	—
96	C₂	D₈×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#89137
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₈	C₂²×#89137
Fitting	Q₃₂×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₄×D₈
prime radical	C₂×D₁₀₉	C₂×D₈
cosocle	C₈×D₁₀₉	C₂³
hypercenter	Q₃₂	#89137
hypocenter	C₁₀₉	#2176
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
12. Dic₈₇₂	C₄	semidirect
13. #16812645	C₄	semidirect
15. Q₃₂×C₁₀₉	C₄	semidirect
49. C₁₀₉	#2176	semidirect
59. Q₃₂	#89137	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×#89137	2
#20295953	2
#20295939	2
#20295913	2
Q₃₂×D₁₀₉	2
#2176	109