Subgroups — #28379435

Order 13952.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20296070	C₂	1	2	yes	maximal, complemented
2	#20295955	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295941	C₂	1	2	yes	maximal, complemented
4	#20295913	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295459	C₂	1	1	yes	maximal, characteristic
6	#16812787	—	2	2	no	—
7	D₁₀₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16812770	C₂²	1	2	yes	—
9	#16812764	—	2	2	no	—
10	C₈×#89137	C₂²	1	1	yes	characteristic
11	#16812722	C₂²	1	2	yes	—
12	#16812642	C₄	1	2	yes	complemented
13	D₁₇₄₄	C₄	1	1	yes	characteristic, complemented
14	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
15	Q₃₂×C₁₀₉	C₄	1	1	yes	characteristic, complemented
16	#13182039	—	2	2	no	—
17	#13182039	—	4	2	no	—
18	D₁₀₉⋊C₈	—	4	2	no	—
19	D₁₀₉⋊C₈	D₄	1	1	yes	characteristic
20	C₄×#89137	D₄	1	1	yes	characteristic
21	#13182032	—	2	1	no	—
22	#13182030	—	2	2	no	—
23	#13182016	—	2	2	no	—
24	D₈₇₂	C₂×C₄	1	2	yes	—
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	—	2	1	no	—
30	Q₈×C₁₀₉	—	2	2	no	—
31	D₄₃₆	—	2	2	no	—
32	D₄₃₆	—	4	2	no	—
33	C₄×D₁₀₉	—	4	2	no	—
34	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
35	#11724786	—	4	2	no	—
36	#11724786	D₈	1	2	yes	—
37	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	#11724784	#41	1	1	yes	characteristic
39	C₂×D₁₀₉	—	4	2	no	—
40	C₂×D₁₀₉	#135	1	1	yes	characteristic
41	#89137	—	4	1	no	—
42	C₄×C₁₀₉	—	4	2	no	—
43	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
44	Dic₁₀₉	C₂×D₈	1	1	yes	characteristic
45	C₂×C₁₀₉	C₄×D₈	1	1	yes	characteristic
46	D₁₀₉	—	2	1	no	—
47	D₁₀₉	—	8	2	no	—
48	#2179	—	109	1	no	maximal, Sylow
49	C₁₀₉	#2179	1	1	yes	characteristic, complemented, Sylow
50	#508	—	109	1	no	—
51	#443	—	109	2	no	—
52	#368	—	109	1	no	—
53	#359	—	109	2	no	—
54	#346	—	109	1	no	—
55	#135	—	109	2	no	—
56	#131	—	218	2	no	—
57	Q₃₂	#89137	1	1	yes	characteristic, complemented
58	SD₃₂	—	109	2	no	—
59	D₁₆	—	109	1	no	—
60	C₂×C₁₆	—	109	1	no	—
61	C₂×C₁₆	—	109	1	no	—
62	#108	—	109	2	no	—
63	#104	—	218	2	no	—
64	C₄×C₈	—	109	1	no	—
65	#41	—	218	2	no	—
66	Q₁₆	C₂×#89137	1	2	yes	—
67	SD₁₆	—	218	2	no	—
68	D₈	—	109	2	no	—
69	M₁₆	—	436	2	no	—
70	M₁₆	—	218	2	no	—
71	C₂×C₈	—	109	1	no	—
72	C₂×C₈	—	109	1	no	—
73	C₂×C₈	—	436	2	no	—
74	C₄²	—	109	1	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	D₄	—	436	2	no	—
80	D₄	—	218	2	no	—
81	C₂×C₄	—	109	1	no	—
82	C₂×C₄	—	436	2	no	—
83	C₂×C₄	—	218	1	no	—
84	C₈	—	109	2	no	—
85	C₈	—	109	1	no	—
86	C₈	—	436	2	no	—
87	C₈	C₂²×#89137	1	1	yes	characteristic
88	C₂²	—	109	1	no	—
89	C₂²	—	436	2	no	—
90	C₄	—	4	2	no	—
91	C₄	—	109	1	no	—
92	C₄	—	436	1	no	—
93	C₄	D₄×#89137	1	1	yes	characteristic
94	C₂	—	872	2	no	—
95	C₂	—	218	1	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	#20295459	C₂
cosocle	C₈×D₁₀₉	C₂³
hypercenter	Q₃₂	#89137
hypocenter	C₁₀₉	#2179
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
1. #20296070	C₂	semidirect
2. #20295955	C₂	semidirect
3. #20295941	C₂	semidirect
4. #20295913	C₂	semidirect
12. #16812642	C₄	semidirect
13. D₁₇₄₄	C₄	semidirect
15. Q₃₂×C₁₀₉	C₄	semidirect
49. C₁₀₉	#2179	semidirect
57. Q₃₂	#89137	semidirect

Maximal subgroups

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

subgroup	index
#20296070	2
#20295955	2
#20295941	2
#20295913	2
#20295459	2
#2179	109