Subgroups — #28379373

Order 13952.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812773	C₂	1	2	yes	maximal
2	#20295948	C₂	1	4	yes	maximal, complemented
3	#20294984	C₂	1	1	yes	maximal, characteristic
4	#16812773	—	2	2	no	—
5	#16812773	C₂²	1	4	yes	—
6	C₂×#13182032	—	2	2	no	—
7	C₂²×#11724784	C₂²	1	1	yes	characteristic
8	#16812658	C₄	1	1	yes	characteristic
9	#16812609	C₄	1	2	yes	—
10	C₈×Dic₁₀₉	C₂²	1	2	yes	—
11	C₂×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
12	#13182032	—	2	4	no	—
13	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
14	C₂×#11724784	D₄	1	2	yes	—
15	C₂×#11724784	Q₈	1	2	yes	—
16	C₂×#11724784	C₂³	1	1	yes	characteristic
17	C₄⋊Dic₁₀₉	—	2	1	no	—
18	C₂²⋊Dic₁₀₉	—	2	1	no	—
19	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	M₁₆×C₁₀₉	C₈	1	4	yes	—
22	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₂³×C₁₀₉	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
26	C₂×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₁₀₉	—	2	2	no	—
29	C₈×C₁₀₉	C₂×C₈	1	4	yes	—
30	#11724784	C₄⋊C₄	1	2	yes	—
31	#11724784	C₂×D₄	1	1	yes	characteristic
32	#11724784	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₁₀₉	—	4	1	no	—
34	C₂²×C₁₀₉	C₄⋊C₈	1	2	yes	—
35	C₂²×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
36	C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
37	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
39	Dic₁₀₉	—	4	2	no	—
40	C₂×C₁₀₉	—	2	1	no	—
41	C₂×C₁₀₉	—	2	2	no	—
42	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
43	#2153	—	109	1	no	maximal, Sylow
44	C₁₀₉	#2153	1	1	yes	characteristic, complemented, Sylow
45	C₂×M₃₂	—	109	2	no	—
46	#405	—	109	1	no	—
47	#364	—	109	4	no	—
48	C₂×M₁₆	#89137	1	1	yes	characteristic
49	C₂²×C₈	—	109	1	no	—
50	#117	—	109	1	no	—
51	M₃₂	—	109	4	no	—
52	M₃₂	—	218	2	no	—
53	C₂×C₁₆	—	218	2	no	—
54	#97	—	109	2	no	—
55	C₄×C₈	—	109	2	no	—
56	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
57	M₁₆	#11724786	1	4	yes	—
58	C₂×C₈	—	109	2	no	—
59	C₂×C₈	—	109	2	no	—
60	C₂×C₈	—	109	1	no	—
61	C₂×C₈	—	109	1	no	—
62	C₂×C₈	C₂×#89137	1	2	yes	—
63	C₄⋊C₄	—	218	1	no	—
64	C₂²⋊C₄	—	218	1	no	—
65	C₄²	—	109	2	no	—
66	C₁₆	—	218	4	no	—
67	C₂³	#13182039	1	1	yes	characteristic
68	C₂×C₄	—	218	2	no	—
69	C₂×C₄	C₄⋊#89137	1	2	yes	—
70	C₂×C₄	C₂×#11724786	1	2	yes	—
71	C₂×C₄	#13182039	1	1	yes	characteristic
72	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
73	C₈	—	109	1	no	—
74	C₈	—	109	2	no	—
75	C₈	—	109	1	no	—
76	C₈	C₂×#11724786	1	4	yes	—
77	C₂²	—	4	1	no	—
78	C₂²	C₄⋊#11724786	1	2	yes	—
79	C₂²	C₂×#13182039	1	1	yes	characteristic
80	C₄	—	436	2	no	—
81	C₄	C₄⋊#11724786	1	2	yes	—
82	C₄	C₂×C₄⋊#89137	1	1	yes	characteristic
83	C₄	C₂²×#11724786	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×C₄⋊#11724786	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₄⋊#89137
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂×M₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×C₄⋊C₈
prime radical	C₂³×C₁₀₉	M₁₆
cosocle	C₂×#11724784	C₂³
hypercenter	C₂×M₁₆	#89137
hypocenter	C₁₀₉	#2153
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20295948	C₂	semidirect
44. C₁₀₉	#2153	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812773	2
#20295948	2
#20294984	2
#2153	109