Subgroups — #28377984

Order 13952.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13182032	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295914	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295104	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182032	—	2	1	no	—
5	C₂×#13182032	—	2	1	no	—
6	C₂×#13182013	—	2	1	no	—
7	C₄⋊#11724784	C₄	1	1	yes	characteristic
8	C₂²⋊#11724784	—	2	1	no	—
9	C₄×#11724784	C₂²	1	1	yes	characteristic
10	C₄×D₄×C₁₀₉	C₄	1	1	yes	characteristic
11	#13182032	—	2	2	no	—
12	#13182032	—	4	1	no	—
13	#13182013	—	4	1	no	—
14	#13182013	—	4	1	no	—
15	C₂×#11724784	—	2	1	no	—
16	C₂×#11724784	D₄	1	1	yes	characteristic
17	C₂×#11724784	D₄	1	1	yes	characteristic
18	C₂×D₄×C₁₀₉	C₈	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₉	—	2	1	no	—
20	C₄⋊C₄×C₁₀₉	C₈	1	1	yes	characteristic
21	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
22	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₁₀₉	—	2	1	no	—
24	D₄×C₁₀₉	—	4	1	no	—
25	C₂×C₄×C₁₀₉	—	2	1	no	—
26	C₂×C₄×C₁₀₉	—	4	1	no	—
27	C₂×C₄×C₁₀₉	—	4	1	no	—
28	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
31	#11724784	—	2	1	no	—
32	#11724784	—	4	1	no	—
33	#11724784	D₈	1	1	yes	characteristic
34	#11724784	SD₁₆	1	1	yes	characteristic
35	C₂²×C₁₀₉	—	4	1	no	—
36	C₂²×C₁₀₉	—	4	1	no	—
37	C₂²×C₁₀₉	—	4	1	no	—
38	C₂²×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₀₉	—	2	1	no	—
40	C₄×C₁₀₉	—	2	1	no	—
41	C₄×C₁₀₉	—	4	1	no	—
42	C₄×C₁₀₉	#102	1	1	yes	characteristic
43	C₄×C₁₀₉	#104	1	1	yes	characteristic
44	C₂×C₁₀₉	—	8	1	no	—
45	C₂×C₁₀₉	D₄⋊C₈	1	1	yes	characteristic
46	C₂×C₁₀₉	#349	1	1	yes	characteristic
47	C₂×C₁₀₉	#350	1	1	yes	characteristic
48	#1330	—	109	1	no	maximal, Sylow
49	C₁₀₉	#1330	1	1	yes	characteristic, complemented, Sylow
50	#436	—	109	1	no	—
51	C₄⋊C₁₆	—	109	1	no	—
52	#346	—	109	1	no	—
53	C₂×M₁₆	—	218	1	no	—
54	C₄×D₄	#89137	1	1	yes	characteristic
55	C₂×C₁₆	—	218	1	no	—
56	C₂×C₁₆	—	218	1	no	—
57	C₄⋊C₈	—	109	1	no	—
58	C₂²⋊C₈	—	218	1	no	—
59	C₄×C₈	—	109	1	no	—
60	C₂×D₄	#11724786	1	1	yes	characteristic
61	C₂²×C₄	—	2	1	no	—
62	M₁₆	—	436	1	no	—
63	M₁₆	—	436	1	no	—
64	C₂×C₈	—	109	1	no	—
65	C₂×C₈	—	109	1	no	—
66	C₂×C₈	—	218	1	no	—
67	C₄⋊C₄	#11724786	1	1	yes	characteristic
68	C₂²⋊C₄	—	2	1	no	—
69	C₄²	C₂×#89137	1	1	yes	characteristic
70	C₁₆	—	436	1	no	—
71	C₁₆	—	218	2	no	—
72	C₂³	—	2	1	no	—
73	D₄	—	4	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
78	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
79	C₂×C₄	#13182038	1	1	yes	characteristic
80	C₈	—	109	1	no	—
81	C₈	—	109	1	no	—
82	C₈	—	436	1	no	—
83	C₈	—	218	1	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	C₂²⋊#11724786	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	#16812761	1	1	yes	characteristic
92	C₄	#16812763	1	1	yes	characteristic
93	C₂	—	8	1	no	—
94	C₂	D₄⋊#11724786	1	1	yes	characteristic
95	C₂	#20295922	1	1	yes	characteristic
96	C₂	#20295924	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂×C₈
Frattini	C₄²	C₂×#89137
Fitting	C₄×D₄×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂²⋊C₈
prime radical	C₂×D₄×C₁₀₉	C₈
cosocle	C₄×#11724784	C₂²
hypercenter	C₄×D₄	#89137
hypocenter	C₁₀₉	#1330
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊#13182032	C₂	semidirect
2. #20295914	C₂	semidirect
49. C₁₀₉	#1330	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13182032	2
#20295914	2
#20295104	2
#1330	109