Subgroups — #28377955

Order 13952.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#13182032	C₂	1	2	yes	maximal, complemented
2	#20294994	C₂	1	1	yes	maximal, characteristic
3	C₂×#13182032	—	2	2	no	—
4	C₂×#13182029	C₄	1	1	yes	characteristic
5	C₂²×#11724784	C₂²	1	1	yes	characteristic
6	#16812611	—	2	2	no	—
7	C₂×M₁₆×C₁₀₉	C₄	1	1	yes	characteristic
8	#13182032	—	4	2	no	—
9	#13182029	—	4	1	no	—
10	#13182029	—	4	1	no	—
11	C₂×Dic₂₁₈	C₈	1	1	yes	characteristic
12	C₂×D₄₃₆	C₈	1	1	yes	characteristic
13	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
14	C₂×C₄×D₁₀₉	—	2	1	no	—
15	C₂×#11724784	—	4	1	no	—
16	C₂×#11724784	D₄	1	2	yes	—
17	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₁₀₉	—	4	1	no	—
19	C₂×C₈×C₁₀₉	—	2	1	no	—
20	C₂³×C₁₀₉	M₁₆	1	1	yes	characteristic
21	C₂²×D₁₀₉	—	2	1	no	—
22	C₂×C₄×C₁₀₉	—	2	1	no	—
23	C₂×C₄×C₁₀₉	—	2	1	no	—
24	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
26	C₂²⋊D₁₀₉	—	4	2	no	—
27	C₂×Dic₁₀₉	—	2	1	no	—
28	D₄₃₆	—	4	1	no	—
29	C₄×D₁₀₉	—	4	1	no	—
30	C₄×D₁₀₉	—	4	1	no	—
31	Dic₂₁₈	—	4	1	no	—
32	C₈×C₁₀₉	—	4	1	no	—
33	#11724784	—	2	2	no	—
34	C₂²×C₁₀₉	—	2	1	no	—
35	C₂²×C₁₀₉	—	2	2	no	—
36	C₂²×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
37	C₂×D₁₀₉	—	4	1	no	—
38	C₂×D₁₀₉	—	4	2	no	—
39	C₄×C₁₀₉	—	2	1	no	—
40	C₄×C₁₀₉	#99	1	1	yes	characteristic
41	C₄×C₁₀₉	#100	1	1	yes	characteristic
42	Dic₁₀₉	—	4	1	no	—
43	C₂×C₁₀₉	—	4	1	no	—
44	C₂×C₁₀₉	#323	1	1	yes	characteristic
45	C₂×C₁₀₉	#350	1	2	yes	—
46	D₁₀₉	—	8	1	no	—
47	#1315	—	109	1	no	maximal, Sylow
48	C₁₀₉	#1315	1	1	yes	characteristic, complemented, Sylow
49	#408	—	109	1	no	—
50	C₂²⋊C₁₆	—	109	2	no	—
51	C₂×#41	—	109	1	no	—
52	C₂×M₁₆	#89137	1	1	yes	characteristic
53	C₂²×C₈	—	109	1	no	—
54	C₂×C₁₆	—	218	2	no	—
55	C₂²⋊C₈	—	218	2	no	—
56	#41	—	436	1	no	—
57	#41	—	436	1	no	—
58	C₂×Q₈	—	109	1	no	—
59	C₂×D₄	—	109	1	no	—
60	C₂×D₄	—	218	1	no	—
61	C₂²×C₄	—	218	1	no	—
62	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
63	M₁₆	—	4	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	109	2	no	—
66	C₂×C₈	—	436	1	no	—
67	C₁₆	—	436	2	no	—
68	C₂³	—	218	1	no	—
69	C₂³	#13182039	1	1	yes	characteristic
70	Q₈	—	436	1	no	—
71	D₄	—	436	1	no	—
72	D₄	—	436	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	436	1	no	—
76	C₂×C₄	—	436	1	no	—
77	C₂×C₄	—	218	1	no	—
78	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
79	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
80	C₈	—	4	1	no	—
81	C₈	—	218	2	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	436	1	no	—
85	C₂²	—	436	2	no	—
86	C₂²	#16812756	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	436	1	no	—
89	C₄	#16812758	1	1	yes	characteristic
90	C₄	#16812759	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	—	872	1	no	—
93	C₂	#20295870	1	1	yes	characteristic
94	C₂	#20295925	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂×C₈
Frattini	C₂²×C₄	C₂×#89137
Fitting	C₂×M₁₆×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂²⋊C₈
prime radical	C₂×#13182029	C₄
cosocle	C₂²×#11724784	C₂²
hypercenter	C₂×M₁₆	#89137
hypocenter	C₁₀₉	#1315
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²⋊#13182032	C₂	semidirect
48. C₁₀₉	#1315	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#13182032	2
#20294994	2
#1315	109