Subgroups — #28379367

Order 13952.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812773	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13182032	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13182032	C₂	1	4	yes	maximal, complemented
4	#20294982	C₂	1	1	yes	maximal, characteristic
5	#16812773	—	2	2	no	—
6	C₂×#13182032	—	2	2	no	—
7	C₂×#13182032	C₂²	1	2	yes	—
8	C₂×#13182032	C₂²	1	2	yes	—
9	C₂×#13182013	C₄	1	1	yes	characteristic
10	C₂²×#11724784	C₂²	1	1	yes	characteristic
11	#16812610	C₄	1	2	yes	—
12	C₄×#11724784	C₂²	1	2	yes	—
13	#117×C₁₀₉	C₄	1	1	yes	characteristic
14	#13182032	—	2	2	no	—
15	#13182032	—	2	2	no	—
16	#13182013	—	2	2	no	—
17	C₂×#11724784	C₂×C₄	1	2	yes	—
18	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
19	C₂×#11724784	D₄	1	2	yes	—
20	C₂×#11724784	Q₈	1	2	yes	—
21	C₂×#11724784	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₁₀₉	C₈	1	2	yes	—
24	C₂²⋊C₄×C₁₀₉	C₈	1	2	yes	—
25	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₀₉	C₂×C₈	1	4	yes	—
29	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	M₁₆	1	2	yes	—
31	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	#11724784	—	2	2	no	—
33	#11724784	C₄⋊C₄	1	2	yes	—
34	#11724784	C₂×D₄	1	1	yes	characteristic
35	#11724784	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₁₀₉	—	2	2	no	—
37	C₂²×C₁₀₉	C₄⋊C₈	1	2	yes	—
38	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₁₀₉	—	2	4	no	—
40	C₄×C₁₀₉	C₄⋊C₈	1	2	yes	—
41	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₁₀₉	—	2	2	no	—
44	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₀₉	#504	1	2	yes	—
46	#2151	—	109	1	no	maximal, Sylow
47	C₁₀₉	#2151	1	1	yes	characteristic, complemented, Sylow
48	C₂×M₃₂	—	109	1	no	—
49	C₂²×C₁₆	—	109	1	no	—
50	#405	—	109	1	no	—
51	C₄⋊C₁₆	—	109	4	no	—
52	C₂×M₁₆	—	109	1	no	—
53	C₂²×C₈	—	109	1	no	—
54	#117	#89137	1	1	yes	characteristic
55	M₃₂	—	218	2	no	—
56	C₂×C₁₆	—	109	2	no	—
57	C₂×C₁₆	—	109	2	no	—
58	C₂×C₁₆	—	218	2	no	—
59	#97	—	109	2	no	—
60	C₄×C₈	—	109	2	no	—
61	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
62	M₁₆	—	218	2	no	—
63	C₂×C₈	—	109	2	no	—
64	C₂×C₈	—	109	2	no	—
65	C₂×C₈	—	109	1	no	—
66	C₂×C₈	—	109	1	no	—
67	C₂×C₈	—	109	2	no	—
68	C₄⋊C₄	#11724786	1	2	yes	—
69	C₂²⋊C₄	#11724786	1	2	yes	—
70	C₄²	C₂×#89137	1	2	yes	—
71	C₁₆	—	218	2	no	—
72	C₁₆	—	218	2	no	—
73	C₂³	C₂×#11724786	1	1	yes	characteristic
74	C₂×C₄	C₄⋊#89137	1	2	yes	—
75	C₂×C₄	C₂×#11724786	1	4	yes	—
76	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
77	C₂×C₄	#13182039	1	2	yes	—
78	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
79	C₈	—	109	1	no	—
80	C₈	—	109	2	no	—
81	C₈	—	109	1	no	—
82	C₈	—	218	2	no	—
83	C₂²	—	2	2	no	—
84	C₂²	C₄⋊#11724786	1	2	yes	—
85	C₂²	C₂²×#11724786	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	C₄⋊#11724786	1	2	yes	—
88	C₄	C₂×C₄⋊#89137	1	1	yes	characteristic
89	C₄	C₂×#13182039	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊#11724786	1	1	yes	characteristic
92	C₂	#20296084	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂²×C₈
Fitting	#117×C₁₀₉	C₄
prime radical	C₂³×C₁₀₉	C₂×C₈
cosocle	C₂×#11724784	C₂³
hypercenter	#117	#89137
hypocenter	C₁₀₉	#2151
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. C₄⋊#13182032	C₂	semidirect
47. C₁₀₉	#2151	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
C₂²×#13182032	2
C₄⋊#13182032	2
#20294982	2
#2151	109