Subgroups — #28361116

Order 13952.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812639	C₂	1	2	yes	maximal
2	C₂×#97×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#16812639	—	2	2	no	—
4	#16812639	C₄	1	4	yes	complemented
5	C₂×#13182004	—	2	2	no	—
6	C₂²×C₈×C₁₀₉	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₀₉	C₄	1	1	yes	characteristic
9	#97×C₁₀₉	—	2	2	no	—
10	#13182004	—	2	4	no	—
11	C₂²×C₄×C₁₀₉	C₈	1	2	yes	—
12	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₁₀₉	—	2	1	no	—
14	C₂×C₈×C₁₀₉	—	2	1	no	—
15	C₂×C₈×C₁₀₉	C₈	1	2	yes	—
16	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
19	C₂×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
20	C₄²×C₁₀₉	—	2	1	no	—
21	C₄²×C₁₀₉	—	2	1	no	—
22	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₁₀₉	—	2	2	no	—
24	C₂×C₄×C₁₀₉	—	4	2	no	—
25	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
31	C₈×C₁₀₉	—	2	2	no	—
32	C₈×C₁₀₉	C₂²⋊C₄	1	2	yes	—
33	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
34	C₂²×C₁₀₉	—	2	2	no	—
35	C₂²×C₁₀₉	C₄×C₈	1	1	yes	characteristic
36	C₂²×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
37	C₂²×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
38	C₄×C₁₀₉	—	4	2	no	—
39	C₄×C₁₀₉	#95	1	1	yes	characteristic
40	C₄×C₁₀₉	#97	1	1	yes	characteristic
41	C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
42	C₄×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
43	C₂×C₁₀₉	—	2	1	no	—
44	C₂×C₁₀₉	—	2	1	no	—
45	C₂×C₁₀₉	#336	1	1	yes	characteristic
46	C₂×C₁₀₉	#349	1	2	yes	—
47	#1377	—	109	1	no	maximal, Sylow
48	C₁₀₉	#1377	1	1	yes	characteristic, complemented, Sylow
49	C₂×M₃₂	—	109	2	no	—
50	C₂×#97	D₁₀₉	1	1	yes	characteristic
51	C₂²×C₈	Dic₁₀₉	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
53	C₂×C₄²	Dic₁₀₉	1	1	yes	characteristic
54	M₃₂	—	109	4	no	—
55	M₃₂	—	218	2	no	—
56	C₂×C₁₆	—	218	2	no	—
57	#97	—	2	2	no	—
58	C₂²×C₄	#11724784	1	2	yes	—
59	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	#11724784	1	2	yes	—
63	C₂×C₈	Dic₂₁₈	1	1	yes	characteristic
64	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
65	C₂×C₈	D₄₃₆	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
67	C₄²	—	2	1	no	—
68	C₄²	—	2	1	no	—
69	C₁₆	—	218	4	no	—
70	C₂³	C₂×#11724784	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	C₄×Dic₁₀₉	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
75	C₂×C₄	C₄⋊Dic₁₀₉	1	1	yes	characteristic
76	C₂×C₄	C₂×#11724784	1	1	yes	characteristic
77	C₂×C₄	#13182013	1	1	yes	characteristic
78	C₂×C₄	#13182013	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	#13182006	1	2	yes	—
81	C₈	#13182008	1	2	yes	—
82	C₂²	—	2	2	no	—
83	C₂²	C₄×#11724784	1	1	yes	characteristic
84	C₂²	C₂²⋊#11724784	1	1	yes	characteristic
85	C₂²	C₄⋊#11724784	1	1	yes	characteristic
86	C₄	—	4	2	no	—
87	C₄	#16812606	1	1	yes	characteristic
88	C₄	#16812610	1	1	yes	characteristic
89	C₄	C₂²⋊#11724784	1	1	yes	characteristic
90	C₄	C₄⋊#11724784	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	2	1	no	—
93	C₂	#20294781	1	1	yes	characteristic
94	C₂	#20294808	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₁₀₉
commutator	C₂²×C₁₀₉	C₄×C₈
Frattini	C₂²×C₈	C₂×D₁₀₉
Fitting	C₂×#97×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²⋊C₈
prime radical	C₂³×C₁₀₉	C₂×C₈
cosocle	C₂²×C₈×C₁₀₉	C₂²
hypercenter	C₂×#97	D₁₀₉
hypocenter	C₁₀₉	#1377
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
4. #16812639	C₄	semidirect
48. C₁₀₉	#1377	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812639	2
C₂×#97×C₁₀₉	2
#1377	109