Subgroups — #28362663

Order 13952.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295424	C₂	1	1	yes	maximal, characteristic
2	#20295422	C₂	1	2	yes	maximal
3	#20295378	C₂	1	2	yes	maximal
4	#20294747	C₂	1	1	yes	maximal, characteristic
5	#494×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812699	C₂²	1	2	yes	—
7	#16812631	—	2	2	no	—
8	#16812630	—	2	4	no	—
9	#16812622	—	2	4	no	—
10	C₄×#11724784	C₂²	1	2	yes	—
11	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
12	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₈	—	2	2	no	—
16	C₂×#11724784	D₄	1	4	yes	—
17	C₄⋊Dic₁₀₉	—	2	2	no	—
18	C₄⋊Dic₁₀₉	—	2	2	no	—
19	C₂×Q₈×C₁₀₉	—	2	1	no	—
20	C₂×Q₈×C₁₀₉	—	2	1	no	—
21	Q₁₆×C₁₀₉	—	4	4	no	—
22	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₀₉	—	2	2	no	—
24	C₄⋊C₄×C₁₀₉	—	2	2	no	—
25	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
26	Q₈×C₁₀₉	—	4	2	no	—
27	Q₈×C₁₀₉	—	4	2	no	—
28	C₂×C₄×C₁₀₉	—	4	1	no	—
29	C₂×C₄×C₁₀₉	—	4	1	no	—
30	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
31	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
32	C₂×Dic₁₀₉	—	4	2	no	—
33	Dic₂₁₈	—	4	4	no	—
34	C₈×C₁₀₉	SD₁₆	1	4	yes	—
35	#11724784	—	2	4	no	—
36	C₂²×C₁₀₉	#127	1	1	yes	characteristic
37	C₄×C₁₀₉	—	8	1	no	—
38	C₄×C₁₀₉	—	8	1	no	—
39	C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
40	C₄×C₁₀₉	#137	1	4	yes	—
41	Dic₁₀₉	—	8	2	no	—
42	C₂×C₁₀₉	#492	1	1	yes	characteristic
43	C₂×C₁₀₉	#497	1	2	yes	—
44	#1716	—	109	1	no	maximal, Sylow
45	C₁₀₉	#1716	1	1	yes	characteristic, complemented, Sylow
46	#498	—	109	2	no	—
47	#498	—	109	1	no	—
48	#494	D₁₀₉	1	1	yes	characteristic
49	#487	—	109	2	no	—
50	#322	—	109	1	no	—
51	C₂×Q₁₆	—	2	2	no	—
52	C₄⋊Q₈	—	109	2	no	—
53	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
54	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
55	#106	—	218	4	no	—
56	#106	—	218	2	no	—
57	#103	—	218	4	no	—
58	C₄×C₈	—	109	2	no	—
59	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	218	2	no	—
63	Q₁₆	—	4	4	no	—
64	C₂×C₈	—	109	4	no	—
65	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	218	2	no	—
69	C₄⋊C₄	—	218	2	no	—
70	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	Q₈	—	436	4	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	436	2	no	—
77	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
78	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
79	C₈	—	218	4	no	—
80	C₈	#13182017	1	4	yes	—
81	C₂²	#16812695	1	1	yes	characteristic
82	C₄	—	8	1	no	—
83	C₄	—	8	1	no	—
84	C₄	—	872	2	no	—
85	C₄	C₂×#13182017	1	2	yes	—
86	C₄	#16812731	1	4	yes	—
87	C₂	#20295398	1	1	yes	characteristic
88	C₂	#20295419	1	2	yes	—

	subgroup	G/H
center	C₂²	#16812695
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#494×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#127
hypercenter	#494	D₁₀₉
hypocenter	C₁₀₉	#1716
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
45. C₁₀₉	#1716	semidirect

Maximal subgroups

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

subgroup	index
#20295424	2
#20295422	2
#20295378	2
#20294747	2
#494×C₁₀₉	2
#1716	109