Subgroups — #28362784

Order 13952.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812610	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic
3	#20294781	C₂	1	4	yes	maximal
4	#376×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724784	C₂²	1	2	yes	—
6	C₂²×#11724784	C₂²	1	2	yes	—
7	#16812610	C₄	1	4	yes	complemented
8	C₄×#11724784	C₄	1	4	yes	complemented
9	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
11	#95×C₁₀₉	C₄	1	4	yes	—
12	C₂×#11724784	—	2	4	no	—
13	C₂×#11724784	—	2	4	no	—
14	C₂×#11724784	C₂×C₄	1	4	yes	—
15	C₂×#11724784	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
19	C₄²×C₁₀₉	—	2	4	no	—
20	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
21	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
22	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₀₉	—	2	4	no	—
24	C₂×C₄×C₁₀₉	—	2	4	no	—
25	C₂×C₄×C₁₀₉	—	2	8	no	—
26	C₂×C₄×C₁₀₉	C₄²	1	4	yes	—
27	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	#41	1	4	yes	—
30	#11724784	—	2	4	no	—
31	#11724784	—	2	4	no	—
32	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
33	C₂²×C₁₀₉	#117	1	2	yes	—
34	C₂²×C₁₀₉	#131	1	4	yes	—
35	C₄×C₁₀₉	—	2	4	no	—
36	C₄×C₁₀₉	—	4	4	no	—
37	C₄×C₁₀₉	#117	1	2	yes	—
38	C₄×C₁₀₉	#117	1	2	yes	—
39	C₂×C₁₀₉	#376	1	1	yes	characteristic
40	C₂×C₁₀₉	#405	1	2	yes	—
41	C₂×C₁₀₉	#433	1	4	yes	—
42	#1746	—	109	1	no	maximal, Sylow
43	C₁₀₉	#1746	1	1	yes	characteristic, complemented, Sylow
44	C₂×#97	—	109	1	no	—
45	C₂×C₄×C₈	—	109	1	no	—
46	#376	D₁₀₉	1	1	yes	characteristic
47	#336	—	109	4	no	—
48	C₂²×C₈	—	109	2	no	—
49	C₂²×C₈	—	109	2	no	—
50	C₂×C₄²	C₂×D₁₀₉	1	2	yes	—
51	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
52	#97	—	109	4	no	—
53	C₄×C₈	—	109	4	no	—
54	#95	Dic₁₀₉	1	4	yes	—
55	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
56	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
57	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
58	C₂×C₈	—	109	4	no	—
59	C₂×C₈	—	109	4	no	—
60	C₂×C₈	—	218	4	no	—
61	C₂×C₈	—	218	4	no	—
62	C₄²	—	2	4	no	—
63	C₄²	C₄×D₁₀₉	1	2	yes	—
64	C₄²	C₄×D₁₀₉	1	2	yes	—
65	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	4	no	—
68	C₂×C₄	—	2	8	no	—
69	C₂×C₄	C₄×Dic₁₀₉	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
71	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
72	C₂×C₄	#13182029	1	4	yes	—
73	C₈	—	218	4	no	—
74	C₈	—	218	4	no	—
75	C₂²	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
76	C₂²	#16812658	1	2	yes	—
77	C₂²	#16812706	1	4	yes	—
78	C₄	—	2	4	no	—
79	C₄	—	4	4	no	—
80	C₄	#16812656	1	2	yes	—
81	C₄	#16812656	1	2	yes	—
82	C₂	#20294872	1	1	yes	characteristic
83	C₂	#20294982	1	2	yes	—
84	C₂	#20295087	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂×C₄²
Fitting	#376×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#376	D₁₀₉
hypocenter	C₁₀₉	#1746
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
7. #16812610	C₄	semidirect
8. C₄×#11724784	C₄	semidirect
43. C₁₀₉	#1746	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812610	2
C₂×C₄×#11724784	2
#20294781	2
#376×C₁₀₉	2
#1746	109