Subgroups — #28363172

Order 13952.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295050	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812610	C₂	1	1	yes	maximal, characteristic
3	#20294790	C₂	1	4	yes	maximal
4	#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	C₂²	1	2	yes	—
6	C₂²×#11724784	C₂²	1	2	yes	—
7	C₄⋊#11724784	—	2	2	no	—
8	#16812610	C₄	1	4	yes	—
9	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₀₉	—	2	2	no	—
12	#13182013	—	2	4	no	—
13	C₂×#11724784	—	2	2	no	—
14	C₂×#11724784	—	2	2	no	—
15	C₂×#11724784	—	2	2	no	—
16	C₂×#11724784	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₁₀₉	Q₈	1	2	yes	—
18	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
19	M₁₆×C₁₀₉	—	2	4	no	—
20	C₂×C₈×C₁₀₉	—	2	2	no	—
21	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
22	C₄²×C₁₀₉	D₄	1	2	yes	—
23	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	—	4	2	no	—
25	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
26	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	#41	1	2	yes	—
29	C₂×C₄×C₁₀₉	#41	1	2	yes	—
30	C₈×C₁₀₉	—	4	2	no	—
31	#11724784	—	2	4	no	—
32	#11724784	—	4	2	no	—
33	C₂²×C₁₀₉	—	2	2	no	—
34	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₁₀₉	#125	1	2	yes	—
36	C₄×C₁₀₉	—	2	4	no	—
37	C₄×C₁₀₉	#117	1	2	yes	—
38	C₄×C₁₀₉	#124	1	1	yes	characteristic
39	C₄×C₁₀₉	#124	1	1	yes	characteristic
40	C₂×C₁₀₉	—	2	2	no	—
41	C₂×C₁₀₉	#382	1	1	yes	characteristic
42	C₂×C₁₀₉	#430	1	2	yes	—
43	#1838	—	109	1	no	maximal, Sylow
44	C₁₀₉	#1838	1	1	yes	characteristic, complemented, Sylow
45	#423	—	109	1	no	—
46	#423	D₁₀₉	1	1	yes	characteristic
47	C₂×#97	—	109	1	no	—
48	#341	—	109	4	no	—
49	C₂×M₁₆	—	109	2	no	—
50	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
51	C₂²×C₈	—	109	2	no	—
52	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
53	C₄⋊C₈	—	2	2	no	—
54	C₄⋊C₈	—	218	2	no	—
55	#97	—	109	4	no	—
56	C₂²×C₄	Dic₂₁₈	1	2	yes	—
57	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
58	M₁₆	—	2	4	no	—
59	M₁₆	—	218	4	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	109	4	no	—
62	C₂×C₈	—	218	2	no	—
63	C₂×C₈	—	218	2	no	—
64	C₂×C₈	—	218	2	no	—
65	C₄²	C₄×D₁₀₉	1	2	yes	—
66	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
67	C₂³	C₂×Dic₂₁₈	1	1	yes	characteristic
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	#13182008	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
71	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
72	C₂×C₄	#13182028	1	2	yes	—
73	C₂×C₄	#13182030	1	2	yes	—
74	C₈	—	4	2	no	—
75	C₈	—	436	2	no	—
76	C₈	—	218	4	no	—
77	C₂²	—	2	2	no	—
78	C₂²	C₂×#13182008	1	1	yes	characteristic
79	C₂²	#16812690	1	2	yes	—
80	C₄	—	2	4	no	—
81	C₄	#16812657	1	2	yes	—
82	C₄	#16812686	1	1	yes	characteristic
83	C₄	#16812688	1	1	yes	characteristic
84	C₂	—	2	2	no	—
85	C₂	#20294891	1	1	yes	characteristic
86	C₂	#20295074	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#423×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#423	D₁₀₉
hypocenter	C₁₀₉	#1838
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
44. C₁₀₉	#1838	semidirect

Maximal subgroups

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

subgroup	index
#20295050	2
C₂×#16812610	2
#20294790	2
#423×C₁₀₉	2
#1838	109