Subgroups — #28363206

Order 13952.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812634	C₂	1	4	yes	maximal
2	#20295050	C₂	1	1	yes	maximal, characteristic
3	C₂×#16812610	C₂	1	1	yes	maximal, characteristic
4	#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	C₂²	1	2	yes	—
6	C₂²×#11724784	C₂²	1	2	yes	—
7	#16812634	—	2	8	no	—
8	C₄⋊#11724784	—	2	2	no	—
9	#16812610	C₄	1	4	yes	—
10	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₀₉	—	2	2	no	—
13	#13182013	—	2	4	no	—
14	C₂×#11724784	—	2	2	no	—
15	C₂×#11724784	C₂×C₄	1	4	yes	—
16	C₂×#11724784	D₄	1	4	yes	—
17	C₂×#11724784	Q₈	1	4	yes	—
18	C₂²×C₄×C₁₀₉	Q₈	1	2	yes	—
19	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
20	M₁₆×C₁₀₉	—	2	4	no	—
21	C₂×C₈×C₁₀₉	—	2	2	no	—
22	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
23	C₄²×C₁₀₉	D₄	1	2	yes	—
24	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	—	4	2	no	—
26	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
27	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
29	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
31	C₈×C₁₀₉	—	4	2	no	—
32	#11724784	—	4	2	no	—
33	#11724784	C₄⋊C₄	1	8	yes	—
34	C₂²×C₁₀₉	—	2	2	no	—
35	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₁₀₉	C₄⋊Q₈	1	2	yes	—
37	C₄×C₁₀₉	—	2	4	no	—
38	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
39	C₄×C₁₀₉	#127	1	1	yes	characteristic
40	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
41	C₂×C₁₀₉	—	2	2	no	—
42	C₂×C₁₀₉	#384	1	1	yes	characteristic
43	C₂×C₁₀₉	#430	1	2	yes	—
44	#1849	—	109	1	no	maximal, Sylow
45	C₁₀₉	#1849	1	1	yes	characteristic, complemented, Sylow
46	C₂×#108	—	109	4	no	—
47	#423	—	109	1	no	—
48	#423	D₁₀₉	1	1	yes	characteristic
49	C₂×#97	—	109	1	no	—
50	C₂×M₁₆	—	109	2	no	—
51	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
52	C₂²×C₈	—	109	2	no	—
53	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
54	#108	—	218	8	no	—
55	C₄⋊C₈	—	2	2	no	—
56	C₄⋊C₈	—	218	2	no	—
57	#97	—	109	4	no	—
58	C₂²×C₄	Dic₂₁₈	1	2	yes	—
59	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
60	M₁₆	—	2	4	no	—
61	M₁₆	—	218	4	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	109	4	no	—
64	C₂×C₈	—	109	4	no	—
65	C₂×C₈	—	109	4	no	—
66	C₂×C₈	—	218	2	no	—
67	C₄²	C₄×D₁₀₉	1	2	yes	—
68	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
69	C₂³	C₂×Dic₂₁₈	1	1	yes	characteristic
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	#13182008	1	4	yes	—
72	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
73	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
74	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
75	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
76	C₈	—	4	2	no	—
77	C₈	—	109	8	no	—
78	C₈	—	436	2	no	—
79	C₂²	—	2	2	no	—
80	C₂²	C₂×#13182008	1	1	yes	characteristic
81	C₂²	#16812698	1	2	yes	—
82	C₄	—	2	4	no	—
83	C₄	C₄⋊C₄×D₁₀₉	1	2	yes	—
84	C₄	#16812695	1	1	yes	characteristic
85	C₄	#16812697	1	1	yes	characteristic
86	C₂	—	2	2	no	—
87	C₂	#20294895	1	1	yes	characteristic
88	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	#1849
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
45. C₁₀₉	#1849	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812634	2
#20295050	2
C₂×#16812610	2
#423×C₁₀₉	2
#1849	109