Subgroups — #28363171

Order 13952.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295049	C₂	1	2	yes	maximal
2	C₂×#16812609	C₂	1	1	yes	maximal, characteristic
3	#20294791	C₂	1	1	yes	maximal, characteristic
4	#20294790	C₂	1	2	yes	maximal
5	#341×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₂²×#11724784	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#16812628	—	2	4	no	—
10	#16812609	C₄	1	4	yes	—
11	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₀₉	Q₈	1	2	yes	—
14	#13182013	—	2	2	no	—
15	#13182013	—	2	2	no	—
16	C₂×#11724784	—	2	2	no	—
17	C₂×#11724784	—	2	2	no	—
18	C₂×#11724784	C₂×C₄	1	2	yes	—
19	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
20	C₄×Dic₁₀₉	D₄	1	2	yes	—
21	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₀₉	—	2	2	no	—
23	M₁₆×C₁₀₉	—	2	2	no	—
24	C₂×C₈×C₁₀₉	—	2	2	no	—
25	C₂×C₈×C₁₀₉	—	2	2	no	—
26	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	#41	1	2	yes	—
31	C₂×C₄×C₁₀₉	#41	1	2	yes	—
32	C₂×Dic₁₀₉	—	4	2	no	—
33	C₂×Dic₁₀₉	C₄⋊C₄	1	4	yes	—
34	C₈×C₁₀₉	—	2	2	no	—
35	C₈×C₁₀₉	—	4	2	no	—
36	#11724784	—	2	2	no	—
37	#11724784	—	4	2	no	—
38	C₂²×C₁₀₉	—	2	2	no	—
39	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₁₀₉	#125	1	2	yes	—
41	C₄×C₁₀₉	#117	1	1	yes	characteristic
42	C₄×C₁₀₉	#117	1	1	yes	characteristic
43	C₄×C₁₀₉	#124	1	2	yes	—
44	Dic₁₀₉	—	2	4	no	—
45	C₂×C₁₀₉	—	2	2	no	—
46	C₂×C₁₀₉	#382	1	1	yes	characteristic
47	C₂×C₁₀₉	#430	1	2	yes	—
48	#1838	—	109	1	no	maximal, Sylow
49	C₁₀₉	#1838	1	1	yes	characteristic, complemented, Sylow
50	#423	—	109	2	no	—
51	C₂×#97	—	109	1	no	—
52	#341	—	109	2	no	—
53	#341	—	109	1	no	—
54	#341	D₁₀₉	1	1	yes	characteristic
55	C₂×M₁₆	—	109	2	no	—
56	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
57	C₂²×C₈	—	109	1	no	—
58	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
59	C₂×C₄²	—	109	1	no	—
60	C₄⋊C₈	—	218	4	no	—
61	#97	—	109	4	no	—
62	C₂²×C₄	—	109	2	no	—
63	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
64	M₁₆	—	2	2	no	—
65	M₁₆	—	2	2	no	—
66	M₁₆	—	218	2	no	—
67	M₁₆	—	218	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	109	2	no	—
71	C₂×C₈	—	218	2	no	—
72	C₂×C₈	—	218	2	no	—
73	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
74	C₄²	—	109	2	no	—
75	C₄²	—	109	2	no	—
76	C₂³	Q₈×D₁₀₉	1	1	yes	characteristic
77	C₂×C₄	—	109	4	no	—
78	C₂×C₄	—	436	2	no	—
79	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
80	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
81	C₂×C₄	#13182028	1	2	yes	—
82	C₂×C₄	#13182029	1	2	yes	—
83	C₈	—	2	2	no	—
84	C₈	—	4	2	no	—
85	C₈	—	436	2	no	—
86	C₈	—	218	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
89	C₂²	#16812689	1	2	yes	—
90	C₄	—	218	4	no	—
91	C₄	#16812655	1	1	yes	characteristic
92	C₄	#16812656	1	1	yes	characteristic
93	C₄	#16812685	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	#20294890	1	1	yes	characteristic
96	C₂	#20295073	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#341×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#341	D₁₀₉
hypocenter	C₁₀₉	#1838
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

Complements

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

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

Maximal subgroups

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

subgroup	index
#20295049	2
C₂×#16812609	2
#20294791	2
#20294790	2
#341×C₁₀₉	2
#1838	109