Subgroups — #27196543

Order 13184.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20267839	C₂	1	3	yes	maximal, complemented
2	C₂×#15692937	C₂	1	3	yes	maximal
3	#338×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₄₁₂	C₂²	1	1	yes	characteristic
5	C₂×#13110089	C₂²	1	3	yes	—
6	#15692956	—	2	6	no	—
7	#15692937	—	2	6	no	—
8	C₂×M₁₆×C₁₀₃	C₂²	1	3	yes	—
9	C₂³×D₁₀₃	Q₈	1	2	yes	—
10	C₂×D₄₁₂	—	2	3	no	—
11	C₂×D₄₁₂	—	2	3	no	—
12	#13110089	—	2	3	no	—
13	#13110089	D₄	1	6	yes	—
14	C₂×C₁₀₃⋊C₈	—	2	3	no	—
15	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
16	M₁₆×C₁₀₃	—	2	3	no	—
17	M₁₆×C₁₀₃	—	2	3	no	—
18	C₂×C₈×C₁₀₃	—	2	3	no	—
19	C₂³×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
20	C₂²×D₁₀₃	—	2	6	no	—
21	C₂²×D₁₀₃	—	8	2	no	—
22	C₂×C₄×C₁₀₃	C₂×D₄	1	3	yes	—
23	C₂×C₄×C₁₀₃	#41	1	3	yes	—
24	D₄₁₂	—	4	3	no	—
25	D₄₁₂	—	4	1	no	—
26	C₈×C₁₀₃	—	4	3	no	—
27	C₁₀₃⋊C₈	—	2	6	no	—
28	C₂²×C₁₀₃	—	4	1	no	—
29	C₂²×C₁₀₃	C₂²⋊Q₈	1	3	yes	—
30	C₂×D₁₀₃	—	4	2	no	—
31	C₂×D₁₀₃	—	8	6	no	—
32	C₄×C₁₀₃	#121	1	3	yes	—
33	C₄×C₁₀₃	#126	1	1	yes	characteristic
34	C₂×C₁₀₃	—	2	3	no	—
35	C₂×C₁₀₃	#396	1	1	yes	characteristic
36	D₁₀₃	—	8	2	no	—
37	#2069	—	103	1	no	maximal, Sylow
38	C₁₀₃	#2069	1	1	yes	characteristic, complemented, Sylow
39	#430	—	103	3	no	—
40	C₂×#100	—	103	3	no	—
41	#338	D₁₀₃	1	1	yes	characteristic, complemented
42	C₂²×D₄	—	103	1	no	—
43	C₂×M₁₆	—	103	3	no	—
44	C₂×M₁₆	C₂×D₁₀₃	1	3	yes	—
45	#108	—	206	6	no	—
46	#100	—	206	6	no	—
47	C₂⁴	—	103	2	no	—
48	C₂×D₄	—	206	3	no	—
49	C₂×D₄	—	206	3	no	—
50	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
51	M₁₆	—	2	3	no	—
52	M₁₆	—	2	3	no	—
53	M₁₆	—	103	6	no	—
54	M₁₆	—	206	3	no	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	—	206	3	no	—
57	C₂³	—	824	2	no	—
58	C₂³	—	206	6	no	—
59	C₂³	Q₈×D₁₀₃	1	1	yes	characteristic
60	D₄	—	412	3	no	—
61	D₄	—	412	1	no	—
62	C₂×C₄	D₄×D₁₀₃	1	3	yes	—
63	C₂×C₄	#13110105	1	3	yes	—
64	C₈	—	4	3	no	—
65	C₈	—	206	6	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	824	6	no	—
68	C₂²	—	412	2	no	—
69	C₂²	#15692999	1	3	yes	—
70	C₄	#15692994	1	3	yes	—
71	C₄	#15693015	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	824	2	no	—
74	C₂	#20267715	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20267715
commutator, cosocle	C₂²×C₄×C₁₀₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#338×C₁₀₃	C₂
socle	C₂×C₁₀₃	#396
prime radical	C₂²×D₄₁₂	C₂²
hypercenter	#338	D₁₀₃
hypocenter	C₁₀₃	#2069
LCS	G ▷ C₂²×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #20267839	C₂	semidirect
3. #338×C₁₀₃	C₂	semidirect
38. C₁₀₃	#2069	semidirect
41. #338	D₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
#20267839	2
C₂×#15692937	2
#338×C₁₀₃	2
#2069	103