Finite Group Databasebeta

Subgroups — #27197044

Order 13184.

154 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20268229	C₂	1	1	yes	maximal, characteristic
2	#20267914	C₂	1	1	yes	maximal, characteristic, complemented
3	#20267904	C₂	1	1	yes	maximal, characteristic, complemented
4	#20267614	C₂	1	1	yes	maximal, characteristic, complemented
5	#20267605	C₂	1	1	yes	maximal, characteristic, complemented
6	#20267602	C₂	1	1	yes	maximal, characteristic, complemented
7	#347×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#15693043	C₂²	1	1	yes	characteristic
9	C₂×D₈₂₄	C₂²	1	1	yes	characteristic
10	#15693027	—	2	1	no	—
11	C₄×D₄₁₂	—	2	1	no	—
12	#15692966	C₄	1	2	yes	complemented
13	D₁₆₄₈	C₄	1	2	yes	complemented
14	#15692957	C₂²	1	1	yes	characteristic
15	#15692953	C₂²	1	1	yes	characteristic
16	#15692949	—	2	1	no	—
17	#15692943	—	2	1	no	—
18	M₃₂×C₁₀₃	C₂²	1	1	yes	characteristic
19	M₃₂×C₁₀₃	C₂²	1	1	yes	characteristic
20	#97×C₁₀₃	C₂²	1	1	yes	characteristic
21	#13110105	—	2	1	no	—
22	C₂×D₄₁₂	—	2	1	no	—
23	C₂×C₄×D₁₀₃	—	4	1	no	—
24	Dic₄₁₂	C₂×C₄	1	1	yes	characteristic, complemented
25	#13110094	—	2	1	no	—
26	D₈₂₄	—	2	1	no	—
27	D₈₂₄	C₂×C₄	1	2	yes	—
28	D₈₂₄	C₂×C₄	1	1	yes	characteristic, complemented
29	#13110089	—	2	1	no	—
30	#13110088	—	4	1	no	—
31	C₈×D₁₀₃	—	4	1	no	—
32	C₄⋊Dic₁₀₃	—	2	1	no	—
33	(C₂×D₁₀₃)⋊C₄	—	4	1	no	—
34	C₂×C₈×C₁₀₃	D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₀₃	C₂³	1	1	yes	characteristic
36	C₄²×C₁₀₃	D₄	1	1	yes	characteristic
37	C₁₆×C₁₀₃	—	2	1	no	—
38	C₁₆×C₁₀₃	C₂×C₄	1	2	yes	complemented
39	C₂²×D₁₀₃	—	4	1	no	—
40	C₂×C₄×C₁₀₃	—	2	1	no	—
41	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
42	C₂²⋊D₁₀₃	—	4	1	no	—
43	C₂×Dic₁₀₃	—	4	1	no	—
44	D₄₁₂	—	2	2	no	—
45	D₄₁₂	—	2	1	no	—
46	D₄₁₂	—	4	1	no	—
47	C₄×D₁₀₃	—	4	1	no	—
48	C₄×D₁₀₃	—	8	2	no	—
49	Dic₂₀₆	—	2	1	no	—
50	C₈×C₁₀₃	—	2	1	no	—
51	C₈×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
52	C₈×C₁₀₃	#41	1	1	yes	characteristic
53	C₁₀₃⋊C₈	—	4	1	no	—
54	C₂²×C₁₀₃	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×D₁₀₃	—	4	1	no	—
56	C₂×D₁₀₃	—	4	2	no	—
57	C₂×D₁₀₃	—	8	1	no	—
58	C₂×D₁₀₃	—	8	1	no	—
59	C₄×C₁₀₃	—	4	1	no	—
60	C₄×C₁₀₃	C₄×D₄	1	1	yes	characteristic
61	C₄×C₁₀₃	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₁₀₃	—	4	1	no	—
63	Dic₁₀₃	—	8	1	no	—
64	C₂×C₁₀₃	—	2	1	no	—
65	C₂×C₁₀₃	#442	1	1	yes	characteristic
66	D₁₀₃	—	8	1	no	—
67	D₁₀₃	—	8	2	no	—
68	AGL(1,ℤ/16)	—	103	1	no	maximal, Sylow
69	C₁₀₃	AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
70	#509	—	103	1	no	—
71	#444	—	103	1	no	—
72	#442	—	103	1	no	—
73	#365	—	103	1	no	—
74	#361	—	103	1	no	—
75	#360	—	103	1	no	—
76	#347	D₁₀₃	1	1	yes	characteristic, complemented
77	#135	—	103	1	no	—
78	C₂×D₈	—	103	1	no	—
79	#131	—	206	1	no	—
80	C₄×D₄	—	206	1	no	—
81	SD₃₂	—	103	2	no	—
82	D₁₆	—	103	2	no	—
83	M₃₂	C₂×D₁₀₃	1	1	yes	characteristic
84	M₃₂	C₂×D₁₀₃	1	1	yes	characteristic
85	#108	—	103	1	no	—
86	#106	—	103	1	no	—
87	#104	—	206	1	no	—
88	#102	—	206	1	no	—
89	#97	C₂×D₁₀₃	1	1	yes	characteristic
90	#41	—	206	1	no	—
91	C₂×D₄	—	206	1	no	—
92	C₂²×C₄	—	412	1	no	—
93	Q₁₆	—	103	1	no	—
94	SD₁₆	—	206	1	no	—
95	D₈	—	103	2	no	—
96	D₈	—	103	1	no	—
97	D₈	—	206	1	no	—
98	M₁₆	—	412	1	no	—
99	M₁₆	—	206	1	no	—
100	C₂×C₈	—	412	1	no	—
101	C₂×C₈	D₄₁₂	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₁₀₃	1	1	yes	characteristic
103	C₄⋊C₄	—	206	1	no	—
104	C₂²⋊C₄	—	412	1	no	—
105	C₄²	D₄₁₂	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	C₄×D₁₀₃	1	2	yes	complemented
108	C₂³	—	412	1	no	—
109	Q₈	—	206	1	no	—
110	D₄	—	412	1	no	—
111	D₄	—	412	1	no	—
112	D₄	—	206	2	no	—
113	D₄	—	206	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	824	2	no	—
116	C₂×C₄	—	412	1	no	—
117	C₂×C₄	—	412	1	no	—
118	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	412	1	no	—
121	C₈	C₂×C₄×D₁₀₃	1	1	yes	characteristic
122	C₈	#13110105	1	1	yes	characteristic
123	C₂²	—	824	1	no	—
124	C₂²	—	824	1	no	—
125	C₂²	—	412	1	no	—
126	C₂²	—	412	2	no	—
127	C₂²	#15693049	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	824	1	no	—
130	C₄	—	412	1	no	—
131	C₄	C₄×D₄₁₂	1	1	yes	characteristic
132	C₄	#15693049	1	1	yes	characteristic
133	C₂	—	2	1	no	—
134	C₂	—	824	2	no	—
135	C₂	—	824	1	no	—
136	C₂	#20267904	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20267904
commutator	C₈×C₁₀₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₃
Fitting	#347×C₁₀₃	C₂
socle	C₂×C₁₀₃	#442
prime radical	#20268229	C₂
cosocle	C₂×C₈×C₁₀₃	C₂³
hypercenter	#347	D₁₀₃
hypocenter	C₁₀₃	AGL(1,ℤ/16)
LCS	G ▷ C₈×C₁₀₃ ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
2. #20267914	C₂	semidirect
3. #20267904	C₂	semidirect
4. #20267614	C₂	semidirect
5. #20267605	C₂	semidirect
6. #20267602	C₂	semidirect
7. #347×C₁₀₃	C₂	semidirect
12. #15692966	C₄	semidirect
13. D₁₆₄₈	C₄	semidirect
24. Dic₄₁₂	C₂×C₄	semidirect
28. D₈₂₄	C₂×C₄	semidirect
38. C₁₆×C₁₀₃	C₂×C₄	semidirect
69. C₁₀₃	AGL(1,ℤ/16)	semidirect
76. #347	D₁₀₃	semidirect
107. C₁₆	C₄×D₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
#20268229	2
#20267914	2
#20267904	2
#20267614	2
#20267605	2
#20267602	2
#347×C₁₀₃	2
AGL(1,ℤ/16)	103