Finite Group Databasebeta

Subgroups — #27208292

Order 13184.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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	#20268608	C₂	1	1	yes	maximal, characteristic, complemented
2	#20268596	C₂	1	1	yes	maximal, characteristic, complemented
3	#20268234	C₂	1	3	yes	maximal, complemented
4	#20268229	C₂	1	3	yes	maximal, complemented
5	#20268223	C₂	1	3	yes	maximal, complemented
6	C₂×#15692966	C₂	1	3	yes	maximal, complemented
7	#504×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#15693068	—	2	1	no	—
9	#15693067	—	2	1	no	—
10	#15693054	—	2	3	no	—
11	#15693049	—	2	3	no	—
12	#15693043	C₂²	1	3	yes	—
13	#15693043	C₂²	1	3	yes	complemented
14	C₂×Dic₄₁₂	C₂²	1	3	yes	complemented
15	C₂×D₈₂₄	C₂²	1	3	yes	—
16	Dic₈₂₄	C₂²	1	3	yes	complemented
17	#15692966	C₂²	1	3	yes	complemented
18	#15692966	C₂²	1	6	yes	complemented
19	#15692966	C₂²	1	1	yes	characteristic, complemented
20	D₁₆₄₈	C₂²	1	3	yes	complemented
21	#131×C₁₀₃	C₂²	1	1	yes	characteristic
22	M₃₂×C₁₀₃	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₁₀₃	C₂²	1	3	yes	complemented
24	#13110106	—	4	1	no	—
25	#13110105	—	2	3	no	—
26	#13110105	—	2	3	no	—
27	#13110104	—	4	3	no	—
28	C₂×Dic₂₀₆	—	2	3	no	—
29	Q₈×D₁₀₃	—	4	1	no	—
30	C₂×D₄₁₂	—	2	3	no	—
31	D₄×D₁₀₃	—	4	3	no	—
32	Dic₄₁₂	—	2	3	no	—
33	Dic₄₁₂	C₂³	1	3	yes	—
34	Dic₄₁₂	C₂³	1	1	yes	characteristic
35	#13110094	—	2	3	no	—
36	#13110094	—	2	3	no	—
37	D₈₂₄	—	2	3	no	—
38	D₈₂₄	C₂³	1	1	yes	characteristic
39	D₈₂₄	C₂³	1	3	yes	—
40	#41×C₁₀₃	D₄	1	1	yes	characteristic
41	M₁₆×C₁₀₃	D₄	1	3	yes	—
42	C₂×C₈×C₁₀₃	C₂³	1	3	yes	—
43	C₁₆×C₁₀₃	C₂³	1	1	yes	characteristic
44	C₁₆×C₁₀₃	C₂³	1	3	yes	—
45	C₂²×D₁₀₃	—	4	3	no	—
46	Q₈×C₁₀₃	D₈	1	1	yes	characteristic
47	D₄×C₁₀₃	D₈	1	3	yes	—
48	C₂×C₄×C₁₀₃	C₂×D₄	1	3	yes	—
49	C₂²⋊D₁₀₃	—	4	3	no	—
50	C₂²⋊D₁₀₃	—	4	3	no	—
51	C₂×Dic₁₀₃	—	4	3	no	—
52	D₄₁₂	—	2	3	no	—
53	D₄₁₂	—	2	1	no	—
54	D₄₁₂	—	4	3	no	—
55	C₄×D₁₀₃	—	4	3	no	—
56	C₄×D₁₀₃	—	4	3	no	—
57	Dic₂₀₆	—	2	1	no	—
58	Dic₂₀₆	—	2	3	no	—
59	Dic₂₀₆	—	4	3	no	—
60	C₈×C₁₀₃	C₂×D₄	1	3	yes	—
61	C₈×C₁₀₃	C₂⁴	1	1	yes	characteristic
62	C₂²×C₁₀₃	C₂×D₈	1	3	yes	—
63	C₂×D₁₀₃	—	4	1	no	—
64	C₂×D₁₀₃	—	4	3	no	—
65	C₂×D₁₀₃	—	16	3	no	—
66	C₄×C₁₀₃	C₂×D₈	1	3	yes	—
67	C₄×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
68	Dic₁₀₃	—	4	3	no	—
69	Dic₁₀₃	—	4	1	no	—
70	C₂×C₁₀₃	—	2	3	no	—
71	C₂×C₁₀₃	C₂²×D₈	1	1	yes	characteristic
72	D₁₀₃	—	8	1	no	—
73	D₁₀₃	—	8	3	no	—
74	#3416	—	103	1	no	maximal, Sylow
75	C₁₀₃	#3416	1	1	yes	characteristic, complemented, Sylow
76	#578	—	103	1	no	—
77	#576	—	103	1	no	—
78	#510	—	103	3	no	—
79	#509	—	103	3	no	—
80	#508	—	103	3	no	—
81	C₂×SD₃₂	—	103	3	no	—
82	#504	D₁₀₃	1	1	yes	characteristic, complemented
83	2₋¹⁺⁴	—	206	1	no	—
84	H(5,𝔽₂)	—	206	1	no	—
85	#137	—	206	3	no	—
86	AGL(1,ℤ/8)	—	206	3	no	—
87	#135	—	103	3	no	—
88	#135	—	103	3	no	—
89	C₂×Q₁₆	—	103	3	no	—
90	C₂×D₈	—	103	3	no	—
91	#131	C₂×D₁₀₃	1	1	yes	characteristic
92	Q₃₂	—	103	3	no	—
93	SD₃₂	—	103	3	no	—
94	SD₃₂	—	103	6	no	—
95	SD₃₂	—	103	1	no	—
96	D₁₆	—	103	3	no	—
97	M₃₂	C₂×D₁₀₃	1	3	yes	complemented
98	C₂×C₁₆	C₂×D₁₀₃	1	3	yes	complemented
99	#41	—	412	1	no	—
100	#41	—	412	3	no	—
101	#41	—	206	3	no	—
102	#41	—	206	3	no	—
103	#41	D₄₁₂	1	1	yes	characteristic
104	C₂×Q₈	—	412	1	no	—
105	C₂×Q₈	—	206	3	no	—
106	C₂×D₄	—	412	3	no	—
107	C₂×D₄	—	206	3	no	—
108	Q₁₆	—	103	3	no	—
109	Q₁₆	—	103	1	no	—
110	Q₁₆	—	206	3	no	—
111	SD₁₆	—	206	3	no	—
112	SD₁₆	—	206	3	no	—
113	D₈	—	103	1	no	—
114	D₈	—	103	3	no	—
115	D₈	—	206	3	no	—
116	M₁₆	D₄₁₂	1	3	yes	—
117	C₂×C₈	C₂²×D₁₀₃	1	3	yes	—
118	C₁₆	C₂²×D₁₀₃	1	1	yes	characteristic
119	C₁₆	C₂²×D₁₀₃	1	3	yes	—
120	C₂³	—	412	3	no	—
121	Q₈	—	412	3	no	—
122	Q₈	—	206	1	no	—
123	Q₈	—	206	3	no	—
124	Q₈	D₈₂₄	1	1	yes	characteristic
125	D₄	—	412	3	no	—
126	D₄	—	412	3	no	—
127	D₄	—	412	3	no	—
128	D₄	—	206	3	no	—
129	D₄	—	206	1	no	—
130	D₄	D₈₂₄	1	3	yes	—
131	C₂×C₄	—	412	3	no	—
132	C₂×C₄	—	412	3	no	—
133	C₂×C₄	—	412	3	no	—
134	C₂×C₄	C₂×D₄₁₂	1	3	yes	—
135	C₈	C₂×D₄₁₂	1	3	yes	—
136	C₈	C₂³×D₁₀₃	1	1	yes	characteristic
137	C₂²	—	1648	3	no	—
138	C₂²	—	412	1	no	—
139	C₂²	—	412	3	no	—
140	C₂²	C₂×D₈₂₄	1	3	yes	—
141	C₄	—	412	1	no	—
142	C₄	—	412	3	no	—
143	C₄	C₂×D₈₂₄	1	3	yes	—
144	C₄	C₂²×D₄₁₂	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	824	3	no	—
147	C₂	—	824	1	no	—
148	C₂	C₂²×D₈₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₈₂₄
commutator, cosocle	C₈×C₁₀₃	C₂⁴
Frattini	C₈	C₂³×D₁₀₃
Fitting	#504×C₁₀₃	C₂
socle	C₂×C₁₀₃	C₂²×D₈
hypercenter	#504	D₁₀₃
hypocenter	C₁₀₃	#3416
LCS	G ▷ C₈×C₁₀₃ ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
1. #20268608	C₂	semidirect
2. #20268596	C₂	semidirect
3. #20268234	C₂	semidirect
4. #20268229	C₂	semidirect
5. #20268223	C₂	semidirect
6. C₂×#15692966	C₂	semidirect
7. #504×C₁₀₃	C₂	semidirect
13. #15693043	C₂²	semidirect
14. C₂×Dic₄₁₂	C₂²	semidirect
16. Dic₈₂₄	C₂²	semidirect
17. #15692966	C₂²	semidirect
18. #15692966	C₂²	semidirect
19. #15692966	C₂²	semidirect
20. D₁₆₄₈	C₂²	semidirect
22. M₃₂×C₁₀₃	C₂²	semidirect
23. C₂×C₁₆×C₁₀₃	C₂²	semidirect
75. C₁₀₃	#3416	semidirect
82. #504	D₁₀₃	semidirect
97. M₃₂	C₂×D₁₀₃	semidirect
98. C₂×C₁₆	C₂×D₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
#20268608	2
#20268596	2
#20268234	2
#20268229	2
#20268223	2
C₂×#15692966	2
#504×C₁₀₃	2
#3416	103