Finite Group Databasebeta

Subgroups — #21458967

Order 7808.

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	#16840579	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840567	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840205	C₂	1	3	yes	maximal, complemented
4	#16840200	C₂	1	3	yes	maximal, complemented
5	#16840194	C₂	1	3	yes	maximal, complemented
6	C₂×#14519722	C₂	1	3	yes	maximal, complemented
7	C₆₁×#504	C₂	1	1	yes	maximal, characteristic, complemented
8	#14519824	—	2	1	no	—
9	#14519823	—	2	1	no	—
10	#14519810	—	2	3	no	—
11	#14519805	—	2	3	no	—
12	#14519799	C₂²	1	3	yes	—
13	#14519799	C₂²	1	3	yes	complemented
14	C₂×Dic₂₄₄	C₂²	1	3	yes	complemented
15	C₂×D₄₈₈	C₂²	1	3	yes	—
16	Dic₄₈₈	C₂²	1	3	yes	complemented
17	#14519722	C₂²	1	1	yes	characteristic, complemented
18	#14519722	C₂²	1	3	yes	complemented
19	#14519722	C₂²	1	6	yes	complemented
20	D₉₇₆	C₂²	1	3	yes	complemented
21	#131×C₆₁	C₂²	1	1	yes	characteristic
22	M₃₂×C₆₁	C₂²	1	3	yes	complemented
23	C₂×C₁₆×C₆₁	C₂²	1	3	yes	complemented
24	#41×C₆₁	D₄	1	1	yes	characteristic
25	#11758226	—	4	1	no	—
26	Q₈×D₆₁	—	4	1	no	—
27	#11758224	—	4	3	no	—
28	D₄×D₆₁	—	4	3	no	—
29	#11758222	—	2	3	no	—
30	#11758222	—	2	3	no	—
31	C₂×D₂₄₄	—	2	3	no	—
32	C₂×Dic₁₂₂	—	2	3	no	—
33	M₁₆×C₆₁	D₄	1	3	yes	—
34	C₂×C₈×C₆₁	C₂³	1	3	yes	—
35	Dic₂₄₄	—	2	3	no	—
36	Dic₂₄₄	C₂³	1	3	yes	—
37	Dic₂₄₄	C₂³	1	1	yes	characteristic
38	D₄₈₈	—	2	3	no	—
39	D₄₈₈	C₂³	1	1	yes	characteristic
40	D₄₈₈	C₂³	1	3	yes	—
41	#11758190	—	2	3	no	—
42	#11758190	—	2	3	no	—
43	C₁₆×C₆₁	C₂³	1	1	yes	characteristic
44	C₁₆×C₆₁	C₂³	1	3	yes	—
45	C₂²×D₆₁	—	4	3	no	—
46	Q₈×C₆₁	D₈	1	1	yes	characteristic
47	D₄×C₆₁	D₈	1	3	yes	—
48	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
49	C₂²⋊D₆₁	—	4	3	no	—
50	C₂²⋊D₆₁	—	4	3	no	—
51	C₂×Dic₆₁	—	4	3	no	—
52	D₂₄₄	—	2	1	no	—
53	D₂₄₄	—	2	3	no	—
54	D₂₄₄	—	4	3	no	—
55	C₄×D₆₁	—	4	3	no	—
56	C₄×D₆₁	—	4	3	no	—
57	Dic₁₂₂	—	2	1	no	—
58	Dic₁₂₂	—	2	3	no	—
59	Dic₁₂₂	—	4	3	no	—
60	C₈×C₆₁	C₂×D₄	1	3	yes	—
61	C₈×C₆₁	C₂⁴	1	1	yes	characteristic
62	C₂²×C₆₁	C₂×D₈	1	3	yes	—
63	C₂×D₆₁	—	4	1	no	—
64	C₂×D₆₁	—	4	3	no	—
65	C₂×D₆₁	—	16	3	no	—
66	C₄×C₆₁	C₂×D₈	1	3	yes	—
67	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
68	Dic₆₁	—	4	3	no	—
69	Dic₆₁	—	4	1	no	—
70	#3416	—	61	1	no	maximal, Sylow
71	C₂×C₆₁	—	2	3	no	—
72	C₂×C₆₁	C₂²×D₈	1	1	yes	characteristic
73	D₆₁	—	8	1	no	—
74	D₆₁	—	8	3	no	—
75	#578	—	61	1	no	—
76	#576	—	61	1	no	—
77	#510	—	61	3	no	—
78	#509	—	61	3	no	—
79	#508	—	61	3	no	—
80	C₂×SD₃₂	—	61	3	no	—
81	#504	D₆₁	1	1	yes	characteristic, complemented
82	C₆₁	#3416	1	1	yes	characteristic, complemented, Sylow
83	2₋¹⁺⁴	—	122	1	no	—
84	H(5,𝔽₂)	—	122	1	no	—
85	#137	—	122	3	no	—
86	AGL(1,ℤ/8)	—	122	3	no	—
87	#135	—	61	3	no	—
88	#135	—	61	3	no	—
89	C₂×Q₁₆	—	61	3	no	—
90	C₂×D₈	—	61	3	no	—
91	#131	C₂×D₆₁	1	1	yes	characteristic
92	Q₃₂	—	61	3	no	—
93	SD₃₂	—	61	3	no	—
94	SD₃₂	—	61	6	no	—
95	SD₃₂	—	61	1	no	—
96	D₁₆	—	61	3	no	—
97	M₃₂	C₂×D₆₁	1	3	yes	complemented
98	C₂×C₁₆	C₂×D₆₁	1	3	yes	complemented
99	#41	—	122	3	no	—
100	#41	—	122	3	no	—
101	#41	—	244	1	no	—
102	#41	—	244	3	no	—
103	#41	D₂₄₄	1	1	yes	characteristic
104	C₂×Q₈	—	122	3	no	—
105	C₂×Q₈	—	244	1	no	—
106	C₂×D₄	—	122	3	no	—
107	C₂×D₄	—	244	3	no	—
108	Q₁₆	—	61	3	no	—
109	Q₁₆	—	61	1	no	—
110	Q₁₆	—	122	3	no	—
111	SD₁₆	—	122	3	no	—
112	SD₁₆	—	122	3	no	—
113	D₈	—	61	1	no	—
114	D₈	—	61	3	no	—
115	D₈	—	122	3	no	—
116	M₁₆	D₂₄₄	1	3	yes	—
117	C₂×C₈	C₂²×D₆₁	1	3	yes	—
118	C₁₆	C₂²×D₆₁	1	1	yes	characteristic
119	C₁₆	C₂²×D₆₁	1	3	yes	—
120	C₂³	—	244	3	no	—
121	Q₈	—	122	1	no	—
122	Q₈	—	122	3	no	—
123	Q₈	—	244	3	no	—
124	Q₈	D₄₈₈	1	1	yes	characteristic
125	D₄	—	122	3	no	—
126	D₄	—	122	1	no	—
127	D₄	—	244	3	no	—
128	D₄	—	244	3	no	—
129	D₄	—	244	3	no	—
130	D₄	D₄₈₈	1	3	yes	—
131	C₂×C₄	—	244	3	no	—
132	C₂×C₄	—	244	3	no	—
133	C₂×C₄	—	244	3	no	—
134	C₂×C₄	C₂×D₂₄₄	1	3	yes	—
135	C₈	C₂×D₂₄₄	1	3	yes	—
136	C₈	C₂³×D₆₁	1	1	yes	characteristic
137	C₂²	—	976	3	no	—
138	C₂²	—	244	1	no	—
139	C₂²	—	244	3	no	—
140	C₂²	C₂×D₄₈₈	1	3	yes	—
141	C₄	—	244	1	no	—
142	C₄	—	244	3	no	—
143	C₄	C₂×D₄₈₈	1	3	yes	—
144	C₄	C₂²×D₂₄₄	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	488	3	no	—
147	C₂	—	488	1	no	—
148	C₂	C₂²×D₄₈₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄₈₈
commutator	C₈×C₆₁	C₂⁴
Frattini	C₈	C₂³×D₆₁
Fitting	C₆₁×#504	C₂
socle	C₂×C₆₁	C₂²×D₈
hypercenter	#504	D₆₁
hypocenter	C₆₁	#3416
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
1. #16840579	C₂	semidirect
2. #16840567	C₂	semidirect
3. #16840205	C₂	semidirect
4. #16840200	C₂	semidirect
5. #16840194	C₂	semidirect
6. C₂×#14519722	C₂	semidirect
7. C₆₁×#504	C₂	semidirect
13. #14519799	C₂²	semidirect
14. C₂×Dic₂₄₄	C₂²	semidirect
16. Dic₄₈₈	C₂²	semidirect
17. #14519722	C₂²	semidirect
18. #14519722	C₂²	semidirect
19. #14519722	C₂²	semidirect
20. D₉₇₆	C₂²	semidirect
22. M₃₂×C₆₁	C₂²	semidirect
23. C₂×C₁₆×C₆₁	C₂²	semidirect
81. #504	D₆₁	semidirect
82. C₆₁	#3416	semidirect
97. M₃₂	C₂×D₆₁	semidirect
98. C₂×C₁₆	C₂×D₆₁	semidirect