Finite Group Databasebeta

Subgroups — #21444622

Order 7808.

180 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 73 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	#16839886	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11758200	C₂	1	1	yes	maximal, characteristic
3	D₄×#92418	C₂	1	1	yes	maximal, characteristic
4	#16839512	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839494	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#92418	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×#11758200	C₄	1	1	yes	characteristic
9	C₂²×#92418	—	2	1	no	—
10	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
11	#14519708	C₄	1	1	yes	characteristic
12	C₄⋊#92418	C₂²	1	1	yes	characteristic
13	#14519706	—	2	1	no	—
14	#14519700	C₄	1	1	yes	characteristic
15	#14519698	C₄	1	1	yes	characteristic
16	C₂²⋊#92418	—	2	1	no	—
17	C₄×#92418	C₂²	1	1	yes	characteristic
18	C₄×#92418	C₂²	1	1	yes	characteristic
19	C₈×Dic₆₁	—	2	1	no	—
20	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₁	—	2	1	no	—
25	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₆₁	—	2	1	no	—
27	C₂×C₈×C₆₁	—	2	1	no	—
28	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂²⋊C₄×C₆₁	—	2	1	no	—
30	C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	#11758200	C₈	1	4	yes	complemented
32	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
33	#11758196	—	2	1	no	—
34	C₄×Dic₆₁	—	2	1	no	—
35	C₂×#92418	—	2	1	no	—
36	C₂×#92418	—	4	2	no	—
37	C₂×#92418	C₂×C₄	1	1	yes	characteristic
38	C₂×#92418	C₂×C₄	1	1	yes	characteristic
39	C₂×#92418	D₄	1	1	yes	characteristic
40	C₂×#92418	D₄	1	1	yes	characteristic
41	C₂³×C₆₁	—	2	1	no	—
42	D₄×C₆₁	—	2	1	no	—
43	D₄×C₆₁	C₂×C₈	1	2	yes	—
44	C₂×C₄×C₆₁	—	2	1	no	—
45	C₂×C₄×C₆₁	—	4	1	no	—
46	C₂×C₄×C₆₁	—	4	1	no	—
47	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
50	C₂×Dic₆₁	—	2	1	no	—
51	C₂×Dic₆₁	—	2	1	no	—
52	Dic₁₂₂	—	2	1	no	—
53	Dic₁₂₂	C₂×C₈	1	2	yes	complemented
54	C₈×C₆₁	—	2	2	no	—
55	C₈×C₆₁	—	4	1	no	—
56	#92418	—	2	1	no	—
57	#92418	—	2	1	no	—
58	#92418	—	4	1	no	—
59	#92418	C₂×C₈	1	2	yes	complemented
60	#92418	SD₁₆	1	2	yes	complemented
61	C₂²×C₆₁	—	2	2	no	—
62	C₂²×C₆₁	—	4	1	no	—
63	C₂²×C₆₁	—	4	1	no	—
64	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
65	C₄×C₆₁	—	2	1	no	—
66	C₄×C₆₁	—	4	1	no	—
67	C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
68	C₄×C₆₁	#131	1	1	yes	characteristic
69	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₆₁	#135	1	1	yes	characteristic
71	Dic₆₁	—	2	2	no	—
72	Dic₆₁	—	4	1	no	—
73	C₈×SD₁₆	—	61	1	no	maximal, Sylow
74	C₂×C₆₁	—	4	2	no	—
75	C₂×C₆₁	C₈×D₄	1	1	yes	characteristic
76	C₂×C₆₁	C₄×SD₁₆	1	1	yes	characteristic
77	C₂×C₆₁	#443	1	1	yes	characteristic
78	C₈×Q₈	—	61	1	no	—
79	C₄×SD₁₆	—	61	1	no	—
80	C₈×D₄	—	61	1	no	—
81	#334	—	61	1	no	—
82	Q₈⋊C₈	—	61	1	no	—
83	D₄⋊C₈	D₆₁	1	1	yes	characteristic
84	C₈²	—	61	1	no	—
85	C₆₁	C₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	61	1	no	—
87	C₂²×C₈	—	122	1	no	—
88	C₄×Q₈	—	61	1	no	—
89	C₄×D₄	C₂×D₆₁	1	1	yes	characteristic
90	#106	—	61	1	no	—
91	C₄⋊C₈	—	61	1	no	—
92	C₄⋊C₈	—	122	1	no	—
93	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
94	#103	—	61	1	no	—
95	#102	—	61	1	no	—
96	C₂²⋊C₈	—	122	1	no	—
97	C₄×C₈	—	61	1	no	—
98	C₄×C₈	—	61	1	no	—
99	C₄×C₈	—	122	1	no	—
100	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
101	C₂×Q₈	—	61	1	no	—
102	C₂×D₄	C₄×D₆₁	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	SD₁₆	—	61	4	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	61	1	no	—
108	C₂×C₈	—	61	1	no	—
109	C₂×C₈	—	61	1	no	—
110	C₂×C₈	—	61	1	no	—
111	C₂×C₈	—	122	1	no	—
112	C₂×C₈	—	244	2	no	—
113	C₄⋊C₄	—	61	1	no	—
114	C₄⋊C₄	—	122	1	no	—
115	C₄⋊C₄	C₄×D₆₁	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	—	122	1	no	—
118	C₄²	C₂²×D₆₁	1	1	yes	characteristic
119	C₂³	—	2	1	no	—
120	Q₈	—	61	2	no	—
121	Q₈	—	122	1	no	—
122	D₄	—	2	1	no	—
123	D₄	C₈×D₆₁	1	2	yes	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	122	1	no	—
128	C₂×C₄	—	122	1	no	—
129	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
130	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
131	C₂×C₄	#11758224	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	4	1	no	—
134	C₈	—	61	2	no	—
135	C₈	—	61	2	no	—
136	C₈	—	122	1	no	—
137	C₈	—	122	1	no	—
138	C₈	—	244	1	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	#14519737	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	122	2	no	—
146	C₄	—	244	1	no	—
147	C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
148	C₄	#14519785	1	1	yes	characteristic
149	C₄	SD₁₆×D₆₁	1	1	yes	characteristic
150	C₄	#14519796	1	1	yes	characteristic
151	C₂	—	4	2	no	—
152	C₂	#16839825	1	1	yes	characteristic
153	C₂	#16839853	1	1	yes	characteristic
154	C₂	#16839876	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₆₁
commutator	C₄×C₆₁	C₂²×C₈
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×D₄⋊C₈	C₂
socle	C₂²×C₆₁	C₄×D₄
hypercenter	D₄⋊C₈	D₆₁
hypocenter	C₆₁	C₈×SD₁₆
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ D₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #16839886	C₂	semidirect
4. #16839512	C₂	semidirect
5. #16839494	C₂	semidirect
6. C₈×#92418	C₂	semidirect
31. #11758200	C₈	semidirect
53. Dic₁₂₂	C₂×C₈	semidirect
59. #92418	C₂×C₈	semidirect
60. #92418	SD₁₆	semidirect
85. C₆₁	C₈×SD₁₆	semidirect