Finite Group Databasebeta

Subgroups — #21444679

Order 7808.

168 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16839893	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11758201	C₂	1	1	yes	maximal, characteristic
3	#16839827	C₂	1	1	yes	maximal, characteristic
4	#16839514	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839491	C₂	1	1	yes	maximal, characteristic
6	#16839483	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758201	C₄	1	1	yes	characteristic
9	C₂×C₈×D₆₁	—	2	1	no	—
10	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
11	#14519708	C₄	1	1	yes	characteristic
12	C₄⋊#92418	—	2	1	no	—
13	C₄⋊#92418	C₂²	1	1	yes	characteristic
14	#14519701	C₄	1	1	yes	characteristic
15	#14519697	C₄	1	1	yes	characteristic
16	#14519689	—	2	1	no	—
17	#14519688	—	2	1	no	—
18	C₄×#92418	C₂²	1	1	yes	characteristic
19	C₄×#92418	C₂²	1	1	yes	characteristic
20	C₄×Q₈×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×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×D₂₄₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₆₁	—	2	1	no	—
26	C₂×C₈×C₆₁	—	2	1	no	—
27	C₂×C₈×C₆₁	—	2	1	no	—
28	C₄⋊C₄×C₆₁	—	2	1	no	—
29	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₆₁	—	2	1	no	—
31	C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	#11758201	—	2	2	no	—
33	#11758198	—	2	1	no	—
34	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×#92418	—	2	1	no	—
36	C₂×#92418	C₂×C₄	1	1	yes	characteristic
37	C₂×#92418	C₂×C₄	1	1	yes	characteristic
38	C₂×#92418	D₄	1	1	yes	characteristic
39	C₂×#92418	D₄	1	1	yes	characteristic
40	C₈×D₆₁	—	4	2	no	—
41	C₂²×D₆₁	—	2	1	no	—
42	Q₈×C₆₁	—	2	1	no	—
43	Q₈×C₆₁	—	2	1	no	—
44	C₂×C₄×C₆₁	—	2	1	no	—
45	C₂×C₄×C₆₁	—	2	1	no	—
46	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
49	C₂×Dic₆₁	—	2	1	no	—
50	D₂₄₄	—	2	1	no	—
51	D₂₄₄	M₁₆	1	2	yes	—
52	C₄×D₆₁	—	4	1	no	—
53	C₄×D₆₁	—	4	1	no	—
54	C₈×C₆₁	—	2	2	no	—
55	C₈×C₆₁	—	4	1	no	—
56	#92418	—	2	1	no	—
57	#92418	—	2	1	no	—
58	#92418	—	2	1	no	—
59	#92418	—	2	1	no	—
60	#92418	—	4	1	no	—
61	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
62	C₂×D₆₁	—	2	2	no	—
63	C₂×D₆₁	—	4	1	no	—
64	C₂×D₆₁	—	4	1	no	—
65	C₄×C₆₁	—	2	1	no	—
66	C₄×C₆₁	—	4	1	no	—
67	C₄×C₆₁	—	4	1	no	—
68	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
69	C₄×C₆₁	#131	1	1	yes	characteristic
70	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
71	C₄×C₆₁	#137	1	1	yes	characteristic
72	Dic₆₁	—	4	1	no	—
73	#1585	—	61	1	no	maximal, Sylow
74	C₂×C₆₁	#435	1	1	yes	characteristic
75	C₂×C₆₁	#440	1	1	yes	characteristic
76	C₂×C₆₁	#443	1	1	yes	characteristic
77	D₆₁	—	4	2	no	—
78	#446	—	61	1	no	—
79	C₄×SD₁₆	—	61	1	no	—
80	C₈×D₄	—	61	1	no	—
81	#335	—	61	1	no	—
82	Q₈⋊C₈	D₆₁	1	1	yes	characteristic, complemented
83	D₄⋊C₈	—	61	1	no	—
84	#322	—	61	1	no	—
85	C₆₁	#1585	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	61	1	no	—
87	C₂²×C₈	—	122	1	no	—
88	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
89	C₄×D₄	—	61	1	no	—
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	#97	—	122	1	no	—
98	C₄×C₈	—	61	1	no	—
99	C₄×C₈	—	61	1	no	—
100	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
101	C₂×Q₈	C₄×D₆₁	1	1	yes	characteristic
102	C₂×D₄	—	61	1	no	—
103	C₂²×C₄	—	122	1	no	—
104	SD₁₆	—	122	2	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	—	2	1	no	—
114	C₄⋊C₄	—	61	1	no	—
115	C₄⋊C₄	C₄×D₆₁	1	1	yes	characteristic
116	C₂²⋊C₄	—	122	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	C₂²×D₆₁	1	1	yes	characteristic
119	C₂³	—	122	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	2	1	no	—
122	D₄	—	61	2	no	—
123	D₄	—	122	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	122	1	no	—
127	C₂×C₄	—	244	1	no	—
128	C₂×C₄	—	244	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	—	122	1	no	—
135	C₈	—	122	1	no	—
136	C₈	—	122	1	no	—
137	C₈	—	122	1	no	—
138	C₈	—	244	1	no	—
139	C₂²	—	122	2	no	—
140	C₂²	—	244	1	no	—
141	C₂²	—	244	1	no	—
142	C₂²	#14519737	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	244	1	no	—
147	C₄	M₁₆×D₆₁	1	1	yes	characteristic
148	C₄	#14519783	1	1	yes	characteristic
149	C₄	#14519804	1	1	yes	characteristic
150	C₄	#14519809	1	1	yes	characteristic
151	C₂	—	244	2	no	—
152	C₂	#16839836	1	1	yes	characteristic
153	C₂	#16839865	1	1	yes	characteristic
154	C₂	#16839876	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×Q₈⋊C₈	C₂
socle	C₂²×C₆₁	C₄×D₄
hypercenter	Q₈⋊C₈	D₆₁
hypocenter	C₆₁	#1585
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
1. #16839893	C₂	semidirect
4. #16839514	C₂	semidirect
6. #16839483	C₂	semidirect
7. C₆₁×Q₈⋊C₈	C₂	semidirect
82. Q₈⋊C₈	D₆₁	semidirect
85. C₆₁	#1585	semidirect