Finite Group Databasebeta

Subgroups — #21445261

Order 7808.

166 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 55 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	#16840108	C₂	1	1	yes	maximal, characteristic
2	#16839972	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×#92418	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839514	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839509	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839506	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#460	C₂	1	1	yes	maximal, characteristic
8	C₂×#11758202	—	2	1	no	—
9	#14519777	C₂²	1	1	yes	characteristic
10	C₄⋊#92418	—	2	1	no	—
11	C₄⋊#92418	C₂²	1	1	yes	characteristic
12	C₄⋊#92418	C₂²	1	1	yes	characteristic
13	#14519700	—	2	1	no	—
14	#14519698	—	2	1	no	—
15	#14519698	—	2	1	no	—
16	C₄×#92418	—	2	1	no	—
17	C₄×#92418	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₆₁	—	2	1	no	—
19	#127×C₆₁	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
22	#102×C₆₁	—	2	1	no	—
23	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
24	C₂×D₄×C₆₁	—	2	1	no	—
25	C₂×D₄×C₆₁	—	2	1	no	—
26	C₂×D₄×C₆₁	—	2	1	no	—
27	C₂×Dic₁₂₂	—	2	1	no	—
28	SD₁₆×C₆₁	—	4	2	no	—
29	C₂×C₈×C₆₁	—	2	1	no	—
30	C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
32	C₄²×C₆₁	—	2	1	no	—
33	C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	#11758202	—	4	2	no	—
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₄⋊Dic₆₁	—	2	1	no	—
37	C₂×#92418	—	2	1	no	—
38	C₂×#92418	—	2	1	no	—
39	C₂×#92418	D₄	1	1	yes	characteristic
40	C₂×#92418	D₄	1	1	yes	characteristic
41	C₂³×C₆₁	—	4	1	no	—
42	Q₈×C₆₁	—	2	1	no	—
43	Q₈×C₆₁	D₈	1	2	yes	—
44	D₄×C₆₁	—	4	1	no	—
45	D₄×C₆₁	—	4	1	no	—
46	D₄×C₆₁	—	4	1	no	—
47	D₄×C₆₁	—	4	1	no	—
48	D₄×C₆₁	—	8	1	no	—
49	C₂×C₄×C₆₁	—	2	1	no	—
50	C₂×C₄×C₆₁	—	2	1	no	—
51	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
54	C₂×Dic₆₁	—	4	1	no	—
55	Dic₁₂₂	—	4	1	no	—
56	Dic₁₂₂	—	4	1	no	—
57	C₈×C₆₁	—	4	1	no	—
58	#92418	—	2	1	no	—
59	#92418	—	2	1	no	—
60	#92418	—	4	1	no	—
61	#92418	—	4	1	no	—
62	C₂²×C₆₁	—	8	1	no	—
63	C₂²×C₆₁	—	8	1	no	—
64	C₂²×C₆₁	—	8	1	no	—
65	C₂²×C₆₁	#121	1	1	yes	characteristic
66	C₄×C₆₁	—	2	2	no	—
67	C₄×C₆₁	—	2	1	no	—
68	C₄×C₆₁	—	4	1	no	—
69	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
70	C₄×C₆₁	#135	1	1	yes	characteristic
71	C₄×C₆₁	#135	1	1	yes	characteristic
72	C₄×C₆₁	#137	1	1	yes	characteristic
73	Dic₆₁	—	8	1	no	—
74	#1682	—	61	1	no	maximal, Sylow
75	C₂×C₆₁	—	16	1	no	—
76	C₂×C₆₁	#464	1	1	yes	characteristic
77	C₂×C₆₁	#466	1	1	yes	characteristic
78	C₂×C₆₁	#471	1	1	yes	characteristic
79	#486	—	61	1	no	—
80	#462	—	61	1	no	—
81	#460	D₆₁	1	1	yes	characteristic
82	C₈×Q₈	—	61	1	no	—
83	#335	—	61	1	no	—
84	#332	—	61	1	no	—
85	#331	—	61	1	no	—
86	C₆₁	#1682	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₁₆	—	122	1	no	—
88	C₂×SD₁₆	—	2	1	no	—
89	C₄⋊Q₈	—	61	1	no	—
90	#127	C₂×D₆₁	1	1	yes	characteristic
91	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
92	C₄⋊C₈	—	61	1	no	—
93	C₄⋊C₈	—	61	1	no	—
94	C₄⋊C₈	—	122	1	no	—
95	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
96	#103	—	122	1	no	—
97	#102	—	2	1	no	—
98	#102	—	122	1	no	—
99	#102	—	122	1	no	—
100	C₄×C₈	—	61	1	no	—
101	C₄×C₈	—	122	1	no	—
102	C₂×Q₈	—	122	1	no	—
103	C₂×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	Q₁₆	—	244	2	no	—
108	SD₁₆	—	4	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	61	1	no	—
111	C₂×C₈	—	61	1	no	—
112	C₂×C₈	—	122	1	no	—
113	C₂×C₈	—	122	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	122	1	no	—
116	C₄⋊C₄	—	122	1	no	—
117	C₄⋊C₄	C₂²⋊D₆₁	1	1	yes	characteristic
118	C₄²	—	2	1	no	—
119	C₄²	C₂²×D₆₁	1	1	yes	characteristic
120	C₂³	—	4	1	no	—
121	Q₈	—	2	1	no	—
122	Q₈	—	244	1	no	—
123	Q₈	—	244	1	no	—
124	Q₈	#11758199	1	2	yes	—
125	D₄	—	4	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	8	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	244	1	no	—
133	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
134	C₂×C₄	#11758224	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
136	C₈	—	4	1	no	—
137	C₈	—	122	1	no	—
138	C₈	—	122	1	no	—
139	C₈	—	244	1	no	—
140	C₈	—	244	1	no	—
141	C₂²	—	8	1	no	—
142	C₂²	—	8	1	no	—
143	C₂²	—	8	1	no	—
144	C₂²	#14519749	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	1	no	—
147	C₄	—	4	1	no	—
148	C₄	—	488	1	no	—
149	C₄	C₂×#11758199	1	1	yes	characteristic
150	C₄	#14519797	1	1	yes	characteristic
151	C₄	#14519798	1	1	yes	characteristic
152	C₄	#14519806	1	1	yes	characteristic
153	C₂	—	16	1	no	—
154	C₂	#16839982	1	1	yes	characteristic
155	C₂	#16839996	1	1	yes	characteristic
156	C₂	#16840023	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14519749
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#460	C₂
socle	C₂²×C₆₁	#121
hypercenter	#460	D₆₁
hypocenter	C₆₁	#1682
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #460

Complements

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

N	H ≅ G/N	type
2. #16839972	C₂	semidirect
3. Q₈×#92418	C₂	semidirect
4. #16839514	C₂	semidirect
5. #16839509	C₂	semidirect
86. C₆₁	#1682	semidirect