Finite Group Databasebeta

Subgroups — #21462140

Order 7808.

224 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 62 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	D₆₁⋊M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840772	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840678	C₂	1	4	yes	maximal, complemented
4	C₂×#14519796	C₂	1	1	yes	maximal, characteristic
5	#14519856	—	2	1	no	—
6	#14519852	C₂²	1	2	yes	complemented
7	#14519851	—	2	1	no	—
8	D₆₁⋊C₁₆	—	2	1	no	—
9	#14519846	—	2	1	no	—
10	#14519845	C₂²	1	2	yes	complemented
11	C₂×#11758224	—	2	1	no	—
12	#14519796	—	2	1	no	—
13	#14519796	—	2	1	no	—
14	#14519796	C₄	1	2	yes	complemented
15	#14519796	C₂²	1	2	yes	—
16	C₂×Dic₂₄₄	C₄	1	1	yes	characteristic, complemented
17	C₂×#11758200	—	2	1	no	—
18	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₆₁	C₄	1	1	yes	characteristic, complemented
20	C₂×D₄×C₆₁	—	2	1	no	—
21	C₂×C₂²⋊D₆₁	—	4	1	no	—
22	C₂²×Dic₆₁	—	4	1	no	—
23	#11758224	—	2	2	no	—
24	#11758224	—	2	2	no	—
25	#11758224	—	4	1	no	—
26	#11758224	—	4	1	no	—
27	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
28	C₂×Dic₁₂₂	—	2	1	no	—
29	C₂²⋊#6923	—	4	1	no	—
30	C₄⋊#6923	—	2	2	no	—
31	C₄×#6923	—	4	1	no	—
32	D₈×C₆₁	—	2	1	no	—
33	D₈×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	#11758200	—	2	2	no	—
36	#11758200	—	2	2	no	—
37	C₂×#92418	D₄	1	1	yes	characteristic
38	Dic₂₄₄	—	2	1	no	—
39	Dic₂₄₄	C₂×C₄	1	2	yes	—
40	C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
41	C₈×D₆₁	D₄	1	1	yes	characteristic
42	C₈×D₆₁	D₄	1	1	yes	characteristic
43	C₈×D₆₁	C₂³	1	1	yes	characteristic
44	#11758187	—	2	2	no	—
45	C₂³×C₆₁	—	4	1	no	—
46	C₂²×D₆₁	D₈	1	1	yes	characteristic
47	C₂×#6923	—	4	2	no	—
48	D₄×C₆₁	—	2	2	no	—
49	D₄×C₆₁	—	4	1	no	—
50	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²⋊D₆₁	—	4	2	no	—
52	C₂²⋊D₆₁	—	4	2	no	—
53	C₂×Dic₆₁	—	4	1	no	—
54	C₂×Dic₆₁	—	4	2	no	—
55	C₂×Dic₆₁	—	4	2	no	—
56	C₂×Dic₆₁	SD₁₆	1	1	yes	characteristic
57	C₄×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₄×D₆₁	D₈	1	1	yes	characteristic
59	C₄×D₆₁	SD₁₆	1	1	yes	characteristic
60	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
61	Dic₁₂₂	—	2	2	no	—
62	Dic₁₂₂	—	4	1	no	—
63	C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
65	#92418	C₂²⋊C₄	1	1	yes	characteristic
66	#92418	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₆₁	—	4	2	no	—
68	C₂²×C₆₁	—	8	2	no	—
69	C₂²×C₆₁	#102	1	1	yes	characteristic
70	C₂×D₆₁	—	4	1	no	—
71	C₂×D₆₁	#102	1	1	yes	characteristic
72	C₂×D₆₁	C₂×D₈	1	1	yes	characteristic
73	#6923	—	8	2	no	—
74	C₄×C₆₁	#102	1	1	yes	characteristic
75	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	Dic₆₁	—	4	2	no	—
77	Dic₆₁	#102	1	1	yes	characteristic
78	Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
79	#2145	—	61	1	no	maximal, Sylow
80	C₂×C₆₁	—	2	1	no	—
81	C₂×C₆₁	—	8	2	no	—
82	C₂×C₆₁	C₂×#102	1	1	yes	characteristic
83	D₆₁	—	2	1	no	—
84	D₆₁	—	2	1	no	—
85	C₂×#135	—	61	1	no	—
86	C₂×M₃₂	—	61	1	no	—
87	#427	—	61	1	no	—
88	#360	—	61	4	no	—
89	C₆₁	#2145	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	122	1	no	—
91	#135	—	61	2	no	—
92	#135	—	61	2	no	—
93	#135	—	122	1	no	—
94	#135	—	122	1	no	—
95	C₂×Q₁₆	—	61	1	no	—
96	C₂×SD₁₆	—	122	1	no	—
97	C₂×D₈	#6923	1	1	yes	characteristic, complemented
98	C₂²×C₈	—	61	1	no	—
99	#117	—	122	1	no	—
100	M₃₂	—	61	2	no	—
101	M₃₂	—	122	1	no	—
102	C₂×C₁₆	—	122	1	no	—
103	#107	—	122	1	no	—
104	#106	—	61	2	no	—
105	#41	—	122	2	no	—
106	#41	—	122	2	no	—
107	#41	—	244	1	no	—
108	#41	—	244	1	no	—
109	C₂×Q₈	—	122	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	244	1	no	—
112	C₂²×C₄	—	61	1	no	—
113	C₂²×C₄	—	244	1	no	—
114	Q₁₆	—	61	2	no	—
115	Q₁₆	—	122	1	no	—
116	SD₁₆	—	122	2	no	—
117	SD₁₆	—	122	2	no	—
118	D₈	—	2	1	no	—
119	D₈	C₂×#6923	1	2	yes	—
120	C₂×C₈	—	61	1	no	—
121	C₂×C₈	—	61	1	no	—
122	C₂×C₈	—	61	1	no	—
123	C₂×C₈	—	61	1	no	—
124	C₂×C₈	—	61	1	no	—
125	C₂×C₈	C₂×#6923	1	1	yes	characteristic
126	C₄⋊C₄	—	122	2	no	—
127	C₂²⋊C₄	—	244	1	no	—
128	C₄²	—	244	1	no	—
129	C₁₆	—	122	2	no	—
130	C₂³	—	4	1	no	—
131	C₂³	—	61	1	no	—
132	Q₈	—	122	2	no	—
133	Q₈	—	244	1	no	—
134	D₄	—	2	2	no	—
135	D₄	—	4	1	no	—
136	D₄	—	244	2	no	—
137	D₄	—	244	2	no	—
138	C₂×C₄	—	61	1	no	—
139	C₂×C₄	—	61	1	no	—
140	C₂×C₄	—	61	1	no	—
141	C₂×C₄	—	61	1	no	—
142	C₂×C₄	—	61	1	no	—
143	C₂×C₄	—	244	2	no	—
144	C₂×C₄	—	244	2	no	—
145	C₂×C₄	—	244	2	no	—
146	C₂×C₄	—	244	1	no	—
147	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
148	C₈	—	61	1	no	—
149	C₈	—	61	1	no	—
150	C₈	C₂²⋊#6923	1	1	yes	characteristic
151	C₈	C₂²×#6923	1	1	yes	characteristic
152	C₂²	—	4	2	no	—
153	C₂²	—	8	2	no	—
154	C₂²	—	61	1	no	—
155	C₂²	—	61	1	no	—
156	C₂²	—	244	1	no	—
157	C₂²	#14519839	1	1	yes	characteristic
158	C₄	—	61	1	no	—
159	C₄	—	61	1	no	—
160	C₄	—	488	2	no	—
161	C₄	—	244	2	no	—
162	C₄	#14519839	1	1	yes	characteristic
163	C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	8	2	no	—
166	C₂	—	122	1	no	—
167	C₂	—	122	1	no	—
168	C₂	C₂×#14519839	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#14519839
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₈	C₂²×#6923
Fitting	C₂×D₈×C₆₁	C₄
socle	C₂×C₆₁	C₂×#102
hypercenter	C₂×D₈	#6923
hypocenter	C₆₁	#2145
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. D₆₁⋊M₃₂	C₂	semidirect
2. #16840772	C₂	semidirect
3. #16840678	C₂	semidirect
6. #14519852	C₂²	semidirect
10. #14519845	C₂²	semidirect
14. #14519796	C₄	semidirect
16. C₂×Dic₂₄₄	C₄	semidirect
19. C₂×D₈×C₆₁	C₄	semidirect
89. C₆₁	#2145	semidirect
97. C₂×D₈	#6923	semidirect