Finite Group Databasebeta

Subgroups — #21462134

Order 7808.

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

normal subgroups only

This group has 220 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	D₆₁⋊#108	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840676	C₂	1	4	yes	maximal, complemented
4	C₂×#14519796	C₂	1	1	yes	maximal, characteristic
5	D₆₁⋊M₁₆	—	2	1	no	—
6	#14519852	—	2	1	no	—
7	#14519851	—	2	1	no	—
8	D₆₁⋊C₁₆	C₂²	1	2	yes	complemented
9	#14519848	C₂²	1	2	yes	complemented
10	#14519847	—	2	1	no	—
11	C₂×#11758224	—	2	1	no	—
12	#14519796	—	2	1	no	—
13	#14519796	—	2	1	no	—
14	#14519796	C₄	1	2	yes	—
15	#14519796	C₂²	1	2	yes	—
16	C₂×Dic₂₄₄	C₄	1	1	yes	characteristic
17	C₂×#11758200	—	2	1	no	—
18	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₆₁	C₄	1	1	yes	characteristic
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	#11758217	—	4	1	no	—
30	#11758213	—	2	2	no	—
31	D₆₁⋊C₈	—	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	SD₁₆	1	1	yes	characteristic
47	D₄×C₆₁	—	2	2	no	—
48	D₄×C₆₁	—	4	1	no	—
49	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²⋊D₆₁	—	4	2	no	—
51	C₂²⋊D₆₁	—	4	2	no	—
52	C₂×Dic₆₁	—	4	1	no	—
53	C₂×Dic₆₁	—	4	2	no	—
54	C₂×Dic₆₁	—	4	2	no	—
55	C₂×Dic₆₁	D₈	1	1	yes	characteristic
56	C₄×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₄×D₆₁	D₈	1	1	yes	characteristic
58	C₄×D₆₁	SD₁₆	1	1	yes	characteristic
59	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
60	Dic₁₂₂	—	2	2	no	—
61	Dic₁₂₂	—	4	1	no	—
62	#92420	—	4	2	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×SD₁₆	1	1	yes	characteristic
73	C₄×C₆₁	#102	1	1	yes	characteristic
74	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	Dic₆₁	—	4	2	no	—
76	Dic₆₁	#102	1	1	yes	characteristic
77	Dic₆₁	C₂×D₈	1	1	yes	characteristic
78	#2143	—	61	1	no	maximal, Sylow
79	C₂×C₆₁	—	2	1	no	—
80	C₂×C₆₁	—	8	2	no	—
81	C₂×C₆₁	C₂×#102	1	1	yes	characteristic
82	D₆₁	—	2	1	no	—
83	D₆₁	—	2	1	no	—
84	C₂×#135	—	61	1	no	—
85	C₂×M₃₂	—	61	1	no	—
86	C₂×#108	—	61	1	no	—
87	#359	—	61	4	no	—
88	C₆₁	#2143	1	1	yes	characteristic, complemented, Sylow
89	C₂×#41	—	122	1	no	—
90	#135	—	61	2	no	—
91	#135	—	61	2	no	—
92	#135	—	122	1	no	—
93	#135	—	122	1	no	—
94	C₂×Q₁₆	—	61	1	no	—
95	C₂×SD₁₆	—	122	1	no	—
96	C₂×D₈	#6923	1	1	yes	characteristic
97	C₂×M₁₆	—	122	1	no	—
98	C₂²×C₈	—	61	1	no	—
99	M₃₂	—	122	1	no	—
100	M₃₂	—	122	1	no	—
101	C₂×C₁₆	—	61	2	no	—
102	#108	—	61	2	no	—
103	#108	—	122	1	no	—
104	#41	—	122	2	no	—
105	#41	—	122	2	no	—
106	#41	—	244	1	no	—
107	#41	—	244	1	no	—
108	C₂×Q₈	—	122	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	244	1	no	—
111	C₂²×C₄	—	61	1	no	—
112	C₂²×C₄	—	244	1	no	—
113	Q₁₆	—	61	2	no	—
114	Q₁₆	—	122	1	no	—
115	SD₁₆	—	122	2	no	—
116	SD₁₆	—	122	2	no	—
117	D₈	—	2	1	no	—
118	D₈	C₂×#6923	1	2	yes	—
119	M₁₆	—	122	2	no	—
120	M₁₆	—	244	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	—	61	1	no	—
126	C₂×C₈	—	244	1	no	—
127	C₂×C₈	C₂×#6923	1	1	yes	characteristic
128	C₁₆	—	122	2	no	—
129	C₂³	—	4	1	no	—
130	C₂³	—	61	1	no	—
131	Q₈	—	122	2	no	—
132	Q₈	—	244	1	no	—
133	D₄	—	2	2	no	—
134	D₄	—	4	1	no	—
135	D₄	—	244	2	no	—
136	D₄	—	244	2	no	—
137	C₂×C₄	—	61	1	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	—	244	2	no	—
143	C₂×C₄	—	244	2	no	—
144	C₂×C₄	—	244	1	no	—
145	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
146	C₈	—	61	1	no	—
147	C₈	—	61	1	no	—
148	C₈	—	244	2	no	—
149	C₈	C₂²⋊#6923	1	1	yes	characteristic
150	C₈	C₂²×#6923	1	1	yes	characteristic
151	C₂²	—	4	2	no	—
152	C₂²	—	8	2	no	—
153	C₂²	—	61	1	no	—
154	C₂²	—	61	1	no	—
155	C₂²	—	244	1	no	—
156	C₂²	#14519839	1	1	yes	characteristic
157	C₄	—	61	1	no	—
158	C₄	—	61	1	no	—
159	C₄	—	244	2	no	—
160	C₄	#14519839	1	1	yes	characteristic
161	C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	8	2	no	—
164	C₂	—	122	1	no	—
165	C₂	—	122	1	no	—
166	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	#2143
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. D₆₁⋊#108	C₂	semidirect
3. #16840676	C₂	semidirect
8. D₆₁⋊C₁₆	C₂²	semidirect
9. #14519848	C₂²	semidirect
88. C₆₁	#2143	semidirect