Finite Group Databasebeta

Subgroups — #21461866

Order 7808.

314 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 66 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	#16840830	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840761	C₂	1	2	yes	maximal
3	D₆₁⋊#101	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840629	C₂	1	2	yes	maximal, complemented
5	#16840337	C₂	1	1	yes	maximal, characteristic
6	D₆₁⋊M₁₆	C₂²	1	2	yes	—
7	Q₈×#6923	—	2	4	no	—
8	#14519856	—	2	2	no	—
9	#14519856	C₂²	1	2	yes	—
10	#14519842	—	2	4	no	—
11	#14519838	—	2	2	no	—
12	C₂×#11758226	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₆₁	C₂²	1	1	yes	characteristic
14	#14519775	C₄	1	1	yes	characteristic, complemented
15	#14519766	C₄	1	1	yes	characteristic, complemented
16	#14519765	C₄	1	1	yes	characteristic, complemented
17	#14519762	—	2	2	no	—
18	#14519755	—	2	2	no	—
19	#14519734	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic, complemented
21	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
23	#11758226	—	2	2	no	—
24	#11758226	—	2	2	no	—
25	Q₈×D₆₁	—	2	2	no	—
26	Q₈×D₆₁	—	2	2	no	—
27	C₂×D₂₄₄	—	2	1	no	—
28	C₂×D₂₄₄	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₆₁	—	2	1	no	—
30	C₂×C₄×D₆₁	D₄	1	2	yes	—
31	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₁₂₂	D₄	1	2	yes	—
34	C₂²⋊#6923	—	2	2	no	—
35	#11758217	—	2	2	no	—
36	C₄⋊#6923	—	2	4	no	—
37	C₄⋊#6923	—	2	2	no	—
38	C₄×#6923	—	2	4	no	—
39	C₄×#6923	D₄	1	4	yes	—
40	#11758213	—	2	2	no	—
41	D₆₁⋊C₈	—	2	2	no	—
42	C₄⋊C₄×C₆₁	—	2	2	no	—
43	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
44	#11758198	—	2	1	no	—
45	#11758198	—	2	2	no	—
46	#11758198	—	2	2	no	—
47	#11758196	—	2	2	no	—
48	#11758196	—	2	1	no	—
49	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₆₁	—	2	1	no	—
51	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
52	C₂×#6923	—	2	4	no	—
53	Q₈×C₆₁	—	2	2	no	—
54	Q₈×C₆₁	—	2	2	no	—
55	C₂×C₄×C₆₁	—	2	1	no	—
56	C₂×C₄×C₆₁	—	2	1	no	—
57	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×Dic₆₁	—	2	1	no	—
60	C₂×Dic₆₁	C₂²⋊C₄	1	2	yes	—
61	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
62	D₂₄₄	—	2	2	no	—
63	D₂₄₄	—	4	1	no	—
64	D₂₄₄	—	4	1	no	—
65	C₄×D₆₁	—	2	2	no	—
66	C₄×D₆₁	—	2	2	no	—
67	C₄×D₆₁	—	4	2	no	—
68	C₄×D₆₁	C₂×D₄	1	2	yes	—
69	C₄×D₆₁	#41	1	2	yes	—
70	Dic₁₂₂	—	2	2	no	—
71	Dic₁₂₂	—	2	4	no	—
72	#92420	—	4	2	no	—
73	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×D₆₁	—	4	1	no	—
75	C₂×D₆₁	—	4	2	no	—
76	C₂×D₆₁	—	4	1	no	—
77	C₂×D₆₁	#120	1	1	yes	characteristic
78	C₂×D₆₁	#123	1	1	yes	characteristic
79	#6923	—	4	4	no	—
80	C₄×C₆₁	—	2	2	no	—
81	C₄×C₆₁	—	4	1	no	—
82	C₄×C₆₁	—	4	1	no	—
83	C₄×C₆₁	C₄×D₄	1	2	yes	—
84	Dic₆₁	—	2	2	no	—
85	Dic₆₁	—	4	1	no	—
86	Dic₆₁	#121	1	2	yes	—
87	#1912	—	61	1	no	maximal, Sylow
88	C₂×C₆₁	—	2	1	no	—
89	C₂×C₆₁	#386	1	1	yes	characteristic
90	D₆₁	—	2	1	no	—
91	D₆₁	—	2	1	no	—
92	D₆₁	—	8	1	no	—
93	#536	—	61	1	no	—
94	#519	—	61	1	no	—
95	#421	—	61	2	no	—
96	C₂×#101	—	61	1	no	—
97	H(3,ℤ/4)	—	61	2	no	—
98	C₆₁	#1912	1	1	yes	characteristic, complemented, Sylow
99	C₂×#41	—	61	1	no	—
100	C₂²×Q₈	—	61	1	no	—
101	C₂×M₁₆	—	61	2	no	—
102	C₄⋊Q₈	—	61	1	no	—
103	C₄⋊Q₈	#6923	1	1	yes	characteristic, complemented
104	#124	—	61	1	no	—
105	#124	—	61	1	no	—
106	#123	—	122	2	no	—
107	C₂²⋊Q₈	—	122	2	no	—
108	C₄×Q₈	—	122	4	no	—
109	#117	—	61	2	no	—
110	#117	—	61	1	no	—
111	#117	—	122	2	no	—
112	#104	—	122	4	no	—
113	#101	—	122	2	no	—
114	#41	—	122	2	no	—
115	#41	—	122	2	no	—
116	C₂×Q₈	—	61	2	no	—
117	C₂×Q₈	—	61	1	no	—
118	C₂×Q₈	—	122	2	no	—
119	C₂×Q₈	—	122	2	no	—
120	C₂×Q₈	C₂×#6923	1	1	yes	characteristic
121	C₂×Q₈	C₂×#6923	1	1	yes	characteristic
122	C₂×D₄	—	61	1	no	—
123	C₂×D₄	—	122	1	no	—
124	C₂²×C₄	—	61	1	no	—
125	C₂²×C₄	—	61	2	no	—
126	C₂²×C₄	—	122	1	no	—
127	M₁₆	—	122	2	no	—
128	M₁₆	—	122	2	no	—
129	C₂×C₈	—	122	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	122	1	no	—
132	C₄⋊C₄	—	122	2	no	—
133	C₄⋊C₄	—	122	2	no	—
134	C₄⋊C₄	—	122	4	no	—
135	C₂²⋊C₄	—	122	2	no	—
136	C₂²⋊C₄	—	122	1	no	—
137	C₂²⋊C₄	—	122	2	no	—
138	C₂²⋊C₄	—	122	2	no	—
139	C₄²	—	61	4	no	—
140	C₄²	—	61	1	no	—
141	C₄²	—	122	4	no	—
142	C₄²	C₂×#6923	1	1	yes	characteristic
143	C₂³	—	61	1	no	—
144	C₂³	—	122	1	no	—
145	Q₈	—	2	2	no	—
146	Q₈	—	2	2	no	—
147	Q₈	—	122	4	no	—
148	Q₈	—	122	2	no	—
149	D₄	—	122	2	no	—
150	D₄	—	244	1	no	—
151	D₄	—	244	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	61	1	no	—
155	C₂×C₄	—	61	2	no	—
156	C₂×C₄	—	61	2	no	—
157	C₂×C₄	—	61	2	no	—
158	C₂×C₄	—	122	2	no	—
159	C₂×C₄	—	122	2	no	—
160	C₂×C₄	—	122	4	no	—
161	C₂×C₄	—	122	1	no	—
162	C₂×C₄	—	244	2	no	—
163	C₂×C₄	C₂²⋊#6923	1	2	yes	—
164	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
165	C₈	—	244	2	no	—
166	C₂²	—	61	1	no	—
167	C₂²	—	61	1	no	—
168	C₂²	—	244	1	no	—
169	C₂²	—	244	2	no	—
170	C₂²	—	244	1	no	—
171	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	4	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	61	2	no	—
176	C₄	—	122	2	no	—
177	C₄	—	244	1	no	—
178	C₄	—	244	4	no	—
179	C₄	D₄×#6923	1	2	yes	—
180	C₂	—	2	1	no	—
181	C₂	—	122	1	no	—
182	C₂	—	122	1	no	—
183	C₂	—	488	1	no	—
184	C₂	#16840710	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16840710
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₄⋊Q₈×C₆₁	C₄
socle	C₂×C₆₁	#386
hypercenter	C₄⋊Q₈	#6923
hypocenter	C₆₁	#1912
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #16840830	C₂	semidirect
3. D₆₁⋊#101	C₂	semidirect
4. #16840629	C₂	semidirect
14. #14519775	C₄	semidirect
15. #14519766	C₄	semidirect
16. #14519765	C₄	semidirect
20. C₄⋊Q₈×C₆₁	C₄	semidirect
98. C₆₁	#1912	semidirect
103. C₄⋊Q₈	#6923	semidirect