Finite Group Databasebeta

Subgroups — #21445940

Order 7808.

172 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₂×#14519712	C₂	1	1	yes	maximal, characteristic
2	#16839798	C₂	1	1	yes	maximal, characteristic
3	#16839720	C₂	1	1	yes	maximal, characteristic
4	#16839533	C₂	1	1	yes	maximal, characteristic
5	#16839519	C₂	1	1	yes	maximal, characteristic
6	#16839518	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#344	C₂	1	1	yes	maximal, characteristic
8	C₂×#11758194	C₂²	1	1	yes	characteristic
9	C₂²×#92418	C₂²	1	1	yes	characteristic
10	#14519736	C₂²	1	1	yes	characteristic
11	#14519736	C₂²	1	1	yes	characteristic
12	#14519712	—	2	1	no	—
13	#14519712	—	2	1	no	—
14	#14519710	—	2	1	no	—
15	#14519708	—	2	1	no	—
16	#14519687	C₄	1	2	yes	complemented
17	C₈×Dic₆₁	C₄	1	2	yes	complemented
18	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
20	#117×C₆₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
22	M₁₆×C₆₁	—	2	1	no	—
23	M₁₆×C₆₁	—	2	1	no	—
24	M₁₆×C₆₁	—	2	1	no	—
25	M₁₆×C₆₁	—	2	1	no	—
26	C₂×C₈×C₆₁	—	2	1	no	—
27	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₆₁	—	2	1	no	—
29	C₂²⋊C₄×C₆₁	—	2	1	no	—
30	C₄²×C₆₁	—	2	1	no	—
31	C₂²⋊Dic₆₁	—	2	1	no	—
32	C₂²⋊Dic₆₁	—	2	1	no	—
33	C₄⋊Dic₆₁	—	2	1	no	—
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₄×Dic₆₁	—	2	1	no	—
36	C₄×Dic₆₁	C₂×C₄	1	2	yes	—
37	#11758194	—	2	1	no	—
38	#11758194	—	2	1	no	—
39	C₂×#92418	—	2	1	no	—
40	C₂×#92418	C₂×C₄	1	1	yes	characteristic
41	C₂×#92418	C₂×C₄	1	1	yes	characteristic
42	C₂×#92418	D₄	1	1	yes	characteristic
43	C₂×#92418	D₄	1	1	yes	characteristic
44	C₂×#92418	Q₈	1	1	yes	characteristic
45	C₂×#92418	Q₈	1	1	yes	characteristic
46	C₂³×C₆₁	#41	1	1	yes	characteristic
47	C₂×C₄×C₆₁	—	4	1	no	—
48	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
54	C₂×Dic₆₁	—	2	2	no	—
55	C₂×Dic₆₁	—	4	1	no	—
56	C₈×C₆₁	—	2	2	no	—
57	C₈×C₆₁	—	4	1	no	—
58	#92418	—	4	1	no	—
59	#92418	C₄⋊C₄	1	1	yes	characteristic
60	#92418	C₄⋊C₄	1	1	yes	characteristic
61	#92418	C₄⋊C₄	1	1	yes	characteristic
62	#92418	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₆₁	—	4	1	no	—
64	C₂²×C₆₁	#117	1	1	yes	characteristic
65	C₂²×C₆₁	#124	1	1	yes	characteristic
66	C₂²×C₆₁	#125	1	1	yes	characteristic
67	C₄×C₆₁	—	8	1	no	—
68	C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₄×C₆₁	#117	1	1	yes	characteristic
70	C₄×C₆₁	#124	1	1	yes	characteristic
71	C₄×C₆₁	#125	1	1	yes	characteristic
72	Dic₆₁	—	4	2	no	—
73	Dic₆₁	—	8	1	no	—
74	#1833	—	61	1	no	maximal, Sylow
75	C₂×C₆₁	—	2	1	no	—
76	C₂×C₆₁	—	2	1	no	—
77	C₂×C₆₁	—	2	1	no	—
78	C₂×C₆₁	#382	1	1	yes	characteristic
79	C₂×#108	—	61	1	no	—
80	#427	—	61	1	no	—
81	#405	—	61	1	no	—
82	#344	—	61	1	no	—
83	#344	D₆₁	1	1	yes	characteristic
84	H(3,ℤ/4)	—	61	1	no	—
85	H(3,ℤ/4)	—	61	1	no	—
86	C₆₁	#1833	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₁₆	—	61	1	no	—
88	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
89	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
90	C₂²×C₈	—	61	1	no	—
91	#117	—	61	1	no	—
92	#117	—	61	1	no	—
93	#117	C₂×D₆₁	1	1	yes	characteristic
94	#108	—	122	1	no	—
95	#108	—	122	1	no	—
96	#107	—	122	1	no	—
97	#106	—	122	1	no	—
98	#97	—	61	2	no	—
99	C₄×C₈	—	61	2	no	—
100	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
101	M₁₆	—	2	1	no	—
102	M₁₆	—	2	1	no	—
103	M₁₆	—	2	1	no	—
104	M₁₆	—	2	1	no	—
105	M₁₆	—	122	1	no	—
106	M₁₆	—	122	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	61	1	no	—
109	C₂×C₈	—	61	1	no	—
110	C₂×C₈	—	61	1	no	—
111	C₂×C₈	—	61	1	no	—
112	C₂×C₈	—	61	1	no	—
113	C₂×C₈	—	61	1	no	—
114	C₂×C₈	—	122	1	no	—
115	C₂×C₈	C₄×D₆₁	1	2	yes	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	122	1	no	—
118	C₄⋊C₄	—	122	1	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	122	1	no	—
121	C₂²⋊C₄	—	122	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	61	2	no	—
124	C₄²	—	122	1	no	—
125	C₂³	#11758222	1	1	yes	characteristic
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	122	2	no	—
128	C₂×C₄	—	244	1	no	—
129	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
130	C₂×C₄	#11758222	1	1	yes	characteristic
131	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
132	C₂×C₄	#11758224	1	1	yes	characteristic
133	C₂×C₄	#11758224	1	1	yes	characteristic
134	C₂×C₄	Q₈×D₆₁	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	4	1	no	—
137	C₈	—	61	1	no	—
138	C₈	—	61	1	no	—
139	C₈	—	61	1	no	—
140	C₈	—	61	1	no	—
141	C₈	—	244	1	no	—
142	C₂²	—	4	1	no	—
143	C₂²	#14519734	1	1	yes	characteristic
144	C₂²	#14519763	1	1	yes	characteristic
145	C₂²	#14519767	1	1	yes	characteristic
146	C₄	—	8	1	no	—
147	C₄	—	488	1	no	—
148	C₄	—	244	2	no	—
149	C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
150	C₄	#14519733	1	1	yes	characteristic
151	C₄	#14519763	1	1	yes	characteristic
152	C₄	#14519767	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#16839626	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14519733
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#344	C₂
socle	C₂×C₆₁	#382
hypercenter	#344	D₆₁
hypocenter	C₆₁	#1833
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

Complements

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

N	H ≅ G/N	type
16. #14519687	C₄	semidirect
17. C₈×Dic₆₁	C₄	semidirect
86. C₆₁	#1833	semidirect