Finite Group Databasebeta

Subgroups — #21461391

Order 7808.

150 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 60 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	#16840784	C₂	1	1	yes	maximal, characteristic
2	#16840781	C₂	1	1	yes	maximal, characteristic
3	#16840626	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840625	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840624	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840623	C₂	1	1	yes	maximal, characteristic, complemented
7	#16839814	C₂	1	1	yes	maximal, characteristic
8	C₄⋊#92420	C₂²	1	1	yes	characteristic
9	C₄⋊#92420	C₂²	1	1	yes	characteristic
10	C₄⋊#92420	C₂²	1	1	yes	characteristic
11	C₄⋊#92420	C₂²	1	1	yes	characteristic
12	#14519834	—	2	1	no	—
13	#14519834	—	2	1	no	—
14	#14519832	—	2	1	no	—
15	C₄×#92420	—	2	1	no	—
16	C₄²×D₆₁	C₂²	1	1	yes	characteristic
17	C₄⋊#92418	C₄	1	1	yes	characteristic
18	#14519689	C₄	1	1	yes	characteristic
19	#14519689	C₄	1	1	yes	characteristic
20	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
21	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic
23	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
25	C₂×C₄×D₆₁	Q₈	1	1	yes	characteristic
26	C₂×#92420	—	2	1	no	—
27	C₂×#92420	—	2	1	no	—
28	C₂×#92420	—	2	1	no	—
29	C₂×#92420	—	2	1	no	—
30	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₆₁	D₄	1	1	yes	characteristic
34	C₄×Dic₆₁	Q₈	1	1	yes	characteristic
35	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
36	C₂×#92418	C₂×C₄	1	1	yes	characteristic
37	C₂×#92418	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₄×D₆₁	—	2	1	no	—
46	C₄×D₆₁	—	2	1	no	—
47	C₄×D₆₁	—	2	1	no	—
48	C₄×D₆₁	—	2	1	no	—
49	C₄×D₆₁	—	4	1	no	—
50	#92420	—	4	1	no	—
51	#92420	—	4	1	no	—
52	#92420	—	4	1	no	—
53	#92420	—	4	1	no	—
54	C₈×C₆₁	—	2	1	no	—
55	C₈×C₆₁	—	2	1	no	—
56	#92418	—	2	1	no	—
57	#92418	—	2	1	no	—
58	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×D₆₁	—	2	1	no	—
60	C₂×D₆₁	—	2	1	no	—
61	C₂×D₆₁	—	2	1	no	—
62	C₄×C₆₁	—	2	1	no	—
63	C₄×C₆₁	—	2	1	no	—
64	C₄×C₆₁	#131	1	1	yes	characteristic
65	C₄×C₆₁	#131	1	1	yes	characteristic
66	Dic₆₁	—	2	1	no	—
67	Dic₆₁	—	2	1	no	—
68	Dic₆₁	#135	1	1	yes	characteristic
69	Dic₆₁	#135	1	1	yes	characteristic
70	#1572	—	61	1	no	maximal, Sylow
71	C₂×C₆₁	#424	1	1	yes	characteristic
72	C₂×C₆₁	#427	1	1	yes	characteristic
73	C₂×C₆₁	#430	1	1	yes	characteristic
74	D₆₁	—	4	1	no	—
75	#432	—	61	1	no	—
76	#431	—	61	1	no	—
77	#431	—	61	1	no	—
78	#335	—	61	1	no	—
79	#335	—	61	1	no	—
80	#334	—	61	1	no	—
81	#334	—	61	1	no	—
82	C₆₁	#1572	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄²	—	61	1	no	—
84	C₄⋊C₈	—	61	1	no	—
85	C₄⋊C₈	—	61	1	no	—
86	C₄⋊C₈	—	61	1	no	—
87	C₄⋊C₈	—	61	1	no	—
88	C₄⋊C₈	—	61	1	no	—
89	C₄⋊C₈	#6923	1	1	yes	characteristic
90	C₂²⋊C₈	—	61	1	no	—
91	C₂²⋊C₈	—	61	1	no	—
92	C₂²⋊C₈	—	122	1	no	—
93	C₂²⋊C₈	—	122	1	no	—
94	#97	—	122	1	no	—
95	C₄×C₈	—	61	1	no	—
96	C₄×C₈	—	61	1	no	—
97	C₄×C₈	—	122	1	no	—
98	C₂²×C₄	—	61	1	no	—
99	C₂²×C₄	—	61	1	no	—
100	C₂²×C₄	—	61	1	no	—
101	C₂×C₈	—	61	1	no	—
102	C₂×C₈	—	61	1	no	—
103	C₂×C₈	—	122	1	no	—
104	C₂×C₈	—	122	1	no	—
105	C₂×C₈	—	122	1	no	—
106	C₂×C₈	—	122	1	no	—
107	C₂×C₈	C₂×#6923	1	1	yes	characteristic
108	C₂×C₈	C₂×#6923	1	1	yes	characteristic
109	C₄²	—	61	1	no	—
110	C₄²	—	61	1	no	—
111	C₄²	—	61	1	no	—
112	C₄²	C₂×#6923	1	1	yes	characteristic
113	C₂³	—	61	1	no	—
114	C₂×C₄	—	61	1	no	—
115	C₂×C₄	—	61	1	no	—
116	C₂×C₄	—	61	1	no	—
117	C₂×C₄	—	122	1	no	—
118	C₂×C₄	—	122	1	no	—
119	C₂×C₄	—	122	1	no	—
120	C₂×C₄	—	122	1	no	—
121	C₂×C₄	—	244	1	no	—
122	C₂×C₄	C₄⋊#6923	1	1	yes	characteristic
123	C₂×C₄	C₄⋊#6923	1	1	yes	characteristic
124	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	122	1	no	—
128	C₈	—	122	1	no	—
129	C₈	—	244	1	no	—
130	C₈	—	244	1	no	—
131	C₈	—	244	1	no	—
132	C₈	—	244	1	no	—
133	C₂²	—	122	1	no	—
134	C₂²	—	122	1	no	—
135	C₂²	—	122	1	no	—
136	C₂²	C₂×C₄⋊#6923	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	61	1	no	—
140	C₄	—	61	1	no	—
141	C₄	—	122	1	no	—
142	C₄	—	122	1	no	—
143	C₄	#14519864	1	1	yes	characteristic
144	C₄	#14519865	1	1	yes	characteristic
145	C₂	—	244	1	no	—
146	C₂	#16840769	1	1	yes	characteristic
147	C₂	#16840772	1	1	yes	characteristic
148	C₂	#16840777	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₄⋊C₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄⋊C₄
hypercenter	C₄⋊C₈	#6923
hypocenter	C₆₁	#1572
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #16840626	C₂	semidirect
4. #16840625	C₂	semidirect
5. #16840624	C₂	semidirect
6. #16840623	C₂	semidirect
82. C₆₁	#1572	semidirect