Finite Group Databasebeta

Subgroups — #21460906

Order 7808.

166 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 38 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	#16840742	C₂	1	1	yes	maximal, characteristic
2	#16840735	C₂	1	1	yes	maximal, characteristic
3	#16839734	C₂	1	1	yes	maximal, characteristic
4	C₂×#11758217	C₄	1	1	yes	characteristic, complemented
5	C₂²×#92420	C₄	1	1	yes	characteristic, complemented
6	#14519856	—	2	1	no	—
7	#14519835	—	2	1	no	—
8	#14519835	C₄	1	2	yes	complemented
9	#14519834	—	2	1	no	—
10	C₂²⋊#92420	—	2	1	no	—
11	C₂×#11758224	C₂²	1	1	yes	characteristic
12	#14519733	—	2	1	no	—
13	#14519692	C₄	1	1	yes	characteristic, complemented
14	#14519691	—	2	1	no	—
15	#99×C₆₁	C₄	1	1	yes	characteristic, complemented
16	C₂×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₂²⋊D₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₂²⋊D₆₁	D₄	1	1	yes	characteristic
19	C₂²×Dic₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₆₁	D₄	1	1	yes	characteristic
21	#11758224	—	4	1	no	—
22	#11758224	—	4	1	no	—
23	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
24	C₂×Dic₁₂₂	Q₈	1	1	yes	characteristic
25	C₂²⋊#6923	—	2	2	no	—
26	#11758217	—	4	1	no	—
27	C₂×#92420	—	2	1	no	—
28	C₂×#92420	—	2	1	no	—
29	C₂×#92420	—	4	1	no	—
30	C₄⋊#6923	—	4	1	no	—
31	C₄×#6923	—	4	1	no	—
32	C₂²⋊C₄×C₆₁	—	2	1	no	—
33	C₂²⋊Dic₆₁	—	2	1	no	—
34	#11758196	—	4	1	no	—
35	C₄×Dic₆₁	—	4	1	no	—
36	C₂³×C₆₁	C₄²	1	1	yes	characteristic
37	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×#6923	—	4	2	no	—
40	D₄×C₆₁	—	4	1	no	—
41	C₂×C₄×C₆₁	—	4	1	no	—
42	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂²⋊D₆₁	—	4	1	no	—
44	C₂²⋊D₆₁	—	4	1	no	—
45	C₂×Dic₆₁	—	2	1	no	—
46	C₂×Dic₆₁	—	2	1	no	—
47	C₂×Dic₆₁	—	2	1	no	—
48	C₂×Dic₆₁	—	2	1	no	—
49	C₂×Dic₆₁	—	4	1	no	—
50	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
53	C₄×D₆₁	—	2	1	no	—
54	C₄×D₆₁	—	2	1	no	—
55	Dic₁₂₂	—	4	1	no	—
56	#92420	—	4	1	no	—
57	#92420	—	4	1	no	—
58	C₂²×C₆₁	—	2	1	no	—
59	C₂²×C₆₁	—	2	1	no	—
60	C₂²×C₆₁	—	4	1	no	—
61	C₂²×C₆₁	—	4	1	no	—
62	C₂²×C₆₁	#95	1	1	yes	characteristic
63	C₂×D₆₁	—	2	1	no	—
64	C₂×D₆₁	—	4	1	no	—
65	#6923	—	8	2	no	—
66	C₄×C₆₁	—	2	1	no	—
67	C₄×C₆₁	—	8	1	no	—
68	Dic₆₁	—	2	1	no	—
69	Dic₆₁	—	2	1	no	—
70	Dic₆₁	—	8	1	no	—
71	Dic₆₁	#99	1	1	yes	characteristic
72	Dic₆₁	#99	1	1	yes	characteristic
73	#1391	—	61	1	no	maximal, Sylow
74	C₂×C₆₁	—	2	1	no	—
75	C₂×C₆₁	—	4	1	no	—
76	C₂×C₆₁	—	4	1	no	—
77	C₂×C₆₁	#342	1	1	yes	characteristic
78	D₆₁	—	4	1	no	—
79	#410	—	61	1	no	—
80	#410	—	61	1	no	—
81	#408	—	61	1	no	—
82	C₆₁	#1391	1	1	yes	characteristic, complemented, Sylow
83	C₂×#41	—	61	1	no	—
84	C₂×M₁₆	—	61	1	no	—
85	C₂²×C₈	—	61	1	no	—
86	#117	—	122	1	no	—
87	#117	—	122	1	no	—
88	#99	—	61	2	no	—
89	#99	—	61	1	no	—
90	#99	—	122	1	no	—
91	#99	—	122	1	no	—
92	#99	#6923	1	1	yes	characteristic, complemented
93	C₂²⋊C₈	—	122	1	no	—
94	C₂²⋊C₈	—	122	1	no	—
95	#41	—	244	1	no	—
96	#41	—	244	1	no	—
97	C₂×Q₈	—	61	1	no	—
98	C₂×D₄	—	61	1	no	—
99	C₂×D₄	—	61	1	no	—
100	C₂×D₄	C₂×#6923	1	1	yes	characteristic
101	C₂²×C₄	—	61	1	no	—
102	C₂²×C₄	—	61	1	no	—
103	C₂²×C₄	—	61	1	no	—
104	M₁₆	—	244	1	no	—
105	C₂×C₈	—	122	1	no	—
106	C₂×C₈	—	122	1	no	—
107	C₂×C₈	—	244	1	no	—
108	C₄⋊C₄	—	244	1	no	—
109	C₄⋊C₄	—	244	1	no	—
110	C₂²⋊C₄	—	2	1	no	—
111	C₂²⋊C₄	—	122	2	no	—
112	C₂²⋊C₄	—	122	1	no	—
113	C₄²	—	244	1	no	—
114	C₄²	—	244	1	no	—
115	C₂³	—	61	1	no	—
116	C₂³	C₄×#6923	1	1	yes	characteristic
117	C₂³	C₂²⋊#6923	1	1	yes	characteristic
118	Q₈	—	244	1	no	—
119	D₄	—	4	1	no	—
120	D₄	—	244	1	no	—
121	D₄	—	244	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	61	1	no	—
124	C₂×C₄	—	61	1	no	—
125	C₂×C₄	—	61	1	no	—
126	C₂×C₄	—	122	1	no	—
127	C₂×C₄	—	122	1	no	—
128	C₂×C₄	—	122	1	no	—
129	C₂×C₄	—	122	1	no	—
130	C₂×C₄	—	122	1	no	—
131	C₂×C₄	—	122	1	no	—
132	C₂×C₄	—	244	1	no	—
133	C₂×C₄	—	244	2	no	—
134	C₂×C₄	C₄⋊#6923	1	1	yes	characteristic
135	C₈	—	244	1	no	—
136	C₈	—	244	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	122	1	no	—
142	C₂²	—	244	1	no	—
143	C₂²	(C₂×#6923)⋊C₄	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	8	1	no	—
146	C₄	—	61	1	no	—
147	C₄	—	61	1	no	—
148	C₄	—	122	1	no	—
149	C₄	—	122	1	no	—
150	C₄	—	488	1	no	—
151	C₄	—	488	2	no	—
152	C₂	—	2	1	no	—
153	C₂	—	4	1	no	—
154	C₂	—	4	1	no	—
155	C₂	—	244	1	no	—
156	C₂	#16840639	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16840639
commutator	C₂³×C₆₁	C₄²
Frattini	C₂×D₄	C₂×#6923
Fitting	#99×C₆₁	C₄
socle	C₂×C₆₁	#342
hypercenter	#99	#6923
hypocenter	C₆₁	#1391
LCS	G ▷ C₂³×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #99

Complements

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

N	H ≅ G/N	type
4. C₂×#11758217	C₄	semidirect
5. C₂²×#92420	C₄	semidirect
8. #14519835	C₄	semidirect
13. #14519692	C₄	semidirect
15. #99×C₆₁	C₄	semidirect
82. C₆₁	#1391	semidirect
92. #99	#6923	semidirect