Finite Group Databasebeta

Subgroups — #35319152

Order 21248.

246 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 99 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	#24909745	C₂	1	2	yes	maximal
2	#24909738	C₂	1	2	yes	maximal
3	#24907081	C₂	1	1	yes	maximal, characteristic
4	#24907073	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1311	C₂	1	1	yes	maximal, characteristic
6	#19128106	—	2	1	no	—
7	#19128106	C₂²	1	1	yes	characteristic
8	#19128106	C₂²	1	1	yes	characteristic
9	#19128106	C₂²	1	2	yes	—
10	#19128081	—	2	1	no	—
11	#19128052	—	2	1	no	—
12	#19128052	—	2	1	no	—
13	#19127529	—	2	4	no	—
14	#19127527	—	2	4	no	—
15	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
16	C₈²×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₃₂	—	4	2	no	—
18	#14581792	—	2	2	no	—
19	#14581792	—	2	1	no	—
20	#14581792	—	2	1	no	—
21	#14581726	—	2	2	no	—
22	#14581726	—	2	2	no	—
23	#14581726	—	2	2	no	—
24	#14581726	—	2	2	no	—
25	#14581726	—	4	2	no	—
26	#14581717	—	4	2	no	—
27	#14581717	—	4	2	no	—
28	C₂×C₁₆×C₈₃	D₄	1	4	yes	—
29	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₆₆	—	4	2	no	—
33	C₂×Dic₁₆₆	—	4	1	no	—
34	C₂×Dic₁₆₆	—	4	1	no	—
35	Dic₃₃₂	—	8	4	no	—
36	C₄⋊Dic₈₃	—	4	2	no	—
37	C₄⋊Dic₈₃	—	4	1	no	—
38	C₄⋊Dic₈₃	—	4	1	no	—
39	C₄⋊Dic₈₃	—	4	1	no	—
40	C₄⋊Dic₈₃	—	4	1	no	—
41	C₄⋊Dic₈₃	—	4	2	no	—
42	C₂×C₈×C₈₃	D₈	1	2	yes	—
43	C₂×C₈×C₈₃	D₈	1	2	yes	—
44	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
45	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₈₃	Q₁₆	1	8	yes	complemented
47	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
48	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₈₃	—	8	1	no	—
51	C₂×Dic₈₃	—	8	2	no	—
52	C₂×Dic₈₃	—	8	1	no	—
53	Dic₁₆₆	—	8	4	no	—
54	Dic₁₆₆	—	8	2	no	—
55	Dic₁₆₆	—	8	2	no	—
56	C₈×C₈₃	—	2	2	no	—
57	C₈×C₈₃	—	2	2	no	—
58	C₈×C₈₃	C₂×Q₁₆	1	4	yes	—
59	C₂²×C₈₃	#493	1	1	yes	characteristic
60	C₄×C₈₃	#494	1	2	yes	—
61	C₄×C₈₃	#509	1	2	yes	—
62	C₄×C₈₃	#510	1	2	yes	—
63	Dic₈₃	—	16	2	no	—
64	Dic₈₃	—	16	1	no	—
65	Dic₈₃	—	16	1	no	—
66	#12524	—	83	1	no	maximal, Sylow
67	C₂×C₈₃	#1713	1	1	yes	characteristic
68	C₂×C₈₃	#2250	1	1	yes	characteristic
69	C₂×C₈₃	#2251	1	1	yes	characteristic
70	#2254	—	83	2	no	—
71	#2252	—	83	2	no	—
72	#1713	—	83	1	no	—
73	#1710	—	83	1	no	—
74	#1311	D₈₃	1	1	yes	characteristic
75	C₈₃	#12524	1	1	yes	characteristic, complemented, Sylow
76	#500	—	83	1	no	—
77	#500	—	83	1	no	—
78	#500	—	83	2	no	—
79	#500	—	166	1	no	—
80	#494	—	166	1	no	—
81	#487	—	166	1	no	—
82	#487	—	166	1	no	—
83	#367	—	166	4	no	—
84	#366	—	166	4	no	—
85	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
86	C₈²	C₂×D₈₃	1	1	yes	characteristic
87	C₂×Q₁₆	—	332	2	no	—
88	C₄⋊Q₈	—	166	1	no	—
89	C₄⋊Q₈	—	166	1	no	—
90	C₄⋊Q₈	—	166	2	no	—
91	C₂×C₁₆	D₃₃₂	1	4	yes	—
92	#107	—	332	2	no	—
93	#107	—	166	2	no	—
94	#107	—	166	2	no	—
95	#107	—	166	2	no	—
96	#107	—	166	2	no	—
97	#103	—	332	2	no	—
98	#103	—	332	2	no	—
99	C₄×C₈	D₃₃₂	1	1	yes	characteristic
100	C₄×C₈	D₃₃₂	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
102	C₂×Q₈	—	332	1	no	—
103	C₂×Q₈	—	332	1	no	—
104	C₂×Q₈	—	332	2	no	—
105	Q₁₆	—	664	4	no	—
106	C₂×C₈	D₆₆₄	1	2	yes	—
107	C₂×C₈	D₆₆₄	1	2	yes	—
108	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
109	C₄⋊C₄	—	332	1	no	—
110	C₄⋊C₄	—	332	2	no	—
111	C₄⋊C₄	—	332	1	no	—
112	C₄⋊C₄	—	332	1	no	—
113	C₄⋊C₄	—	332	1	no	—
114	C₄⋊C₄	—	332	2	no	—
115	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
116	C₁₆	Dic₃₃₂	1	8	yes	complemented
117	Q₈	—	664	4	no	—
118	Q₈	—	664	2	no	—
119	Q₈	—	664	2	no	—
120	C₂×C₄	—	664	1	no	—
121	C₂×C₄	—	664	2	no	—
122	C₂×C₄	—	664	1	no	—
123	C₂×C₄	#14581789	1	1	yes	characteristic
124	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
125	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	C₂×Dic₃₃₂	1	4	yes	—
129	C₂²	#19128078	1	1	yes	characteristic
130	C₄	—	1328	2	no	—
131	C₄	—	1328	1	no	—
132	C₄	—	1328	1	no	—
133	C₄	#19128081	1	2	yes	—
134	C₄	#19128139	1	2	yes	—
135	C₄	#19128144	1	2	yes	—
136	C₂	#24907081	1	1	yes	characteristic
137	C₂	#24909729	1	1	yes	characteristic
138	C₂	#24909734	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19128078
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1311	C₂
socle, prime radical	C₂²×C₈₃	#493
hypercenter	#1311	D₈₃
hypocenter	C₈₃	#12524
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

Complements

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

N	H ≅ G/N	type
46. C₁₆×C₈₃	Q₁₆	semidirect
75. C₈₃	#12524	semidirect
116. C₁₆	Dic₃₃₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#24909745	2
#24909738	2
#24907081	2
#24907073	2
C₈₃×#1311	2
#12524	83