Finite Group Databasebeta

Subgroups — #35325299

Order 21248.

164 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 59 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	#24909737	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	#24905663	C₂	1	1	yes	maximal, characteristic
4	#24905662	C₂	1	1	yes	maximal, characteristic
5	Dic₈₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	#24905644	C₂	1	1	yes	maximal, characteristic
7	C₈₃×#1416	C₂	1	1	yes	maximal, characteristic
8	#19128105	—	2	1	no	—
9	Q₁₆×Dic₈₃	—	2	1	no	—
10	C₂×C₈₃⋊C₃₂	C₂²	1	1	yes	characteristic
11	#19127527	C₂²	1	1	yes	characteristic
12	#19127527	C₂²	1	1	yes	characteristic
13	#19127526	—	2	1	no	—
14	#19127505	—	2	1	no	—
15	C₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₈₃	C₂²	1	1	yes	characteristic
18	#366×C₈₃	C₂²	1	1	yes	characteristic
19	#14581791	—	4	1	no	—
20	Q₈×Dic₈₃	—	4	1	no	—
21	C₂×C₈₃⋊C₁₆	D₄	1	1	yes	characteristic
22	#14581726	—	2	1	no	—
23	#14581726	—	2	1	no	—
24	#14581725	—	4	1	no	—
25	#14581716	—	4	1	no	—
26	C₈×Dic₈₃	D₄	1	1	yes	characteristic
27	C₈₃⋊C₃₂	—	2	1	no	—
28	C₂×Q₁₆×C₈₃	—	2	1	no	—
29	Q₃₂×C₈₃	—	2	1	no	—
30	Q₃₂×C₈₃	Q₈	1	2	yes	complemented
31	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
32	#107×C₈₃	—	2	1	no	—
33	C₃₂×C₈₃	—	2	1	no	—
34	C₂×Dic₁₆₆	—	8	1	no	—
35	C₂×C₈₃⋊C₈	D₈	1	1	yes	characteristic
36	C₄⋊Dic₈₃	—	4	1	no	—
37	C₄⋊Dic₈₃	—	4	1	no	—
38	C₄⋊Dic₈₃	—	8	1	no	—
39	Dic₈₃⋊C₄	—	8	1	no	—
40	C₄×Dic₈₃	—	8	1	no	—
41	C₄×Dic₈₃	D₈	1	1	yes	characteristic
42	C₈₃⋊C₁₆	—	2	1	no	—
43	C₂×Q₈×C₈₃	—	4	1	no	—
44	Q₁₆×C₈₃	—	2	2	no	—
45	Q₁₆×C₈₃	—	4	1	no	—
46	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₄⋊C₄×C₈₃	—	4	1	no	—
48	C₁₆×C₈₃	C₂×Q₈	1	1	yes	characteristic
49	C₁₆×C₈₃	#41	1	1	yes	characteristic
50	Q₈×C₈₃	—	4	2	no	—
51	Q₈×C₈₃	—	8	1	no	—
52	C₂×C₄×C₈₃	—	8	1	no	—
53	C₂×C₄×C₈₃	—	8	1	no	—
54	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₈₃	—	8	1	no	—
56	C₂×Dic₈₃	—	8	1	no	—
57	C₂×Dic₈₃	D₁₆	1	1	yes	characteristic
58	C₂×Dic₈₃	D₁₆	1	1	yes	characteristic
59	Dic₁₆₆	—	16	2	no	—
60	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₈×C₈₃	#137	1	1	yes	characteristic
62	C₈₃⋊C₈	—	2	1	no	—
63	C₂²×C₈₃	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₈₃	—	8	2	no	—
65	C₄×C₈₃	—	16	1	no	—
66	C₄×C₈₃	#474	1	1	yes	characteristic
67	C₄×C₈₃	#510	1	1	yes	characteristic
68	Dic₈₃	—	2	1	no	—
69	Dic₈₃	—	16	1	no	—
70	Dic₈₃	—	16	1	no	—
71	Dic₈₃	Q₆₄	1	2	yes	—
72	#13706	—	83	1	no	maximal, Sylow
73	C₂×C₈₃	#2224	1	1	yes	characteristic
74	C₂×C₈₃	C₂×Q₆₄	1	1	yes	characteristic
75	C₂×C₈₃	#2263	1	1	yes	characteristic
76	#2252	—	83	1	no	—
77	C₄×Q₃₂	—	83	1	no	—
78	#1423	—	83	1	no	—
79	#1423	—	83	1	no	—
80	C₄⋊C₃₂	—	83	1	no	—
81	#1416	—	83	1	no	—
82	#1416	D₈₃	1	1	yes	characteristic
83	C₈₃	#13706	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₃₂	C₂×D₈₃	1	1	yes	characteristic
85	#500	—	166	1	no	—
86	C₄×Q₁₆	—	166	1	no	—
87	C₂×C₃₂	—	83	1	no	—
88	C₂×C₃₂	C₂×D₈₃	1	1	yes	characteristic
89	#366	—	83	1	no	—
90	#366	—	83	1	no	—
91	#366	—	166	1	no	—
92	#366	C₂×D₈₃	1	1	yes	characteristic
93	#358	—	166	1	no	—
94	C₄×C₁₆	—	83	1	no	—
95	C₂×Q₁₆	—	2	1	no	—
96	C₄⋊Q₈	—	332	1	no	—
97	C₄×Q₈	—	332	1	no	—
98	Q₃₂	—	2	1	no	—
99	Q₃₂	Dic₁₆₆	1	2	yes	complemented
100	C₂×C₁₆	—	83	1	no	—
101	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
102	#107	—	2	1	no	—
103	#107	—	332	1	no	—
104	#107	—	166	1	no	—
105	#107	—	166	1	no	—
106	#103	—	332	1	no	—
107	C₄×C₈	—	83	1	no	—
108	C₃₂	—	2	1	no	—
109	C₃₂	—	166	1	no	—
110	C₂×Q₈	—	4	1	no	—
111	C₂×Q₈	—	664	1	no	—
112	Q₁₆	—	2	2	no	—
113	Q₁₆	—	4	1	no	—
114	C₂×C₈	—	83	1	no	—
115	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	332	1	no	—
118	C₄⋊C₄	—	332	1	no	—
119	C₄⋊C₄	—	664	1	no	—
120	C₄⋊C₄	—	664	1	no	—
121	C₄²	—	83	1	no	—
122	C₄²	—	664	1	no	—
123	C₁₆	—	166	1	no	—
124	C₁₆	C₂×Dic₁₆₆	1	1	yes	characteristic
125	C₁₆	#13050589	1	1	yes	characteristic
126	Q₈	—	4	2	no	—
127	Q₈	—	8	1	no	—
128	Q₈	—	1328	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	83	1	no	—
132	C₂×C₄	—	83	1	no	—
133	C₂×C₄	—	664	1	no	—
134	C₂×C₄	—	664	1	no	—
135	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
136	C₈	—	166	1	no	—
137	C₈	#14581769	1	1	yes	characteristic
138	C₈	#14581821	1	1	yes	characteristic
139	C₂²	D₁₆×D₈₃	1	1	yes	characteristic
140	C₄	—	8	2	no	—
141	C₄	—	16	1	no	—
142	C₄	—	1328	1	no	—
143	C₄	—	1328	1	no	—
144	C₄	—	83	2	no	—
145	C₄	—	166	1	no	—
146	C₄	#19127979	1	1	yes	characteristic
147	C₄	#19128140	1	1	yes	characteristic
148	C₂	#24909598	1	1	yes	characteristic
149	C₂	Q₆₄×D₈₃	1	1	yes	characteristic
150	C₂	#24909777	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₈₃
commutator, cosocle	C₂×C₁₆×C₈₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₃
Fitting	C₈₃×#1416	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₁₆
hypercenter	#1416	D₈₃
hypocenter	C₈₃	#13706
LCS	G ▷ C₂×C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

Complements

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

N	H ≅ G/N	type
30. Q₃₂×C₈₃	Q₈	semidirect
83. C₈₃	#13706	semidirect
99. Q₃₂	Dic₁₆₆	semidirect

Maximal subgroups

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

subgroup	index
#24909737	2
Q₃₂×Dic₈₃	2
#24905663	2
#24905662	2
Dic₈₃⋊C₃₂	2
#24905644	2
C₈₃×#1416	2
#13706	83