Finite Group Databasebeta

← #24913402

Subgroups — #24913402

Order 10624.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581791C211yesmaximal, characteristic, complemented
2C2×#14581781C211yesmaximal, characteristic
3#19127630C214yesmaximal, complemented
4#19127614C214yesmaximal, complemented
5#19127608C211yesmaximal, characteristic, complemented
6#19127603C211yesmaximal, characteristic
7#19127602C211yesmaximal, characteristic
8#19127547C211yesmaximal, characteristic, complemented
9#391×C83C211yesmaximal, characteristic, complemented
10C22×Dic166C2212yes—
11C22×D332C2211yescharacteristic
12#14581791—24no—
13#14581781—24no—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×Dic83⋊C4C2212yes—
16C2×(C2×D83)⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2212yes—
18C2×(C2×D83)⋊C4C2214yes—
19C2×(C2×D83)⋊C4C2214yes—
20C2×C4×Dic83C2211yescharacteristic
21C2×C4×Dic83C2211yescharacteristic
22C83⋊#95C2214yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C22×Q8×C83C2211yescharacteristic
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C42×C83C2211yescharacteristic
28#95×C83C2214yes—
29C23×D83C2312yes—
30C2×Dic166—24no—
31C2×Dic166—44no—
32C2×D332—22no—
33C2×D332—22no—
34C2×D332—22no—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C22×Dic83C2311yescharacteristic
38C22×Dic83C2311yescharacteristic
39C4⋊Dic83—22no—
40Dic83⋊C4—24no—
41(C2×D83)⋊C4—24no—
42(C2×D83)⋊C4—24no—
43(C2×D83)⋊C4—44no—
44(C2×D83)⋊C4—44no—
45C4×Dic83—22no—
46C4×Dic83D414yes—
47C2×Q8×C83—22no—
48C2×Q8×C83—22no—
49C2×Q8×C83—22no—
50C22×C4×C83C2312yes—
51C22×C4×C83C2312yes—
52C22×C4×C83C2312yes—
53C22×C4×C83C2311yescharacteristic
54C4⋊C4×C83—22no—
55C42×C83—22no—
56C23×C83C2411yescharacteristic
57C22×D83—22no—
58C22×D83—22no—
59C22×D83—22no—
60C22×D83—24no—
61C22×D83—24no—
62Q8×C83—44no—
63C2×C4×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83—22no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—24no—
70C2×C4×C83—22no—
71C2×C4×C83C2×D412yes—
72C2×C4×C83#4112yes—
73C2×C4×C83#4112yes—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—22no—
79C2×Dic83—24no—
80C2×Dic83—22no—
81C2×Dic83C2×D412yes—
82C2×Dic83C2×D412yes—
83D332—44no—
84Dic166—48no—
85C22×C83C22×D411yescharacteristic
86C22×C83C2×#4111yescharacteristic
87C22×C83C2×#4111yescharacteristic
88C22×C83H(5,𝔽2)12yes—
89C22×C832−1+412yes—
90C2×D83—42no—
91C2×D83—42no—
92C2×D83—42no—
93C2×D83—44no—
94C2×D83—44no—
95C4×C83—42no—
96C4×C83—42no—
97C4×C83—42no—
98C4×C83#12414yes—
99Dic83—22no—
100Dic83—22no—
101Dic83—42no—
102Dic83—42no—
103C2×C83C2×#12411yescharacteristic
104C2×C83#53511yescharacteristic
105C2×C83#53611yescharacteristic
106C2×C83#54212yes—
107C2×C83#54212yes—
108D83—82no—
109#2640—831nomaximal, Sylow
110C83#264011yescharacteristic, complemented, Sylow
111C2×C4⋊Q8—831no—
112C2×#124—831no—
113#397—834no—
114#393—834no—
115#391—831no—
116#391D8311yescharacteristic, complemented
117#390—831no—
118#390—831no—
119#376—831no—
120C22×Q8—832no—
121C22×Q8C2×D8311yescharacteristic
122C22×D4—831no—
123C4⋊Q8—1664no—
124#124—1664no—
125C2×C4⋊C4—831no—
126C2×C4⋊C4—832no—
127C2×C4⋊C4C2×D8311yescharacteristic
128C2×C22⋊C4—834no—
129C2×C22⋊C4—834no—
130C2×C22⋊C4—832no—
131C2×C22⋊C4—832no—
132C2×C42—831no—
133C2×C42—831no—
134C2×C42C2×D8311yescharacteristic
135#95—832no—
136#95—832no—
137#95—834no—
138#95C2×D8314yes—
139C24—832no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—3324no—
144C2×Q8—1664no—
145C2×D4—1662no—
146C2×D4—1662no—
147C2×D4—1662no—
148C22×C4—831no—
149C22×C4—831no—
150C22×C4—832no—
151C22×C4—832no—
152C22×C4C22×D8312yes—
153C22×C4C22×D8312yes—
154C22×C4C22×D8312yes—
155C22×C4C22×D8311yescharacteristic
156C4⋊C4—22no—
157C4⋊C4—1664no—
158C4⋊C4—1662no—
159C22⋊C4—3324no—
160C22⋊C4—3324no—
161C22⋊C4—1664no—
162C22⋊C4—1664no—
163C42—22no—
164C42—834no—
165C42—1662no—
166C23—1662no—
167C23—1662no—
168C23—1662no—
169C23—1664no—
170C23—1664no—
171C23C23×D8311yescharacteristic
172Q8—44no—
173Q8—3328no—
174D4—3324no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—832no—
184C2×C4—832no—
185C2×C4—1662no—
186C2×C4—1662no—
187C2×C4—1662no—
188C2×C4—1664no—
189C2×C4—1662no—
190C2×C4—1662no—
191C2×C4—1662no—
192C2×C4D4×D8312yes—
193C2×C4#1305058912yes—
194C2×C4#1305059012yes—
195C22—3324no—
196C22—3324no—
197C22—3322no—
198C22—3322no—
199C22—3322no—
200C22C2×D4×D8311yescharacteristic
201C22C2×#1305058911yescharacteristic
202C22C2×#1305059011yescharacteristic
203C22#1458183812yes—
204C22#1458184012yes—
205C4—42no—
206C4—42no—
207C4—42no—
208C4—3322no—
209C4—3322no—
210C4—1662no—
211C4—1662no—
212C4#1458177814yes—
213C2—6642no—
214C2C2×#1458177811yescharacteristic
215C2#1912826911yescharacteristic
216C2#1912827611yescharacteristic
217C2#1912832312yes—
218C2#1912832412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#391×C83C2
prime radicalC22×D332C22
hypercenter#391D83
hypocenterC83#2640
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#14581791C2semidirect
3. #19127630C2semidirect
4. #19127614C2semidirect
5. #19127608C2semidirect
8. #19127547C2semidirect
9. #391×C83C2semidirect
110. C83#2640semidirect
116. #391D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817912
C2×#145817812
#191276302
#191276142
#191276082
#191276032
#191276022
#191275472
#391×C832
#264083