Finite Group Databasebeta

← #24915630

Subgroups — #24915630

Order 10624.

416 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#19127642C212yesmaximal, complemented
2#19127630C212yesmaximal, complemented
3#19127621C211yesmaximal, characteristic, complemented
4#19127619C212yesmaximal
5#19127618C211yesmaximal, characteristic
6#19127614C212yesmaximal, complemented
7#19127612C212yesmaximal, complemented
8#19127592C212yesmaximal, complemented
9#391×C83C211yesmaximal, characteristic, complemented
10C22×Dic166C2211yescharacteristic
11C22×D332C2211yescharacteristic
12C2×Dic83⋊C4C2212yes—
13C2×Dic83⋊C4C2212yes—
14C2×(C2×D83)⋊C4C2212yes—
15C2×(C2×D83)⋊C4C2212yes—
16C2×(C2×D83)⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2212yes—
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2211yescharacteristic
25C83⋊#95C2211yescharacteristic
26C22×Q8×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2211yescharacteristic
28C2×C42×C83C2211yescharacteristic
29#95×C83C2212yes—
30#95×C83C2212yes—
31C23×D83C2312yes—
32C2×Dic166—41no—
33C2×Dic166—42no—
34C2×D332—41no—
35C2×D332—42no—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C22×Dic83C2312yes—
39Dic83⋊C4—42no—
40Dic83⋊C4—42no—
41(C2×D83)⋊C4—24no—
42(C2×D83)⋊C4—42no—
43(C2×D83)⋊C4—42no—
44(C2×D83)⋊C4—42no—
45(C2×D83)⋊C4—42no—
46(C2×D83)⋊C4—42no—
47C2×Q8×C83—22no—
48C2×Q8×C83—41no—
49C2×Q8×C83—41no—
50C22×C4×C83C2311yescharacteristic
51C22×C4×C83C2311yescharacteristic
52C22×C4×C83C2312yes—
53C22×C4×C83C2312yes—
54C22×C4×C83C2311yescharacteristic
55C4⋊C4×C83—41no—
56C42×C83—41no—
57C23×C83C2411yescharacteristic
58C22×D83—22no—
59C22×D83—22no—
60C22×D83—22no—
61C22×D83—22no—
62C22×D83—22no—
63C22×D83—22no—
64C22×D83—22no—
65Q8×C83—44no—
66C2×C4×C83—21no—
67C2×C4×C83—21no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—21no—
71C2×C4×C83—21no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83#4112yes—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88D332—44no—
89Dic166—44no—
90C22×C83C2×#4111yescharacteristic
91C22×C83H(5,𝔽2)11yescharacteristic
92C22×C83H(5,𝔽2)11yescharacteristic
93C22×C83H(5,𝔽2)12yes—
94C22×C832−1+412yes—
95C2×D83—42no—
96C2×D83—42no—
97C2×D83—42no—
98C2×D83—42no—
99C2×D83—42no—
100C2×D83—42no—
101C2×D83—42no—
102C4×C83—22no—
103C4×C83—42no—
104C4×C83—41no—
105C4×C83—41no—
106C4×C83—42no—
107Dic83—42no—
108Dic83—42no—
109Dic83—42no—
110C2×C83#53811yescharacteristic
111C2×C83#54212yes—
112C2×C83H(3,𝔽4)11yescharacteristic
113C2×C83#56212yes—
114C2×C83#56311yescharacteristic
115D83—82no—
116#2832—831nomaximal, Sylow
117C83#283211yescharacteristic, complemented, Sylow
118#400—832no—
119#397—832no—
120#395—831no—
121#394—831no—
122#394—832no—
123#393—832no—
124#393—832no—
125#391D8311yescharacteristic, complemented
126#388—832no—
127C22×Q8—831no—
128C22×Q8C2×D8311yescharacteristic
129C22×D4—831no—
130C2×C4⋊C4—832no—
131C2×C4⋊C4—832no—
132C2×C4⋊C4C2×D8311yescharacteristic
133C2×C22⋊C4—832no—
134C2×C22⋊C4—832no—
135C2×C22⋊C4—832no—
136C2×C22⋊C4—832no—
137C2×C22⋊C4—832no—
138C2×C22⋊C4—832no—
139C2×C42C2×D8311yescharacteristic
140#95—832no—
141#95—832no—
142#95—832no—
143#95—832no—
144#95—831no—
145#95—831no—
146#95C2×D8312yes—
147#95C2×D8312yes—
148C24—832no—
149C2×Q8—22no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—3321no—
153C2×Q8—3322no—
154C2×D4—3321no—
155C2×D4—3322no—
156C22×C4—832no—
157C22×C4—832no—
158C22×C4—832no—
159C22×C4C22×D8312yes—
160C22×C4C22×D8312yes—
161C22×C4C22×D8311yescharacteristic
162C22×C4C22×D8311yescharacteristic
163C22×C4C22×D8311yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—3322no—
166C4⋊C4—3322no—
167C22⋊C4—3322no—
168C22⋊C4—3322no—
169C22⋊C4—3322no—
170C22⋊C4—3322no—
171C22⋊C4—3322no—
172C22⋊C4—1664no—
173C42—41no—
174C23—1662no—
175C23—1662no—
176C23—1662no—
177C23—1662no—
178C23—1662no—
179C23—1662no—
180C23—1662no—
181C23C23×D8311yescharacteristic
182Q8—44no—
183Q8—3324no—
184D4—3324no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—1662no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4—1662no—
206C2×C4#1305059012yes—
207C22—3322no—
208C22—3322no—
209C22—3322no—
210C22—3322no—
211C22—3322no—
212C22—3322no—
213C22—3322no—
214C22C2×#1305059011yescharacteristic
215C22#1458183711yescharacteristic
216C22#1458183711yescharacteristic
217C22#1458183812yes—
218C22#1458184012yes—
219C4—22no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4—42no—
224C4—3322no—
225C4—3322no—
226C4—3322no—
227C2—6642no—
228C2#1912828911yescharacteristic
229C2#1912832312yes—
230C2#1912844311yescharacteristic
231C2#1912844712yes—
232C2#1912845011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#391×C83C2
prime radicalC22×D332C22
hypercenter#391D83
hypocenterC83#2832
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #19127642C2semidirect
2. #19127630C2semidirect
3. #19127621C2semidirect
6. #19127614C2semidirect
7. #19127612C2semidirect
8. #19127592C2semidirect
9. #391×C83C2semidirect
117. C83#2832semidirect
125. #391D83semidirect

Maximal subgroups

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

subgroupindex
#191276422
#191276302
#191276212
#191276192
#191276182
#191276142
#191276122
#191275922
#391×C832
#283283