Finite Group Databasebeta

← #24913778

Subgroups — #24913778

Order 10624.

544 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14581791C212yesmaximal, complemented
2C2×#14581771C212yesmaximal
3#19127637C211yesmaximal, characteristic
4#19127634C212yesmaximal, complemented
5#19127622C212yesmaximal, complemented
6#19127613C211yesmaximal, characteristic
7#19127594C212yesmaximal
8#19127588C212yesmaximal, complemented
9#385×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11C22×Dic166C2212yes—
12#14581791—28no—
13#14581771—28no—
14C2×C4⋊Dic83C2212yes—
15C2×C4⋊Dic83C2212yes—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×C22⋊Dic83C2212yes—
21C2×C22⋊Dic83C2212yes—
22C2×C4×Dic83C2212yes—
23C83⋊#95C2211yescharacteristic
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2211yescharacteristic
28C23×C4×C83C2211yescharacteristic
29C2×C4⋊C4×C83C2212yes—
30C2×C22⋊C4×C83C2211yescharacteristic
31C2×C22⋊C4×C83C2211yescharacteristic
32#95×C83C2212yes—
33C2×Dic166—24no—
34C2×Dic166—24no—
35C2×Dic166—24no—
36C2×Dic166—41no—
37C2×Dic166—42no—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C4⋊Dic83—24no—
43C4⋊Dic83—24no—
44C4⋊Dic83—41no—
45Dic83⋊C4—24no—
46Dic83⋊C4—24no—
47Dic83⋊C4D418yes—
48C22⋊Dic83—24no—
49C22⋊Dic83—24no—
50C4×Dic83—24no—
51C24×C83C2311yescharacteristic
52C22×C4×C83—24no—
53C22×C4×C83—41no—
54C22×C4×C83C2311yescharacteristic
55C22×C4×C83C2311yescharacteristic
56C22×C4×C83C2312yes—
57C22×C4×C83C2312yes—
58C4⋊C4×C83—24no—
59C22⋊C4×C83—41no—
60C22⋊C4×C83Q814yescomplemented
61C23×C83—22no—
62C23×C83—21no—
63C23×C83—22no—
64C23×C83—21no—
65C23×C83C2×Q812yes—
66C23×C83C2411yescharacteristic
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—21no—
73C2×C4×C83—21no—
74C2×C4×C83—22no—
75C2×C4×C83—44no—
76C2×C4×C83C2×D414yes—
77C2×C4×C83C2×Q814yes—
78C2×Dic83—22no—
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—
88C2×Dic83C2×D414yes—
89C2×Dic83C2×D414yes—
90Dic166—44no—
91Dic166—48no—
92C22×C83—22no—
93C22×C83—24no—
94C22×C83—42no—
95C22×C83—41no—
96C22×C83—41no—
97C22×C83C22×D412yes—
98C22×C83C22×Q811yescharacteristic
99C22×C83H(5,𝔽2)11yescharacteristic
100C22×C832−1+412yes—
101C22×C832−1+411yescharacteristic
102C4×C83—24no—
103C4×C83—24no—
104C4×C83—41no—
105C4×C83—41no—
106Dic83—24no—
107Dic83—24no—
108Dic83—42no—
109Dic83—42no—
110C2×C83—42no—
111C2×C83#53612yes—
112C2×C83#54411yescharacteristic
113C2×C83D4211yescharacteristic
114C2×C83D4×Q812yes—
115C2×C83#56311yescharacteristic
116#2677—831nomaximal, Sylow
117C83#267711yescharacteristic, complemented, Sylow
118C2×C4⋊Q8—832no—
119C2×C22⋊Q8—832no—
120#399—831no—
121#398—832no—
122#395—832no—
123#393—831no—
124#388—832no—
125#387—832no—
126#385D8311yescharacteristic
127C22×Q8—832no—
128C22×Q8—831no—
129C23×C4C2×D8311yescharacteristic
130C4⋊Q8—1668no—
131C22⋊Q8—1668no—
132C2×C4⋊C4—831no—
133C2×C4⋊C4—832no—
134C2×C4⋊C4—832no—
135C2×C4⋊C4—832no—
136C2×C4⋊C4—832no—
137C2×C4⋊C4—832no—
138C2×C4⋊C4C2×D8312yes—
139C2×C22⋊C4—832no—
140C2×C22⋊C4—832no—
141C2×C22⋊C4C2×D8311yescharacteristic
142C2×C22⋊C4C2×D8311yescharacteristic
143C2×C42—832no—
144#95—831no—
145#95—832no—
146#95—832no—
147#95—832no—
148#95—831no—
149#95C2×D8312yes—
150C24C22×D8311yescharacteristic
151C2×Q8—3322no—
152C2×Q8—3321no—
153C2×Q8—1664no—
154C2×Q8—1664no—
155C2×Q8—1664no—
156C22×C4—24no—
157C22×C4—41no—
158C22×C4—832no—
159C22×C4—832no—
160C22×C4—832no—
161C22×C4—832no—
162C22×C4C22×D8311yescharacteristic
163C22×C4C22×D8311yescharacteristic
164C22×C4C22×D8312yes—
165C22×C4C22×D8312yes—
166C4⋊C4—24no—
167C4⋊C4—3321no—
168C4⋊C4—838no—
169C4⋊C4—1664no—
170C4⋊C4—1664no—
171C4⋊C4—1664no—
172C4⋊C4—1664no—
173C22⋊C4—41no—
174C22⋊C4—1664no—
175C22⋊C4—1664no—
176C22⋊C4Dic16614yescomplemented
177C42—1664no—
178C23—22no—
179C23—21no—
180C23—22no—
181C23—21no—
182C23C2×Dic16612yes—
183C23C23×D8311yescharacteristic
184Q8—3328no—
185Q8—3324no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—44no—
195C2×C4—834no—
196C2×C4—834no—
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—1662no—
207C2×C4D4×D8314yes—
208C2×C4C2×Dic16614yes—
209C22—22no—
210C22—24no—
211C22—41no—
212C22—41no—
213C22—42no—
214C22C2×D4×D8312yes—
215C22C22×Dic16611yescharacteristic
216C22#1458183711yescharacteristic
217C22#1458183912yes—
218C22#1458184011yescharacteristic
219C4—24no—
220C4—24no—
221C4—41no—
222C4—41no—
223C4—3322no—
224C4—3322no—
225C4—1664no—
226C4—1664no—
227C2—42no—
228C2#1912827412yes—
229C2#1912833211yescharacteristic
230C2#1912833311yescharacteristic
231C2D4×Dic16612yes—
232C2#1912845411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#385×C83C2
prime radicalC24×C83C23
hypercenter#385D83
hypocenterC83#2677
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#14581791C2semidirect
4. #19127634C2semidirect
5. #19127622C2semidirect
8. #19127588C2semidirect
60. C22⋊C4×C83Q8semidirect
117. C83#2677semidirect
176. C22⋊C4Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#145817912
C2×#145817712
#191276372
#191276342
#191276222
#191276132
#191275942
#191275882
#385×C832
#267783