Finite Group Databasebeta

← #24915523

Subgroups — #24915523

Order 10624.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14581787C211yesmaximal, characteristic
3#19127643C212yesmaximal, complemented
4#19127636C212yesmaximal
5#19127631C211yesmaximal, characteristic, complemented
6#19127628C211yesmaximal, characteristic
7#19127627C211yesmaximal, characteristic
8#19127621C212yesmaximal, complemented
9#19127614C212yesmaximal
10#19127568C211yesmaximal, characteristic, complemented
11#398×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2212yes—
13#14581791—24no—
14#14581787—24no—
15C2×C4⋊Dic83C2211yescharacteristic
16C2×C4⋊Dic83C2211yescharacteristic
17C2×Dic83⋊C4C2211yescharacteristic
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2211yescharacteristic
21C2×(C2×D83)⋊C4C2211yescharacteristic
22C2×(C2×D83)⋊C4C2211yescharacteristic
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×C4×Dic83C2211yescharacteristic
26C83⋊#95C2212yes—
27C83⋊#95C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2212yes—
35#95×C83C2211yescharacteristic
36#95×C83C2212yes—
37C23×D83C2311yescharacteristic
38C2×Dic166—24no—
39C2×Dic166—42no—
40C2×Dic166—42no—
41C22×Dic83C2312yes—
42C22×Dic83C2312yes—
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2311yescharacteristic
46C4⋊Dic83—22no—
47C4⋊Dic83—22no—
48Dic83⋊C4—22no—
49Dic83⋊C4—24no—
50Dic83⋊C4—42no—
51(C2×D83)⋊C4—22no—
52(C2×D83)⋊C4—22no—
53(C2×D83)⋊C4—22no—
54(C2×D83)⋊C4—42no—
55(C2×D83)⋊C4—42no—
56C4×Dic83D414yes—
57C22×C4×C83C2312yes—
58C22×C4×C83C2312yes—
59C22×C4×C83C2311yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C4⋊C4×C83—22no—
63C4⋊C4×C83—22no—
64C4⋊C4×C83—42no—
65C23×C83C2411yescharacteristic
66C22×D83—21no—
67C22×D83—21no—
68C22×D83—21no—
69C22×D83—22no—
70C22×D83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—21no—
73C2×C4×C83—21no—
74C2×C4×C83—22no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—21no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83C2×D412yes—
84C2×Dic83—21no—
85C2×Dic83—21no—
86C2×Dic83—22no—
87C2×Dic83—21no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83C2×D412yes—
96C2×Dic83C2×D412yes—
97Dic166—48no—
98C22×C83C22×D411yescharacteristic
99C22×C83H(5,𝔽2)11yescharacteristic
100C22×C83H(5,𝔽2)11yescharacteristic
101C22×C832−1+412yes—
102C22×C832−1+412yes—
103C2×D83—41no—
104C2×D83—41no—
105C2×D83—41no—
106C2×D83—42no—
107C2×D83—42no—
108C4×C83—22no—
109C4×C83—42no—
110C4×C83—41no—
111C4×C83—41no—
112C4×C83—42no—
113Dic83—22no—
114Dic83—22no—
115Dic83—42no—
116Dic83—41no—
117Dic83—42no—
118C2×C83#53511yescharacteristic
119C2×C83#53611yescharacteristic
120C2×C83#53611yescharacteristic
121C2×C83#56312yes—
122C2×C83#56312yes—
123D83—81no—
124#2821—831nomaximal, Sylow
125C83#282111yescharacteristic, complemented, Sylow
126C2×C4⋊Q8—831no—
127C2×#126—831no—
128#400—832no—
129#399—832no—
130#398—831no—
131#398D8311yescharacteristic, complemented
132#397—831no—
133#397—831no—
134#395—832no—
135#393—832no—
136#383—831no—
137C22×Q8—832no—
138C4⋊Q8—1664no—
139#126—1664no—
140C2×C4⋊C4—832no—
141C2×C4⋊C4—831no—
142C2×C4⋊C4—831no—
143C2×C4⋊C4—831no—
144C2×C4⋊C4—832no—
145C2×C4⋊C4C2×D8311yescharacteristic
146C2×C4⋊C4C2×D8311yescharacteristic
147C2×C4⋊C4C2×D8312yes—
148C2×C22⋊C4—832no—
149C2×C22⋊C4—832no—
150C2×C22⋊C4—831no—
151C2×C22⋊C4—831no—
152C2×C22⋊C4—831no—
153C2×C42—831no—
154#95—832no—
155#95—831no—
156#95—832no—
157#95—832no—
158#95—832no—
159#95—832no—
160#95C2×D8311yescharacteristic
161#95C2×D8312yes—
162C24—831no—
163C2×Q8—3322no—
164C2×Q8—3322no—
165C2×Q8—1664no—
166C22×C4—831no—
167C22×C4—831no—
168C22×C4—832no—
169C22×C4—831no—
170C22×C4—832no—
171C22×C4C22×D8312yes—
172C22×C4C22×D8311yescharacteristic
173C22×C4C22×D8312yes—
174C22×C4C22×D8311yescharacteristic
175C22×C4C22×D8311yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—3322no—
180C4⋊C4—1662no—
181C4⋊C4—1662no—
182C4⋊C4—1664no—
183C4⋊C4—1662no—
184C22⋊C4—3322no—
185C22⋊C4—3322no—
186C22⋊C4—1662no—
187C22⋊C4—1662no—
188C22⋊C4—1662no—
189C42—834no—
190C23—1661no—
191C23—1661no—
192C23—1661no—
193C23—1662no—
194C23—1662no—
195C23C23×D8311yescharacteristic
196Q8—3328no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—832no—
210C2×C4—832no—
211C2×C4—1661no—
212C2×C4—1662no—
213C2×C4—1662no—
214C2×C4—1662no—
215C2×C4—1662no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1661no—
220C2×C4—1661no—
221C2×C4—1662no—
222C2×C4D4×D8312yes—
223C22—3322no—
224C22—3322no—
225C22—3321no—
226C22—3321no—
227C22—3321no—
228C22C2×D4×D8311yescharacteristic
229C22#1458183711yescharacteristic
230C22#1458183811yescharacteristic
231C22#1458183912yes—
232C22#1458184012yes—
233C4—22no—
234C4—42no—
235C4—42no—
236C4—41no—
237C4—41no—
238C4—3322no—
239C4—3322no—
240C4—3321no—
241C4—1662no—
242C4—1662no—
243C2—6641no—
244C2#1912826711yescharacteristic
245C2#1912827311yescharacteristic
246C2#1912827611yescharacteristic
247C2#1912845212yes—
248C2#1912845512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#398×C83C2
prime radicalC23×D83C23
hypercenter#398D83
hypocenterC83#2821
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#14581791C2semidirect
3. #19127643C2semidirect
5. #19127631C2semidirect
8. #19127621C2semidirect
10. #19127568C2semidirect
11. #398×C83C2semidirect
125. C83#2821semidirect
131. #398D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817912
C2×#145817872
#191276432
#191276362
#191276312
#191276282
#191276272
#191276212
#191276142
#191275682
#398×C832
#282183