Finite Group Databasebeta

← #24913291

Subgroups — #24913291

Order 10624.

488 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 143 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×#14581750C211yesmaximal, characteristic
3#19127640C211yesmaximal, characteristic, complemented
4#19127638C211yesmaximal, characteristic
5#19127636C211yesmaximal, characteristic
6#19127622C212yesmaximal, complemented
7#19127614C212yesmaximal
8#19127598C212yesmaximal, complemented
9#19127565C211yesmaximal, characteristic, complemented
10#19127560C212yesmaximal
11#400×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2212yes—
13C22×C4×D83C2211yescharacteristic
14#14581791—24no—
15#14581750—24no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×C4⋊Dic83C2211yescharacteristic
18C2×C4⋊Dic83C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2212yes—
21C2×Dic83⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2211yescharacteristic
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2212yes—
26C2×C4×Dic83C2211yescharacteristic
27C2×C4×Dic83C2211yescharacteristic
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2211yescharacteristic
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34#95×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
46C2×C4×D83—22no—
47C2×C4×D83—22no—
48C2×C4×D83Q814yescomplemented
49C4⋊Dic83—22no—
50C4⋊Dic83—22no—
51C4⋊Dic83—42no—
52Dic83⋊C4—24no—
53Dic83⋊C4—24no—
54Dic83⋊C4—42no—
55(C2×D83)⋊C4—22no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—42no—
58(C2×D83)⋊C4—42no—
59C4×Dic83—22no—
60C4×Dic83D414yes—
61C22×C4×C83C2312yes—
62C22×C4×C83C2312yes—
63C22×C4×C83C2311yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2311yescharacteristic
66C4⋊C4×C83—22no—
67C4⋊C4×C83—22no—
68C23×C83C2411yescharacteristic
69C22×D83—21no—
70C22×D83—21no—
71C22×D83—22no—
72C22×D83—22no—
73C22×D83C2×Q812yes—
74C2×C4×C83—21no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83—21no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83C2×D412yes—
86C2×C4×C83C2×Q812yes—
87C2×Dic83—22no—
88C2×Dic83—21no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83C2×D412yes—
97C2×Dic83C2×D412yes—
98C2×Dic83C2×Q812yes—
99C2×Dic83#4112yes—
100C4×D83—28no—
101Dic166—48no—
102C22×C83C22×D411yescharacteristic
103C22×C83C22×Q811yescharacteristic
104C22×C83C2×#4111yescharacteristic
105C22×C83H(5,𝔽2)12yes—
106C22×C832−1+412yes—
107C2×D83—22no—
108C2×D83—24no—
109C2×D83—41no—
110C2×D83—41no—
111C2×D83—42no—
112C4×C83—22no—
113C4×C83—22no—
114C4×C83—42no—
115C4×C83—41no—
116C4×C83—42no—
117Dic83—22no—
118Dic83—42no—
119Dic83—41no—
120Dic83—42no—
121Dic83C22⋊Q814yes—
122C2×C83C2×C22⋊Q811yescharacteristic
123C2×C83#53511yescharacteristic
124C2×C83#53611yescharacteristic
125C2×C83#54212yes—
126C2×C83#54412yes—
127D83—42no—
128#2628—831nomaximal, Sylow
129C83#262811yescharacteristic, complemented, Sylow
130C2×C4⋊Q8—831no—
131C2×#117—831no—
132#400—831no—
133#400D8311yescharacteristic, complemented
134#399—831no—
135#399—831no—
136#395—832no—
137#393—832no—
138#389—832no—
139#382—831no—
140#380—832no—
141C22×Q8—832no—
142C23×C4—831no—
143C4⋊Q8—1664no—
144#117—1664no—
145C2×C4⋊C4—832no—
146C2×C4⋊C4—832no—
147C2×C4⋊C4—831no—
148C2×C4⋊C4—832no—
149C2×C4⋊C4—831no—
150C2×C4⋊C4—832no—
151C2×C4⋊C4C2×D8311yescharacteristic
152C2×C4⋊C4C2×D8311yescharacteristic
153C2×C22⋊C4—832no—
154C2×C22⋊C4—832no—
155C2×C22⋊C4—831no—
156C2×C22⋊C4—831no—
157C2×C42—831no—
158C2×C42—831no—
159#95—832no—
160#95—832no—
161#95—831no—
162#95—832no—
163#95C2×D8312yes—
164#95C2×D8311yescharacteristic
165#95C2×D8312yes—
166C24—831no—
167C2×Q8—3322no—
168C2×Q8—3322no—
169C2×Q8—1664no—
170C22×C4—831no—
171C22×C4—831no—
172C22×C4—834no—
173C22×C4—832no—
174C22×C4—832no—
175C22×C4—831no—
176C22×C4—1662no—
177C22×C4—1662no—
178C22×C4C22×D8311yescharacteristic
179C22×C4C22×D8312yes—
180C22×C4C22×D8312yes—
181C22×C4C22×D8311yescharacteristic
182C22×C4C22×D8311yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—3322no—
186C4⋊C4—3322no—
187C4⋊C4—1662no—
188C4⋊C4—1664no—
189C4⋊C4—1664no—
190C4⋊C4—1662no—
191C22⋊C4—3322no—
192C22⋊C4—3322no—
193C22⋊C4—1662no—
194C22⋊C4—1662no—
195C42—834no—
196C42—1662no—
197C23—832no—
198C23—1661no—
199C23—1661no—
200C23—1662no—
201C23—1662no—
202C23C23×D8311yescharacteristic
203Q8—3328no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—832no—
216C2×C4—832no—
217C2×C4—832no—
218C2×C4—832no—
219C2×C4—1668no—
220C2×C4—1661no—
221C2×C4—1662no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4—1662no—
225C2×C4—1662no—
226C2×C4—1662no—
227C2×C4—1662no—
228C2×C4—1662no—
229C2×C4D4×D8312yes—
230C2×C4Q8×D8312yes—
231C22—3321no—
232C22—3321no—
233C22—3322no—
234C22—1664no—
235C22—1662no—
236C22C2×D4×D8311yescharacteristic
237C22C2×Q8×D8311yescharacteristic
238C22#41×D8311yescharacteristic
239C22#1458183812yes—
240C22#1458183912yes—
241C4—22no—
242C4—22no—
243C4—42no—
244C4—41no—
245C4—42no—
246C4—3322no—
247C4—3321no—
248C4—3322no—
249C4—834no—
250C4—1662no—
251C2—3322no—
252C2C22⋊Q8×D8311yescharacteristic
253C2#1912826911yescharacteristic
254C2#1912827311yescharacteristic
255C2#1912831712yes—
256C2#1912833112yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#400×C83C2
prime radicalC23×D83C23
hypercenter#400D83
hypocenterC83#2628
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#14581791C2semidirect
3. #19127640C2semidirect
6. #19127622C2semidirect
8. #19127598C2semidirect
9. #19127565C2semidirect
11. #400×C83C2semidirect
48. C2×C4×D83Q8semidirect
129. C83#2628semidirect
133. #400D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817912
C2×#145817502
#191276402
#191276382
#191276362
#191276222
#191276142
#191275982
#191275652
#191275602
#400×C832
#262883