Finite Group Databasebeta

← #24913283

Subgroups — #24913283

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×#14581790C211yesmaximal, characteristic, complemented
2C2×#14581748C211yesmaximal, characteristic
3#19127642C212yesmaximal, complemented
4#19127637C212yesmaximal
5#19127622C211yesmaximal, characteristic, complemented
6#19127620C211yesmaximal, characteristic, complemented
7#19127613C211yesmaximal, characteristic
8#19127598C212yesmaximal, complemented
9#19127565C211yesmaximal, characteristic, complemented
10#19127559C212yesmaximal
11#393×C83C211yesmaximal, characteristic
12C22×Dic166C2211yescharacteristic
13C23×Dic83C2211yescharacteristic
14#14581790—24no—
15#14581748—24no—
16C2×C4⋊Dic83C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2212yes—
21C2×Dic83⋊C4C2211yescharacteristic
22C2×Dic83⋊C4C2212yes—
23C2×C22⋊Dic83C2211yescharacteristic
24C2×C22⋊Dic83C2212yes—
25C2×C4×Dic83C2211yescharacteristic
26C2×C4×Dic83C2211yescharacteristic
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2211yescharacteristic
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C22×Q8×C83C2211yescharacteristic
33C2×C22⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35#95×C83C2211yescharacteristic
36#95×C83C2212yes—
37C2×Dic166—22no—
38C2×Dic166—42no—
39C22×Dic83—22no—
40C22×Dic83—22no—
41C22×Dic83Q814yescomplemented
42C22×Dic83C2312yes—
43C22×Dic83C2312yes—
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2311yescharacteristic
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2311yescharacteristic
48C4⋊Dic83—42no—
49Dic83⋊C4—22no—
50Dic83⋊C4—24no—
51Dic83⋊C4—24no—
52Dic83⋊C4—22no—
53Dic83⋊C4—24no—
54Dic83⋊C4—42no—
55C22⋊Dic83—22no—
56C22⋊Dic83—42no—
57C4×Dic83—22no—
58C4×Dic83D414yes—
59C24×C83C2311yescharacteristic
60C2×Q8×C83—22no—
61C2×Q8×C83—42no—
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2312yes—
64C22×C4×C83C2312yes—
65C22×C4×C83C2311yescharacteristic
66C22⋊C4×C83—22no—
67C22⋊C4×C83—42no—
68C23×C83—21no—
69C23×C83—21no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83C2×Q812yes—
73C23×C83C2411yescharacteristic
74Q8×C83—44no—
75C2×C4×C83—22no—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
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×C83C2×D412yes—
85C2×Dic83—21no—
86C2×Dic83—21no—
87C2×Dic83—22no—
88C2×Dic83—28no—
89C2×Dic83—21no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83C2×D412yes—
98C2×Dic83C2×D412yes—
99C2×Dic83C2×Q812yes—
100C2×Dic83C2×Q812yes—
101C2×Dic83#4112yes—
102Dic166—44no—
103C22×C83—22no—
104C22×C83—24no—
105C22×C83—41no—
106C22×C83—41no—
107C22×C83—42no—
108C22×C83C22×D411yescharacteristic
109C22×C83C22×Q811yescharacteristic
110C22×C83C2×#4111yescharacteristic
111C22×C83H(5,𝔽2)12yes—
112C22×C832−1+412yes—
113C4×C83—22no—
114C4×C83—42no—
115C4×C83—41no—
116C4×C83—42no—
117Dic83—22no—
118Dic83—22no—
119Dic83—42no—
120Dic83—41no—
121Dic83—42no—
122Dic83C22⋊Q814yes—
123C2×C83—42no—
124C2×C83C2×C22⋊Q811yescharacteristic
125C2×C83#53511yescharacteristic
126C2×C83#53611yescharacteristic
127C2×C83#54212yes—
128C2×C83#54412yes—
129#2628—831nomaximal, Sylow
130C83#262811yescharacteristic, complemented, Sylow
131C2×C4⋊Q8—831no—
132C2×#117—831no—
133#400—832no—
134#399—832no—
135#395—831no—
136#395—831no—
137#393—831no—
138#393D8311yescharacteristic
139#389—832no—
140#382—831no—
141#380—832no—
142C22×Q8—831no—
143C22×Q8C2×D8311yescharacteristic
144C23×C4—831no—
145C4⋊Q8—1664no—
146#117—1664no—
147C2×C4⋊C4—832no—
148C2×C4⋊C4—832no—
149C2×C4⋊C4—831no—
150C2×C4⋊C4—831no—
151C2×C4⋊C4—832no—
152C2×C4⋊C4—832no—
153C2×C4⋊C4—832no—
154C2×C22⋊C4—832no—
155C2×C22⋊C4—831no—
156C2×C22⋊C4C2×D8311yescharacteristic
157C2×C22⋊C4C2×D8312yes—
158C2×C42—831no—
159C2×C42—831no—
160#95—832no—
161#95—832no—
162#95—831no—
163#95—832no—
164#95—832no—
165#95C2×D8311yescharacteristic
166#95C2×D8312yes—
167C24C22×D8311yescharacteristic
168C2×Q8—22no—
169C2×Q8—42no—
170C2×Q8—3322no—
171C2×Q8—1662no—
172C22×C4—831no—
173C22×C4—834no—
174C22×C4—831no—
175C22×C4—831no—
176C22×C4—832no—
177C22×C4—832no—
178C22×C4—831no—
179C22×C4—1662no—
180C22×C4—1662no—
181C22×C4C22×D8311yescharacteristic
182C22×C4C22×D8312yes—
183C22×C4C22×D8312yes—
184C22×C4C22×D8311yescharacteristic
185C4⋊C4—3322no—
186C4⋊C4—3322no—
187C4⋊C4—1662no—
188C4⋊C4—1664no—
189C4⋊C4—1664no—
190C4⋊C4—1664no—
191C4⋊C4—1662no—
192C22⋊C4—22no—
193C22⋊C4—42no—
194C22⋊C4—3322no—
195C22⋊C4—1662no—
196C42—834no—
197C42—1662no—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—22no—
202C23Q8×D8312yes—
203C23C23×D8311yescharacteristic
204Q8—44no—
205Q8—3324no—
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—832no—
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—1668no—
229C2×C4—1661no—
230C2×C4—1661no—
231C2×C4—1662no—
232C2×C4D4×D8312yes—
233C22—24no—
234C22—22no—
235C22—41no—
236C22—41no—
237C22—42no—
238C22C2×D4×D8311yescharacteristic
239C22C2×Q8×D8311yescharacteristic
240C22#41×D8311yescharacteristic
241C22#1458183712yes—
242C22#1458183912yes—
243C4—22no—
244C4—42no—
245C4—41no—
246C4—42no—
247C4—3322no—
248C4—3321no—
249C4—3322no—
250C4—834no—
251C4—1662no—
252C4—1662no—
253C2—42no—
254C2C22⋊Q8×D8311yescharacteristic
255C2#1912826611yescharacteristic
256C2#1912827111yescharacteristic
257C2#1912831512yes—
258C2#1912833012yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#393×C83C2
prime radicalC24×C83C23
hypercenter#393D83
hypocenterC83#2628
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#14581790C2semidirect
3. #19127642C2semidirect
5. #19127622C2semidirect
6. #19127620C2semidirect
8. #19127598C2semidirect
9. #19127565C2semidirect
41. C22×Dic83Q8semidirect
130. C83#2628semidirect

Maximal subgroups

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

subgroupindex
C2×#145817902
C2×#145817482
#191276422
#191276372
#191276222
#191276202
#191276132
#191275982
#191275652
#191275592
#393×C832
#262883