Finite Group Databasebeta

← #24915507

Subgroups — #24915507

Order 10624.

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

normal subgroups only

This group has 580 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×#14581778C212yesmaximal, complemented
2#19127618C211yesmaximal, characteristic, complemented
3#19127617C212yesmaximal, complemented
4#19127614C212yesmaximal, complemented
5#19127613C212yesmaximal, complemented
6#19127610C212yesmaximal, complemented
7#19127609C212yesmaximal, complemented
8#19127566C211yesmaximal, characteristic, complemented
9#394×C83C211yesmaximal, characteristic, complemented
10C22×Dic166C2212yes—
11C22×D332C2211yescharacteristic
12C22×C22⋊D83C2212yes—
13C22×C22⋊D83C2212yes—
14#14581778—28no—
15C2×C4⋊Dic83C2211yescharacteristic
16C2×C22⋊Dic83C2212yes—
17C2×C22⋊Dic83C2212yes—
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×C4×Dic83C2211yescharacteristic
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2211yescharacteristic
26C83⋊#95C2212yes—
27C22×D4×C83C2211yescharacteristic
28C2×C4⋊C4×C83C2211yescharacteristic
29C2×C22⋊C4×C83C2212yes—
30C2×C22⋊C4×C83C2212yes—
31#95×C83C2211yescharacteristic
32C23×D83C2312yes—
33C2×Dic166—24no—
34C2×Dic166—42no—
35C2×Dic166—42no—
36C2×D332—42no—
37C2×D332—41no—
38C2×C22⋊D83—24no—
39C2×C22⋊D83—42no—
40C2×C22⋊D83—42no—
41C2×C22⋊D83—42no—
42C2×C22⋊D83—42no—
43C2×C22⋊D83—42no—
44C22×Dic83C2312yes—
45C22×Dic83C2312yes—
46C22×Dic83C2312yes—
47C4⋊Dic83—41no—
48C22⋊Dic83—24no—
49C22⋊Dic83—42no—
50(C2×D83)⋊C4—24no—
51(C2×D83)⋊C4—24no—
52(C2×D83)⋊C4—42no—
53(C2×D83)⋊C4—42no—
54C4×Dic83D414yescomplemented
55C24×C83C2312yes—
56C2×D4×C83—41no—
57C2×D4×C83—42no—
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2312yes—
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C4⋊C4×C83—41no—
63C22⋊C4×C83—24no—
64C22⋊C4×C83—42no—
65C23×C83—22no—
66C23×C83—22no—
67C23×C83—22no—
68C23×C83—22no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83C2411yescharacteristic
73C22×D83—22no—
74C22×D83—22no—
75C22×D83—22no—
76C22×D83—22no—
77C22×D83—22no—
78C22×D83—22no—
79C22×D83—22no—
80D4×C83—44no—
81C2×C4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—21no—
87C2×C4×C83—21no—
88C2×C4×C83—22no—
89C2×C4×C83—22no—
90C2×C4×C83C2×D412yes—
91C22⋊D83—48no—
92C22⋊D83—48no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83C2×D414yes—
102D332—44no—
103Dic166—48no—
104C22×C83—42no—
105C22×C83—42no—
106C22×C83—42no—
107C22×C83—42no—
108C22×C83—42no—
109C22×C83—42no—
110C22×C83—42no—
111C22×C83C22×D411yescharacteristic
112C22×C83H(5,𝔽2)12yes—
113C22×C83H(5,𝔽2)11yescharacteristic
114C22×C83H(5,𝔽2)12yes—
115C22×C832−1+411yescharacteristic
116C2×D83—42no—
117C2×D83—42no—
118C2×D83—42no—
119C2×D83—42no—
120C2×D83—42no—
121C2×D83—42no—
122C2×D83—42no—
123C4×C83—22no—
124C4×C83—42no—
125C4×C83—41no—
126C4×C83—41no—
127Dic83—24no—
128Dic83—42no—
129Dic83—42no—
130C2×C83—82no—
131C2×C83#53512yes—
132C2×C83#53711yescharacteristic
133C2×C83H(3,𝔽4)11yescharacteristic
134C2×C83H(3,𝔽4)11yescharacteristic
135C2×C83#56212yes—
136D83—82no—
137#2819—831nomaximal, Sylow
138C83#281911yescharacteristic, complemented, Sylow
139C2×#124—832no—
140#394—831no—
141#394—832no—
142#394D8311yescharacteristic, complemented
143#393—832no—
144#393—832no—
145#392—832no—
146#392—832no—
147#382—831no—
148C22×Q8—832no—
149C22×D4—832no—
150C22×D4—832no—
151C22×D4—831no—
152C22×D4C2×D8311yescharacteristic
153#124—1668no—
154C2×C4⋊C4—831no—
155C2×C4⋊C4C2×D8311yescharacteristic
156C2×C22⋊C4—832no—
157C2×C22⋊C4—832no—
158C2×C22⋊C4—832no—
159C2×C22⋊C4—832no—
160C2×C22⋊C4—832no—
161C2×C22⋊C4—832no—
162C2×C22⋊C4C2×D8312yes—
163C2×C22⋊C4C2×D8312yes—
164C2×C42—831no—
165#95—832no—
166#95—832no—
167#95—832no—
168#95—831no—
169#95C2×D8311yescharacteristic
170C24—832no—
171C24C22×D8312yes—
172C2×Q8—3322no—
173C2×Q8—3322no—
174C2×Q8—1664no—
175C2×D4—42no—
176C2×D4—41no—
177C2×D4—3322no—
178C2×D4—3321no—
179C2×D4—3322no—
180C2×D4—3322no—
181C2×D4—3322no—
182C2×D4—3322no—
183C2×D4—3322no—
184C2×D4—1664no—
185C22×C4—832no—
186C22×C4—832no—
187C22×C4—832no—
188C22×C4C22×D8311yescharacteristic
189C22×C4C22×D8311yescharacteristic
190C22×C4C22×D8312yes—
191C22×C4C22×D8311yescharacteristic
192C4⋊C4—41no—
193C4⋊C4—3321no—
194C22⋊C4—24no—
195C22⋊C4—42no—
196C22⋊C4—3322no—
197C22⋊C4—3322no—
198C22⋊C4—3322no—
199C22⋊C4—1664no—
200C22⋊C4—1664no—
201C22⋊C4—1664no—
202C42—834no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—1662no—
211C23—1662no—
212C23—1662no—
213C23—1662no—
214C23—1662no—
215C23—1662no—
216C23—1662no—
217C23C23×D8311yescharacteristic
218Q8—3328no—
219D4—44no—
220D4—3324no—
221D4—3328no—
222D4—3328no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—834no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4—1662no—
240C2×C4—1662no—
241C2×C4D4×D8312yes—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—3322no—
250C22—3322no—
251C22—3322no—
252C22—3322no—
253C22—3322no—
254C22—3322no—
255C22—3322no—
256C22C2×D4×D8311yescharacteristic
257C22#1458183711yescharacteristic
258C22#1458183712yes—
259C22#1458183812yes—
260C22#1458183911yescharacteristic
261C4—22no—
262C4—41no—
263C4—41no—
264C4—42no—
265C4—3322no—
266C4—3322no—
267C4—1664no—
268C2—82no—
269C2—6642no—
270C2#1912826712yes—
271C2#1912827811yescharacteristic
272C2#1912844311yescharacteristic
273C2#1912844411yescharacteristic
274C2#1912844812yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
hypercenter#394D83
hypocenterC83#2819
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#14581778C2semidirect
2. #19127618C2semidirect
3. #19127617C2semidirect
4. #19127614C2semidirect
5. #19127613C2semidirect
6. #19127610C2semidirect
7. #19127609C2semidirect
8. #19127566C2semidirect
9. #394×C83C2semidirect
54. C4×Dic83D4semidirect
138. C83#2819semidirect
142. #394D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817782
#191276182
#191276172
#191276142
#191276132
#191276102
#191276092
#191275662
#394×C832
#281983