Finite Group Databasebeta

← #24913401

Subgroups — #24913401

Order 10624.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14581778C211yesmaximal, characteristic
3#19127630C212yesmaximal, complemented
4#19127629C212yesmaximal, complemented
5#19127614C211yesmaximal, characteristic, complemented
6#19127613C211yesmaximal, characteristic, complemented
7#19127612C211yesmaximal, characteristic, complemented
8#19127605C212yesmaximal, complemented
9#19127601C212yesmaximal
10#19127546C211yesmaximal, characteristic, complemented
11#393×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×Dic166C2211yescharacteristic
14C22×C22⋊D83C2211yescharacteristic
15#14581790—24no—
16#14581778—24no—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×C22⋊Dic83C2211yescharacteristic
20C2×C22⋊Dic83C2212yes—
21C2×(C2×D83)⋊C4C2211yescharacteristic
22C2×(C2×D83)⋊C4C2211yescharacteristic
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×C4×Dic83C2212yes—
26C2×C4×Dic83C2211yescharacteristic
27C83⋊#95C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2211yescharacteristic
31C83⋊#95C2212yes—
32C22×Q8×C83C2211yescharacteristic
33C2×C22⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35#95×C83C2212yes—
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C2×Dic166—22no—
39C2×Dic166—22no—
40C2×Dic166—24no—
41C2×Dic166—42no—
42C2×C22⋊D83—22no—
43C2×C22⋊D83—24no—
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2312yes—
46C22×Dic83C2312yes—
47C22×Dic83C2311yescharacteristic
48C22×Dic83C2311yescharacteristic
49Dic83⋊C4—24no—
50Dic83⋊C4—24no—
51C22⋊Dic83—22no—
52C22⋊Dic83—42no—
53(C2×D83)⋊C4—22no—
54(C2×D83)⋊C4—22no—
55(C2×D83)⋊C4—42no—
56(C2×D83)⋊C4—42no—
57C4×Dic83—24no—
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—22no—
70C23×C83—22no—
71C23×C83—21no—
72C23×C83—21no—
73C23×C83C2411yescharacteristic
74C22×D83—21no—
75C22×D83—22no—
76C22×D83—22no—
77C22×D83—21no—
78C22×D83—21no—
79Q8×C83—44no—
80C2×C4×C83—22no—
81C2×C4×C83—21no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83C2×D412yes—
90C22⋊D83—44no—
91C2×Dic83—21no—
92C2×Dic83—21no—
93C2×Dic83—21no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83C2×D412yes—
102C2×Dic83C2×D412yes—
103C2×Dic83#4114yes—
104Dic166—44no—
105Dic166—44no—
106C22×C83—41no—
107C22×C83—41no—
108C22×C83—41no—
109C22×C83—42no—
110C22×C83—42no—
111C22×C83C22×D411yescharacteristic
112C22×C83C2×#4112yes—
113C22×C83H(5,𝔽2)12yes—
114C22×C832−1+411yescharacteristic
115C22×C832−1+411yescharacteristic
116C2×D83—41no—
117C2×D83—41no—
118C2×D83—41no—
119C2×D83—42no—
120C2×D83—42no—
121C4×C83—22no—
122C4×C83—41no—
123C4×C83—42no—
124C4×C83—42no—
125Dic83—22no—
126Dic83—41no—
127Dic83—42no—
128Dic83—42no—
129Dic83#12414yes—
130C2×C83—81no—
131C2×C83C2×#12411yescharacteristic
132C2×C83#53511yescharacteristic
133C2×C83#53611yescharacteristic
134C2×C83#54212yes—
135C2×C83#54212yes—
136D83—81no—
137#2640—831nomaximal, Sylow
138C83#264011yescharacteristic, complemented, Sylow
139C2×C4⋊Q8—831no—
140C2×#124—831no—
141#397—832no—
142#397—832no—
143#393—831no—
144#393—831no—
145#393—831no—
146#393D8311yescharacteristic, complemented
147#391—832no—
148#390—832no—
149#376—831no—
150C22×Q8—831no—
151C22×Q8—831no—
152C22×Q8C2×D8311yescharacteristic
153C22×D4—831no—
154C4⋊Q8—1664no—
155#124—1664no—
156C2×C4⋊C4—832no—
157C2×C4⋊C4—832no—
158C2×C22⋊C4—832no—
159C2×C22⋊C4—832no—
160C2×C22⋊C4—832no—
161C2×C22⋊C4—831no—
162C2×C22⋊C4—831no—
163C2×C22⋊C4—831no—
164C2×C22⋊C4C2×D8311yescharacteristic
165C2×C22⋊C4C2×D8312yes—
166C2×C42—831no—
167C2×C42—832no—
168#95—831no—
169#95—832no—
170#95—832no—
171#95—832no—
172#95—832no—
173#95C2×D8311yescharacteristic
174#95C2×D8312yes—
175C24—831no—
176C24C22×D8311yescharacteristic
177C2×Q8—22no—
178C2×Q8—42no—
179C2×Q8—3322no—
180C2×Q8—1662no—
181C2×Q8—1664no—
182C2×Q8—1662no—
183C2×D4—1662no—
184C2×D4—1664no—
185C22×C4—831no—
186C22×C4—831no—
187C22×C4—832no—
188C22×C4—831no—
189C22×C4—832no—
190C22×C4C22×D8312yes—
191C22×C4C22×D8311yescharacteristic
192C22×C4C22×D8312yes—
193C22×C4C22×D8311yescharacteristic
194C4⋊C4—1664no—
195C4⋊C4—1664no—
196C22⋊C4—22no—
197C22⋊C4—42no—
198C22⋊C4—3322no—
199C22⋊C4—3322no—
200C22⋊C4—3322no—
201C22⋊C4—1662no—
202C22⋊C4—1662no—
203C22⋊C4—1662no—
204C42—834no—
205C42—1664no—
206C23—21no—
207C23—22no—
208C23—22no—
209C23—21no—
210C23—21no—
211C23—1661no—
212C23—1662no—
213C23—1662no—
214C23—1661no—
215C23—1661no—
216C23C23×D8311yescharacteristic
217Q8—44no—
218Q8—3324no—
219Q8—3324no—
220D4—3324no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—832no—
231C2×C4—834no—
232C2×C4—832no—
233C2×C4—1661no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4—1662no—
240C2×C4—1662no—
241C2×C4—1661no—
242C2×C4—1661no—
243C2×C4D4×D8312yes—
244C22—42no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22—42no—
249C22—3322no—
250C22—3321no—
251C22—3321no—
252C22—3321no—
253C22—3322no—
254C22C2×D4×D8311yescharacteristic
255C22#41×D8312yes—
256C22#1458183712yes—
257C22#1458183911yescharacteristic
258C22#1458184011yescharacteristic
259C4—22no—
260C4—41no—
261C4—42no—
262C4—42no—
263C4—3322no—
264C4—3321no—
265C4—3322no—
266C4—834no—
267C4—1662no—
268C2—81no—
269C2—6641no—
270C2#124×D8311yescharacteristic
271C2#1912826611yescharacteristic
272C2#1912827411yescharacteristic
273C2#1912831512yes—
274C2#1912832112yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#393×C83C2
prime radicalC22×C22⋊D83C22
hypercenter#393D83
hypocenterC83#2640
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. #19127630C2semidirect
4. #19127629C2semidirect
5. #19127614C2semidirect
6. #19127613C2semidirect
7. #19127612C2semidirect
8. #19127605C2semidirect
10. #19127546C2semidirect
11. #393×C83C2semidirect
138. C83#2640semidirect
146. #393D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817902
C2×#145817782
#191276302
#191276292
#191276142
#191276132
#191276122
#191276052
#191276012
#191275462
#393×C832
#264083