Finite Group Databasebeta

← #24911909

Subgroups — #24911909

Order 10624.

546 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 153 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×#14581751C211yesmaximal, characteristic, complemented
2#19127640C211yesmaximal, characteristic, complemented
3#19127637C211yesmaximal, characteristic, complemented
4#19127631C211yesmaximal, characteristic, complemented
5#19127626C211yesmaximal, characteristic, complemented
6#19127589C212yesmaximal, complemented
7#19127582C212yesmaximal, complemented
8#19127581C212yesmaximal
9#19127559C212yesmaximal, complemented
10C2×C83⋊#95C211yesmaximal, characteristic, complemented
11C2×#123×C83C211yesmaximal, characteristic
12C23×Dic83C2212yes—
13#14581751—24no—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×C4⋊Dic83C2211yescharacteristic
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2212yes—
21C2×C4×Dic83C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2211yescharacteristic
28C83⋊#95C2211yescharacteristic
29C22×D4×C83C2211yescharacteristic
30C23×C4×C83C2211yescharacteristic
31#123×C83—24no—
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2211yescharacteristic
35C2×C22⋊C4×C83C2211yescharacteristic
36C2×C22⋊C4×C83C2211yescharacteristic
37C22×Dic83—24no—
38C22×Dic83—24no—
39C22×Dic83—42no—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C22×Dic83C2312yes—
43C22×Dic83C2312yes—
44C4⋊Dic83—22no—
45C4⋊Dic83—22no—
46Dic83⋊C4—24no—
47Dic83⋊C4—42no—
48C22⋊Dic83—24no—
49C22⋊Dic83—24no—
50C22⋊Dic83—42no—
51C4×Dic83—24no—
52C24×C83C2311yescharacteristic
53C24×C83C2311yescharacteristic
54C2×D4×C83—22no—
55C2×D4×C83—24no—
56C22×C4×C83—24no—
57C22×C4×C83D414yes—
58C22×C4×C83C2311yescharacteristic
59C22×C4×C83C2311yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2311yescharacteristic
63C4⋊C4×C83—22no—
64C4⋊C4×C83—22no—
65C22⋊C4×C83—22no—
66C22⋊C4×C83—22no—
67C22⋊C4×C83—22no—
68C23×C83—22no—
69C23×C83—21no—
70C23×C83—21no—
71C23×C83—21no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—21no—
75C23×C83—21no—
76C23×C83C2×D412yes—
77C23×C83#4114yes—
78C23×C83C2411yescharacteristic
79D4×C83—44no—
80C2×C4×C83—28no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—21no—
84C2×C4×C83—21no—
85C2×C4×C83—21no—
86C2×C4×C83—22no—
87C2×C4×C83—22no—
88C2×C4×C83—21no—
89C2×C4×C83—21no—
90C2×C4×C83C2×D412yes—
91C2×C4×C83C2×D412yes—
92C2×Dic83—22no—
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×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—48no—
104C2×Dic83#4114yes—
105C22×C83—22no—
106C22×C83—22no—
107C22×C83—24no—
108C22×C83—24no—
109C22×C83—41no—
110C22×C83—41no—
111C22×C83—41no—
112C22×C83—42no—
113C22×C83—42no—
114C22×C83#12314yes—
115C22×C83C22×D411yescharacteristic
116C22×C83C2×#4112yes—
117C22×C83C2×#4112yes—
118C22×C83H(5,𝔽2)11yescharacteristic
119C22×C832−1+411yescharacteristic
120C4×C83—22no—
121C4×C83—22no—
122C4×C83—41no—
123C4×C83—41no—
124C4×C83—41no—
125Dic83—24no—
126Dic83—42no—
127Dic83—42no—
128Dic83—42no—
129C2×C83—24no—
130C2×C83—81no—
131C2×C83C2×#12311yescharacteristic
132C2×C83#53211yescharacteristic
133C2×C83#53711yescharacteristic
134C2×C83#55112yes—
135C2×C83#55212yes—
136#2498—831nomaximal, Sylow
137C83#249811yescharacteristic, complemented, Sylow
138C2×#123D8311yescharacteristic
139C2×#117—831no—
140#400—831no—
141#399—831no—
142#398—831no—
143#397—831no—
144#388—832no—
145#387—832no—
146#386—832no—
147#380—832no—
148C2×#95—831no—
149C22×D4C2×D8311yescharacteristic
150C23×C4—832no—
151C23×C4C2×D8311yescharacteristic
152#123—24no—
153#117—1664no—
154C2×C4⋊C4—832no—
155C2×C4⋊C4—831no—
156C2×C4⋊C4—831no—
157C2×C4⋊C4—832no—
158C2×C4⋊C4C2×D8311yescharacteristic
159C2×C4⋊C4C2×D8311yescharacteristic
160C2×C22⋊C4—832no—
161C2×C22⋊C4—832no—
162C2×C22⋊C4—832no—
163C2×C22⋊C4C2×D8311yescharacteristic
164C2×C22⋊C4C2×D8311yescharacteristic
165C2×C22⋊C4C2×D8311yescharacteristic
166C2×C42—832no—
167#95—832no—
168#95—831no—
169#95—831no—
170#95—832no—
171#95—832no—
172#95—832no—
173#95—832no—
174C24C22×D8311yescharacteristic
175C24C22×D8311yescharacteristic
176C2×D4—22no—
177C2×D4—24no—
178C22×C4—24no—
179C22×C4—3322no—
180C22×C4—832no—
181C22×C4—832no—
182C22×C4—832no—
183C22×C4—832no—
184C22×C4—1664no—
185C22×C4—1664no—
186C22×C4C22⋊D8314yes—
187C22×C4C22×D8311yescharacteristic
188C22×C4C22×D8311yescharacteristic
189C22×C4C22×D8311yescharacteristic
190C22×C4C22×D8311yescharacteristic
191C22×C4C22×D8311yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—3322no—
195C4⋊C4—1664no—
196C4⋊C4—1662no—
197C4⋊C4—1662no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—3322no—
202C22⋊C4—1664no—
203C22⋊C4—1664no—
204C42—1664no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23C2×C22⋊D8312yes—
214C23#1305058914yes—
215C23C23×D8311yescharacteristic
216D4—44no—
217C2×C4—28no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—3328no—
228C2×C4—834no—
229C2×C4—1662no—
230C2×C4—1662no—
231C2×C4—1662no—
232C2×C4—1662no—
233C2×C4—1662no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4—1662no—
237C2×C4—1662no—
238C2×C4—1662no—
239C2×C4—1662no—
240C2×C4C2×C22⋊D8312yes—
241C2×C4C2×C22⋊D8312yes—
242C22—24no—
243C22—24no—
244C22—22no—
245C22—22no—
246C22—42no—
247C22—42no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22#1458177314yes—
252C22C22×C22⋊D8311yescharacteristic
253C22#41×D8312yes—
254C22C2×#1305058912yes—
255C22#1458183811yescharacteristic
256C22#1458183911yescharacteristic
257C4—22no—
258C4—22no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—3322no—
263C4—3322no—
264C4—3322no—
265C4—1664no—
266C2—24no—
267C2—81no—
268C2C2×#1458177311yescharacteristic
269C2#1912824711yescharacteristic
270C2#1912827911yescharacteristic
271C2#1912837412yes—
272C2#1912838812yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×#123×C83C2
prime radicalC22×D4×C83C22
hypercenterC2×#123D83
hypocenterC83#2498
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#14581751C2semidirect
2. #19127640C2semidirect
3. #19127637C2semidirect
4. #19127631C2semidirect
5. #19127626C2semidirect
6. #19127589C2semidirect
7. #19127582C2semidirect
9. #19127559C2semidirect
10. C2×C83⋊#95C2semidirect
137. C83#2498semidirect

Maximal subgroups

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

subgroupindex
C2×#145817512
#191276402
#191276372
#191276312
#191276262
#191275892
#191275822
#191275812
#191275592
C2×C83⋊#952
C2×#123×C832
#249883