Finite Group Databasebeta

← #24915494

Subgroups — #24915494

Order 10624.

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

normal subgroups only

This group has 528 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×#14581770C211yesmaximal, characteristic, complemented
2C2×#14581763C211yesmaximal, characteristic
3#19127637C212yesmaximal, complemented
4#19127633C211yesmaximal, characteristic, complemented
5#19127631C211yesmaximal, characteristic, complemented
6#19127617C212yesmaximal, complemented
7#19127616C212yesmaximal, complemented
8#19127615C211yesmaximal, characteristic, complemented
9#19127613C212yesmaximal, complemented
10#19127563C211yesmaximal, characteristic, complemented
11#394×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×C22⋊D83C2212yes—
14C22×C4×D83C2211yescharacteristic
15#14581770—24no—
16#14581763—24no—
17C2×C4⋊Dic83C2211yescharacteristic
18C2×C4⋊Dic83C2212yes—
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2212yes—
22C2×C22⋊Dic83C2212yes—
23C2×C22⋊Dic83C2212yes—
24C2×C22⋊Dic83C2212yes—
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2211yescharacteristic
27C2×(C2×D83)⋊C4C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2211yescharacteristic
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C22×D4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35C2×C22⋊C4×C83C2212yes—
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C2×Dic166—22no—
39C2×Dic166—42no—
40C2×C22⋊D83—24no—
41C2×C22⋊D83—42no—
42C2×C22⋊D83—42no—
43C22×Dic83C2312yes—
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2312yes—
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2311yescharacteristic
48C2×C4×D83—42no—
49C2×C4×D83D414yes—
50C4⋊Dic83—22no—
51C4⋊Dic83—42no—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54Dic83⋊C4—42no—
55C22⋊Dic83—24no—
56C22⋊Dic83—42no—
57C22⋊Dic83—42no—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4—42no—
61C24×C83C2312yes—
62C2×D4×C83—22no—
63C2×D4×C83—42no—
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2312yes—
66C22×C4×C83C2311yescharacteristic
67C22×C4×C83C2311yescharacteristic
68C4⋊C4×C83—22no—
69C22⋊C4×C83—42no—
70C22⋊C4×C83—42no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83—22no—
76C23×C83—22no—
77C23×C83—22no—
78C23×C83C2411yescharacteristic
79C22×D83—22no—
80C22×D83—21no—
81C22×D83—21no—
82C22×D83—22no—
83C22×D83C2×D412yes—
84D4×C83—44no—
85C2×C4×C83—22no—
86C2×C4×C83—21no—
87C2×C4×C83—21no—
88C2×C4×C83—21no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—21no—
93C2×C4×C83—22no—
94C2×C4×C83C2×D412yes—
95C22⋊D83—48no—
96C2×Dic83—22no—
97C2×Dic83—21no—
98C2×Dic83—21no—
99C2×Dic83—22no—
100C2×Dic83—22no—
101C2×Dic83—21no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—21no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83C2×D412yes—
109C4×D83—44no—
110Dic166—44no—
111C22×C83—42no—
112C22×C83—42no—
113C22×C83—42no—
114C22×C83—42no—
115C22×C83—42no—
116C22×C83—42no—
117C22×C83—42no—
118C22×C83C22×D411yescharacteristic
119C22×C83H(5,𝔽2)12yes—
120C22×C83H(5,𝔽2)11yescharacteristic
121C22×C83H(5,𝔽2)11yescharacteristic
122C22×C832−1+412yes—
123C2×D83—22no—
124C2×D83—22no—
125C2×D83—22no—
126C2×D83—42no—
127C2×D83—42no—
128C4×C83—22no—
129C4×C83—41no—
130C4×C83—41no—
131C4×C83—42no—
132Dic83—22no—
133Dic83—41no—
134Dic83—41no—
135Dic83—42no—
136Dic83—42no—
137C2×C83—82no—
138C2×C83#53411yescharacteristic
139C2×C83#53511yescharacteristic
140C2×C83#53611yescharacteristic
141C2×C83#56212yes—
142C2×C83#56212yes—
143D83—42no—
144#2816—831nomaximal, Sylow
145C83#281611yescharacteristic, complemented, Sylow
146C2×C22⋊Q8—831no—
147C2×#121—831no—
148#399—832no—
149#398—831no—
150#398—831no—
151#394—831no—
152#394—832no—
153#394—832no—
154#394D8311yescharacteristic, complemented
155#393—832no—
156#381—831no—
157C22×Q8—831no—
158C22×D4—832no—
159C22×D4C2×D8311yescharacteristic
160C23×C4—831no—
161C22⋊Q8—1664no—
162#121—1664no—
163C2×C4⋊C4—832no—
164C2×C4⋊C4—832no—
165C2×C4⋊C4—831no—
166C2×C4⋊C4—831no—
167C2×C4⋊C4—831no—
168C2×C4⋊C4C2×D8311yescharacteristic
169C2×C22⋊C4—832no—
170C2×C22⋊C4—832no—
171C2×C22⋊C4—832no—
172C2×C22⋊C4—832no—
173C2×C22⋊C4—831no—
174C2×C22⋊C4—831no—
175C2×C22⋊C4C2×D8312yes—
176C2×C22⋊C4C2×D8312yes—
177#95—832no—
178#95—831no—
179#95—832no—
180#95—832no—
181#95C2×D8311yescharacteristic
182C24—831no—
183C24C22×D8312yes—
184C2×Q8—3322no—
185C2×Q8—1662no—
186C2×D4—22no—
187C2×D4—42no—
188C2×D4—3322no—
189C2×D4—3322no—
190C2×D4—1664no—
191C22×C4—3322no—
192C22×C4—834no—
193C22×C4—831no—
194C22×C4—832no—
195C22×C4—831no—
196C22×C4—832no—
197C22×C4—831no—
198C22×C4C22×D8312yes—
199C22×C4C22×D8311yescharacteristic
200C22×C4C22×D8311yescharacteristic
201C22×C4C22×D8311yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—3322no—
204C4⋊C4—3322no—
205C4⋊C4—1662no—
206C4⋊C4—1662no—
207C4⋊C4—1662no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—3322no—
211C22⋊C4—3322no—
212C22⋊C4—3322no—
213C22⋊C4—1662no—
214C22⋊C4—1662no—
215C22⋊C4—1664no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—832no—
224C23—1662no—
225C23—1661no—
226C23—1661no—
227C23—1662no—
228C23C23×D8311yescharacteristic
229Q8—3324no—
230D4—44no—
231D4—3328no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—3324no—
242C2×C4—832no—
243C2×C4—1661no—
244C2×C4—1662no—
245C2×C4—1661no—
246C2×C4—1662no—
247C2×C4—1662no—
248C2×C4—1661no—
249C2×C4—1662no—
250C2×C4—1661no—
251C2×C4—1662no—
252C2×C4—1662no—
253C2×C4—1662no—
254C2×C4—1662no—
255C2×C4D4×D8312yes—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—3322no—
264C22—3322no—
265C22—1662no—
266C22—1662no—
267C22—1662no—
268C22C2×D4×D8311yescharacteristic
269C22#1458183711yescharacteristic
270C22#1458183712yes—
271C22#1458183811yescharacteristic
272C22#1458183912yes—
273C4—22no—
274C4—42no—
275C4—41no—
276C4—41no—
277C4—3322no—
278C4—3321no—
279C4—3322no—
280C4—3321no—
281C4—1662no—
282C2—82no—
283C2—3322no—
284C2#1912826111yescharacteristic
285C2#1912826711yescharacteristic
286C2#1912827111yescharacteristic
287C2#1912844612yes—
288C2#1912844812yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
prime radicalC2×#14581763C2
hypercenter#394D83
hypocenterC83#2816
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#14581770C2semidirect
3. #19127637C2semidirect
4. #19127633C2semidirect
5. #19127631C2semidirect
6. #19127617C2semidirect
7. #19127616C2semidirect
8. #19127615C2semidirect
9. #19127613C2semidirect
10. #19127563C2semidirect
11. #394×C83C2semidirect
145. C83#2816semidirect
154. #394D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817702
C2×#145817632
#191276372
#191276332
#191276312
#191276172
#191276162
#191276152
#191276132
#191275632
#394×C832
#281683