Finite Group Databasebeta

← #24913405

Subgroups — #24913405

Order 10624.

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

normal subgroups only

This group has 584 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×#14581788C211yesmaximal, characteristic
2C2×#14581782C211yesmaximal, characteristic, complemented
3#19127619C211yesmaximal, characteristic, complemented
4#19127618C211yesmaximal, characteristic, complemented
5#19127617C212yesmaximal, complemented
6#19127616C212yesmaximal, complemented
7#19127615C211yesmaximal, characteristic, complemented
8#19127601C212yesmaximal, complemented
9#19127601C212yesmaximal, complemented
10#19127546C211yesmaximal, characteristic, complemented
11#394×C83C211yesmaximal, characteristic, complemented
12C22×D332C2211yescharacteristic
13C22×C22⋊D83C2212yes—
14C22×C22⋊D83C2212yes—
15#14581788—24no—
16#14581782—24no—
17C2×C4⋊Dic83C2211yescharacteristic
18C2×Dic83⋊C4C2211yescharacteristic
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2211yescharacteristic
22C2×C22⋊Dic83C2212yes—
23C2×C22⋊Dic83C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2212yes—
27C2×(C2×D83)⋊C4C2212yes—
28C2×C4×Dic83C2212yes—
29C2×C4×Dic83C2211yescharacteristic
30C83⋊#95C2211yescharacteristic
31C83⋊#95C2212yes—
32C22×D4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35C2×C22⋊C4×C83C2212yes—
36#95×C83C2211yescharacteristic
37C23×D83C2312yes—
38C2×D332—22no—
39C2×D332—42no—
40C2×C22⋊D83—24no—
41C2×C22⋊D83—24no—
42C2×C22⋊D83—24no—
43C2×C22⋊D83—24no—
44C2×C22⋊D83—42no—
45C2×C22⋊D83—42no—
46C22×Dic83C2312yes—
47C22×Dic83C2311yescharacteristic
48C22×Dic83C2311yescharacteristic
49C22×Dic83C2311yescharacteristic
50C22×Dic83C2311yescharacteristic
51C4⋊Dic83—22no—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54Dic83⋊C4—22no—
55Dic83⋊C4—22no—
56C22⋊Dic83—42no—
57C22⋊Dic83—42no—
58(C2×D83)⋊C4—42no—
59(C2×D83)⋊C4—42no—
60(C2×D83)⋊C4—42no—
61(C2×D83)⋊C4—42no—
62C4×Dic83—24no—
63C4×Dic83D414yes—
64C24×C83C2312yes—
65C2×D4×C83—22no—
66C2×D4×C83—42no—
67C22×C4×C83C2312yes—
68C22×C4×C83C2311yescharacteristic
69C22×C4×C83C2311yescharacteristic
70C22×C4×C83C2311yescharacteristic
71C4⋊C4×C83—22no—
72C22⋊C4×C83—42no—
73C22⋊C4×C83—42no—
74C23×C83—22no—
75C23×C83—22no—
76C23×C83—22no—
77C23×C83—22no—
78C23×C83—22no—
79C23×C83—22no—
80C23×C83—22no—
81C23×C83C2411yescharacteristic
82C22×D83—22no—
83C22×D83—22no—
84C22×D83—22no—
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—22no—
88C22×D83—22no—
89D4×C83—44no—
90C2×C4×C83—22no—
91C2×C4×C83—21no—
92C2×C4×C83—21no—
93C2×C4×C83—22no—
94C2×C4×C83—21no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—21no—
99C2×C4×C83C2×D412yes—
100C22⋊D83—48no—
101C22⋊D83—48no—
102C2×Dic83—21no—
103C2×Dic83—21no—
104C2×Dic83—21no—
105C2×Dic83—21no—
106C2×Dic83—22no—
107C2×Dic83—21no—
108C2×Dic83—22no—
109C2×Dic83—22no—
110C2×Dic83—22no—
111C2×Dic83—21no—
112C2×Dic83C2×D412yes—
113C2×Dic83C2×D412yes—
114C2×Dic83#4114yes—
115D332—44no—
116C22×C83—42no—
117C22×C83—42no—
118C22×C83—42no—
119C22×C83—42no—
120C22×C83—42no—
121C22×C83—42no—
122C22×C83—42no—
123C22×C83C22×D411yescharacteristic
124C22×C83C2×#4112yes—
125C22×C83H(5,𝔽2)12yes—
126C22×C83H(5,𝔽2)11yescharacteristic
127C22×C83H(5,𝔽2)11yescharacteristic
128C2×D83—42no—
129C2×D83—42no—
130C2×D83—42no—
131C2×D83—42no—
132C2×D83—42no—
133C2×D83—42no—
134C2×D83—42no—
135C4×C83—22no—
136C4×C83—41no—
137C4×C83—41no—
138C4×C83—42no—
139Dic83—22no—
140Dic83—41no—
141Dic83—41no—
142Dic83—42no—
143Dic83#12414yes—
144C2×C83—82no—
145C2×C83C2×#12411yescharacteristic
146C2×C83#53511yescharacteristic
147C2×C83#53511yescharacteristic
148C2×C83#54012yes—
149C2×C83#54012yes—
150D83—82no—
151#2641—831nomaximal, Sylow
152C83#264111yescharacteristic, complemented, Sylow
153C2×#127—831no—
154C2×#125—831no—
155#394—831no—
156#394—831no—
157#394—832no—
158#394—832no—
159#394—831no—
160#394D8311yescharacteristic, complemented
161#390—832no—
162#390—832no—
163#376—831no—
164C22×D4—832no—
165C22×D4—832no—
166C22×D4—831no—
167C22×D4C2×D8311yescharacteristic
168#127—1664no—
169#125—1664no—
170C2×C4⋊C4—831no—
171C2×C4⋊C4—831no—
172C2×C4⋊C4—831no—
173C2×C4⋊C4—831no—
174C2×C4⋊C4—831no—
175C2×C4⋊C4C2×D8311yescharacteristic
176C2×C22⋊C4—832no—
177C2×C22⋊C4—832no—
178C2×C22⋊C4—832no—
179C2×C22⋊C4—832no—
180C2×C22⋊C4—832no—
181C2×C22⋊C4—832no—
182C2×C22⋊C4C2×D8312yes—
183C2×C22⋊C4C2×D8312yes—
184C2×C42—831no—
185C2×C42—832no—
186#95—832no—
187#95—831no—
188#95C2×D8311yescharacteristic
189C24—832no—
190C24C22×D8312yes—
191C2×D4—22no—
192C2×D4—42no—
193C2×D4—3322no—
194C2×D4—3322no—
195C2×D4—3322no—
196C2×D4—1664no—
197C2×D4—1664no—
198C2×D4—1664no—
199C2×D4—1664no—
200C2×D4—1662no—
201C22×C4—831no—
202C22×C4—831no—
203C22×C4—832no—
204C22×C4—831no—
205C22×C4—831no—
206C22×C4C22×D8312yes—
207C22×C4C22×D8311yescharacteristic
208C22×C4C22×D8311yescharacteristic
209C22×C4C22×D8311yescharacteristic
210C4⋊C4—22no—
211C4⋊C4—1662no—
212C4⋊C4—1662no—
213C4⋊C4—1662no—
214C4⋊C4—1662no—
215C4⋊C4—1662no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—3322no—
219C22⋊C4—3322no—
220C22⋊C4—3322no—
221C22⋊C4—3322no—
222C22⋊C4—3322no—
223C22⋊C4—3322no—
224C42—834no—
225C42—1664no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—1662no—
234C23—1662no—
235C23—1662no—
236C23—1662no—
237C23—1662no—
238C23—1662no—
239C23—1662no—
240C23C23×D8311yescharacteristic
241D4—44no—
242D4—3324no—
243D4—3328no—
244D4—3328no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—832no—
255C2×C4—834no—
256C2×C4—832no—
257C2×C4—1661no—
258C2×C4—1661no—
259C2×C4—1662no—
260C2×C4—1662no—
261C2×C4—1661no—
262C2×C4—1662no—
263C2×C4—1661no—
264C2×C4—1662no—
265C2×C4—1661no—
266C2×C4—1661no—
267C2×C4D4×D8312yes—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—3322no—
276C22—3322no—
277C22—3322no—
278C22—3322no—
279C22—3322no—
280C22—3322no—
281C22—3322no—
282C22C2×D4×D8311yescharacteristic
283C22#41×D8312yes—
284C22#1458183712yes—
285C22#1458183711yescharacteristic
286C22#1458183811yescharacteristic
287C4—22no—
288C4—41no—
289C4—41no—
290C4—42no—
291C4—3321no—
292C4—3321no—
293C4—3322no—
294C4—834no—
295C4—1662no—
296C2—82no—
297C2—6642no—
298C2#124×D8311yescharacteristic
299C2#1912826611yescharacteristic
300C2#1912826711yescharacteristic
301C2#1912830112yes—
302C2#1912830312yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
prime radicalC2×#14581788C2
hypercenter#394D83
hypocenterC83#2641
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#14581782C2semidirect
3. #19127619C2semidirect
4. #19127618C2semidirect
5. #19127617C2semidirect
6. #19127616C2semidirect
7. #19127615C2semidirect
8. #19127601C2semidirect
9. #19127601C2semidirect
10. #19127546C2semidirect
11. #394×C83C2semidirect
152. C83#2641semidirect
160. #394D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817882
C2×#145817822
#191276192
#191276182
#191276172
#191276162
#191276152
#191276012
#191275462
#394×C832
#264183