Finite Group Databasebeta

← #24912837

Subgroups — #24912837

Order 10624.

544 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 159 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×#14581772C211yesmaximal, characteristic
2C2×C4⋊C4×D83C211yesmaximal, characteristic
3#19127632C211yesmaximal, characteristic, complemented
4#19127622C211yesmaximal, characteristic, complemented
5#19127621C211yesmaximal, characteristic, complemented
6#19127600C212yesmaximal, complemented
7#19127598C212yesmaximal, complemented
8#19127575C212yesmaximal
9#19127572C212yesmaximal
10#19127548C211yesmaximal, characteristic
11#398×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14C22×C4×D83C2211yescharacteristic
15C22×C4×D83C2211yescharacteristic
16#14581772—24no—
17C2×C4⋊Dic83C2212yes—
18C2×C4⋊Dic83C2211yescharacteristic
19C2×C4⋊Dic83C2212yes—
20C2×Dic83⋊C4C2212yes—
21C2×Dic83⋊C4C2211yescharacteristic
22C2×Dic83⋊C4C2211yescharacteristic
23C2×Dic83⋊C4C2212yes—
24C4⋊C4×D83—24no—
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2212yes—
27C2×C4×Dic83C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2211yescharacteristic
32C2×C4⋊C4×C83C2212yes—
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2211yescharacteristic
35#95×C83C2212yes—
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C2×Dic166—22no—
39C2×Dic166—42no—
40C22×Dic83C2311yescharacteristic
41C22×Dic83C2312yes—
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2312yes—
44C22×Dic83C2311yescharacteristic
45C2×C4×D83—22no—
46C2×C4×D83—22no—
47C2×C4×D83—22no—
48C2×C4×D83—22no—
49C2×C4×D83—42no—
50C2×C4×D83—42no—
51C2×C4×D83D414yes—
52C4⋊Dic83—24no—
53C4⋊Dic83—22no—
54C4⋊Dic83—42no—
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4—24no—
58Dic83⋊C4—42no—
59(C2×D83)⋊C4—24no—
60(C2×D83)⋊C4Q818yescomplemented
61C4×Dic83—42no—
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2312yes—
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2312yes—
66C22×C4×C83C2311yescharacteristic
67C4⋊C4×C83—22no—
68C4⋊C4×C83—22no—
69C4⋊C4×C83—24no—
70C23×C83C2411yescharacteristic
71C22×D83—21no—
72C22×D83—21no—
73C22×D83—22no—
74C22×D83C2×D412yes—
75C22×D83C2×Q814yes—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83—21no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83C2×D412yes—
87C2×C4×C83C2×Q814yes—
88C2×C4×C83#4112yes—
89C2×Dic83—21no—
90C2×Dic83—22no—
91C2×Dic83—21no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—21no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83C2×D412yes—
100C2×Dic83C2×Q814yes—
101C2×Dic83#4112yes—
102C4×D83—28no—
103C4×D83—44no—
104C4×D83—44no—
105Dic166—44no—
106C22×C83C22×D411yescharacteristic
107C22×C83C22×Q812yes—
108C22×C83C2×#4111yescharacteristic
109C22×C83C2×#4111yescharacteristic
110C22×C83H(5,𝔽2)12yes—
111C2×D83—22no—
112C2×D83—22no—
113C2×D83—24no—
114C2×D83—24no—
115C2×D83C4⋊Q814yes—
116C4×C83—22no—
117C4×C83—22no—
118C4×C83—24no—
119C4×C83—42no—
120C4×C83—41no—
121Dic83—22no—
122Dic83—22no—
123Dic83—24no—
124Dic83—42no—
125Dic83—41no—
126C2×C83#52511yescharacteristic
127C2×C83C2×C4⋊Q811yescharacteristic
128C2×C83#53411yescharacteristic
129C2×C83#55412yes—
130C2×C83#55412yes—
131D83—24no—
132#2583—831nomaximal, Sylow
133C83#258311yescharacteristic, complemented, Sylow
134C2×C22⋊Q8—831no—
135C22×C4⋊C4—831no—
136#398—831no—
137#398D8311yescharacteristic, complemented
138#395—831no—
139#395—831no—
140#389—832no—
141#389—832no—
142#385—832no—
143#385—832no—
144C4×C22⋊C4—831no—
145C22×Q8—831no—
146C23×C4—831no—
147C23×C4—831no—
148C23×C4—831no—
149C22⋊Q8—1664no—
150C2×C4⋊C4—832no—
151C2×C4⋊C4—832no—
152C2×C4⋊C4—831no—
153C2×C4⋊C4—831no—
154C2×C4⋊C4—831no—
155C2×C4⋊C4—832no—
156C2×C4⋊C4—832no—
157C2×C4⋊C4—1664no—
158C2×C4⋊C4C2×D8311yescharacteristic
159C2×C4⋊C4C2×D8311yescharacteristic
160C2×C4⋊C4C2×D8312yes—
161C2×C22⋊C4—832no—
162C2×C22⋊C4—832no—
163C2×C42—832no—
164#95—831no—
165#95—832no—
166#95—832no—
167#95—832no—
168#95C2×D8311yescharacteristic
169#95C2×D8312yes—
170C24—831no—
171C2×Q8—3322no—
172C2×Q8—1662no—
173C22×C4—3322no—
174C22×C4—3322no—
175C22×C4—834no—
176C22×C4—832no—
177C22×C4—831no—
178C22×C4—831no—
179C22×C4—832no—
180C22×C4—831no—
181C22×C4—1662no—
182C22×C4—1662no—
183C22×C4—1662no—
184C22×C4—1662no—
185C22×C4C22×D8312yes—
186C22×C4C22×D8311yescharacteristic
187C22×C4C22×D8311yescharacteristic
188C22×C4C22×D8312yes—
189C22×C4C22×D8311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—3322no—
194C4⋊C4—3322no—
195C4⋊C4—1662no—
196C4⋊C4—1662no—
197C4⋊C4—1664no—
198C4⋊C4—1664no—
199C4⋊C4—1662no—
200C22⋊C4—838no—
201C22⋊C4—1664no—
202C42—3322no—
203C23—832no—
204C23—834no—
205C23—1661no—
206C23—1661no—
207C23—1662no—
208C23C23×D8311yescharacteristic
209Q8—3324no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—3324no—
221C2×C4—3324no—
222C2×C4—832no—
223C2×C4—834no—
224C2×C4—832no—
225C2×C4—1668no—
226C2×C4—1662no—
227C2×C4—1661no—
228C2×C4—1662no—
229C2×C4—1662no—
230C2×C4—1662no—
231C2×C4—1661no—
232C2×C4—1662no—
233C2×C4—1661no—
234C2×C4—1662no—
235C2×C4—1662no—
236C2×C4D4×D8312yes—
237C2×C4Q8×D8314yes—
238C2×C4#1305058912yes—
239C22—834no—
240C22—1664no—
241C22—1664no—
242C22—1662no—
243C22—1662no—
244C22C2×D4×D8311yescharacteristic
245C22C2×Q8×D8312yes—
246C22#41×D8311yescharacteristic
247C22C2×#1305058911yescharacteristic
248C22#1458183812yes—
249C4—24no—
250C4—22no—
251C4—22no—
252C4—42no—
253C4—41no—
254C4—3321no—
255C4—3322no—
256C4—1664no—
257C4—1662no—
258C4—1662no—
259C2—1664no—
260C2#1912821111yescharacteristic
261C2C4⋊Q8×D8311yescharacteristic
262C2#1912826211yescharacteristic
263C2#1912840712yes—
264C2#1912840912yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#398×C83C2
prime radicalC23×D83C23
hypercenter#398D83
hypocenterC83#2583
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #19127632C2semidirect
4. #19127622C2semidirect
5. #19127621C2semidirect
6. #19127600C2semidirect
7. #19127598C2semidirect
11. #398×C83C2semidirect
60. (C2×D83)⋊C4Q8semidirect
133. C83#2583semidirect
137. #398D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817722
C2×C4⋊C4×D832
#191276322
#191276222
#191276212
#191276002
#191275982
#191275752
#191275722
#191275482
#398×C832
#258383