Finite Group Databasebeta

← #24911936

Subgroups — #24911936

Order 10624.

544 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 161 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
2C2×C4⋊C4×D83C211yesmaximal, characteristic
3#19127633C211yesmaximal, characteristic, complemented
4#19127632C211yesmaximal, characteristic, complemented
5#19127631C211yesmaximal, characteristic, complemented
6#19127605C212yesmaximal, complemented
7#19127575C212yesmaximal
8#19127572C212yesmaximal
9#19127570C212yesmaximal, complemented
10#19127548C211yesmaximal, characteristic
11#398×C83C211yesmaximal, characteristic, complemented
12C22×Dic166C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14C22×C4×D83C2212yes—
15#14581770—24no—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×C4⋊Dic83C2212yes—
18C2×Dic83⋊C4C2211yescharacteristic
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2212yes—
22C2×Dic83⋊C4C2212yes—
23C4⋊C4×D83—24no—
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2211yescharacteristic
26C2×(C2×D83)⋊C4C2212yes—
27C2×C4×Dic83C2212yes—
28C83⋊#95C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2211yescharacteristic
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2212yes—
35#95×C83C2212yes—
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C2×Dic166—22no—
39C2×Dic166—24no—
40C22×Dic83C2312yes—
41C22×Dic83C2311yescharacteristic
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2312yes—
44C22×Dic83C2311yescharacteristic
45C2×C4×D83—24no—
46C2×C4×D83—24no—
47C2×C4×D83—42no—
48C2×C4×D83—42no—
49C2×C4×D83D414yes—
50C4⋊Dic83—22no—
51C4⋊Dic83—42no—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54Dic83⋊C4—24no—
55Dic83⋊C4—42no—
56Dic83⋊C4—42no—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4—22no—
59(C2×D83)⋊C4Q818yescomplemented
60C4×Dic83—24no—
61C22×C4×C83C2312yes—
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2311yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C22×C4×C83C2312yes—
66C4⋊C4×C83—22no—
67C4⋊C4×C83—22no—
68C4⋊C4×C83—42no—
69C23×C83C2411yescharacteristic
70C22×D83—22no—
71C22×D83—21no—
72C22×D83—21no—
73C22×D83C2×D412yes—
74C22×D83C2×Q814yes—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—21no—
85C2×C4×C83—22no—
86C2×C4×C83C2×D412yes—
87C2×C4×C83C2×Q814yes—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—21no—
91C2×Dic83—21no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—21no—
95C2×Dic83—22no—
96C2×Dic83—21no—
97C2×Dic83—22no—
98C2×Dic83C2×D412yes—
99C2×Dic83C2×Q814yes—
100C2×Dic83#4114yes—
101C4×D83—28no—
102C4×D83—48no—
103Dic166—44no—
104C22×C83C22×D411yescharacteristic
105C22×C83C22×Q812yes—
106C22×C83C2×#4112yes—
107C22×C83H(5,𝔽2)11yescharacteristic
108C22×C83H(5,𝔽2)11yescharacteristic
109C2×D83—22no—
110C2×D83—22no—
111C2×D83—24no—
112C2×D83—24no—
113C2×D83C4⋊Q814yes—
114C4×C83—22no—
115C4×C83—24no—
116C4×C83—41no—
117C4×C83—42no—
118C4×C83—41no—
119Dic83—24no—
120Dic83—41no—
121Dic83—42no—
122Dic83—41no—
123Dic83#12314yes—
124C2×C83C2×#12311yescharacteristic
125C2×C83C2×C4⋊Q811yescharacteristic
126C2×C83#53511yescharacteristic
127C2×C83#55412yes—
128C2×C83#55412yes—
129D83—24no—
130#2502—831nomaximal, Sylow
131C83#250211yescharacteristic, complemented, Sylow
132C2×C22⋊Q8—831no—
133C22×C4⋊C4—831no—
134#398—831no—
135#398—831no—
136#398—831no—
137#398D8311yescharacteristic, complemented
138#391—832no—
139#385—832no—
140#385—832no—
141#384—832no—
142C4×C22⋊C4—831no—
143C22×Q8—831no—
144C23×C4—832no—
145C23×C4—831no—
146C22⋊Q8—1664no—
147C2×C4⋊C4—832no—
148C2×C4⋊C4—832no—
149C2×C4⋊C4—832no—
150C2×C4⋊C4—832no—
151C2×C4⋊C4—831no—
152C2×C4⋊C4—831no—
153C2×C4⋊C4—831no—
154C2×C4⋊C4—1664no—
155C2×C4⋊C4C2×D8311yescharacteristic
156C2×C4⋊C4C2×D8311yescharacteristic
157C2×C4⋊C4C2×D8312yes—
158C2×C22⋊C4—832no—
159C2×C22⋊C4—831no—
160C2×C22⋊C4—831no—
161C2×C42—832no—
162#95—831no—
163#95—832no—
164#95—832no—
165#95—832no—
166#95C2×D8311yescharacteristic
167#95C2×D8312yes—
168C24—831no—
169C2×Q8—1662no—
170C2×Q8—1664no—
171C22×C4—3322no—
172C22×C4—3322no—
173C22×C4—831no—
174C22×C4—832no—
175C22×C4—834no—
176C22×C4—831no—
177C22×C4—832no—
178C22×C4—831no—
179C22×C4—1664no—
180C22×C4—1664no—
181C22×C4C22×D8311yescharacteristic
182C22×C4C22×D8312yes—
183C22×C4C22×D8311yescharacteristic
184C22×C4C22×D8311yescharacteristic
185C22×C4C22×D8312yes—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—3322no—
190C4⋊C4—3322no—
191C4⋊C4—3322no—
192C4⋊C4—1664no—
193C4⋊C4—1662no—
194C4⋊C4—1662no—
195C4⋊C4—1662no—
196C22⋊C4—838no—
197C22⋊C4—1662no—
198C22⋊C4—1662no—
199C42—1664no—
200C23—832no—
201C23—834no—
202C23—1662no—
203C23—1661no—
204C23—1661no—
205C23C23×D8311yescharacteristic
206Q8—3324no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—3328no—
219C2×C4—832no—
220C2×C4—834no—
221C2×C4—834no—
222C2×C4—1668no—
223C2×C4—1661no—
224C2×C4—1661no—
225C2×C4—1662no—
226C2×C4—1662no—
227C2×C4—1661no—
228C2×C4—1662no—
229C2×C4—1661no—
230C2×C4—1662no—
231C2×C4—1662no—
232C2×C4—1662no—
233C2×C4D4×D8312yes—
234C2×C4Q8×D8314yes—
235C22—834no—
236C22—1664no—
237C22—1664no—
238C22—1662no—
239C22—1662no—
240C22C2×D4×D8311yescharacteristic
241C22C2×Q8×D8312yes—
242C22#41×D8312yes—
243C22#1458183711yescharacteristic
244C22#1458183811yescharacteristic
245C4—24no—
246C4—22no—
247C4—41no—
248C4—42no—
249C4—41no—
250C4—3321no—
251C4—3322no—
252C4—3321no—
253C4—834no—
254C4—1664no—
255C2—1664no—
256C2#123×D8311yescharacteristic
257C2C4⋊Q8×D8311yescharacteristic
258C2#1912826711yescharacteristic
259C2#1912840512yes—
260C2#1912840712yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#398×C83C2
prime radicalC23×D83C23
hypercenter#398D83
hypocenterC83#2502
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. #19127633C2semidirect
4. #19127632C2semidirect
5. #19127631C2semidirect
6. #19127605C2semidirect
9. #19127570C2semidirect
11. #398×C83C2semidirect
59. (C2×D83)⋊C4Q8semidirect
131. C83#2502semidirect
137. #398D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817702
C2×C4⋊C4×D832
#191276332
#191276322
#191276312
#191276052
#191275752
#191275722
#191275702
#191275482
#398×C832
#250283