Finite Group Databasebeta

← #24912839

Subgroups — #24912839

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×#14581768C211yesmaximal, characteristic
2C2×C4⋊C4×D83C211yesmaximal, characteristic
3#19127633C211yesmaximal, characteristic, complemented
4#19127631C211yesmaximal, characteristic, complemented
5#19127620C211yesmaximal, characteristic, complemented
6#19127598C212yesmaximal, complemented
7#19127597C212yesmaximal, complemented
8#19127575C212yesmaximal
9#19127572C212yesmaximal
10#19127548C211yesmaximal, characteristic
11#395×C83C211yesmaximal, characteristic, complemented
12C22×C4×D83C2211yescharacteristic
13C22×C4×D83C2211yescharacteristic
14C22×C4×D83C2211yescharacteristic
15#14581768—24no—
16C2×C4⋊Dic83C2212yes—
17C2×C4⋊Dic83C2211yescharacteristic
18C2×C4⋊Dic83C2211yescharacteristic
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2212yes—
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
32C22×Q8×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2212yes—
35#95×C83C2212yes—
36#95×C83C2211yescharacteristic
37C23×D83C2311yescharacteristic
38C22×Dic83C2311yescharacteristic
39C22×Dic83C2312yes—
40C22×Dic83C2311yescharacteristic
41C22×Dic83C2312yes—
42C22×Dic83C2311yescharacteristic
43C2×C4×D83—22no—
44C2×C4×D83—22no—
45C2×C4×D83—22no—
46C2×C4×D83—22no—
47C2×C4×D83—42no—
48C2×C4×D83—42no—
49C2×C4×D83D414yes—
50C4⋊Dic83—22no—
51C4⋊Dic83—22no—
52C4⋊Dic83—24no—
53Dic83⋊C4—22no—
54Dic83⋊C4—24no—
55Dic83⋊C4—24no—
56Dic83⋊C4—22no—
57Dic83⋊C4—42no—
58(C2×D83)⋊C4—24no—
59(C2×D83)⋊C4Q818yescomplemented
60C4×Dic83—42no—
61C2×Q8×C83—22no—
62C2×Q8×C83—42no—
63C22×C4×C83C2311yescharacteristic
64C22×C4×C83C2312yes—
65C22×C4×C83C2311yescharacteristic
66C22×C4×C83C2312yes—
67C22×C4×C83C2311yescharacteristic
68C4⋊C4×C83—22no—
69C4⋊C4×C83—42no—
70C23×C83C2411yescharacteristic
71C22×D83—21no—
72C22×D83—21no—
73C22×D83—22no—
74C22×D83C2×D412yes—
75C22×D83C2×Q814yes—
76Q8×C83—44no—
77C2×C4×C83—21no—
78C2×C4×C83—22no—
79C2×C4×C83—21no—
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×C83—22no—
87C2×C4×C83C2×D412yes—
88C2×C4×C83C2×Q814yes—
89C2×C4×C83#4112yes—
90C2×Dic83—21no—
91C2×Dic83—22no—
92C2×Dic83—21no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—21no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—22no—
100C2×Dic83C2×D412yes—
101C2×Dic83C2×Q814yes—
102C2×Dic83#4112yes—
103C4×D83—28no—
104C4×D83—44no—
105C4×D83—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—41no—
120C4×C83—42no—
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#398—831no—
138#395—831no—
139#395D8311yescharacteristic, complemented
140#389—832no—
141#389—832no—
142#385—832no—
143#385—832no—
144C4×C22⋊C4—831no—
145C22×Q8C2×D8311yescharacteristic
146C23×C4—831no—
147C23×C4—831no—
148C23×C4—831no—
149C22⋊Q8—1664no—
150C2×C4⋊C4—832no—
151C2×C4⋊C4—831no—
152C2×C4⋊C4—831no—
153C2×C4⋊C4—831no—
154C2×C4⋊C4—832no—
155C2×C4⋊C4—831no—
156C2×C4⋊C4—832no—
157C2×C4⋊C4—832no—
158C2×C4⋊C4—1664no—
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—22no—
172C2×Q8—42no—
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×D8311yescharacteristic
186C22×C4C22×D8312yes—
187C22×C4C22×D8311yescharacteristic
188C22×C4C22×D8312yes—
189C22×C4C22×D8311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—3322no—
193C4⋊C4—1662no—
194C4⋊C4—1662no—
195C4⋊C4—1664no—
196C4⋊C4—1664no—
197C4⋊C4—1662no—
198C4⋊C4—1662no—
199C4⋊C4—1664no—
200C22⋊C4—838no—
201C22⋊C4—1664no—
202C42—3322no—
203C23—832no—
204C23—834no—
205C23—1661no—
206C23—1661no—
207C23—1662no—
208C23C23×D8311yescharacteristic
209Q8—44no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
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#1305059112yes—
239C22—834no—
240C22—1664no—
241C22—1664no—
242C22—1662no—
243C22—1662no—
244C22C2×D4×D8311yescharacteristic
245C22C2×Q8×D8312yes—
246C22#41×D8311yescharacteristic
247C22C2×#1305059111yescharacteristic
248C22#1458183712yes—
249C4—24no—
250C4—22no—
251C4—22no—
252C4—41no—
253C4—42no—
254C4—3322no—
255C4—3321no—
256C4—1664no—
257C4—1662no—
258C4—1662no—
259C2—1664no—
260C2#1912821411yescharacteristic
261C2C4⋊Q8×D8311yescharacteristic
262C2#1912826111yescharacteristic
263C2#1912840512yes—
264C2#1912841212yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#395×C83C2
prime radicalC23×D83C23
hypercenter#395D83
hypocenterC83#2583
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
3. #19127633C2semidirect
4. #19127631C2semidirect
5. #19127620C2semidirect
6. #19127598C2semidirect
7. #19127597C2semidirect
11. #395×C83C2semidirect
59. (C2×D83)⋊C4Q8semidirect
133. C83#2583semidirect
139. #395D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817682
C2×C4⋊C4×D832
#191276332
#191276312
#191276202
#191275982
#191275972
#191275752
#191275722
#191275482
#395×C832
#258383