Finite Group Databasebeta

← #24907910

Subgroups — #24907910

Order 10624.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#19128213C211yesmaximal, characteristic
2C2×#14581749C211yesmaximal, characteristic, complemented
3C2×#14581720C211yesmaximal, characteristic, complemented
4C2×#14581718C211yesmaximal, characteristic, complemented
5#19127664C211yesmaximal, characteristic, complemented
6#19127460C211yesmaximal, characteristic, complemented
7#339×C83C211yesmaximal, characteristic, complemented
8C2×#13050590C2211yescharacteristic
9C22×C4×D83C2211yescharacteristic
10C2×#13050574C2211yescharacteristic
11#14581776—21no—
12#14581772C412yescomplemented
13#14581767C412yescomplemented
14#14581762—21no—
15C4×D332—22no—
16#14581749—22no—
17#14581749—22no—
18C2×Dic83⋊C4—21no—
19C2×(C2×D83)⋊C4—21no—
20C2×C4×Dic83C2211yescharacteristic
21#14581720—22no—
22#14581718—22no—
23(C2×D83)⋊C8—22no—
24C2×M16×C83C2211yescharacteristic
25#117×C83C2211yescharacteristic
26C2×C42×C83C2211yescharacteristic
27C23×D83D411yescharacteristic
28#13050590—21no—
29#13050590—21no—
30#13050590—21no—
31#13050590—21no—
32C2×Dic166C2×C411yescharacteristic
33C2×D332—22no—
34C2×D332C2×C411yescharacteristic
35C2×C22⋊D83—21no—
36C22×Dic83—21no—
37C22×Dic83D411yescharacteristic
38C2×C4×D83—21no—
39C2×C4×D83—21no—
40C2×C4×D83—21no—
41C2×C4×D83—21no—
42C2×C4×D83—21no—
43C2×C4×D83C2×C412yes—
44C2×C4×D83D412yes—
45#13050574—21no—
46#13050574—21no—
47C2×C83⋊C8—21no—
48C4⋊Dic83—22no—
49Dic83⋊C4—24no—
50(C2×D83)⋊C4—24no—
51(C2×D83)⋊C4—22no—
52(C2×D83)⋊C4—22no—
53C4×Dic83—21no—
54C4×Dic83D412yescomplemented
55C22×C4×C83—21no—
56C22×C4×C83C2311yescharacteristic
57M16×C83—21no—
58M16×C83—21no—
59C2×C8×C83—21no—
60C4⋊C4×C83C2×C412yes—
61C22⋊C4×C83—21no—
62C42×C83—21no—
63C42×C83—21no—
64C42×C83D412yescomplemented
65C23×C83C2×D411yescharacteristic
66C22×D83—21no—
67C22×D83—21no—
68C22×D83—21no—
69C22×D83—42no—
70C22×D83C22⋊C412yes—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—42no—
74C2×C4×C83C22×C411yescharacteristic
75C2×C4×C83C2×D411yescharacteristic
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83#4111yescharacteristic
79C2×C4×C83#4111yescharacteristic
80C22⋊D83—41no—
81C22⋊D83—41no—
82C2×Dic83—21no—
83C2×Dic83—21no—
84C2×Dic83—21no—
85C2×Dic83—22no—
86C2×Dic83—42no—
87C2×Dic83C22⋊C412yes—
88D332—21no—
89D332—21no—
90D332—44no—
91C4×D83—22no—
92C4×D83—22no—
93C4×D83—22no—
94C4×D83—22no—
95C4×D83—41no—
96C4×D83—41no—
97Dic166—21no—
98Dic166—21no—
99C8×C83—41no—
100C83⋊C8—41no—
101C22×C83—22no—
102C22×C83C2×C22⋊C411yescharacteristic
103C22×C83#12011yescharacteristic
104C22×C83#12311yescharacteristic
105C2×D83—22no—
106C2×D83—22no—
107C2×D83—22no—
108C2×D83—42no—
109C2×D83—42no—
110C2×D83—41no—
111C2×D83—42no—
112C4×C83—42no—
113C4×C83—42no—
114C4×C83C4×D411yescharacteristic
115C4×C83C4×D411yescharacteristic
116C4×C83#12111yescharacteristic
117C4×C83#12111yescharacteristic
118Dic83—22no—
119Dic83—42no—
120Dic83—41no—
121C2×C83—21no—
122C2×C83—21no—
123C2×C83#38611yescharacteristic
124C2×C83#42112yes—
125D83—42no—
126D83—81no—
127#1897—831nomaximal, Sylow
128C83#189711yescharacteristic, complemented, Sylow
129#525—831no—
130C2×#117—831no—
131C2×#104—831no—
132C2×#104—831no—
133#407—831no—
134#339—831no—
135#339D8311yescharacteristic, complemented
136C2×#41—831no—
137C23×C4—831no—
138C2×M16—831no—
139C2×M16C2×D8311yescharacteristic
140#123—1661no—
141C22⋊Q8—832no—
142#121—832no—
143#120—1661no—
144C4×D4—1662no—
145#117—1662no—
146#117—1662no—
147#117C2×D8311yescharacteristic
148C2×C4⋊C4—1661no—
149C2×C22⋊C4—1661no—
150C2×C42—831no—
151C2×C42C2×D8311yescharacteristic
152#104—1662no—
153#104—1662no—
154C22⋊C8—1662no—
155C24—831no—
156#41—1661no—
157#41—1661no—
158#41—1661no—
159#41—1661no—
160C2×Q8—831no—
161C2×D4—831no—
162C2×D4—1662no—
163C2×D4—1661no—
164C22×C4—21no—
165C22×C4—831no—
166C22×C4—832no—
167C22×C4—832no—
168C22×C4—1661no—
169C22×C4—1661no—
170C22×C4—1661no—
171C22×C4—1661no—
172C22×C4—1661no—
173C22×C4—1661no—
174C22×C4C22×D8311yescharacteristic
175M16—21no—
176M16—21no—
177M16—1661no—
178M16—1661no—
179C2×C8—21no—
180C2×C8—1661no—
181C4⋊C4—1662no—
182C4⋊C4—1664no—
183C4⋊C4C4×D8312yes—
184C22⋊C4—21no—
185C22⋊C4—1662no—
186C22⋊C4—1662no—
187C22⋊C4—1664no—
188C42—21no—
189C42—21no—
190C42—832no—
191C42—1661no—
192C42D33212yescomplemented
193C23—3322no—
194C23—832no—
195C23—1661no—
196C23—1661no—
197C23—1661no—
198C23D4×D8311yescharacteristic
199Q8—1661no—
200Q8—1661no—
201D4—3324no—
202D4—3321no—
203D4—3321no—
204D4—1661no—
205D4—1661no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—3322no—
210C2×C4—3321no—
211C2×C4—3321no—
212C2×C4—832no—
213C2×C4—1662no—
214C2×C4—1661no—
215C2×C4—1661no—
216C2×C4—1661no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4C2×C4×D8311yescharacteristic
222C2×C4D4×D8311yescharacteristic
223C2×C4D4×D8311yescharacteristic
224C2×C4C2×D33211yescharacteristic
225C2×C4#1305059011yescharacteristic
226C2×C4#1305059111yescharacteristic
227C8—41no—
228C8—3321no—
229C22—22no—
230C22—3322no—
231C22—3322no—
232C22—3321no—
233C22—3322no—
234C22—1662no—
235C22—1662no—
236C22—1662no—
237C22C22⋊C4×D8311yescharacteristic
238C22#1458176211yescharacteristic
239C22#1458177711yescharacteristic
240C4—42no—
241C4—42no—
242C4—3322no—
243C4—3321no—
244C4—1662no—
245C4#1458175411yescharacteristic
246C4C4×D33211yescharacteristic
247C4#1458176511yescharacteristic
248C4#1458176711yescharacteristic
249C2—21no—
250C2—21no—
251C2—3322no—
252C2—6641no—
253C2#1912757911yescharacteristic
254C2#1912771612yes—
subgroupG/H
centerC2×C4D4×D83
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#339×C83C2
socleC22×C83C2×C22⋊C4
prime radical#19128213C2
cosocleC22×C4×C83C23
hypercenter#339D83
hypocenterC83#1897
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×#14581749C2semidirect
3. C2×#14581720C2semidirect
4. C2×#14581718C2semidirect
5. #19127664C2semidirect
6. #19127460C2semidirect
7. #339×C83C2semidirect
12. #14581772C4semidirect
13. #14581767C4semidirect
54. C4×Dic83D4semidirect
64. C42×C83D4semidirect
128. C83#1897semidirect
135. #339D83semidirect
192. C42D332semidirect

Maximal subgroups

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

subgroupindex
#191282132
C2×#145817492
C2×#145817202
C2×#145817182
#191276642
#191274602
#339×C832
#189783