Finite Group Databasebeta

← #24907494

Subgroups — #24907494

Order 10624.

316 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C22×#13050574C211yesmaximal, characteristic
2C2×#14581725C211yesmaximal, characteristic, complemented
3C2×#14581723C211yesmaximal, characteristic, complemented
4#19127561C211yesmaximal, characteristic
5#19127463C211yesmaximal, characteristic, complemented
6#19127463C211yesmaximal, characteristic, complemented
7#380×C83C211yesmaximal, characteristic
8C2×#13050574—22no—
9C2×#13050574—22no—
10C2×#13050574C414yescomplemented
11C22×C83⋊C8C2211yescharacteristic
12C22×C83⋊C8C2211yescharacteristic
13C2×C4⋊Dic83C2211yescharacteristic
14C2×C4⋊Dic83C2211yescharacteristic
15C2×C22⋊Dic83—21no—
16#14581725—22no—
17#14581723—22no—
18C83⋊#95—21no—
19C23×C4×C83C2211yescharacteristic
20C2×C4⋊C4×C83C2211yescharacteristic
21C2×C4⋊C4×C83C2211yescharacteristic
22C2×C22⋊C4×C83—21no—
23#95×C83—21no—
24C22×Dic83—21no—
25C22×Dic83—21no—
26#13050574—28no—
27C2×C83⋊C8—21no—
28C2×C83⋊C8—21no—
29C2×C83⋊C8C2×C412yes—
30C2×C83⋊C8C2×C412yes—
31C2×C83⋊C8D412yes—
32C2×C83⋊C8D412yes—
33C4⋊Dic83—21no—
34C4⋊Dic83—21no—
35C4⋊Dic83—21no—
36C4⋊Dic83—21no—
37C22⋊Dic83—42no—
38C24×C83D411yescharacteristic
39C22×C4×C83—21no—
40C22×C4×C83—21no—
41C22×C4×C83—21no—
42C22×C4×C83—21no—
43C22×C4×C83—21no—
44C22×C4×C83—21no—
45C22×C4×C83C2×C412yes—
46C22×C4×C83D411yescharacteristic
47C22×C4×C83Q812yes—
48C22×C4×C83C2311yescharacteristic
49C4⋊C4×C83—21no—
50C4⋊C4×C83—21no—
51C4⋊C4×C83—21no—
52C4⋊C4×C83—21no—
53C22⋊C4×C83—42no—
54C23×C83—21no—
55C23×C83—21no—
56C23×C83—21no—
57C23×C83—21no—
58C23×C83—21no—
59C23×C83—21no—
60C23×C83C4⋊C412yes—
61C23×C83C2×D411yescharacteristic
62C2×C4×C83—21no—
63C2×C4×C83—21no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83—22no—
67C2×C4×C83—22no—
68C2×C4×C83—41no—
69C2×C4×C83—41no—
70C2×C4×C83—41no—
71C2×C4×C83—41no—
72C2×C4×C83—41no—
73C2×C4×C83—41no—
74C2×C4×C83C4⋊C412yes—
75C2×C4×C83C22×C411yescharacteristic
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2×Q811yescharacteristic
79C2×C4×C83#4111yescharacteristic
80C2×C4×C83#4111yescharacteristic
81C2×Dic83—41no—
82C2×Dic83—41no—
83C2×Dic83—41no—
84C2×Dic83—41no—
85C2×Dic83—41no—
86C2×Dic83—41no—
87C83⋊C8—22no—
88C83⋊C8C22⋊C414yescomplemented
89C22×C83—22no—
90C22×C83—22no—
91C22×C83—22no—
92C22×C83—41no—
93C22×C83—41no—
94C22×C83—41no—
95C22×C83—41no—
96C22×C83C2×C4⋊C411yescharacteristic
97C22×C83#12111yescharacteristic
98C22×C83#12111yescharacteristic
99C22×C83AGL(1,ℤ/8)11yescharacteristic
100C22×C83AGL(1,ℤ/8)11yescharacteristic
101C22×C83#13711yescharacteristic
102C22×C83#13711yescharacteristic
103C4×C83—22no—
104C4×C83—81no—
105C4×C83—81no—
106C4×C83C2×C22⋊C411yescharacteristic
107C4×C83#11711yescharacteristic
108C4×C83C22⋊Q811yescharacteristic
109C4×C83C22⋊Q811yescharacteristic
110Dic83—81no—
111Dic83—81no—
112C2×C83—42no—
113C2×C83#38011yescharacteristic
114C2×C83#42811yescharacteristic
115C2×C83#42811yescharacteristic
116C2×C83#46811yescharacteristic
117C2×C83#46811yescharacteristic
118C2×C83#46911yescharacteristic
119C2×C83#47011yescharacteristic
120#1809—831nomaximal, Sylow
121C83#180911yescharacteristic, complemented, Sylow
122C22×M16—831no—
123C2×#107—831no—
124C2×#106—831no—
125#380—831no—
126#380D8311yescharacteristic
127#340—831no—
128#340—831no—
129C23×C4C2×D8311yescharacteristic
130C2×M16—834no—
131C2×M16—1662no—
132C2×M16—1662no—
133C22×C8—831no—
134C22×C8—831no—
135C2×C4⋊C4—831no—
136C2×C4⋊C4—831no—
137C2×C4⋊C4C2×D8311yescharacteristic
138C2×C4⋊C4C2×D8311yescharacteristic
139C2×C22⋊C4—21no—
140C2×C22⋊C4—1661no—
141#107—1662no—
142#106—1662no—
143#95—21no—
144#95—1661no—
145C24C22⋊D8311yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—1661no—
153C22×C4—1661no—
154C22×C4Dic16612yes—
155C22×C4C4×D8312yes—
156C22×C4C22⋊D8311yescharacteristic
157C22×C4C22×D8311yescharacteristic
158M16—1668no—
159C2×C8—832no—
160C2×C8—832no—
161C2×C8—832no—
162C2×C8—832no—
163C2×C8—1661no—
164C2×C8—1661no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—1661no—
170C4⋊C4—1661no—
171C4⋊C4—1661no—
172C4⋊C4—1661no—
173C22⋊C4—42no—
174C22⋊C4—3322no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23Dic83⋊C412yes—
182C23C2×C22⋊D8311yescharacteristic
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—3321no—
196C2×C4—3321no—
197C2×C4—3321no—
198C2×C4—3321no—
199C2×C4—3321no—
200C2×C4—3321no—
201C2×C4Dic83⋊C412yes—
202C2×C4C2×C4×D8311yescharacteristic
203C2×C4D4×D8311yescharacteristic
204C2×C4D4×D8311yescharacteristic
205C2×C4C2×Dic16611yescharacteristic
206C2×C4#1305058911yescharacteristic
207C2×C4#1305058911yescharacteristic
208C8—834no—
209C8—1662no—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22C2×Dic83⋊C411yescharacteristic
218C22#1458176411yescharacteristic
219C22#1458176411yescharacteristic
220C22#1458181711yescharacteristic
221C22#1458181811yescharacteristic
222C22#1458182211yescharacteristic
223C22#1458182311yescharacteristic
224C4—22no—
225C4—81no—
226C4—81no—
227C4—6641no—
228C4—6641no—
229C4C22⋊C4×D8311yescharacteristic
230C4#1458174811yescharacteristic
231C4#1458176911yescharacteristic
232C4#1458176911yescharacteristic
233C2—42no—
234C2#1912755911yescharacteristic
235C2#1912774111yescharacteristic
236C2#1912774411yescharacteristic
237C2#1912794511yescharacteristic
238C2#1912794911yescharacteristic
239C2#1912795211yescharacteristic
240C2#1912795711yescharacteristic
subgroupG/H
centerC23C2×C22⋊D83
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#380×C83C2
socleC23×C83C2×D4
prime radicalC24×C83D4
cosocleC22×C4×C83C23
hypercenter#380D83
hypocenterC83#1809
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#14581725C2semidirect
3. C2×#14581723C2semidirect
5. #19127463C2semidirect
6. #19127463C2semidirect
10. C2×#13050574C4semidirect
88. C83⋊C8C22⋊C4semidirect
121. C83#1809semidirect

Maximal subgroups

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

subgroupindex
C22×#130505742
C2×#145817252
C2×#145817232
#191275612
#191274632
#380×C832
#180983