Finite Group Databasebeta

← #24907495

Subgroups — #24907495

Order 10624.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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×#13050573C211yesmaximal, characteristic
2C2×#14581725C211yesmaximal, characteristic, complemented
3C2×#14581723C211yesmaximal, characteristic, complemented
4#19127558C211yesmaximal, characteristic
5#19127558C211yesmaximal, characteristic
6#19127464C211yesmaximal, characteristic, complemented
7#340×C83C211yesmaximal, characteristic, complemented
8C22×C4×D83C2211yescharacteristic
9C2×#13050573—22no—
10C2×#13050573—22no—
11C2×#13050573C414yescomplemented
12C22×C83⋊C8C2211yescharacteristic
13C2×C4⋊Dic83C2211yescharacteristic
14C2×C4⋊Dic83C2211yescharacteristic
15C2×(C2×D83)⋊C4—21no—
16C2×(C2×D83)⋊C4—21no—
17#14581725—22no—
18#14581723—22no—
19C83⋊#95—21no—
20C83⋊#95—21no—
21C22×C8×C83C2211yescharacteristic
22C2×C4⋊C4×C83C2211yescharacteristic
23C2×C4⋊C4×C83C2211yescharacteristic
24C23×D83D411yescharacteristic
25C22×Dic83—21no—
26C22×Dic83—21no—
27C22×Dic83D411yescharacteristic
28C2×C4×D83—21no—
29C2×C4×D83—21no—
30C2×C4×D83—21no—
31C2×C4×D83—21no—
32C2×C4×D83C2×C412yes—
33C2×C4×D83Q812yes—
34#13050573—28no—
35C2×C83⋊C8C2×C412yes—
36C2×C83⋊C8D412yes—
37C2×C83⋊C8D412yes—
38C4⋊Dic83—21no—
39C4⋊Dic83—21no—
40C4⋊Dic83—21no—
41C4⋊Dic83—21no—
42(C2×D83)⋊C4—42no—
43(C2×D83)⋊C4—42no—
44C22×C4×C83—21no—
45C22×C4×C83—21no—
46C22×C4×C83C2311yescharacteristic
47C2×C8×C83—21no—
48C2×C8×C83—21no—
49C2×C8×C83C2×C412yes—
50C4⋊C4×C83—21no—
51C4⋊C4×C83—21no—
52C4⋊C4×C83—21no—
53C4⋊C4×C83—21no—
54C23×C83C2×D411yescharacteristic
55C22×D83—21no—
56C22×D83—21no—
57C22×D83—21no—
58C22×D83—21no—
59C22×D83—21no—
60C22×D83—21no—
61C22×D83C4⋊C412yes—
62C2×C4×C83—41no—
63C2×C4×C83—41no—
64C2×C4×C83—41no—
65C2×C4×C83—41no—
66C2×C4×C83—41no—
67C2×C4×C83—41no—
68C2×C4×C83C22×C411yescharacteristic
69C2×C4×C83C2×D411yescharacteristic
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83C2×Q811yescharacteristic
72C2×C4×C83#4111yescharacteristic
73C2×C4×C83#4111yescharacteristic
74C2×Dic83—21no—
75C2×Dic83—21no—
76C2×Dic83—41no—
77C2×Dic83—41no—
78C2×Dic83—41no—
79C2×Dic83—41no—
80C2×Dic83—41no—
81C2×Dic83—41no—
82C2×Dic83C4⋊C412yes—
83C4×D83—22no—
84C4×D83—22no—
85C4×D83—22no—
86C4×D83—22no—
87C8×C83—22no—
88C83⋊C8C22⋊C414yescomplemented
89C22×C83C2×C4⋊C411yescharacteristic
90C22×C83#12111yescharacteristic
91C22×C83#12111yescharacteristic
92C22×C83AGL(1,ℤ/8)11yescharacteristic
93C22×C83AGL(1,ℤ/8)11yescharacteristic
94C22×C83#13711yescharacteristic
95C22×C83#13711yescharacteristic
96C2×D83—22no—
97C2×D83—22no—
98C2×D83—22no—
99C2×D83—41no—
100C2×D83—41no—
101C2×D83—41no—
102C2×D83—41no—
103C4×C83—81no—
104C4×C83—81no—
105C4×C83C2×C22⋊C411yescharacteristic
106C4×C83#11711yescharacteristic
107C4×C83C22⋊Q811yescharacteristic
108C4×C83C22⋊Q811yescharacteristic
109Dic83—22no—
110Dic83—81no—
111Dic83—81no—
112C2×C83#38011yescharacteristic
113C2×C83#42811yescharacteristic
114C2×C83#42811yescharacteristic
115C2×C83#46811yescharacteristic
116C2×C83#46811yescharacteristic
117C2×C83#46911yescharacteristic
118C2×C83#47011yescharacteristic
119D83—42no—
120#1809—831nomaximal, Sylow
121C83#180911yescharacteristic, complemented, Sylow
122C22×M16—831no—
123C2×#107—831no—
124C2×#106—831no—
125#380—831no—
126#380—831no—
127#340—831no—
128#340D8311yescharacteristic, complemented
129C23×C4—831no—
130C2×M16—834no—
131C2×M16—1662no—
132C2×M16—1662no—
133C22×C8—831no—
134C22×C8C2×D8311yescharacteristic
135C2×C4⋊C4—831no—
136C2×C4⋊C4—831no—
137C2×C4⋊C4C2×D8311yescharacteristic
138C2×C4⋊C4C2×D8311yescharacteristic
139C2×C22⋊C4—1661no—
140C2×C22⋊C4—1661no—
141#107—1662no—
142#106—1662no—
143#95—1661no—
144#95—1661no—
145C24—831no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—831no—
149C22×C4—832no—
150C22×C4—832no—
151C22×C4—1661no—
152C22×C4—1661no—
153C22×C4—1661no—
154C22×C4—1661no—
155C22×C4—1661no—
156C22×C4—1661no—
157C22×C4C22×D8311yescharacteristic
158M16—1668no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—832no—
162C2×C8—832no—
163C2×C8—832no—
164C2×C8C4×D8312yes—
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—3322no—
174C22⋊C4—3322no—
175C23—832no—
176C23—1661no—
177C23—1661no—
178C23—1661no—
179C23—1661no—
180C23—1661no—
181C23—1661no—
182C23D4×D8311yescharacteristic
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—3321no—
190C2×C4—3321no—
191C2×C4—3321no—
192C2×C4—3321no—
193C2×C4—3321no—
194C2×C4—3321no—
195C2×C4—832no—
196C2×C4—1661no—
197C2×C4—1661no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4C2×C4×D8311yescharacteristic
203C2×C4D4×D8311yescharacteristic
204C2×C4D4×D8311yescharacteristic
205C2×C4Q8×D8311yescharacteristic
206C2×C4#1305059011yescharacteristic
207C2×C4#1305059011yescharacteristic
208C8—22no—
209C8—834no—
210C22—3321no—
211C22—3321no—
212C22—3321no—
213C22—3321no—
214C22—1662no—
215C22—1662no—
216C22—1662no—
217C22C4⋊C4×D8311yescharacteristic
218C22#1458176511yescharacteristic
219C22#1458176511yescharacteristic
220C22#1458181611yescharacteristic
221C22#1458181911yescharacteristic
222C22#1458182111yescharacteristic
223C22#1458182411yescharacteristic
224C4—81no—
225C4—81no—
226C4—6641no—
227C4—6641no—
228C4—1662no—
229C4C22⋊C4×D8311yescharacteristic
230C4#1458174911yescharacteristic
231C4#1458177011yescharacteristic
232C4#1458177011yescharacteristic
233C2—3322no—
234C2#1912756011yescharacteristic
235C2#1912774011yescharacteristic
236C2#1912774211yescharacteristic
237C2#1912794611yescharacteristic
238C2#1912795011yescharacteristic
239C2#1912795311yescharacteristic
240C2#1912795811yescharacteristic
subgroupG/H
centerC23D4×D83
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#340×C83C2
socleC23×C83C2×D4
prime radicalC23×D83D4
cosocleC22×C4×C83C23
hypercenter#340D83
hypocenterC83#1809
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#14581725C2semidirect
3. C2×#14581723C2semidirect
6. #19127464C2semidirect
7. #340×C83C2semidirect
11. C2×#13050573C4semidirect
88. C83⋊C8C22⋊C4semidirect
121. C83#1809semidirect
128. #340D83semidirect

Maximal subgroups

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

subgroupindex
C22×#130505732
C2×#145817252
C2×#145817232
#191275582
#191274642
#340×C832
#180983