Finite Group Databasebeta

← #24908338

Subgroups — #24908338

Order 10624.

376 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 99 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×#13050575C211yesmaximal, characteristic
2C2×#14581723C211yesmaximal, characteristic, complemented
3C2×#14581713C211yesmaximal, characteristic, complemented
4C2×#14581712C211yesmaximal, characteristic, complemented
5C2×#14581703C211yesmaximal, characteristic, complemented
6#19127603C211yesmaximal, characteristic, complemented
7#390×C83C211yesmaximal, characteristic, complemented
8C22×D332C2211yescharacteristic
9C2×#13050575—22no—
10C2×#13050575—22no—
11C2×#13050575C414yescomplemented
12C22×C83⋊C8C2211yescharacteristic
13C22×C83⋊C8C2211yescharacteristic
14C2×C4⋊Dic83C2211yescharacteristic
15C2×(C2×D83)⋊C4—22no—
16#14581723—22no—
17#14581713—22no—
18#14581712—22no—
19#14581703—22no—
20C22×D4×C83C2211yescharacteristic
21C2×C4⋊C4×C83C2211yescharacteristic
22C2×C22⋊C4×C83—22no—
23C2×C42×C83C2211yescharacteristic
24C23×D83—21no—
25C2×D332—21no—
26C2×D332—21no—
27C2×D332—21no—
28C2×D332—21no—
29C2×D332—21no—
30C2×D332C2×C412yes—
31C22×Dic83—21no—
32#13050575—28no—
33C2×C83⋊C8—21no—
34C2×C83⋊C8—21no—
35C2×C83⋊C8C2×C412yes—
36C2×C83⋊C8D412yes—
37C2×C83⋊C8D412yes—
38C2×C83⋊C8D412yes—
39C4⋊Dic83—21no—
40C4⋊Dic83—21no—
41(C2×D83)⋊C4—44no—
42C24×C83—21no—
43C2×D4×C83—21no—
44C2×D4×C83—21no—
45C2×D4×C83—21no—
46C2×D4×C83—21no—
47C2×D4×C83—21no—
48C2×D4×C83C2×C412yes—
49C22×C4×C83—21no—
50C22×C4×C83D412yes—
51C22×C4×C83C2311yescharacteristic
52C4⋊C4×C83—21no—
53C4⋊C4×C83—21no—
54C22⋊C4×C83—44no—
55C42×C83—21no—
56C42×C83—21no—
57C23×C83—22no—
58C23×C83—41no—
59C23×C83—41no—
60C23×C83—42no—
61C23×C83—42no—
62C23×C83C2×D411yescharacteristic
63C22×D83—22no—
64C22×D83—41no—
65C22×D83—41no—
66C22×D83—42no—
67C22×D83—42no—
68D4×C83—22no—
69D4×C83—22no—
70D4×C83—41no—
71D4×C83—41no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—41no—
76C2×C4×C83—42no—
77C2×C4×C83C22×C411yescharacteristic
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83#4111yescharacteristic
82C2×C4×C83#4111yescharacteristic
83C2×Dic83—41no—
84C2×Dic83—42no—
85D332—22no—
86D332—22no—
87D332—41no—
88D332—41no—
89C83⋊C8—22no—
90C83⋊C8C22⋊C414yescomplemented
91C22×C83—42no—
92C22×C83—44no—
93C22×C83—82no—
94C22×C83—81no—
95C22×C83—81no—
96C22×C83C4×D411yescharacteristic
97C22×C83#12111yescharacteristic
98C22×C83#12711yescharacteristic
99C22×C83AGL(1,ℤ/8)12yes—
100C22×C83AGL(1,ℤ/8)12yes—
101C2×D83—42no—
102C2×D83—44no—
103C2×D83—82no—
104C2×D83—81no—
105C2×D83—81no—
106C4×C83—42no—
107C4×C83—81no—
108C4×C83C2×C22⋊C411yescharacteristic
109C4×C83C4×D411yescharacteristic
110C4×C83#12111yescharacteristic
111C4×C83#12411yescharacteristic
112Dic83—81no—
113C2×C83—82no—
114C2×C83#39011yescharacteristic
115C2×C83#44212yes—
116C2×C83#46912yes—
117C2×C83#49611yescharacteristic
118C2×C83#49611yescharacteristic
119D83—82no—
120#1972—831nomaximal, Sylow
121C83#197211yescharacteristic, complemented, Sylow
122C22×D8—831no—
123C2×#106—831no—
124C2×#102—831no—
125C2×#102—831no—
126C2×#97—831no—
127#390—831no—
128#390D8311yescharacteristic, complemented
129C22×D4—831no—
130C22×D4C2×D8311yescharacteristic
131C2×D8—834no—
132C2×D8—1662no—
133C2×D8—1662no—
134C22×C8—831no—
135C22×C8—831no—
136C2×C4⋊C4—831no—
137C2×C4⋊C4C2×D8311yescharacteristic
138C2×C22⋊C4—22no—
139C2×C22⋊C4—1662no—
140C2×C42C2×D8311yescharacteristic
141#106—1662no—
142#102—1662no—
143#102—1662no—
144#97—1662no—
145C24—21no—
146C24—1661no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—832no—
153C2×D4—1661no—
154C2×D4—1661no—
155C2×D4—1661no—
156C2×D4—1661no—
157C2×D4—1661no—
158C2×D4C4×D8312yes—
159C22×C4—21no—
160C22×C4—1661no—
161C22×C4C22⋊D8312yes—
162C22×C4C22×D8311yescharacteristic
163D8—1668no—
164C2×C8—832no—
165C2×C8—832no—
166C2×C8—832no—
167C2×C8—832no—
168C2×C8—1661no—
169C2×C8—1661no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—1661no—
173C4⋊C4—1661no—
174C22⋊C4—44no—
175C22⋊C4—3324no—
176C42—21no—
177C42—21no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—42no—
182C23—42no—
183C23—3321no—
184C23—3321no—
185C23—3322no—
186C23—3322no—
187C23—1662no—
188C23C2×C22⋊D8311yescharacteristic
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193D4—3321no—
194D4—3321no—
195D4—1662no—
196D4—1662no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—3321no—
203C2×C4—3322no—
204C2×C4C2×C4×D8311yescharacteristic
205C2×C4D4×D8311yescharacteristic
206C2×C4D4×D8311yescharacteristic
207C2×C4D4×D8311yescharacteristic
208C2×C4#1305058911yescharacteristic
209C2×C4#1305059011yescharacteristic
210C8—834no—
211C8—1662no—
212C22—44no—
213C22—42no—
214C22—81no—
215C22—81no—
216C22—82no—
217C22—3324no—
218C22—3322no—
219C22—6641no—
220C22—6641no—
221C22—6642no—
222C22C4×C22⋊D8311yescharacteristic
223C22#1458176411yescharacteristic
224C22#1458178811yescharacteristic
225C22#1458181712yes—
226C22#1458181812yes—
227C4—42no—
228C4—81no—
229C4—6641no—
230C4C22⋊C4×D8311yescharacteristic
231C4(C22⋊D83)⋊C411yescharacteristic
232C4#1458176511yescharacteristic
233C4#1458177811yescharacteristic
234C2—82no—
235C2—6642no—
236C2#1912760111yescharacteristic
237C2#1912781112yes—
238C2#1912795212yes—
239C2#1912808811yescharacteristic
240C2#1912809111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D83
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#390×C83C2
socleC23×C83C2×D4
prime radicalC22×#13050575C2
cosocleC22×C4×C83C23
hypercenter#390D83
hypocenterC83#1972
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#14581723C2semidirect
3. C2×#14581713C2semidirect
4. C2×#14581712C2semidirect
5. C2×#14581703C2semidirect
6. #19127603C2semidirect
7. #390×C83C2semidirect
11. C2×#13050575C4semidirect
90. C83⋊C8C22⋊C4semidirect
121. C83#1972semidirect
128. #390D83semidirect

Maximal subgroups

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

subgroupindex
C22×#130505752
C2×#145817232
C2×#145817132
C2×#145817122
C2×#145817032
#191276032
#390×C832
#197283