Finite Group Databasebeta

← #24920180

Subgroups — #24920180

Order 10624.

372 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 115 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#19128421C211yesmaximal, characteristic, complemented
2#19128366C211yesmaximal, characteristic, complemented
3#19128059C212yesmaximal, complemented
4#19128053C211yesmaximal, characteristic, complemented
5#19127995C212yesmaximal, complemented
6#19127915C212yesmaximal
7#19127907C211yesmaximal, characteristic, complemented
8#19127828C211yesmaximal, characteristic, complemented
9#19127800C212yesmaximal, complemented
10C4×#13050575C211yesmaximal, characteristic
11#474×C83C211yesmaximal, characteristic, complemented
12C2×Q8×D83—21no—
13C2×#13050579C2212yes—
14C2×#13050575C2211yescharacteristic
15#14581792C2211yescharacteristic
16#14581791C2212yes—
17#14581785—22no—
18#14581779C2212yes—
19#14581771—21no—
20#14581770—22no—
21#14581768—21no—
22C4×Dic166C2211yescharacteristic
23#14581757C2212yes—
24C4×D332C2211yescharacteristic
25#14581754C2212yes—
26#14581751—21no—
27#14581725C2211yescharacteristic
28Dic83⋊C8C2212yes—
29#14581717C2212yes—
30#14581715C2212yes—
31#14581715C2212yes—
32#14581713C2211yescharacteristic
33#14581712C2211yescharacteristic
34#14581702C2212yes—
35C4×C83⋊C8C2211yescharacteristic
36C4⋊Q8×C83C2211yescharacteristic
37C4×D4×C83C2211yescharacteristic
38#107×C83C2212yes—
39C4⋊C8×C83C2211yescharacteristic
40#102×C83C2212yes—
41C2×Dic166—22no—
42C2×Dic166D411yescharacteristic
43C2×Dic166C2312yes—
44Q8×D83—44no—
45C2×D332C2311yescharacteristic
46C2×C4×D83—21no—
47C2×C4×D83—22no—
48#13050579—24no—
49#13050575—22no—
50C2×C83⋊C8C2311yescharacteristic
51C2×C83⋊C8C2311yescharacteristic
52C4⋊Dic83—21no—
53C4⋊Dic83—21no—
54C4⋊Dic83D411yescharacteristic
55C4⋊Dic83C2311yescharacteristic
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4D412yes—
59C22⋊Dic83—22no—
60C22⋊Dic83—22no—
61(C2×D83)⋊C4—21no—
62(C2×D83)⋊C4—22no—
63C4×Dic83—22no—
64C4×Dic83C2312yes—
65C2×Q8×C83—21no—
66C2×D4×C83C2311yescharacteristic
67C22×C4×C83—21no—
68C2×C8×C83C2312yes—
69C4⋊C4×C83—21no—
70C4⋊C4×C83C2312yes—
71C4⋊C4×C83C2311yescharacteristic
72C22⋊C4×C83—21no—
73C42×C83C2311yescharacteristic
74C23×C83—21no—
75C22×D83—21no—
76Q8×C83—42no—
77D4×C83—21no—
78D4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—22no—
81C2×C4×C83—42no—
82C2×C4×C83C2×D411yescharacteristic
83C2×C4×C83C2×D411yescharacteristic
84C2×C4×C83C2411yescharacteristic
85C2×Dic83—22no—
86C2×Dic83—21no—
87C2×Dic83C2×D412yes—
88C2×Dic83C2×D412yes—
89D332—21no—
90D332#4112yes—
91C4×D83—44no—
92C4×D83—42no—
93Dic166—22no—
94Dic166—22no—
95Dic166—22no—
96Dic166—44no—
97C8×C83—22no—
98C83⋊C8—21no—
99C83⋊C8—21no—
100C22×C83—41no—
101C22×C83—41no—
102C22×C83—41no—
103C22×C83C22×D411yescharacteristic
104C2×D83—22no—
105C2×D83—41no—
106C2×D83—41no—
107C4×C83—21no—
108C4×C83—42no—
109C4×C83—41no—
110C4×C83AGL(1,ℤ/8)12yes—
111C4×C83C2×#4111yescharacteristic
112C4×C832−1+411yescharacteristic
113Dic83—22no—
114Dic83—22no—
115Dic83—42no—
116Dic83—41no—
117C2×C83—81no—
118C2×C83#54811yescharacteristic
119C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C83#57811yescharacteristic
121D83—42no—
122#3367—831nomaximal, Sylow
123C83#336711yescharacteristic, complemented, Sylow
124#556—831no—
125D4×Q8—831no—
126#489—832no—
127#487—831no—
128#477—832no—
129#474D8311yescharacteristic, complemented
130#463—832no—
131#461—831no—
132#446—831no—
133#440—832no—
134C4×D8—831no—
135C22×Q8—1661no—
136C2×SD16—832no—
137C2×D8—831no—
138C4⋊Q8—832no—
139C4⋊Q8—831no—
140C4⋊Q8C2×D8311yescharacteristic
141#126—1662no—
142#124—832no—
143C22⋊Q8—1662no—
144C22⋊Q8—1661no—
145C22⋊Q8—1661no—
146C4×Q8—831no—
147C4×Q8—832no—
148C4×D4—832no—
149C4×D4—831no—
150C4×D4C2×D8311yescharacteristic
151#117—1661no—
152#107—831no—
153#107C2×D8312yes—
154C4⋊C8—832no—
155C4⋊C8C2×D8311yescharacteristic
156#103—832no—
157#103—832no—
158#103—832no—
159#102—831no—
160#102—831no—
161#102C2×D8312yes—
162#97—832no—
163C4×C8—831no—
164C2×Q8—21no—
165C2×Q8—3324no—
166C2×Q8—831no—
167C2×Q8—832no—
168C2×Q8—1662no—
169C2×D4—831no—
170C2×D4C22×D8311yescharacteristic
171C22×C4—21no—
172C22×C4—1662no—
173C22×C4—1661no—
174SD16—1664no—
175D8—1662no—
176C2×C8—831no—
177C2×C8—831no—
178C2×C8C22×D8312yes—
179C4⋊C4—21no—
180C4⋊C4—831no—
181C4⋊C4—832no—
182C4⋊C4—831no—
183C4⋊C4—1662no—
184C4⋊C4—1662no—
185C4⋊C4—1661no—
186C4⋊C4—1661no—
187C4⋊C4C22×D8311yescharacteristic
188C4⋊C4C22×D8312yes—
189C22⋊C4—21no—
190C22⋊C4—1662no—
191C22⋊C4—1662no—
192C22⋊C4—1662no—
193C22⋊C4—1661no—
194C42—832no—
195C42—1662no—
196C42C22×D8311yescharacteristic
197C23—21no—
198C23—1661no—
199Q8—42no—
200Q8—3324no—
201Q8—1662no—
202Q8—1662no—
203Q8—1662no—
204D4—21no—
205D4—21no—
206D4—832no—
207D4—1661no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—3324no—
212C2×C4—3322no—
213C2×C4—832no—
214C2×C4—832no—
215C2×C4—1662no—
216C2×C4—1661no—
217C2×C4D4×D8311yescharacteristic
218C2×C4D4×D8311yescharacteristic
219C2×C4C23×D8311yescharacteristic
220C8—22no—
221C8—1661no—
222C8—1661no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—3321no—
227C22—3321no—
228C22—1662no—
229C22C2×D4×D8311yescharacteristic
230C4—21no—
231C4—41no—
232C4—42no—
233C4—3322no—
234C4—3321no—
235C4—1662no—
236C4—1662no—
237C4#1458181612yes—
238C4#41×D8311yescharacteristic
239C4#1458184011yescharacteristic
240C2—81no—
241C2—3322no—
242C2#1912836111yescharacteristic
243C2C2×#1458181611yescharacteristic
244C2#1912851711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#474×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050575C22
hypercenter#474D83
hypocenterC83#3367
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #19128421C2semidirect
2. #19128366C2semidirect
3. #19128059C2semidirect
4. #19128053C2semidirect
5. #19127995C2semidirect
7. #19127907C2semidirect
8. #19127828C2semidirect
9. #19127800C2semidirect
11. #474×C83C2semidirect
123. C83#3367semidirect
129. #474D83semidirect

Maximal subgroups

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

subgroupindex
#191284212
#191283662
#191280592
#191280532
#191279952
#191279152
#191279072
#191278282
#191278002
C4×#130505752
#474×C832
#336783