Finite Group Databasebeta

← #24920514

Subgroups — #24920514

Order 10624.

362 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128436C211yesmaximal, characteristic
2#19128429C211yesmaximal, characteristic, complemented
3#19128094C212yesmaximal, complemented
4#19128074C211yesmaximal, characteristic
5#19127926C212yesmaximal, complemented
6#19127911C211yesmaximal, characteristic, complemented
7#19127905C212yesmaximal, complemented
8#19127901C211yesmaximal, characteristic
9#19127831C211yesmaximal, characteristic, complemented
10#19127800C212yesmaximal, complemented
11C4×Q16×C83C211yesmaximal, characteristic, complemented
12C2×#13050580C2212yes—
13C2×#13050579C2212yes—
14C2×#13050579C2212yes—
15C2×#13050577C2212yes—
16#14581792C2211yescharacteristic
17#14581790C2212yes—
18#14581789C2211yescharacteristic
19#14581784—21no—
20#14581782—22no—
21#14581781C2212yes—
22#14581780—21no—
23#14581777—22no—
24C4×Dic166—21no—
25C4×Dic166C2211yescharacteristic
26#14581757C2212yes—
27C4×D332—21no—
28#14581754C2212yes—
29C4⋊(C83⋊C8)C2211yescharacteristic
30Dic83⋊C8C2212yes—
31#14581715C2212yes—
32#14581712C2212yes—
33#14581702C2212yes—
34C2×Q16×C83C2211yescharacteristic
35C4×Q8×C83C2211yescharacteristic
36C4×Q8×C83C2211yescharacteristic
37#107×C83C2211yescharacteristic
38#103×C83C2211yescharacteristic
39#103×C83C2211yescharacteristic
40C4×C8×C83C2211yescharacteristic
41C2×Dic166D411yescharacteristic
42C2×Dic166C2312yes—
43C2×D332—21no—
44C2×D332—21no—
45C2×D332C2312yes—
46C2×C4×D83—22no—
47#13050580—42no—
48#13050579—24no—
49#13050579—42no—
50#13050577—42no—
51C2×C83⋊C8C2312yes—
52C4⋊Dic83—21no—
53C4⋊Dic83—21no—
54C4⋊Dic83D411yescharacteristic
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4D412yes—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—22no—
62C4×Dic83—22no—
63C4×Dic83C2312yes—
64C2×Q8×C83C2311yescharacteristic
65C2×Q8×C83C2311yescharacteristic
66Q16×C83—22no—
67C2×C8×C83C2311yescharacteristic
68C2×C8×C83C2311yescharacteristic
69C4⋊C4×C83—21no—
70C4⋊C4×C83—21no—
71C4⋊C4×C83C2311yescharacteristic
72C4⋊C4×C83C2311yescharacteristic
73C42×C83—21no—
74C42×C83—21no—
75C42×C83C2311yescharacteristic
76C22×D83—22no—
77Q8×C83—21no—
78Q8×C83—21no—
79Q8×C83—21no—
80Q8×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—21no—
84C2×C4×C83—21no—
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2411yescharacteristic
88C2×Dic83—22no—
89C2×Dic83C2×D412yes—
90C2×Dic83C2×D412yes—
91D332—22no—
92D332—22no—
93D332—42no—
94D332—81no—
95C4×D83—82no—
96Dic166—22no—
97Dic166—22no—
98Dic166—22no—
99C8×C83—21no—
100C8×C83#4112yes—
101C83⋊C8—22no—
102C22×C83C22×D411yescharacteristic
103C2×D83—42no—
104C2×D83—42no—
105C2×D83—42no—
106C4×C83—21no—
107C4×C83—41no—
108C4×C83—41no—
109C4×C83—41no—
110C4×C83—41no—
111C4×C83#13712yes—
112C4×C83C2×#4111yescharacteristic
113C4×C83H(5,𝔽2)11yescharacteristic
114Dic83—22no—
115Dic83—22no—
116Dic83—42no—
117C2×C83#55011yescharacteristic
118C2×C83C2×#13711yescharacteristic
119C2×C83#57711yescharacteristic
120D83—82no—
121#3400—831nomaximal, Sylow
122C83#340011yescharacteristic, complemented, Sylow
123#559—831no—
124#557—831no—
125#497—832no—
126#492—831no—
127#464—832no—
128#462—831no—
129#461—832no—
130#460—831no—
131#446—831no—
132#440—832no—
133C4×Q16D8311yescharacteristic, complemented
134C2×Q16—832no—
135C2×Q16C2×D8311yescharacteristic
136C2×SD16—832no—
137C2×SD16—832no—
138C2×SD16—832no—
139C4⋊Q8—832no—
140C4⋊Q8—831no—
141#127—831no—
142#125—1661no—
143#125—1662no—
144#124—832no—
145#124—1661no—
146#123—1662no—
147C4×Q8—831no—
148C4×Q8—832no—
149C4×Q8—1661no—
150C4×Q8C2×D8311yescharacteristic
151C4×Q8C2×D8311yescharacteristic
152C4×D4—832no—
153C4×D4—1661no—
154#107C2×D8311yescharacteristic
155C4⋊C8—832no—
156C4⋊C8—831no—
157#103—832no—
158#103C2×D8311yescharacteristic
159#103C2×D8311yescharacteristic
160#102—832no—
161#97—832no—
162C4×C8C2×D8311yescharacteristic
163C2×Q8—831no—
164C2×Q8—832no—
165C2×Q8C22×D8311yescharacteristic
166C2×Q8C22×D8311yescharacteristic
167C2×D4—832no—
168C2×D4—1661no—
169C2×D4—1661no—
170C22×C4—1662no—
171Q16—22no—
172Q16—3322no—
173SD16—3322no—
174SD16—3322no—
175SD16—1664no—
176C2×C8—832no—
177C2×C8C22×D8311yescharacteristic
178C2×C8C22×D8311yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—831no—
182C4⋊C4—832no—
183C4⋊C4—1662no—
184C4⋊C4—1662no—
185C4⋊C4—1662no—
186C4⋊C4—1661no—
187C4⋊C4—1661no—
188C4⋊C4C22×D8311yescharacteristic
189C4⋊C4C22×D8311yescharacteristic
190C22⋊C4—1662no—
191C22⋊C4—1662no—
192C22⋊C4—1662no—
193C42—21no—
194C42—21no—
195C42—832no—
196C42—1662no—
197C42C22×D8311yescharacteristic
198C23—1662no—
199Q8—21no—
200Q8—21no—
201Q8—21no—
202Q8—21no—
203Q8—1662no—
204Q8—1662no—
205Q8—1662no—
206D4—3322no—
207D4—6641no—
208D4—1662no—
209D4—1662no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—832no—
215C2×C4—832no—
216C2×C4—6642no—
217C2×C4—1662no—
218C2×C4D4×D8311yescharacteristic
219C2×C4D4×D8311yescharacteristic
220C2×C4C23×D8311yescharacteristic
221C8—21no—
222C8—1662no—
223C8#1305059012yes—
224C22—3322no—
225C22—3322no—
226C22—3322no—
227C22C2×D4×D8311yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—3322no—
234C4—1662no—
235C4—1662no—
236C4#1458182412yes—
237C4C2×#1305059011yescharacteristic
238C4#1458183811yescharacteristic
239C2—6642no—
240C2#1912837211yescharacteristic
241C2C2×#1458182411yescharacteristic
242C2#1912850811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC4×Q16×C83C2
socleC22×C83C22×D4
prime radical#14581789C22
hypercenterC4×Q16D83
hypocenterC83#3400
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
2. #19128429C2semidirect
3. #19128094C2semidirect
5. #19127926C2semidirect
6. #19127911C2semidirect
7. #19127905C2semidirect
9. #19127831C2semidirect
10. #19127800C2semidirect
11. C4×Q16×C83C2semidirect
122. C83#3400semidirect
133. C4×Q16D83semidirect

Maximal subgroups

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

subgroupindex
#191284362
#191284292
#191280942
#191280742
#191279262
#191279112
#191279052
#191279012
#191278312
#191278002
C4×Q16×C832
#340083