Finite Group Databasebeta

← #24920498

Subgroups — #24920498

Order 10624.

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

normal subgroups only

This group has 350 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#19128426C211yesmaximal, characteristic, complemented
2#19128421C211yesmaximal, characteristic
3#19128092C212yesmaximal
4#19128080C211yesmaximal, characteristic, complemented
5#19127921C212yesmaximal
6#19127912C211yesmaximal, characteristic, complemented
7#19127910C212yesmaximal, complemented
8#19127828C211yesmaximal, characteristic, complemented
9#19127805C212yesmaximal, complemented
10C4×#13050578C211yesmaximal, characteristic
11#461×C83C211yesmaximal, characteristic
12C2×Dic332C2212yes—
13C2×#13050580C2212yes—
14C2×#13050580C2212yes—
15C2×#13050578C2211yescharacteristic
16#14581792C2211yescharacteristic
17#14581790C2212yes—
18#14581785—22no—
19#14581783—21no—
20#14581782—22no—
21#14581779C2212yes—
22#14581771—21no—
23Q8×Dic83C2212yes—
24C4×Dic166C2211yescharacteristic
25C4×Dic166C2211yescharacteristic
26#14581757—21no—
27#14581757C2212yes—
28#14581751—21no—
29#14581723C2211yescharacteristic
30Dic83⋊C8C2212yes—
31#14581717C2212yes—
32#14581715C2211yescharacteristic
33#14581713C2211yescharacteristic
34#14581702C2212yes—
35C4×C83⋊C8C2211yescharacteristic
36C2×SD16×C83C2212yes—
37C4⋊Q8×C83C2211yescharacteristic
38C4×D4×C83C2211yescharacteristic
39C4⋊C8×C83C2211yescharacteristic
40#103×C83C2212yes—
41C2×Dic166D411yescharacteristic
42C2×Dic166C2312yes—
43C2×Dic166C2311yescharacteristic
44Dic332—42no—
45#13050580—24no—
46#13050580—42no—
47#13050578—22no—
48C2×C83⋊C8C2311yescharacteristic
49C2×C83⋊C8C2311yescharacteristic
50C4⋊Dic83—21no—
51C4⋊Dic83—21no—
52C4⋊Dic83—22no—
53C4⋊Dic83D411yescharacteristic
54C4⋊Dic83C2311yescharacteristic
55Dic83⋊C4—21no—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4—22no—
59Dic83⋊C4D412yes—
60C22⋊Dic83—22no—
61C22⋊Dic83—22no—
62C4×Dic83—21no—
63C4×Dic83—22no—
64C4×Dic83—22no—
65C4×Dic83C2312yes—
66C2×Q8×C83C2312yes—
67C2×D4×C83C2311yescharacteristic
68C22×C4×C83—21no—
69SD16×C83—42no—
70C2×C8×C83C2312yes—
71C4⋊C4×C83—21no—
72C4⋊C4×C83—21no—
73C4⋊C4×C83C2311yescharacteristic
74C22⋊C4×C83—21no—
75C42×C83C2311yescharacteristic
76C23×C83—21no—
77Q8×C83—22no—
78Q8×C83—22no—
79D4×C83—21no—
80D4×C83—21no—
81C2×C4×C83—21no—
82C2×C4×C83—22no—
83C2×C4×C83—42no—
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2411yescharacteristic
87C2×Dic83—21no—
88C2×Dic83—22no—
89C2×Dic83—21no—
90C2×Dic83C2×D412yes—
91C2×Dic83C2×D412yes—
92Dic166—22no—
93Dic166—21no—
94Dic166—22no—
95Dic166—22no—
96Dic166—21no—
97C8×C83—22no—
98C83⋊C8—21no—
99C83⋊C8#4112yes—
100C22×C83—41no—
101C22×C83—41no—
102C22×C83—41no—
103C22×C83C22×D411yescharacteristic
104C4×C83—21no—
105C4×C83—42no—
106C4×C83—41no—
107C4×C83#13712yes—
108C4×C83C2×#4111yescharacteristic
109C4×C83H(5,𝔽2)11yescharacteristic
110Dic83—22no—
111Dic83—22no—
112Dic83—41no—
113Dic83—42no—
114Dic83—41no—
115C2×C83—81no—
116C2×C83#55011yescharacteristic
117C2×C83C2×#13711yescharacteristic
118C2×C83#57811yescharacteristic
119#3398—831nomaximal, Sylow
120C83#339811yescharacteristic, complemented, Sylow
121#557—831no—
122#556—831no—
123#497—832no—
124#494—831no—
125#464—832no—
126#462—831no—
127#462—832no—
128#461D8311yescharacteristic
129#446—831no—
130#441—832no—
131C4×SD16—831no—
132C2×Q16—832no—
133C2×Q16—832no—
134C2×Q16—832no—
135C2×SD16—831no—
136C2×SD16C2×D8312yes—
137C4⋊Q8—832no—
138C4⋊Q8—831no—
139C4⋊Q8C2×D8311yescharacteristic
140#126—1662no—
141#125—1662no—
142#125—1661no—
143#124—832no—
144C22⋊Q8—1661no—
145C4×Q8—831no—
146C4×Q8—832no—
147C4×Q8—832no—
148C4×Q8—831no—
149C4×Q8—1661no—
150C4×D4C2×D8311yescharacteristic
151#117—1661no—
152#106—831no—
153C4⋊C8—832no—
154C4⋊C8C2×D8311yescharacteristic
155#103—832no—
156#103—831no—
157#103C2×D8312yes—
158#102—831no—
159#97—832no—
160C4×C8—831no—
161C2×Q8—831no—
162C2×Q8—832no—
163C2×Q8—831no—
164C2×Q8C22×D8312yes—
165C2×D4C22×D8311yescharacteristic
166C22×C4—21no—
167Q16—3322no—
168Q16—3322no—
169Q16—1664no—
170SD16—42no—
171SD16—1662no—
172C2×C8—831no—
173C2×C8—831no—
174C2×C8C22×D8312yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—831no—
178C4⋊C4—832no—
179C4⋊C4—831no—
180C4⋊C4—1662no—
181C4⋊C4—1662no—
182C4⋊C4—1662no—
183C4⋊C4—1662no—
184C4⋊C4—1661no—
185C4⋊C4—1661no—
186C4⋊C4—1661no—
187C4⋊C4C22×D8311yescharacteristic
188C22⋊C4—21no—
189C22⋊C4—1662no—
190C22⋊C4—1662no—
191C42—832no—
192C42—1662no—
193C42—1662no—
194C42—1661no—
195C42C22×D8311yescharacteristic
196C23—21no—
197Q8—22no—
198Q8—22no—
199Q8—1661no—
200Q8—1662no—
201Q8—1661no—
202Q8—1662no—
203Q8—1662no—
204D4—21no—
205D4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—832no—
210C2×C4—832no—
211C2×C4—1661no—
212C2×C4—1662no—
213C2×C4—1661no—
214C2×C4D4×D8311yescharacteristic
215C2×C4D4×D8311yescharacteristic
216C2×C4C23×D8311yescharacteristic
217C8—22no—
218C8—832no—
219C8—1661no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22C2×D4×D8311yescharacteristic
224C4—21no—
225C4—42no—
226C4—41no—
227C4—3322no—
228C4—3321no—
229C4—3321no—
230C4—1662no—
231C4—1662no—
232C4#1458182112yes—
233C4#41×D8311yescharacteristic
234C4#1458183711yescharacteristic
235C2—81no—
236C2#1912836911yescharacteristic
237C2C2×#1458182111yescharacteristic
238C2#1912851711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#461×C83C2
socleC22×C83C22×D4
prime radicalC2×D4×C83C23
hypercenter#461D83
hypocenterC83#3398
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #19128426C2semidirect
4. #19128080C2semidirect
6. #19127912C2semidirect
7. #19127910C2semidirect
8. #19127828C2semidirect
9. #19127805C2semidirect
120. C83#3398semidirect

Maximal subgroups

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

subgroupindex
#191284262
#191284212
#191280922
#191280802
#191279212
#191279122
#191279102
#191278282
#191278052
C4×#130505782
#461×C832
#339883