Finite Group Databasebeta

← #24914432

Subgroups — #24914432

Order 10624.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#14581768C212yesmaximal, complemented
2#19127641C212yesmaximal, complemented
3#19127631C211yesmaximal, characteristic, complemented
4#19127628C212yesmaximal, complemented
5#19127618C211yesmaximal, characteristic
6#19127592C212yesmaximal, complemented
7#19127579C212yesmaximal
8#19127575C212yesmaximal, complemented
9#391×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C22×C4×D83C2212yes—
12#14581768—28no—
13C2×C4⋊Dic83C2212yes—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×Dic83⋊C4C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2212yes—
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2211yescharacteristic
26C83⋊#95C2211yescharacteristic
27C22×Q8×C83C2211yescharacteristic
28C2×C4⋊C4×C83C2211yescharacteristic
29C2×C42×C83C2211yescharacteristic
30#95×C83C2212yes—
31#95×C83C2212yes—
32C23×D83C2312yes—
33C2×D332—24no—
34C2×D332—41no—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C2×C4×D83—24no—
39C2×C4×D83—42no—
40C2×C4×D83—42no—
41C4⋊Dic83—24no—
42C4⋊Dic83—41no—
43Dic83⋊C4—24no—
44Dic83⋊C4—24no—
45(C2×D83)⋊C4—24no—
46(C2×D83)⋊C4—24no—
47(C2×D83)⋊C4—24no—
48(C2×D83)⋊C4—24no—
49(C2×D83)⋊C4—42no—
50C2×Q8×C83—41no—
51C2×Q8×C83—41no—
52C2×Q8×C83D414yescomplemented
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2312yes—
55C22×C4×C83C2311yescharacteristic
56C22×C4×C83C2312yes—
57C22×C4×C83C2311yescharacteristic
58C4⋊C4×C83—41no—
59C42×C83—41no—
60C23×C83C2411yescharacteristic
61C22×D83—22no—
62C22×D83—22no—
63C22×D83—22no—
64C22×D83—22no—
65C22×D83—22no—
66C22×D83—22no—
67C22×D83#4114yes—
68Q8×C83—44no—
69C2×C4×C83—21no—
70C2×C4×C83—21no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—21no—
78C2×C4×C83—21no—
79C2×C4×C83—22no—
80C2×C4×C83C2×D412yes—
81C2×C4×C83C2×D414yes—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91D332—44no—
92C4×D83—48no—
93C22×C83C22×D411yescharacteristic
94C22×C83C2×#4112yes—
95C22×C83H(5,𝔽2)11yescharacteristic
96C22×C83H(5,𝔽2)11yescharacteristic
97C22×C832−1+412yes—
98C2×D83—24no—
99C2×D83—24no—
100C2×D83—24no—
101C2×D83—42no—
102C2×D83—42no—
103C2×D83—42no—
104C2×D83—42no—
105C4×C83—24no—
106C4×C83—22no—
107C4×C83—42no—
108C4×C83—41no—
109C4×C83—41no—
110Dic83—42no—
111Dic83—42no—
112Dic83—42no—
113C2×C83#53411yescharacteristic
114C2×C83#53912yes—
115C2×C83#54812yes—
116C2×C83#55111yescharacteristic
117C2×C83#56311yescharacteristic
118D83—44no—
119#2728—831nomaximal, Sylow
120C83#272811yescharacteristic, complemented, Sylow
121C2×C22⋊Q8—832no—
122#400—832no—
123#398—831no—
124#397—832no—
125#394—831no—
126#391D8311yescharacteristic, complemented
127#388—832no—
128#386—832no—
129#385—832no—
130C22×Q8C2×D8311yescharacteristic
131C22×D4—831no—
132C23×C4—832no—
133C22⋊Q8—1668no—
134C2×C4⋊C4—831no—
135C2×C4⋊C4—832no—
136C2×C4⋊C4—832no—
137C2×C4⋊C4—832no—
138C2×C4⋊C4C2×D8311yescharacteristic
139C2×C22⋊C4—832no—
140C2×C22⋊C4—832no—
141C2×C22⋊C4—832no—
142C2×C22⋊C4—832no—
143C2×C22⋊C4—832no—
144C2×C42C2×D8311yescharacteristic
145#95—832no—
146#95—832no—
147#95—832no—
148#95—831no—
149#95—831no—
150#95C2×D8312yes—
151#95C2×D8312yes—
152C24—832no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8D33214yescomplemented
156C2×D4—3321no—
157C2×D4—1664no—
158C22×C4—3322no—
159C22×C4—3322no—
160C22×C4—832no—
161C22×C4—832no—
162C22×C4—832no—
163C22×C4—1664no—
164C22×C4C22×D8311yescharacteristic
165C22×C4C22×D8312yes—
166C22×C4C22×D8311yescharacteristic
167C22×C4C22×D8312yes—
168C22×C4C22×D8311yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—3321no—
171C4⋊C4—1664no—
172C4⋊C4—1664no—
173C4⋊C4—1664no—
174C22⋊C4—3322no—
175C22⋊C4—1664no—
176C22⋊C4—1664no—
177C22⋊C4—1664no—
178C22⋊C4—1664no—
179C42—41no—
180C23—834no—
181C23—1662no—
182C23—1662no—
183C23—1662no—
184C23—1662no—
185C23—1662no—
186C23—1662no—
187C23C23×D8311yescharacteristic
188Q8—44no—
189D4—3324no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—3328no—
202C2×C4—1662no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4—1662no—
206C2×C4—1662no—
207C2×C4—1662no—
208C2×C4—1662no—
209C2×C4—1662no—
210C2×C4—1662no—
211C2×C4C2×D33212yes—
212C2×C4C2×D33214yes—
213C22—3322no—
214C22—3322no—
215C22—3322no—
216C22—3322no—
217C22—1664no—
218C22—1664no—
219C22—1664no—
220C22C22×D33211yescharacteristic
221C22#41×D8312yes—
222C22#1458183711yescharacteristic
223C22#1458183711yescharacteristic
224C22#1458184012yes—
225C4—24no—
226C4—22no—
227C4—41no—
228C4—41no—
229C4—42no—
230C4—3322no—
231C4—3322no—
232C4—3322no—
233C2—3324no—
234C2#1912826311yescharacteristic
235C2#1912829712yes—
236C2#1912836312yes—
237C2#1912838111yescharacteristic
238C2#1912845011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#391×C83C2
prime radicalC22×D332C22
hypercenter#391D83
hypocenterC83#2728
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#14581768C2semidirect
2. #19127641C2semidirect
3. #19127631C2semidirect
4. #19127628C2semidirect
6. #19127592C2semidirect
8. #19127575C2semidirect
9. #391×C83C2semidirect
52. C2×Q8×C83D4semidirect
120. C83#2728semidirect
126. #391D83semidirect
155. C2×Q8D332semidirect

Maximal subgroups

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

subgroupindex
C2×#145817682
#191276412
#191276312
#191276282
#191276182
#191275922
#191275792
#191275752
#391×C832
#272883