Finite Group Databasebeta

← #24915166

Subgroups — #24915166

Order 10624.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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#19127640C211yesmaximal, characteristic, complemented
2#19127623C212yesmaximal, complemented
3#19127612C212yesmaximal, complemented
4#19127602C211yesmaximal, characteristic
5#19127588C212yesmaximal, complemented
6#19127579C212yesmaximal
7#19127575C212yesmaximal, complemented
8#19127558C212yesmaximal, complemented
9#395×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C22×C4×D83C2212yes—
12C2×C4⋊Dic83C2212yes—
13C2×C4⋊Dic83C2211yescharacteristic
14C2×Dic83⋊C4C2212yes—
15C2×(C2×D83)⋊C4C2212yes—
16C2×(C2×D83)⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2212yes—
18C2×(C2×D83)⋊C4C2212yes—
19C2×(C2×D83)⋊C4C2212yes—
20C2×C4×Dic83C2211yescharacteristic
21C83⋊#95C2211yescharacteristic
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C22×Q8×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2212yes—
28C2×C4⋊C4×C83C2211yescharacteristic
29#95×C83C2211yescharacteristic
30#95×C83C2212yes—
31C23×D83C2312yes—
32C2×D332—41no—
33C2×D332—41no—
34C2×D332Q814yescomplemented
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C2×C4×D83—24no—
39C2×C4×D83—42no—
40C2×C4×D83—42no—
41C4⋊Dic83—42no—
42C4⋊Dic83—41no—
43Dic83⋊C4—24no—
44(C2×D83)⋊C4—24no—
45(C2×D83)⋊C4—42no—
46(C2×D83)⋊C4—42no—
47(C2×D83)⋊C4—42no—
48(C2×D83)⋊C4—42no—
49C4×Dic83—41no—
50C2×Q8×C83—42no—
51C2×Q8×C83—41no—
52C22×C4×C83C2311yescharacteristic
53C22×C4×C83C2312yes—
54C22×C4×C83C2311yescharacteristic
55C22×C4×C83C2311yescharacteristic
56C22×C4×C83C2312yes—
57C4⋊C4×C83—42no—
58C4⋊C4×C83—41no—
59C23×C83C2411yescharacteristic
60C22×D83—22no—
61C22×D83—22no—
62C22×D83—22no—
63C22×D83—22no—
64C22×D83—22no—
65C22×D83—22no—
66C22×D83C2×Q814yes—
67Q8×C83—44no—
68C2×C4×C83—22no—
69C2×C4×C83—21no—
70C2×C4×C83—21no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—21no—
74C2×C4×C83—22no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83C2×Q812yes—
80C2×C4×C83#4114yes—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90D332—44no—
91C4×D83—48no—
92C22×C83C22×Q811yescharacteristic
93C22×C83C2×#4112yes—
94C22×C83H(5,𝔽2)12yes—
95C22×C83H(5,𝔽2)11yescharacteristic
96C22×C83H(5,𝔽2)11yescharacteristic
97C2×D83—24no—
98C2×D83—24no—
99C2×D83—24no—
100C2×D83—42no—
101C2×D83—42no—
102C2×D83—42no—
103C2×D83—42no—
104C4×C83—24no—
105C4×C83—22no—
106C4×C83—41no—
107C4×C83—42no—
108C4×C83—41no—
109Dic83—42no—
110Dic83—42no—
111Dic83—42no—
112C2×C83#53812yes—
113C2×C83#54311yescharacteristic
114C2×C83#55412yes—
115C2×C83#55511yescharacteristic
116C2×C83#56011yescharacteristic
117D83—44no—
118#2784—831nomaximal, Sylow
119C83#278411yescharacteristic, complemented, Sylow
120#400—831no—
121#396—832no—
122#395D8311yescharacteristic, complemented
123#393—832no—
124#390—831no—
125#387—832no—
126#386—832no—
127#385—832no—
128#380—832no—
129C22×Q8C2×D8311yescharacteristic
130C22×D4—831no—
131C23×C4—832no—
132C2×C4⋊C4—832no—
133C2×C4⋊C4—831no—
134C2×C4⋊C4—832no—
135C2×C4⋊C4C2×D8312yes—
136C2×C4⋊C4C2×D8311yescharacteristic
137C2×C22⋊C4—832no—
138C2×C22⋊C4—832no—
139C2×C22⋊C4—832no—
140C2×C22⋊C4—832no—
141C2×C22⋊C4—832no—
142C2×C42—831no—
143#95—831no—
144#95—832no—
145#95—832no—
146#95—832no—
147#95—832no—
148#95C2×D8311yescharacteristic
149#95C2×D8312yes—
150C24—832no—
151C2×Q8—41no—
152C2×Q8—42no—
153C2×D4—3321no—
154C2×D4—3321no—
155C2×D4—834no—
156C22×C4—3322no—
157C22×C4—3322no—
158C22×C4—832no—
159C22×C4—832no—
160C22×C4—832no—
161C22×C4—1664no—
162C22×C4C22×D8312yes—
163C22×C4C22×D8311yescharacteristic
164C22×C4C22×D8311yescharacteristic
165C22×C4C22×D8311yescharacteristic
166C22×C4C22×D8312yes—
167C4⋊C4—42no—
168C4⋊C4—41no—
169C4⋊C4—3322no—
170C4⋊C4—3321no—
171C4⋊C4—1664no—
172C22⋊C4—3322no—
173C22⋊C4—3322no—
174C22⋊C4—3322no—
175C22⋊C4—3322no—
176C22⋊C4—1664no—
177C42—3321no—
178C23—834no—
179C23—1662no—
180C23—1662no—
181C23—1662no—
182C23—1662no—
183C23—1662no—
184C23—1662no—
185C23C23×D8311yescharacteristic
186Q8—44no—
187D4—3324no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—3328no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4—1662no—
204C2×C4—1662no—
205C2×C4—1662no—
206C2×C4—1662no—
207C2×C4—1662no—
208C2×C4—1662no—
209C2×C4Q8×D8312yes—
210C2×C4#1305059114yes—
211C22—3322no—
212C22—3322no—
213C22—3322no—
214C22—3322no—
215C22—1664no—
216C22—1664no—
217C22—1664no—
218C22C2×Q8×D8311yescharacteristic
219C22C2×#1305059112yes—
220C22#1458183711yescharacteristic
221C22#1458183712yes—
222C22#1458183811yescharacteristic
223C4—22no—
224C4—24no—
225C4—42no—
226C4—41no—
227C4—41no—
228C4—3322no—
229C4—3322no—
230C4—3322no—
231C2—3324no—
232C2#1912829112yes—
233C2#1912832611yescharacteristic
234C2#1912841212yes—
235C2#1912841711yescharacteristic
236C2#1912844011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#395×C83C2
prime radicalC22×D332C22
hypercenter#395D83
hypocenterC83#2784
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. #19127640C2semidirect
2. #19127623C2semidirect
3. #19127612C2semidirect
5. #19127588C2semidirect
7. #19127575C2semidirect
8. #19127558C2semidirect
9. #395×C83C2semidirect
34. C2×D332Q8semidirect
119. C83#2784semidirect
122. #395D83semidirect

Maximal subgroups

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

subgroupindex
#191276402
#191276232
#191276122
#191276022
#191275882
#191275792
#191275752
#191275582
#395×C832
#278483