Finite Group Databasebeta

← #24913233

Subgroups — #24913233

Order 10624.

528 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14581790C211yesmaximal, characteristic, complemented
2C2×#14581771C211yesmaximal, characteristic
3C2×#14581751C211yesmaximal, characteristic
4#19127634C214yesmaximal, complemented
5#19127629C214yesmaximal
6#19127608C211yesmaximal, characteristic, complemented
7#19127606C211yesmaximal, characteristic, complemented
8#19127566C211yesmaximal, characteristic, complemented
9C2×C22⋊Q8×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11#14581790—24no—
12#14581771—24no—
13#14581751—24no—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×C4⋊Dic83C2212yes—
16C2×Dic83⋊C4C2214yes—
17C2×Dic83⋊C4C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C4×Dic83C2211yescharacteristic
21C2×C4×Dic83C2211yescharacteristic
22C83⋊#95C2214yes—
23C83⋊#95C2214yes—
24C83⋊#95C2214yes—
25C22×Q8×C83C2211yescharacteristic
26C23×C4×C83C2211yescharacteristic
27C22⋊Q8×C83—24no—
28C2×C4⋊C4×C83C2211yescharacteristic
29C2×C4⋊C4×C83C2212yes—
30C2×C22⋊C4×C83C2212yes—
31C2×Dic166—22no—
32C2×Dic166—22no—
33C2×Dic166—22no—
34C22×Dic83C2312yes—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C4⋊Dic83—22no—
39C4⋊Dic83—24no—
40Dic83⋊C4—24no—
41Dic83⋊C4—28no—
42C22⋊Dic83—24no—
43C22⋊Dic83—24no—
44C4×Dic83—22no—
45C4×Dic83D414yes—
46C24×C83C2311yescharacteristic
47C2×Q8×C83—22no—
48C2×Q8×C83—22no—
49C2×Q8×C83—22no—
50C22×C4×C83—22no—
51C22×C4×C83—22no—
52C22×C4×C83D414yes—
53C22×C4×C83C2312yes—
54C22×C4×C83C2312yes—
55C22×C4×C83C2311yescharacteristic
56C22×C4×C83C2311yescharacteristic
57C4⋊C4×C83—22no—
58C4⋊C4×C83—24no—
59C22⋊C4×C83—24no—
60C23×C83—21no—
61C23×C83—21no—
62C23×C83—22no—
63C23×C83—22no—
64C23×C83C2×D412yes—
65C23×C83C2411yescharacteristic
66Q8×C83—44no—
67C2×C4×C83—21no—
68C2×C4×C83—21no—
69C2×C4×C83—28no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—24no—
75C2×C4×C83C2×D412yes—
76C2×C4×C83C2×D412yes—
77C2×C4×C83C2×D412yes—
78C2×C4×C83#4112yes—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—24no—
89C2×Dic83C2×D414yes—
90Dic166—44no—
91C22×C83—22no—
92C22×C83—22no—
93C22×C83—22no—
94C22×C83—42no—
95C22×C83—42no—
96C22×C83C22×D411yescharacteristic
97C22×C83C22×D411yescharacteristic
98C22×C83C2×#4111yescharacteristic
99C22×C832−1+412yes—
100C22×C832−1+412yes—
101C4×C83—22no—
102C4×C83—42no—
103C4×C83—42no—
104C4×C83#12114yes—
105Dic83—24no—
106Dic83—42no—
107Dic83—42no—
108Dic83—42no—
109C2×C83—42no—
110C2×C83C2×#12111yescharacteristic
111C2×C83#53611yescharacteristic
112C2×C83#53612yes—
113C2×C83#53611yescharacteristic
114C2×C83#54112yes—
115#2621—831nomaximal, Sylow
116C83#262111yescharacteristic, complemented, Sylow
117C2×C4⋊Q8—831no—
118C2×C22⋊Q8—831no—
119C2×C22⋊Q8D8311yescharacteristic
120C2×#117—831no—
121#398—834no—
122#397—834no—
123#391—831no—
124#391—831no—
125#382—831no—
126C22×Q8—831no—
127C22×Q8C2×D8311yescharacteristic
128C23×C4C2×D8311yescharacteristic
129C4⋊Q8—1664no—
130C22⋊Q8—24no—
131C22⋊Q8—1664no—
132#117—1664no—
133C2×C4⋊C4—832no—
134C2×C4⋊C4—831no—
135C2×C4⋊C4—832no—
136C2×C4⋊C4—834no—
137C2×C4⋊C4C2×D8311yescharacteristic
138C2×C4⋊C4C2×D8312yes—
139C2×C22⋊C4—832no—
140C2×C22⋊C4—832no—
141C2×C22⋊C4C2×D8312yes—
142C2×C42—831no—
143C2×C42—831no—
144#95—834no—
145#95—834no—
146#95—834no—
147C24C22×D8311yescharacteristic
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—1662no—
152C2×Q8—1662no—
153C2×Q8—1662no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—832no—
157C22×C4—832no—
158C22×C4—832no—
159C22×C4—832no—
160C22×C4C22⋊D8314yes—
161C22×C4C22×D8312yes—
162C22×C4C22×D8312yes—
163C22×C4C22×D8311yescharacteristic
164C22×C4C22×D8311yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—24no—
167C4⋊C4—1664no—
168C4⋊C4—1668no—
169C4⋊C4—1664no—
170C4⋊C4—1662no—
171C22⋊C4—24no—
172C22⋊C4—1664no—
173C22⋊C4—1664no—
174C42—834no—
175C42—1662no—
176C23—21no—
177C23—21no—
178C23—22no—
179C23—22no—
180C23C2×C22⋊D8312yes—
181C23C23×D8311yescharacteristic
182Q8—44no—
183Q8—3324no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—28no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—834no—
193C2×C4—1662no—
194C2×C4—1662no—
195C2×C4—1662no—
196C2×C4—1662no—
197C2×C4—1662no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4—1664no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4D4×D8312yes—
204C2×C4C2×C22⋊D8312yes—
205C2×C4C2×C22⋊D8312yes—
206C2×C4#1305058912yes—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22C2×D4×D8311yescharacteristic
213C22C22×C22⋊D8311yescharacteristic
214C22C2×#1305058911yescharacteristic
215C22#1458183912yes—
216C22#1458184012yes—
217C4—22no—
218C4—42no—
219C4—42no—
220C4—3322no—
221C4—3322no—
222C4—3322no—
223C4—1664no—
224C4#1458176414yes—
225C2—42no—
226C2C2×#1458176411yescharacteristic
227C2#1912827211yescharacteristic
228C2#1912827412yes—
229C2#1912827511yescharacteristic
230C2#1912831112yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C22⋊Q8×C83C2
prime radicalC24×C83C23
hypercenterC2×C22⋊Q8D83
hypocenterC83#2621
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14581790C2semidirect
4. #19127634C2semidirect
6. #19127608C2semidirect
7. #19127606C2semidirect
8. #19127566C2semidirect
116. C83#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#145817902
C2×#145817712
C2×#145817512
#191276342
#191276292
#191276082
#191276062
#191275662
C2×C22⋊Q8×C832
#262183