Finite Group Databasebeta

← #24912372

Subgroups — #24912372

Order 10624.

604 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14581771C212yesmaximal, complemented
2C2×C4×Dic166C211yesmaximal, characteristic, complemented
3#19127632C212yesmaximal, complemented
4#19127615C212yesmaximal
5#19127589C214yesmaximal, complemented
6#19127567C211yesmaximal, characteristic, complemented
7#19127561C212yesmaximal, complemented
8C2×C4×D4×C83C211yesmaximal, characteristic
9C22×Dic166C2211yescharacteristic
10#14581771—28no—
11C4×Dic166—24no—
12C2×C4⋊Dic83C2211yescharacteristic
13C2×C4⋊Dic83C2212yes—
14C2×Dic83⋊C4C2214yes—
15C2×Dic83⋊C4C2212yes—
16C2×C22⋊Dic83C2214yes—
17C2×C22⋊Dic83C2214yes—
18C2×C4×Dic83C2212yes—
19C83⋊#95C2214yes—
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C22×D4×C83C2211yescharacteristic
23C23×C4×C83C2212yes—
24C4×D4×C83—24no—
25C2×C4⋊C4×C83C2211yescharacteristic
26C2×C22⋊C4×C83C2212yes—
27C2×C42×C83C2211yescharacteristic
28C2×Dic166—22no—
29C2×Dic166—22no—
30C2×Dic166D414yescomplemented
31C22×Dic83C2314yes—
32C22×Dic83C2312yes—
33C22×Dic83C2312yes—
34C4⋊Dic83—24no—
35C4⋊Dic83—22no—
36Dic83⋊C4—24no—
37Dic83⋊C4—28no—
38C22⋊Dic83—28no—
39C22⋊Dic83—28no—
40C4×Dic83—24no—
41C24×C83C2312yes—
42C2×D4×C83—24no—
43C2×D4×C83—22no—
44C22×C4×C83—24no—
45C22×C4×C83—24no—
46C22×C4×C83—24no—
47C22×C4×C83C2312yes—
48C22×C4×C83C2311yescharacteristic
49C22×C4×C83C2311yescharacteristic
50C22×C4×C83C2311yescharacteristic
51C4⋊C4×C83—22no—
52C22⋊C4×C83—24no—
53C42×C83D414yes—
54C23×C83—22no—
55C23×C83—22no—
56C23×C83—22no—
57C23×C83—22no—
58C23×C83—22no—
59C23×C83—22no—
60C23×C83#4114yes—
61C23×C83C2411yescharacteristic
62D4×C83—28no—
63C2×C4×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83—22no—
67C2×C4×C83—22no—
68C2×C4×C83—48no—
69C2×C4×C83C2×D412yes—
70C2×C4×C83C2×D412yes—
71C2×C4×C83C2×D412yes—
72C2×C4×C83C2×D412yes—
73C2×C4×C83#4112yes—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—24no—
78C2×Dic83—24no—
79C2×Dic83—24no—
80C2×Dic83—24no—
81C2×Dic83C2×D414yes—
82Dic166—28no—
83C22×C83—24no—
84C22×C83—24no—
85C22×C83—24no—
86C22×C83—42no—
87C22×C83—42no—
88C22×C83—42no—
89C22×C83—42no—
90C22×C83C22×D411yescharacteristic
91C22×C83C22×D411yescharacteristic
92C22×C83C2×#4111yescharacteristic
93C22×C83C2×#4112yes—
94C22×C832−1+412yes—
95C4×C83—22no—
96C4×C83—22no—
97C4×C83—42no—
98C4×C83#12114yes—
99Dic83—24no—
100Dic83—42no—
101Dic83—44no—
102C2×C83—44no—
103C2×C83C2×#12111yescharacteristic
104C2×C83#52511yescharacteristic
105C2×C83#53611yescharacteristic
106C2×C83#54812yes—
107C2×C83#55212yes—
108#2536—831nomaximal, Sylow
109C83#253611yescharacteristic, complemented, Sylow
110C2×C22⋊Q8—832no—
111C2×C4×Q8—831no—
112C2×C4×D4D8311yescharacteristic
113#398—832no—
114#394—832no—
115#388—834no—
116#382—831no—
117#380—832no—
118C22×Q8—831no—
119C22×D4C2×D8311yescharacteristic
120C23×C4C2×D8312yes—
121C22⋊Q8—1668no—
122C4×Q8—1664no—
123C4×D4—24no—
124C2×C4⋊C4—831no—
125C2×C4⋊C4—832no—
126C2×C4⋊C4—832no—
127C2×C4⋊C4—834no—
128C2×C4⋊C4C2×D8311yescharacteristic
129C2×C22⋊C4—834no—
130C2×C22⋊C4—834no—
131C2×C22⋊C4C2×D8312yes—
132C2×C42—832no—
133C2×C42C2×D8311yescharacteristic
134#95—834no—
135#95—832no—
136#95—832no—
137C24C22×D8312yes—
138C2×Q8—834no—
139C2×Q8—1662no—
140C2×Q8—1662no—
141C2×D4—24no—
142C2×D4—22no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—832no—
147C22×C4—834no—
148C22×C4—832no—
149C22×C4C22×D8312yes—
150C22×C4C22×D8311yescharacteristic
151C22×C4C22×D8311yescharacteristic
152C22×C4C22×D8311yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—1668no—
155C4⋊C4—1664no—
156C4⋊C4—1664no—
157C4⋊C4—1662no—
158C22⋊C4—24no—
159C22⋊C4—1668no—
160C22⋊C4—1668no—
161C42—1664no—
162C42C22⋊D8314yes—
163C23—22no—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23#1305059014yes—
170C23C23×D8311yescharacteristic
171Q8—1668no—
172D4—28no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—48no—
179C2×C4—834no—
180C2×C4—1664no—
181C2×C4—1662no—
182C2×C4—1664no—
183C2×C4—1664no—
184C2×C4—1664no—
185C2×C4—1662no—
186C2×C4—1662no—
187C2×C4D4×D8312yes—
188C2×C4C2×C22⋊D8312yes—
189C2×C4C2×C22⋊D8312yes—
190C2×C4C2×C22⋊D8312yes—
191C2×C4#1305058912yes—
192C22—24no—
193C22—24no—
194C22—24no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22C2×D4×D8311yescharacteristic
200C22C22×C22⋊D8311yescharacteristic
201C22C2×#1305058911yescharacteristic
202C22C2×#1305059012yes—
203C22#1458183912yes—
204C4—22no—
205C4—22no—
206C4—42no—
207C4—3324no—
208C4—3322no—
209C4—1664no—
210C4#1458176314yes—
211C2—44no—
212C2C2×#1458176311yescharacteristic
213C2#1912821211yescharacteristic
214C2#1912827211yescharacteristic
215C2#1912836012yes—
216C2#1912839112yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C4×D4×C83C2
prime radicalC22×D4×C83C22
hypercenterC2×C4×D4D83
hypocenterC83#2536
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#14581771C2semidirect
2. C2×C4×Dic166C2semidirect
3. #19127632C2semidirect
5. #19127589C2semidirect
6. #19127567C2semidirect
7. #19127561C2semidirect
30. C2×Dic166D4semidirect
109. C83#2536semidirect

Maximal subgroups

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

subgroupindex
C2×#145817712
C2×C4×Dic1662
#191276322
#191276152
#191275892
#191275672
#191275612
C2×C4×D4×C832
#253683