Finite Group Databasebeta

← #24913224

Subgroups — #24913224

Order 10624.

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

normal subgroups only

This group has 612 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×#14581779C211yesmaximal, characteristic, complemented
2C2×#14581771C211yesmaximal, characteristic, complemented
3C2×#14581751C211yesmaximal, characteristic, complemented
4#19127624C214yesmaximal, complemented
5#19127615C214yesmaximal, complemented
6#19127606C211yesmaximal, characteristic, complemented
7#19127604C211yesmaximal, characteristic, complemented
8#19127570C211yesmaximal, characteristic, complemented
9C2×#121×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11#14581779—24no—
12#14581771—24no—
13#14581751—24no—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×C4⋊Dic83C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2214yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2212yes—
20C2×C22⋊Dic83C2214yes—
21C2×C22⋊Dic83C2214yes—
22C2×C4×Dic83C2211yescharacteristic
23C2×C4×Dic83C2211yescharacteristic
24C83⋊#95C2214yes—
25C22×D4×C83C2211yescharacteristic
26C22×D4×C83C2212yes—
27C23×C4×C83C2211yescharacteristic
28#121×C83—24no—
29C2×C4⋊C4×C83C2211yescharacteristic
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—44no—
42C22⋊Dic83—24no—
43C22⋊Dic83—28no—
44C22⋊Dic83—24no—
45C22⋊Dic83—44no—
46C4×Dic83—22no—
47C4×Dic83D414yescomplemented
48C24×C83C2312yes—
49C24×C83C2311yescharacteristic
50C2×D4×C83—24no—
51C2×D4×C83—22no—
52C2×D4×C83—22no—
53C2×D4×C83—22no—
54C2×D4×C83—44no—
55C22×C4×C83—22no—
56C22×C4×C83—22no—
57C22×C4×C83D414yes—
58C22×C4×C83C2312yes—
59C22×C4×C83C2311yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C4⋊C4×C83—22no—
62C22⋊C4×C83—24no—
63C23×C83—21no—
64C23×C83—21no—
65C23×C83—22no—
66C23×C83—22no—
67C23×C83—22no—
68C23×C83—22no—
69C23×C83—22no—
70C23×C83—24no—
71C23×C83—24no—
72C23×C83C2×D412yes—
73C23×C83C2411yescharacteristic
74D4×C83—48no—
75D4×C83—44no—
76C2×C4×C83—21no—
77C2×C4×C83—21no—
78C2×C4×C83—28no—
79C2×C4×C83—22no—
80C2×C4×C83—24no—
81C2×C4×C83C2×D412yes—
82C2×C4×C83C2×D412yes—
83C2×C4×C83C2×D412yes—
84C2×C4×C83#4112yes—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—24no—
94C2×Dic83—22no—
95C2×Dic83C2×D414yes—
96Dic166—44no—
97C22×C83—22no—
98C22×C83—22no—
99C22×C83—22no—
100C22×C83—42no—
101C22×C83—42no—
102C22×C83—42no—
103C22×C83—42no—
104C22×C83—42no—
105C22×C83—44no—
106C22×C83—44no—
107C22×C83C22×D411yescharacteristic
108C22×C83C22×D411yescharacteristic
109C22×C83C2×#4111yescharacteristic
110C22×C83H(5,𝔽2)12yes—
111C22×C832−1+412yes—
112C4×C83—22no—
113C4×C83—42no—
114C4×C83#12114yes—
115Dic83—24no—
116Dic83—42no—
117Dic83—42no—
118Dic83—42no—
119C2×C83—42no—
120C2×C83—82no—
121C2×C83C2×#12111yescharacteristic
122C2×C83#53511yescharacteristic
123C2×C83#53611yescharacteristic
124C2×C83#53712yes—
125C2×C83#54212yes—
126#2620—831nomaximal, Sylow
127C83#262011yescharacteristic, complemented, Sylow
128C2×#124—831no—
129C2×C22⋊Q8—831no—
130C2×#121D8311yescharacteristic
131C2×#117—831no—
132#396—834no—
133#394—834no—
134#391—831no—
135#390—831no—
136#384—831no—
137C22×Q8—831no—
138C22×D4C2×D8311yescharacteristic
139C22×D4C2×D8312yes—
140C23×C4C2×D8311yescharacteristic
141#124—1664no—
142C22⋊Q8—1664no—
143#121—24no—
144#117—1664no—
145C2×C4⋊C4—834no—
146C2×C4⋊C4—832no—
147C2×C4⋊C4—831no—
148C2×C4⋊C4—832no—
149C2×C4⋊C4C2×D8311yescharacteristic
150C2×C22⋊C4—834no—
151C2×C22⋊C4—832no—
152C2×C22⋊C4—832no—
153C2×C22⋊C4—834no—
154C2×C22⋊C4C2×D8312yes—
155C2×C42—831no—
156C2×C42—831no—
157#95—834no—
158C24C22×D8312yes—
159C24C22×D8311yescharacteristic
160C2×Q8—1662no—
161C2×Q8—1662no—
162C2×Q8—1662no—
163C2×D4—24no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—44no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—832no—
171C22×C4—832no—
172C22×C4—832no—
173C22×C4—832no—
174C22×C4C22⋊D8314yes—
175C22×C4C22×D8312yes—
176C22×C4C22×D8311yescharacteristic
177C22×C4C22×D8311yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—3324no—
180C4⋊C4—1664no—
181C4⋊C4—1664no—
182C4⋊C4—1662no—
183C22⋊C4—24no—
184C22⋊C4—3324no—
185C22⋊C4—1664no—
186C22⋊C4—1668no—
187C22⋊C4—1664no—
188C42—834no—
189C42—1662no—
190C23—21no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—24no—
198C23—24no—
199C23C2×C22⋊D8312yes—
200C23C23×D8311yescharacteristic
201Q8—3324no—
202D4—48no—
203D4—44no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—28no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—834no—
210C2×C4—1662no—
211C2×C4—1662no—
212C2×C4—1662no—
213C2×C4—1662no—
214C2×C4—1662no—
215C2×C4—1662no—
216C2×C4—1664no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4—1662no—
220C2×C4D4×D8312yes—
221C2×C4C2×C22⋊D8312yes—
222C2×C4C2×C22⋊D8312yes—
223C2×C4#1305058912yes—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22—44no—
233C22—44no—
234C22C2×D4×D8311yescharacteristic
235C22C22×C22⋊D8311yescharacteristic
236C22C2×#1305058911yescharacteristic
237C22#1458183712yes—
238C22#1458183912yes—
239C4—22no—
240C4—42no—
241C4—3322no—
242C4—3322no—
243C4—3322no—
244C4—1664no—
245C4#1458176414yes—
246C2—42no—
247C2—82no—
248C2C2×#1458176411yescharacteristic
249C2#1912826811yescharacteristic
250C2#1912827211yescharacteristic
251C2#1912828012yes—
252C2#1912831812yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting, prime radicalC2×#121×C83C2
hypercenterC2×#121D83
hypocenterC83#2620
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#14581779C2semidirect
2. C2×#14581771C2semidirect
3. C2×#14581751C2semidirect
4. #19127624C2semidirect
5. #19127615C2semidirect
6. #19127606C2semidirect
7. #19127604C2semidirect
8. #19127570C2semidirect
47. C4×Dic83D4semidirect
127. C83#2620semidirect

Maximal subgroups

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

subgroupindex
C2×#145817792
C2×#145817712
C2×#145817512
#191276242
#191276152
#191276062
#191276042
#191275702
C2×#121×C832
#262083