Finite Group Databasebeta

← #24913629

Subgroups — #24913629

Order 10624.

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

normal subgroups only

This group has 434 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
1C2×#14581787C211yesmaximal, characteristic
2C2×#14581783C211yesmaximal, characteristic, complemented
3#19127640C211yesmaximal, characteristic, complemented
4#19127638C211yesmaximal, characteristic
5#19127633C212yesmaximal, complemented
6#19127627C211yesmaximal, characteristic
7#19127623C212yesmaximal
8#19127597C212yesmaximal, complemented
9#19127590C212yesmaximal
10#19127552C211yesmaximal, characteristic, complemented
11#398×C83C211yesmaximal, characteristic, complemented
12#14581787—24no—
13#14581783—24no—
14C2×C4⋊Dic83C2212yes—
15C2×C4⋊Dic83C2212yes—
16C2×C4⋊Dic83C2211yescharacteristic
17C2×C4⋊Dic83C2211yescharacteristic
18C2×Dic83⋊C4C2212yes—
19C2×Dic83⋊C4C2212yes—
20C2×Dic83⋊C4C2211yescharacteristic
21C2×(C2×D83)⋊C4C2211yescharacteristic
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2211yescharacteristic
24C2×(C2×D83)⋊C4C2211yescharacteristic
25C2×(C2×D83)⋊C4C2212yes—
26C2×C4×Dic83C2212yes—
27C2×C4×Dic83C2211yescharacteristic
28C83⋊#95C2211yescharacteristic
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C4⋊C4×C83C2212yes—
35#95×C83C2211yescharacteristic
36#95×C83C2212yes—
37C23×D83C2311yescharacteristic
38C22×Dic83C2312yes—
39C22×Dic83C2311yescharacteristic
40C22×Dic83C2312yes—
41C22×Dic83C2311yescharacteristic
42C22×Dic83C2311yescharacteristic
43C4⋊Dic83—22no—
44C4⋊Dic83—24no—
45C4⋊Dic83—24no—
46C4⋊Dic83—22no—
47Dic83⋊C4—24no—
48Dic83⋊C4—24no—
49Dic83⋊C4—22no—
50(C2×D83)⋊C4—22no—
51(C2×D83)⋊C4—24no—
52(C2×D83)⋊C4—22no—
53(C2×D83)⋊C4—22no—
54(C2×D83)⋊C4—42no—
55C4×Dic83—42no—
56C4×Dic83D414yes—
57C22×C4×C83C2311yescharacteristic
58C22×C4×C83C2312yes—
59C22×C4×C83C2311yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2312yes—
62C4⋊C4×C83—22no—
63C4⋊C4×C83—22no—
64C4⋊C4×C83—42no—
65C23×C83C2411yescharacteristic
66C22×D83—21no—
67C22×D83—22no—
68C22×D83—21no—
69C22×D83—22no—
70C22×D83—21no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—21no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83—21no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83—22no—
81C2×C4×C83—21no—
82C2×C4×C83C2×D412yes—
83C2×C4×C83#4114yes—
84C2×Dic83—22no—
85C2×Dic83—21no—
86C2×Dic83—21no—
87C2×Dic83—21no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—22no—
94C2×Dic83—22no—
95C2×Dic83C2×D412yes—
96C2×Dic83C2×D412yes—
97C22×C83C22×D411yescharacteristic
98C22×C83C2×#4112yes—
99C22×C83H(5,𝔽2)11yescharacteristic
100C22×C832−1+412yes—
101C22×C832−1+411yescharacteristic
102C2×D83—41no—
103C2×D83—41no—
104C2×D83—41no—
105C2×D83—42no—
106C2×D83—42no—
107C4×C83—22no—
108C4×C83—24no—
109C4×C83—41no—
110C4×C83—41no—
111C4×C83—42no—
112Dic83—22no—
113Dic83—22no—
114Dic83—41no—
115Dic83—42no—
116Dic83—42no—
117C2×C83#53211yescharacteristic
118C2×C83#53611yescharacteristic
119C2×C83#53711yescharacteristic
120C2×C83#53912yes—
121C2×C83#54112yes—
122D83—81no—
123#2664—831nomaximal, Sylow
124C83#266411yescharacteristic, complemented, Sylow
125C2×#126—831no—
126C2×#125—831no—
127#400—831no—
128#399—831no—
129#398—832no—
130#398D8311yescharacteristic, complemented
131#397—831no—
132#396—832no—
133#389—832no—
134#388—832no—
135C4×C4⋊C4—831no—
136#126—1664no—
137#125—1664no—
138C2×C4⋊C4—832no—
139C2×C4⋊C4—832no—
140C2×C4⋊C4—831no—
141C2×C4⋊C4—831no—
142C2×C4⋊C4—831no—
143C2×C4⋊C4—832no—
144C2×C4⋊C4—832no—
145C2×C4⋊C4C2×D8311yescharacteristic
146C2×C4⋊C4C2×D8311yescharacteristic
147C2×C4⋊C4C2×D8312yes—
148C2×C22⋊C4—832no—
149C2×C22⋊C4—831no—
150C2×C22⋊C4—831no—
151C2×C22⋊C4—831no—
152C2×C22⋊C4—832no—
153C2×C42—832no—
154C2×C42—831no—
155#95—831no—
156#95—832no—
157#95—832no—
158#95—832no—
159#95C2×D8311yescharacteristic
160#95C2×D8312yes—
161C24—831no—
162C22×C4—831no—
163C22×C4—831no—
164C22×C4—832no—
165C22×C4—831no—
166C22×C4—832no—
167C22×C4C22×D8311yescharacteristic
168C22×C4C22×D8311yescharacteristic
169C22×C4C22×D8312yes—
170C22×C4C22×D8311yescharacteristic
171C22×C4C22×D8312yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—1664no—
176C4⋊C4—1664no—
177C4⋊C4—1662no—
178C4⋊C4—1662no—
179C4⋊C4—1662no—
180C4⋊C4—1664no—
181C4⋊C4—1664no—
182C22⋊C4—3322no—
183C22⋊C4—1662no—
184C22⋊C4—1662no—
185C22⋊C4—1662no—
186C22⋊C4—1664no—
187C42—3322no—
188C42—834no—
189C23—1661no—
190C23—1662no—
191C23—1661no—
192C23—1662no—
193C23—1661no—
194C23C23×D8311yescharacteristic
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—832no—
207C2×C4—832no—
208C2×C4—1661no—
209C2×C4—1662no—
210C2×C4—1662no—
211C2×C4—1662no—
212C2×C4—1662no—
213C2×C4—1662no—
214C2×C4—1662no—
215C2×C4—1662no—
216C2×C4—1662no—
217C2×C4—1661no—
218C2×C4—1661no—
219C2×C4D4×D8312yes—
220C2×C4#1305059114yes—
221C22—3321no—
222C22—3321no—
223C22—3321no—
224C22—3322no—
225C22—3322no—
226C22C2×D4×D8311yescharacteristic
227C22C2×#1305059112yes—
228C22#1458183711yescharacteristic
229C22#1458183912yes—
230C22#1458183911yescharacteristic
231C4—22no—
232C4—24no—
233C4—41no—
234C4—41no—
235C4—42no—
236C4—3321no—
237C4—3322no—
238C4—3322no—
239C4—1662no—
240C4—1662no—
241C2—6641no—
242C2#1912825211yescharacteristic
243C2#1912827311yescharacteristic
244C2#1912828011yescharacteristic
245C2#1912830012yes—
246C2#1912831412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#398×C83C2
prime radicalC23×D83C23
hypercenter#398D83
hypocenterC83#2664
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#14581783C2semidirect
3. #19127640C2semidirect
5. #19127633C2semidirect
8. #19127597C2semidirect
10. #19127552C2semidirect
11. #398×C83C2semidirect
124. C83#2664semidirect
130. #398D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817872
C2×#145817832
#191276402
#191276382
#191276332
#191276272
#191276232
#191275972
#191275902
#191275522
#398×C832
#266483