Finite Group Databasebeta

← #24914193

Subgroups — #24914193

Order 10624.

612 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14581788C211yesmaximal, characteristic
2C2×#14581773C211yesmaximal, characteristic, complemented
3#19127619C212yesmaximal, complemented
4#19127617C212yesmaximal, complemented
5#19127617C212yesmaximal, complemented
6#19127586C212yesmaximal, complemented
7#19127580C212yesmaximal, complemented
8#19127578C212yesmaximal, complemented
9#386×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C22×C22⋊D83C2212yes—
12C22×C22⋊D83C2212yes—
13C23×Dic83C2211yescharacteristic
14#14581788—24no—
15#14581773—24no—
16C2×C4⋊Dic83C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×C22⋊Dic83C2212yes—
19C2×C22⋊Dic83C2211yescharacteristic
20C2×(C2×D83)⋊C4C2212yes—
21C2×(C2×D83)⋊C4C2212yes—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2212yes—
24C2×C4×Dic83C2211yescharacteristic
25C83⋊#95C2212yes—
26C83⋊#95C2212yes—
27C83⋊#95C2212yes—
28C22×D4×C83C2211yescharacteristic
29C23×C4×C83C2211yescharacteristic
30C2×C22⋊C4×C83C2212yes—
31C2×C22⋊C4×C83C2211yescharacteristic
32#95×C83C2212yes—
33C23×D83C2312yes—
34C2×D332—22no—
35C2×D332—41no—
36C2×D332—41no—
37C2×C22⋊D83—24no—
38C2×C22⋊D83—24no—
39C2×C22⋊D83—24no—
40C2×C22⋊D83—24no—
41C2×C22⋊D83—42no—
42C2×C22⋊D83—42no—
43C22×Dic83—22no—
44C22×Dic83—41no—
45C22×Dic83—41no—
46C22×Dic83C2312yes—
47C22×Dic83C2312yes—
48C22×Dic83C2312yes—
49C4⋊Dic83—24no—
50Dic83⋊C4—24no—
51C22⋊Dic83—22no—
52C22⋊Dic83—24no—
53(C2×D83)⋊C4—42no—
54(C2×D83)⋊C4—42no—
55(C2×D83)⋊C4—42no—
56(C2×D83)⋊C4—42no—
57C4×Dic83—22no—
58C24×C83C2311yescharacteristic
59C24×C83C2311yescharacteristic
60C2×D4×C83—41no—
61C2×D4×C83—41no—
62C2×D4×C83D414yes—
63C22×C4×C83—42no—
64C22×C4×C83—41no—
65C22×C4×C83C2312yes—
66C22×C4×C83C2312yes—
67C22×C4×C83C2311yescharacteristic
68C22⋊C4×C83—42no—
69C22⋊C4×C83—41no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83—21no—
73C23×C83—22no—
74C23×C83—21no—
75C23×C83—22no—
76C23×C83—21no—
77C23×C83—21no—
78C23×C83C2×D412yes—
79C23×C83C2×D412yes—
80C23×C83C2411yescharacteristic
81C22×D83—22no—
82C22×D83—22no—
83C22×D83—22no—
84C22×D83—22no—
85C22×D83—22no—
86C22×D83—22no—
87C22×D83—22no—
88D4×C83—44no—
89C2×C4×C83—21no—
90C2×C4×C83—21no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—44no—
98C2×C4×C83C2×D412yes—
99C22⋊D83—48no—
100C22⋊D83—48no—
101C2×Dic83—22no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83—22no—
107C2×Dic83—22no—
108C2×Dic83—22no—
109C2×Dic83—44no—
110C2×Dic83#4114yes—
111D332—44no—
112C22×C83—22no—
113C22×C83—22no—
114C22×C83—24no—
115C22×C83—24no—
116C22×C83—42no—
117C22×C83—42no—
118C22×C83—41no—
119C22×C83—41no—
120C22×C83—41no—
121C22×C83—41no—
122C22×C83C22×D411yescharacteristic
123C22×C83C2×#4112yes—
124C22×C83H(5,𝔽2)11yescharacteristic
125C22×C83H(5,𝔽2)11yescharacteristic
126C22×C83H(5,𝔽2)12yes—
127C2×D83—42no—
128C2×D83—42no—
129C2×D83—42no—
130C2×D83—42no—
131C2×D83—42no—
132C2×D83—42no—
133C2×D83—42no—
134C4×C83—22no—
135C4×C83—42no—
136C4×C83—42no—
137Dic83—24no—
138Dic83—42no—
139Dic83—42no—
140C2×C83—42no—
141C2×C83—42no—
142C2×C83#53411yescharacteristic
143C2×C83#54012yes—
144C2×C83#54612yes—
145C2×C83#55911yescharacteristic
146C2×C83#56011yescharacteristic
147D83—82no—
148#2709—831nomaximal, Sylow
149C83#270911yescharacteristic, complemented, Sylow
150C2×#127—831no—
151C2×#123—831no—
152#394—832no—
153#394—832no—
154#394—832no—
155#387—832no—
156#386—832no—
157#386—832no—
158#386D8311yescharacteristic, complemented
159C22×D4—832no—
160C22×D4—832no—
161C22×D4—831no—
162C22×D4C2×D8311yescharacteristic
163C23×C4—831no—
164C23×C4C2×D8311yescharacteristic
165#127—1664no—
166#123—1664no—
167C2×C4⋊C4—832no—
168C2×C4⋊C4—832no—
169C2×C22⋊C4—832no—
170C2×C22⋊C4—832no—
171C2×C22⋊C4—832no—
172C2×C22⋊C4—832no—
173C2×C22⋊C4—832no—
174C2×C22⋊C4—831no—
175C2×C22⋊C4C2×D8312yes—
176C2×C22⋊C4C2×D8311yescharacteristic
177C2×C42—831no—
178#95—832no—
179#95—832no—
180#95—832no—
181#95C2×D8312yes—
182C24—832no—
183C24C22×D8311yescharacteristic
184C24C22×D8311yescharacteristic
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—3321no—
188C2×D4—3321no—
189C2×D4—3322no—
190C2×D4—3322no—
191C2×D4—1664no—
192C2×D4—1664no—
193C2×D4—1664no—
194C2×D4—1664no—
195C2×D4—1662no—
196C2×D4D33214yes—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—3321no—
200C22×C4—3321no—
201C22×C4—832no—
202C22×C4—832no—
203C22×C4—832no—
204C22×C4—1662no—
205C22×C4C22×D8312yes—
206C22×C4C22×D8312yes—
207C22×C4C22×D8311yescharacteristic
208C4⋊C4—1664no—
209C4⋊C4—1664no—
210C22⋊C4—41no—
211C22⋊C4—42no—
212C22⋊C4—3322no—
213C22⋊C4—3322no—
214C22⋊C4—3322no—
215C22⋊C4—3322no—
216C22⋊C4—1662no—
217C22⋊C4—1664no—
218C42—1662no—
219C23—22no—
220C23—22no—
221C23—21no—
222C23—22no—
223C23—21no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—1662no—
228C23—1662no—
229C23—1662no—
230C23—1662no—
231C23—1662no—
232C23—1662no—
233C23—1662no—
234C23C2×D33212yes—
235C23C2×D33212yes—
236C23C23×D8311yescharacteristic
237D4—44no—
238D4—3324no—
239D4—3328no—
240D4—3328no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—44no—
250C2×C4—3324no—
251C2×C4—834no—
252C2×C4—1662no—
253C2×C4—1662no—
254C2×C4—1662no—
255C2×C4—1662no—
256C2×C4—1662no—
257C2×C4—1662no—
258C2×C4—1662no—
259C2×C4—1662no—
260C2×C4C2×D33212yes—
261C22—24no—
262C22—24no—
263C22—22no—
264C22—22no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—3322no—
272C22—3322no—
273C22—3322no—
274C22—3322no—
275C22—3322no—
276C22—3322no—
277C22—3322no—
278C22C22×D33211yescharacteristic
279C22#41×D8312yes—
280C22#1458183712yes—
281C22#1458183711yescharacteristic
282C22#1458183711yescharacteristic
283C4—22no—
284C4—42no—
285C4—42no—
286C4—3322no—
287C4—3322no—
288C4—1664no—
289C2—42no—
290C2—42no—
291C2—6642no—
292C2#1912826311yescharacteristic
293C2#1912830112yes—
294C2#1912834612yes—
295C2#1912843511yescharacteristic
296C2#1912844011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#386×C83C2
prime radicalC2×#14581788C2
hypercenter#386D83
hypocenterC83#2709
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#14581773C2semidirect
3. #19127619C2semidirect
4. #19127617C2semidirect
5. #19127617C2semidirect
6. #19127586C2semidirect
7. #19127580C2semidirect
8. #19127578C2semidirect
9. #386×C83C2semidirect
149. C83#2709semidirect
158. #386D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817882
C2×#145817732
#191276192
#191276172
#191275862
#191275802
#191275782
#386×C832
#270983