Finite Group Databasebeta

← #37575809

Subgroups — #37575809

Order 24448.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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×#20259287C211yesmaximal, characteristic, complemented
2C2×#20259284C211yesmaximal, characteristic
3#27132601C211yesmaximal, characteristic, complemented
4#27132592C212yesmaximal, complemented
5#27132581C211yesmaximal, characteristic
6#27132580C211yesmaximal, characteristic
7#27132576C212yesmaximal, complemented
8#27132510C212yesmaximal, complemented
9#27132510C212yesmaximal, complemented
10#27132508C211yesmaximal, characteristic, complemented
11#400×C191C211yesmaximal, characteristic, complemented
12C22×Dic382C2211yescharacteristic
13C22×D764C2211yescharacteristic
14C22×C4×D191C2212yes—
15#20259287—24no—
16#20259284—24no—
17C2×Dic191⋊C4C2211yescharacteristic
18C2×Dic191⋊C4C2211yescharacteristic
19C2×(C2×D191)⋊C4C2212yes—
20C2×(C2×D191)⋊C4C2212yes—
21C2×(C2×D191)⋊C4C2212yes—
22C2×(C2×D191)⋊C4C2212yes—
23C2×(C2×D191)⋊C4C2212yes—
24C2×C4×Dic191C2212yes—
25C2×C4×Dic191C2211yescharacteristic
26C2×C4×Dic191C2211yescharacteristic
27C2×C4×Dic191C2211yescharacteristic
28C191⋊#95C2212yes—
29C191⋊#95C2211yescharacteristic
30C191⋊#95C2212yes—
31C191⋊#95C2212yes—
32C2×C4⋊C4×C191C2211yescharacteristic
33C2×C4⋊C4×C191C2211yescharacteristic
34#95×C191C2211yescharacteristic
35#95×C191C2212yes—
36#95×C191C2212yes—
37C23×D191C2312yes—
38C2×Dic382—22no—
39C2×Dic382—22no—
40C2×Dic382—22no—
41C2×D764—24no—
42C2×D764—22no—
43C22×Dic191C2312yes—
44C22×Dic191C2311yescharacteristic
45C22×Dic191C2311yescharacteristic
46C22×Dic191C2311yescharacteristic
47C22×Dic191C2311yescharacteristic
48C2×C4×D191—24no—
49C2×C4×D191—24no—
50C2×C4×D191—24no—
51Dic191⋊C4—22no—
52Dic191⋊C4—22no—
53(C2×D191)⋊C4—24no—
54(C2×D191)⋊C4—24no—
55(C2×D191)⋊C4—24no—
56(C2×D191)⋊C4—24no—
57(C2×D191)⋊C4—24no—
58C4×Dic191—22no—
59C4×Dic191—22no—
60C4×Dic191—24no—
61C4×Dic191D414yes—
62C22×C4×C191C2312yes—
63C22×C4×C191C2312yes—
64C22×C4×C191C2311yescharacteristic
65C22×C4×C191C2311yescharacteristic
66C22×C4×C191C2311yescharacteristic
67C4⋊C4×C191—22no—
68C4⋊C4×C191—22no—
69C22×D191—22no—
70C22×D191—22no—
71C22×D191—22no—
72C22×D191—22no—
73C22×D191—22no—
74C22×D191—22no—
75C22×D191#4114yes—
76Dic382—28no—
77D764—28no—
78C2×Dic191—21no—
79C2×Dic191—21no—
80C2×Dic191—21no—
81C2×Dic191—21no—
82C2×Dic191—21no—
83C2×Dic191—22no—
84C2×Dic191—22no—
85C2×Dic191—22no—
86C2×Dic191—22no—
87C2×Dic191C2×D412yes—
88C2×Dic191C2×D412yes—
89C2×Dic191#4112yes—
90C2×Dic191#4114yes—
91C4×D191—48no—
92C23×C191C2411yescharacteristic
93C2×C4×C191—21no—
94C2×C4×C191—22no—
95C2×C4×C191—21no—
96C2×C4×C191—21no—
97C2×C4×C191—22no—
98C2×C4×C191—22no—
99C2×C4×C191—22no—
100C2×C4×C191—22no—
101C2×C4×C191—22no—
102C2×C4×C191—22no—
103C2×C4×C191—22no—
104C2×C4×C191C2×D412yes—
105C2×C4×C191#4112yes—
106C22×C191C22×D411yescharacteristic
107C22×C191C2×#4112yes—
108C22×C191C2×#4111yescharacteristic
109C22×C191C2×#4112yes—
110C22×C191C2×#4111yescharacteristic
111C2×D191—24no—
112C2×D191—24no—
113C2×D191—24no—
114C2×D191—42no—
115C2×D191—42no—
116C2×D191—42no—
117C2×D191—42no—
118C4×C191—22no—
119C4×C191—22no—
120C4×C191—41no—
121C4×C191—42no—
122C4×C191—42no—
123Dic191—22no—
124Dic191—22no—
125Dic191—41no—
126Dic191—42no—
127Dic191#12414yes—
128C2×C191#52511yescharacteristic
129C2×C191#52511yescharacteristic
130C2×C191C2×#12411yescharacteristic
131C2×C191#52912yes—
132C2×C191#52912yes—
133D191—44no—
134C191#239811yescharacteristic, complemented, Sylow
135#2398—1911nomaximal, Sylow
136C2×C4×Q8—1911no—
137C2×C4×D4—1911no—
138#400—1911no—
139#400D19111yescharacteristic, complemented
140#397—1912no—
141#394—1911no—
142#394—1911no—
143#393—1912no—
144C4×C22⋊C4—1912no—
145C4×C22⋊C4—1912no—
146#376—1911no—
147C22×Q8—1911no—
148C22×D4—1911no—
149C23×C4—1912no—
150C4×Q8—3824no—
151C4×D4—3824no—
152C2×C4⋊C4—1911no—
153C2×C4⋊C4—1911no—
154C2×C4⋊C4C2×D19111yescharacteristic
155C2×C4⋊C4C2×D19111yescharacteristic
156C2×C22⋊C4—1912no—
157C2×C22⋊C4—1912no—
158C2×C22⋊C4—1912no—
159C2×C22⋊C4—1912no—
160C2×C22⋊C4—1912no—
161C2×C42—1911no—
162C2×C42—1911no—
163C2×C42—1912no—
164C2×C42—1911no—
165#95—1912no—
166#95—1911no—
167#95—1912no—
168#95—1912no—
169#95C2×D19111yescharacteristic
170#95C2×D19112yes—
171#95C2×D19112yes—
172C24—1912no—
173C2×Q8—3822no—
174C2×Q8—3822no—
175C2×Q8—3822no—
176C2×D4—3824no—
177C2×D4—3822no—
178C22×C4—3824no—
179C22×C4—3824no—
180C22×C4—3824no—
181C22×C4—1912no—
182C22×C4—1911no—
183C22×C4—1911no—
184C22×C4—1911no—
185C22×C4—1911no—
186C22×C4C22×D19112yes—
187C22×C4C22×D19112yes—
188C22×C4C22×D19111yescharacteristic
189C22×C4C22×D19111yescharacteristic
190C22×C4C22×D19111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—3822no—
194C4⋊C4—3822no—
195C22⋊C4—3824no—
196C22⋊C4—3824no—
197C22⋊C4—3824no—
198C22⋊C4—3824no—
199C22⋊C4—3824no—
200C42—3824no—
201C42—3822no—
202C42—3822no—
203C42—1914no—
204C23—3822no—
205C23—3822no—
206C23—3822no—
207C23—3822no—
208C23—3822no—
209C23—3822no—
210C23—1914no—
211C23C23×D19111yescharacteristic
212Q8—3828no—
213D4—3828no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—3821no—
226C2×C4—3821no—
227C2×C4—3821no—
228C2×C4—3821no—
229C2×C4—3821no—
230C2×C4—3822no—
231C2×C4—3822no—
232C2×C4—3822no—
233C2×C4—3822no—
234C2×C4—1912no—
235C2×C4—1912no—
236C2×C4—1912no—
237C2×C4—1914no—
238C2×C4—7648no—
239C2×C4D4×D19112yes—
240C2×C4#1569028812yes—
241C22—3824no—
242C22—3824no—
243C22—3824no—
244C22—7642no—
245C22—7642no—
246C22—7642no—
247C22—7642no—
248C22C2×D4×D19111yescharacteristic
249C22#41×D19111yescharacteristic
250C22#41×D19112yes—
251C22#41×D19112yes—
252C22C2×#1569028811yescharacteristic
253C4—22no—
254C4—22no—
255C4—42no—
256C4—41no—
257C4—42no—
258C4—3822no—
259C4—3822no—
260C4—1914no—
261C4—7642no—
262C4—7641no—
263C2—7644no—
264C2#2713317111yescharacteristic
265C2#2713317611yescharacteristic
266C2#124×D19111yescharacteristic
267C2#2713319612yes—
268C2#2713319812yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#400×C191C2
prime radicalC22×D764C22
hypercenter#400D191
hypocenterC191#2398
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#20259287C2semidirect
3. #27132601C2semidirect
4. #27132592C2semidirect
7. #27132576C2semidirect
8. #27132510C2semidirect
9. #27132510C2semidirect
10. #27132508C2semidirect
11. #400×C191C2semidirect
134. C191#2398semidirect
139. #400D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202592872
C2×#202592842
#271326012
#271325922
#271325812
#271325802
#271325762
#271325102
#271325082
#400×C1912
#2398191