Finite Group Databasebeta

← #37575784

Subgroups — #37575784

Order 24448.

560 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 147 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×#20259284C211yesmaximal, characteristic
2C2×#20259280C211yesmaximal, characteristic, complemented
3#27132602C211yesmaximal, characteristic, complemented
4#27132600C211yesmaximal, characteristic, complemented
5#27132590C212yesmaximal, complemented
6#27132589C211yesmaximal, characteristic, complemented
7#27132580C212yesmaximal
8C4⋊C4×Dic191C212yesmaximal, complemented
9#27132510C212yesmaximal, complemented
10#27132510C211yesmaximal, characteristic, complemented
11#398×C191C211yesmaximal, characteristic, complemented
12C22×D764C2211yescharacteristic
13C22×C4×D191C2211yescharacteristic
14C22×C4×D191C2211yescharacteristic
15#20259284—24no—
16#20259280—24no—
17C2×C4⋊Dic191C2212yes—
18C2×C4⋊Dic191C2211yescharacteristic
19C2×(C2×D191)⋊C4C2212yes—
20C2×(C2×D191)⋊C4C2212yes—
21C2×(C2×D191)⋊C4C2212yes—
22C2×(C2×D191)⋊C4C2212yes—
23C2×(C2×D191)⋊C4C2211yescharacteristic
24C2×(C2×D191)⋊C4C2211yescharacteristic
25C2×C4×Dic191C2212yes—
26C2×C4×Dic191C2212yes—
27C2×C4×Dic191C2211yescharacteristic
28C191⋊#95C2211yescharacteristic
29C191⋊#95C2212yes—
30C191⋊#95C2212yes—
31C191⋊#95C2212yes—
32C2×C4⋊C4×C191C2212yes—
33C2×C4⋊C4×C191C2211yescharacteristic
34C2×C4⋊C4×C191C2211yescharacteristic
35#95×C191C2212yes—
36#95×C191C2211yescharacteristic
37C23×D191C2311yescharacteristic
38C23×D191C2311yescharacteristic
39C2×D764—22no—
40C2×D764—22no—
41C2×D764—22no—
42C22×Dic191C2312yes—
43C22×Dic191C2311yescharacteristic
44C22×Dic191C2312yes—
45C22×Dic191C2311yescharacteristic
46C2×C4×D191—24no—
47C2×C4×D191—22no—
48C2×C4×D191—24no—
49C2×C4×D191D414yes—
50C4⋊Dic191—22no—
51C4⋊Dic191—24no—
52(C2×D191)⋊C4—24no—
53(C2×D191)⋊C4—22no—
54(C2×D191)⋊C4—24no—
55(C2×D191)⋊C4—22no—
56(C2×D191)⋊C4—24no—
57(C2×D191)⋊C4—24no—
58C4×Dic191—24no—
59C4×Dic191—22no—
60C4×Dic191—24no—
61C22×C4×C191C2311yescharacteristic
62C22×C4×C191C2312yes—
63C22×C4×C191C2311yescharacteristic
64C22×C4×C191C2311yescharacteristic
65C22×C4×C191C2312yes—
66C4⋊C4×C191—22no—
67C4⋊C4×C191—22no—
68C4⋊C4×C191—24no—
69C22×D191—21no—
70C22×D191—22no—
71C22×D191—22no—
72C22×D191—22no—
73C22×D191—22no—
74C22×D191—21no—
75C22×D191—21no—
76C22×D191—21no—
77C22×D191C2×D412yes—
78C22×D191#4112yes—
79D764—28no—
80C2×Dic191—21no—
81C2×Dic191—22no—
82C2×Dic191—22no—
83C2×Dic191—22no—
84C2×Dic191—22no—
85C2×Dic191—22no—
86C2×Dic191—22no—
87C2×Dic191—22no—
88C2×Dic191C2×D412yes—
89C2×Dic191#4114yes—
90C4×D191—28no—
91C4×D191—44no—
92C23×C191C2411yescharacteristic
93C2×C4×C191—21no—
94C2×C4×C191—22no—
95C2×C4×C191—22no—
96C2×C4×C191—22no—
97C2×C4×C191—21no—
98C2×C4×C191—22no—
99C2×C4×C191—21no—
100C2×C4×C191—22no—
101C2×C4×C191—22no—
102C2×C4×C191—22no—
103C2×C4×C191C2×D412yes—
104C2×C4×C191#4112yes—
105C2×C4×C191#4114yes—
106C22×C191C22×D411yescharacteristic
107C22×C191C2×#4112yes—
108C22×C191C2×#4111yescharacteristic
109C22×C191C2×#4111yescharacteristic
110C22×C191C2×#4112yes—
111C2×D191—22no—
112C2×D191—22no—
113C2×D191—22no—
114C2×D191—24no—
115C2×D191—22no—
116C2×D191—41no—
117C2×D191—42no—
118C2×D191—42no—
119C2×D191—42no—
120C2×D191—41no—
121C4×C191—22no—
122C4×C191—22no—
123C4×C191—24no—
124C4×C191—41no—
125C4×C191—42no—
126Dic191—42no—
127Dic191—41no—
128Dic191—42no—
129Dic191#12314yes—
130C2×C191C2×#12311yescharacteristic
131C2×C191#52511yescharacteristic
132C2×C191#52912yes—
133C2×C191#52912yes—
134C2×C191#53211yescharacteristic
135D191—42no—
136D191—42no—
137C191#239511yescharacteristic, complemented, Sylow
138#2395—1911nomaximal, Sylow
139C2×C4×D4—1911no—
140C2×#117—1911no—
141#400—1911no—
142#399—1911no—
143#398D19111yescharacteristic, complemented
144#397—1911no—
145#397—1912no—
146#394—1912no—
147C4×C4⋊C4—1912no—
148C4×C22⋊C4—1912no—
149C4×C22⋊C4—1911no—
150C22×D4—1911no—
151C23×C4—1911no—
152C23×C4—1911no—
153C4×D4—3824no—
154#117—3824no—
155C2×C4⋊C4—1911no—
156C2×C4⋊C4—1912no—
157C2×C4⋊C4C2×D19111yescharacteristic
158C2×C4⋊C4C2×D19111yescharacteristic
159C2×C4⋊C4C2×D19112yes—
160C2×C22⋊C4—1912no—
161C2×C22⋊C4—1911no—
162C2×C22⋊C4—1911no—
163C2×C22⋊C4—1912no—
164C2×C22⋊C4—1912no—
165C2×C22⋊C4—1912no—
166C2×C42—1912no—
167C2×C42—1911no—
168C2×C42—1912no—
169#95—1911no—
170#95—1912no—
171#95—1912no—
172#95—1912no—
173#95C2×D19112yes—
174#95C2×D19111yescharacteristic
175C24—1911no—
176C24—1911no—
177C2×D4—3822no—
178C2×D4—3822no—
179C2×D4—3822no—
180C22×C4—3824no—
181C22×C4—3822no—
182C22×C4—3824no—
183C22×C4—1912no—
184C22×C4—1911no—
185C22×C4—1912no—
186C22×C4—1914no—
187C22×C4—1911no—
188C22×C4C22×D19111yescharacteristic
189C22×C4C22×D19112yes—
190C22×C4C22×D19111yescharacteristic
191C22×C4C22×D19111yescharacteristic
192C22×C4C22×D19112yes—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—3822no—
197C4⋊C4—3824no—
198C22⋊C4—3824no—
199C22⋊C4—3824no—
200C22⋊C4—3822no—
201C22⋊C4—3822no—
202C22⋊C4—3824no—
203C22⋊C4—3824no—
204C42—3824no—
205C42—3824no—
206C42—3822no—
207C23—3821no—
208C23—3822no—
209C23—3822no—
210C23—3821no—
211C23—3821no—
212C23—3821no—
213C23—3822no—
214C23—3822no—
215C23—1912no—
216C23—1912no—
217C23C23×D19111yescharacteristic
218D4—3828no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—3828no—
230C2×C4—3821no—
231C2×C4—3822no—
232C2×C4—3822no—
233C2×C4—3822no—
234C2×C4—3822no—
235C2×C4—3822no—
236C2×C4—3822no—
237C2×C4—3822no—
238C2×C4—1912no—
239C2×C4—1914no—
240C2×C4—7644no—
241C2×C4D4×D19112yes—
242C2×C4#1569028612yes—
243C2×C4#1569028814yes—
244C22—3824no—
245C22—3822no—
246C22—3822no—
247C22—3822no—
248C22—3822no—
249C22—7642no—
250C22—7642no—
251C22—7642no—
252C22—7641no—
253C22—7641no—
254C22C2×D4×D19111yescharacteristic
255C22#41×D19111yescharacteristic
256C22#41×D19112yes—
257C22C2×#1569028611yescharacteristic
258C22C2×#1569028812yes—
259C4—24no—
260C4—22no—
261C4—22no—
262C4—41no—
263C4—42no—
264C4—1914no—
265C4—7642no—
266C4—7641no—
267C4—7642no—
268C2—7642no—
269C2—7642no—
270C2#123×D19111yescharacteristic
271C2#2713317311yescharacteristic
272C2#2713319812yes—
273C2#2713319912yes—
274C2#2713321411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#398×C191C2
prime radicalC22×D764C22
hypercenter#398D191
hypocenterC191#2395
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#20259280C2semidirect
3. #27132602C2semidirect
4. #27132600C2semidirect
5. #27132590C2semidirect
6. #27132589C2semidirect
8. C4⋊C4×Dic191C2semidirect
9. #27132510C2semidirect
10. #27132510C2semidirect
11. #398×C191C2semidirect
137. C191#2395semidirect
143. #398D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202592842
C2×#202592802
#271326022
#271326002
#271325902
#271325892
#271325802
C4⋊C4×Dic1912
#271325102
#398×C1912
#2395191