Finite Group Databasebeta

← #37575805

Subgroups — #37575805

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×(C22⋊D191)⋊C4C211yesmaximal, characteristic
3#27132604C212yesmaximal, complemented
4#27132589C211yesmaximal, characteristic, complemented
5#27132578C212yesmaximal
6#27132576C211yesmaximal, characteristic, complemented
7#27132575C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic191C212yesmaximal, complemented
9#27132510C212yesmaximal, complemented
10#27132508C211yesmaximal, characteristic, complemented
11#397×C191C211yesmaximal, characteristic, complemented
12C22×Dic382C2211yescharacteristic
13C22×C22⋊D191C2211yescharacteristic
14C23×Dic191C2211yescharacteristic
15C22×C4×D191C2211yescharacteristic
16#20259287—24no—
17(C22⋊D191)⋊C4—24no—
18C2×Dic191⋊C4C2212yes—
19C2×Dic191⋊C4C2211yescharacteristic
20C2×C22⋊Dic191C2212yes—
21C2×(C2×D191)⋊C4C2212yes—
22C2×(C2×D191)⋊C4C2212yes—
23C2×(C2×D191)⋊C4C2211yescharacteristic
24C2×C4×Dic191C2212yes—
25C2×C4×Dic191C2212yes—
26C2×C4×Dic191C2211yescharacteristic
27C191⋊#95C2211yescharacteristic
28C191⋊#95C2212yes—
29C191⋊#95C2212yes—
30C191⋊#95C2212yes—
31C191⋊#95C2212yes—
32C2×C4⋊C4×C191C2211yescharacteristic
33C2×C22⋊C4×C191C2212yes—
34C2×C22⋊C4×C191C2211yescharacteristic
35#95×C191C2211yescharacteristic
36#95×C191C2212yes—
37C23×D191C2311yescharacteristic
38C2×Dic382—24no—
39C2×Dic382—22no—
40C2×C22⋊D191—22no—
41C2×C22⋊D191—22no—
42C2×C22⋊D191—22no—
43C22×Dic191—22no—
44C22×Dic191—24no—
45C22×Dic191C2311yescharacteristic
46C22×Dic191C2312yes—
47C22×Dic191C2311yescharacteristic
48C22×Dic191C2312yes—
49C22×Dic191C2311yescharacteristic
50C2×C4×D191—22no—
51C2×C4×D191—24no—
52Dic191⋊C4—22no—
53Dic191⋊C4—24no—
54C22⋊Dic191—24no—
55(C2×D191)⋊C4—24no—
56(C2×D191)⋊C4—24no—
57(C2×D191)⋊C4—22no—
58C4×Dic191—24no—
59C4×Dic191—24no—
60C4×Dic191D414yes—
61C24×C191C2311yescharacteristic
62C22×C4×C191C2311yescharacteristic
63C22×C4×C191C2312yes—
64C22×C4×C191C2312yes—
65C22×C4×C191C2311yescharacteristic
66C4⋊C4×C191—22no—
67C22⋊C4×C191—24no—
68C22⋊C4×C191—22no—
69C22×D191—21no—
70C22×D191—22no—
71C22×D191—22no—
72C22×D191—21no—
73C22×D191#4112yes—
74Dic382—28no—
75C22⋊D191—28no—
76C2×Dic191—22no—
77C2×Dic191—21no—
78C2×Dic191—21no—
79C2×Dic191—22no—
80C2×Dic191—21no—
81C2×Dic191—22no—
82C2×Dic191—22no—
83C2×Dic191—22no—
84C2×Dic191—22no—
85C2×Dic191—44no—
86C2×Dic191C2×D412yes—
87C2×Dic191C2×D412yes—
88C2×Dic191#4114yes—
89C2×Dic191#4114yes—
90C4×D191—44no—
91C23×C191—21no—
92C23×C191—22no—
93C23×C191—22no—
94C23×C191—21no—
95C23×C191#4112yes—
96C23×C191C2411yescharacteristic
97C2×C4×C191—22no—
98C2×C4×C191—21no—
99C2×C4×C191—22no—
100C2×C4×C191—22no—
101C2×C4×C191—22no—
102C2×C4×C191—22no—
103C2×C4×C191—22no—
104C2×C4×C191—22no—
105C2×C4×C191—22no—
106C2×C4×C191C2×D412yes—
107C22×C191—22no—
108C22×C191—24no—
109C22×C191—41no—
110C22×C191—41no—
111C22×C191—42no—
112C22×C191C22×D411yescharacteristic
113C22×C191C2×#4111yescharacteristic
114C22×C191C2×#4112yes—
115C22×C191C2×#4112yes—
116C22×C191C2×#4111yescharacteristic
117C2×D191—22no—
118C2×D191—24no—
119C2×D191—41no—
120C2×D191—41no—
121C2×D191—42no—
122C4×C191—22no—
123C4×C191—42no—
124C4×C191—41no—
125C4×C191—42no—
126Dic191—22no—
127Dic191—24no—
128Dic191—41no—
129Dic191—42no—
130Dic191#12414yes—
131C2×C191—42no—
132C2×C191#52511yescharacteristic
133C2×C191#52511yescharacteristic
134C2×C191C2×#12411yescharacteristic
135C2×C191#52912yes—
136C2×C191#52912yes—
137D191—42no—
138C191#239811yescharacteristic, complemented, Sylow
139#2398—1911nomaximal, Sylow
140C2×C4×Q8—1911no—
141C2×C4×D4—1911no—
142#400—1912no—
143#397—1911no—
144#397D19111yescharacteristic, complemented
145#394—1912no—
146#393—1911no—
147#393—1911no—
148C4×C22⋊C4—1912no—
149C4×C22⋊C4—1912no—
150#376—1911no—
151C22×Q8—1911no—
152C22×D4—1911no—
153C23×C4—1911no—
154C23×C4—1911no—
155C4×Q8—3824no—
156C4×D4—3824no—
157C2×C4⋊C4—1911no—
158C2×C4⋊C4—1912no—
159C2×C4⋊C4C2×D19111yescharacteristic
160C2×C22⋊C4—1912no—
161C2×C22⋊C4—1911no—
162C2×C22⋊C4—1912no—
163C2×C22⋊C4—1912no—
164C2×C22⋊C4C2×D19111yescharacteristic
165C2×C22⋊C4C2×D19112yes—
166C2×C42—1912no—
167C2×C42—1912no—
168C2×C42—1911no—
169#95—1911no—
170#95—1912no—
171#95—1912no—
172#95—1912no—
173#95—1912no—
174#95C2×D19111yescharacteristic
175#95C2×D19112yes—
176C24—1911no—
177C24C22×D19111yescharacteristic
178C2×Q8—3824no—
179C2×Q8—3822no—
180C2×D4—3822no—
181C2×D4—3822no—
182C2×D4—3822no—
183C22×C4—3822no—
184C22×C4—3824no—
185C22×C4—3822no—
186C22×C4—3824no—
187C22×C4—1912no—
188C22×C4—1911no—
189C22×C4—1912no—
190C22×C4—1911no—
191C22×C4—1911no—
192C22×C4C22×D19112yes—
193C22×C4C22×D19111yescharacteristic
194C22×C4C22×D19112yes—
195C22×C4C22×D19111yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—3822no—
198C4⋊C4—3824no—
199C22⋊C4—24no—
200C22⋊C4—22no—
201C22⋊C4—3824no—
202C22⋊C4—3824no—
203C22⋊C4—3824no—
204C22⋊C4—3822no—
205C42—3824no—
206C42—3824no—
207C42—1914no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—3821no—
213C23—3822no—
214C23—3822no—
215C23—3821no—
216C23—1912no—
217C23#1569028612yes—
218C23C23×D19111yescharacteristic
219Q8—3828no—
220D4—3828no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—3822no—
231C2×C4—3821no—
232C2×C4—3822no—
233C2×C4—3821no—
234C2×C4—3821no—
235C2×C4—3822no—
236C2×C4—3822no—
237C2×C4—3822no—
238C2×C4—3822no—
239C2×C4—1912no—
240C2×C4—1914no—
241C2×C4—1912no—
242C2×C4—1914no—
243C2×C4—7644no—
244C2×C4—7644no—
245C2×C4D4×D19112yes—
246C22—24no—
247C22—22no—
248C22—42no—
249C22—41no—
250C22—41no—
251C22—3824no—
252C22—3822no—
253C22—7642no—
254C22—7641no—
255C22—7641no—
256C22C2×D4×D19111yescharacteristic
257C22#41×D19112yes—
258C22#41×D19112yes—
259C22#41×D19111yescharacteristic
260C22C2×#1569028611yescharacteristic
261C4—22no—
262C4—42no—
263C4—42no—
264C4—41no—
265C4—3824no—
266C4—3822no—
267C4—1914no—
268C4—7642no—
269C4—7641no—
270C2—42no—
271C2—7642no—
272C2#2713317111yescharacteristic
273C2#2713317311yescharacteristic
274C2#124×D19111yescharacteristic
275C2#2713319012yes—
276C2#2713319612yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#397×C191C2
prime radicalC22×C22⋊D191C22
hypercenter#397D191
hypocenterC191#2398
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#20259287C2semidirect
3. #27132604C2semidirect
4. #27132589C2semidirect
6. #27132576C2semidirect
7. #27132575C2semidirect
8. C22⋊C4×Dic191C2semidirect
9. #27132510C2semidirect
10. #27132508C2semidirect
11. #397×C191C2semidirect
138. C191#2398semidirect
144. #397D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202592872
C2×(C22⋊D191)⋊C42
#271326042
#271325892
#271325782
#271325762
#271325752
C22⋊C4×Dic1912
#271325102
#271325082
#397×C1912
#2398191