Finite Group Databasebeta

← #37583128

Subgroups — #37583128

Order 24448.

364 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 119 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
1#27133218C211yesmaximal, characteristic, complemented
2#27133198C211yesmaximal, characteristic
3#27133057C211yesmaximal, characteristic, complemented
4#27133033C211yesmaximal, characteristic, complemented
5#27133025C211yesmaximal, characteristic, complemented
6#27132922C212yesmaximal, complemented
7#27132910C212yesmaximal
8#27132757C212yesmaximal, complemented
9#27132750C212yesmaximal, complemented
10#27132615C211yesmaximal, characteristic, complemented
11#501×C191C211yesmaximal, characteristic, complemented
12C2×Dic764C2212yes—
13C2×#15690276C2212yes—
14C2×#15690270C2212yes—
15#20259322C2211yescharacteristic
16#20259317—21no—
17#20259313—21no—
18#20259310C2211yescharacteristic
19#20259307—21no—
20#20259302C2211yescharacteristic
21#20259302C2212yes—
22#20259298—21no—
23#20259297C2211yescharacteristic
24Q8×Dic191—21no—
25#20259287C2211yescharacteristic
26#20259287C2212yes—
27#20259284—21no—
28#20259284C2211yescharacteristic
29#20259280—21no—
30#20259280—21no—
31C42×D191C2211yescharacteristic
32#20259245C2212yes—
33#20259245C2212yes—
34#20259245C2212yes—
35#20259242C2212yes—
36#20259233C2211yescharacteristic
37C8×Dic191C2212yes—
38C4⋊Q8×C191C2211yescharacteristic
39#125×C191C2211yescharacteristic
40#107×C191C2212yes—
41#106×C191C2212yes—
42#97×C191C2211yescharacteristic
43C2×Dic382C2312yes—
44C2×Dic382C2311yescharacteristic
45C2×D764—21no—
46C2×D764C2311yescharacteristic
47C2×C4×D191—21no—
48C2×C4×D191D411yescharacteristic
49C2×C4×D191D411yescharacteristic
50C2×C4×D191C2311yescharacteristic
51Dic764—24no—
52#15690276—24no—
53#15690270—24no—
54C2×C191⋊C8C2312yes—
55C4⋊Dic191—21no—
56C4⋊Dic191—22no—
57C4⋊Dic191—21no—
58C4⋊Dic191—21no—
59Dic191⋊C4—22no—
60Dic191⋊C4—21no—
61(C2×D191)⋊C4—22no—
62(C2×D191)⋊C4—21no—
63(C2×D191)⋊C4—21no—
64(C2×D191)⋊C4—21no—
65(C2×D191)⋊C4—21no—
66(C2×D191)⋊C4—21no—
67C4×Dic191—21no—
68C4×Dic191—22no—
69C4×Dic191D411yescharacteristic
70C4×Dic191D411yescharacteristic
71C4×Dic191C2311yescharacteristic
72C2×Q8×C191—21no—
73C2×C8×C191C2312yes—
74C4⋊C4×C191—21no—
75C4⋊C4×C191—21no—
76C4⋊C4×C191—21no—
77C4⋊C4×C191C2311yescharacteristic
78C4⋊C4×C191C2312yes—
79C4⋊C4×C191C2311yescharacteristic
80C42×C191C2311yescharacteristic
81C22×D191—21no—
82C22×D191C2×D411yescharacteristic
83Dic382—22no—
84Dic382—22no—
85Dic382—22no—
86D764—22no—
87D764—42no—
88C2×Dic191—21no—
89C2×Dic191—22no—
90C2×Dic191C2×D411yescharacteristic
91C2×Dic191C2×D411yescharacteristic
92C2×Dic191C2×D411yescharacteristic
93C4×D191—22no—
94C4×D191—22no—
95C4×D191—41no—
96C4×D191—42no—
97C191⋊C8—22no—
98Q8×C191—42no—
99C2×C4×C191—21no—
100C2×C4×C191—21no—
101C2×C4×C191—22no—
102C2×C4×C191C2×D411yescharacteristic
103C2×C4×C191C2×D411yescharacteristic
104C2×C4×C191C2411yescharacteristic
105C8×C191#4114yes—
106C22×C191C22×D411yescharacteristic
107C2×D191—21no—
108C2×D191—21no—
109C2×D191—21no—
110C2×D191—41no—
111C2×D191—41no—
112C2×D191—41no—
113C4×C191—21no—
114C4×C191—21no—
115C4×C191—41no—
116C4×C191—42no—
117C4×C191—41no—
118C4×C191C2×#4112yes—
119Dic191—21no—
120Dic191—21no—
121Dic191—41no—
122Dic191—42no—
123Dic191#13712yes—
124C2×C191#53211yescharacteristic
125C2×C191C2×#13711yescharacteristic
126C2×C191#57511yescharacteristic
127D191—41no—
128D191—81no—
129C191#317211yescharacteristic, complemented, Sylow
130#3172—1911nomaximal, Sylow
131#533—1911no—
132#529—1911no—
133#501D19111yescharacteristic, complemented
134#497—1911no—
135#491—1911no—
136#489—1911no—
137#470—1912no—
138#468—1912no—
139C4×Q16—1912no—
140C4×SD16—1912no—
141C4×M16—1911no—
142C2×Q16—1912no—
143C2×SD16—1912no—
144C2×M16—1912no—
145C4⋊Q8—1911no—
146C4⋊Q8C2×D19111yescharacteristic
147#126—3821no—
148#125—3821no—
149#125C2×D19111yescharacteristic
150#124—1911no—
151#123—3821no—
152C22⋊Q8—3821no—
153C22⋊Q8—1911no—
154C22⋊Q8—1912no—
155#121—1911no—
156C4×Q8—3821no—
157C4×Q8—1911no—
158C4×Q8—1912no—
159C4×D4—3821no—
160C4×D4—1911no—
161#117—3821no—
162#117—3821no—
163C2×C42—1911no—
164#107C2×D19112yes—
165#106C2×D19112yes—
166#103—1912no—
167#103—1912no—
168#103—1912no—
169#102—1912no—
170#97—1911no—
171#97C2×D19111yescharacteristic
172C4×C8—1912no—
173C2×Q8—21no—
174C2×Q8—1911no—
175C2×Q8—1912no—
176C2×D4—3821no—
177C2×D4—1911no—
178C22×C4—3821no—
179C22×C4—1911no—
180C22×C4—1911no—
181C22×C4—1911no—
182Q16—3824no—
183SD16—3824no—
184M16—3824no—
185C2×C8—1912no—
186C2×C8C22×D19112yes—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—3822no—
191C4⋊C4—3821no—
192C4⋊C4—3821no—
193C4⋊C4—3822no—
194C4⋊C4—3821no—
195C4⋊C4—3821no—
196C4⋊C4C22×D19112yes—
197C4⋊C4C22×D19111yescharacteristic
198C4⋊C4C22×D19111yescharacteristic
199C22⋊C4—3822no—
200C22⋊C4—3821no—
201C22⋊C4—3821no—
202C22⋊C4—3821no—
203C22⋊C4—3821no—
204C22⋊C4—3821no—
205C42—3821no—
206C42—3822no—
207C42—1911no—
208C42—1911no—
209C42—1911no—
210C42C22×D19111yescharacteristic
211C23—3821no—
212C23—1911no—
213Q8—42no—
214Q8—3822no—
215Q8—3822no—
216Q8—3822no—
217D4—3822no—
218D4—7642no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—3822no—
223C2×C4—3822no—
224C2×C4—3821no—
225C2×C4—3822no—
226C2×C4—1911no—
227C2×C4—1911no—
228C2×C4—1911no—
229C2×C4—7641no—
230C2×C4—7642no—
231C2×C4D4×D19111yescharacteristic
232C2×C4D4×D19111yescharacteristic
233C2×C4C23×D19111yescharacteristic
234C8—3822no—
235C8#1569028814yes—
236C22—3821no—
237C22—3821no—
238C22—3821no—
239C22—7641no—
240C22—7641no—
241C22—7641no—
242C22C2×D4×D19111yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—41no—
247C4—42no—
248C4—3821no—
249C4—3821no—
250C4—1912no—
251C4—7641no—
252C4—7642no—
253C4C2×#1569028812yes—
254C2—15281no—
255C2—7641no—
256C2#2713321411yescharacteristic
257C2#137×D19111yescharacteristic
258C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#501×C191C2
socleC22×C191C22×D4
prime radical#20259297C22
hypercenter#501D191
hypocenterC191#3172
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. #27133218C2semidirect
3. #27133057C2semidirect
4. #27133033C2semidirect
5. #27133025C2semidirect
6. #27132922C2semidirect
8. #27132757C2semidirect
9. #27132750C2semidirect
10. #27132615C2semidirect
11. #501×C191C2semidirect
129. C191#3172semidirect
133. #501D191semidirect

Maximal subgroups

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

subgroupindex
#271332182
#271331982
#271330572
#271330332
#271330252
#271329222
#271329102
#271327572
#271327502
#271326152
#501×C1912
#3172191