Finite Group Databasebeta

← #37583129

Subgroups — #37583129

Order 24448.

364 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 115 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#27133192C211yesmaximal, characteristic
3#27133069C211yesmaximal, characteristic, complemented
4#27133058C211yesmaximal, characteristic, complemented
5#27133032C211yesmaximal, characteristic, complemented
6#27132920C212yesmaximal, complemented
7#27132908C212yesmaximal
8#27132755C212yesmaximal, complemented
9#27132754C212yesmaximal, complemented
10#27132615C211yesmaximal, characteristic, complemented
11#489×C191C211yesmaximal, characteristic, complemented
12C2×#15690277C2212yes—
13C2×#15690275C2212yes—
14C2×#15690270C2212yes—
15#20259322C2211yescharacteristic
16#20259315—21no—
17#20259314C2211yescharacteristic
18#20259313—21no—
19#20259304—21no—
20#20259302C2212yes—
21#20259298—21no—
22#20259298C2211yescharacteristic
23#20259293C2211yescharacteristic
24Q8×Dic191—21no—
25Q8×Dic191C2211yescharacteristic
26#20259287C2212yes—
27D4×Dic191C2211yescharacteristic
28(C22⋊D191)⋊C4—21no—
29#20259280—21no—
30#20259278—21no—
31C42×D191C2211yescharacteristic
32#20259255C2212yes—
33#20259253C2212yes—
34#20259247C2212yes—
35#20259247C2212yes—
36#20259233C2211yescharacteristic
37C8×Dic191C2212yes—
38C4⋊Q8×C191C2211yescharacteristic
39#124×C191C2211yescharacteristic
40#103×C191C2212yes—
41#102×C191C2212yes—
42#97×C191C2211yescharacteristic
43C2×Dic382C2312yes—
44C2×C22⋊D191—21no—
45C22×Dic191—21no—
46C2×C4×D191D411yescharacteristic
47C2×C4×D191D411yescharacteristic
48C2×C4×D191C2311yescharacteristic
49#15690277—24no—
50#15690275—24no—
51#15690270—24no—
52C2×C191⋊C8C2312yes—
53C4⋊Dic191—22no—
54C4⋊Dic191—21no—
55C4⋊Dic191—21no—
56C4⋊Dic191—21no—
57C4⋊Dic191C2311yescharacteristic
58C4⋊Dic191C2311yescharacteristic
59Dic191⋊C4—21no—
60Dic191⋊C4—22no—
61Dic191⋊C4—21no—
62Dic191⋊C4—21no—
63C22⋊Dic191—21no—
64C22⋊Dic191—21no—
65(C2×D191)⋊C4—22no—
66(C2×D191)⋊C4—21no—
67C4×Dic191—21no—
68C4×Dic191—22no—
69C4×Dic191D411yescharacteristic
70C4×Dic191D411yescharacteristic
71C4×Dic191C2311yescharacteristic
72C2×Q8×C191—21no—
73C2×Q8×C191C2311yescharacteristic
74C2×D4×C191C2311yescharacteristic
75C2×C8×C191C2312yes—
76C4⋊C4×C191—21no—
77C4⋊C4×C191C2312yes—
78C22⋊C4×C191—21no—
79C22⋊C4×C191—21no—
80C42×C191C2311yescharacteristic
81C22×D191C2×D411yescharacteristic
82Dic382—22no—
83Dic382—22no—
84C22⋊D191—42no—
85C2×Dic191—22no—
86C2×Dic191—21no—
87C2×Dic191—21no—
88C2×Dic191—42no—
89C2×Dic191C2×D411yescharacteristic
90C2×Dic191C2×D411yescharacteristic
91C2×Dic191C2×D411yescharacteristic
92C4×D191—22no—
93C4×D191—22no—
94C4×D191—41no—
95C191⋊C8#4114yes—
96C23×C191—21no—
97Q8×C191—22no—
98Q8×C191—42no—
99D4×C191—22no—
100C2×C4×C191—21no—
101C2×C4×C191—22no—
102C2×C4×C191C2×D411yescharacteristic
103C2×C4×C191C2×D411yescharacteristic
104C2×C4×C191C2411yescharacteristic
105C8×C191—22no—
106C22×C191—41no—
107C22×C191—41no—
108C22×C191—41no—
109C22×C191C22×D411yescharacteristic
110C2×D191—21no—
111C2×D191—21no—
112C2×D191—21no—
113C4×C191—21no—
114C4×C191—21no—
115C4×C191—41no—
116C4×C191—42no—
117C4×C191C2×#4112yes—
118Dic191—21no—
119Dic191—21no—
120Dic191—42no—
121Dic191—41no—
122Dic191—41no—
123Dic191#13712yes—
124C2×C191—81no—
125C2×C191#53211yescharacteristic
126C2×C191C2×#13711yescharacteristic
127C2×C191#57511yescharacteristic
128D191—41no—
129C191#317211yescharacteristic, complemented, Sylow
130#3172—1911nomaximal, Sylow
131#533—1911no—
132#529—1911no—
133#501—1911no—
134#497—1911no—
135#491—1911no—
136#489D19111yescharacteristic, complemented
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#125—1911no—
150#124C2×D19111yescharacteristic
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#107—1912no—
165#106—1912no—
166#103—1912no—
167#103—1912no—
168#103C2×D19112yes—
169#102C2×D19112yes—
170#97—1911no—
171#97C2×D19111yescharacteristic
172C4×C8—1912no—
173C2×Q8—21no—
174C2×Q8—1912no—
175C2×Q8C22×D19111yescharacteristic
176C2×D4—3821no—
177C2×D4C22×D19111yescharacteristic
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—3821no—
189C4⋊C4—3822no—
190C4⋊C4—3822no—
191C4⋊C4—3821no—
192C4⋊C4—3821no—
193C4⋊C4—3821no—
194C4⋊C4—3821no—
195C4⋊C4—3821no—
196C4⋊C4—1911no—
197C4⋊C4—1911no—
198C4⋊C4C22×D19112yes—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—3822no—
202C22⋊C4—3821no—
203C22⋊C4—3821no—
204C22⋊C4—3821no—
205C42—3821no—
206C42—3822no—
207C42—1911no—
208C42—1911no—
209C42—1911no—
210C42C22×D19111yescharacteristic
211C23—21no—
212C23—1911no—
213Q8—22no—
214Q8—42no—
215Q8—3822no—
216Q8—3822no—
217D4—22no—
218D4—7642no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—3822no—
222C2×C4—3822no—
223C2×C4—3822no—
224C2×C4—3821no—
225C2×C4—3821no—
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—22no—
235C8—1914no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—3821no—
240C22—3821no—
241C22—3821no—
242C22C2×D4×D19111yescharacteristic
243C4—21no—
244C4—21no—
245C4—42no—
246C4—41no—
247C4—3821no—
248C4—3821no—
249C4—1912no—
250C4—7641no—
251C4—7641no—
252C4—7642no—
253C4#41×D19112yes—
254C2—81no—
255C2—7641no—
256C2#2713321211yescharacteristic
257C2#137×D19111yescharacteristic
258C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#489×C191C2
socleC22×C191C22×D4
prime radical#20259293C22
hypercenter#489D191
hypocenterC191#3172
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #27133218C2semidirect
3. #27133069C2semidirect
4. #27133058C2semidirect
5. #27133032C2semidirect
6. #27132920C2semidirect
8. #27132755C2semidirect
9. #27132754C2semidirect
10. #27132615C2semidirect
11. #489×C191C2semidirect
129. C191#3172semidirect
136. #489D191semidirect

Maximal subgroups

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

subgroupindex
#271332182
#271331922
#271330692
#271330582
#271330322
#271329202
#271329082
#271327552
#271327542
#271326152
#489×C1912
#3172191