Finite Group Databasebeta

← #37583230

Subgroups — #37583230

Order 24448.

366 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 125 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#27133220C211yesmaximal, characteristic, complemented
2#27133073C211yesmaximal, characteristic, complemented
3#27133058C211yesmaximal, characteristic, complemented
4#27133033C211yesmaximal, characteristic, complemented
5#27133022C211yesmaximal, characteristic, complemented
6#27132902C212yesmaximal, complemented
7#27132890C212yesmaximal
8#27132768C212yesmaximal, complemented
9#27132763C212yesmaximal, complemented
10C2×#20259233C211yesmaximal, characteristic, complemented
11#529×C191C211yesmaximal, characteristic
12C2×#15690277C2212yes—
13C2×#15690275C2212yes—
14C22×C191⋊C8C2212yes—
15#20259322C2211yescharacteristic
16#20259322C2211yescharacteristic
17#20259314—21no—
18#20259301—21no—
19#20259301C2212yes—
20C4×Dic382—21no—
21C4×Dic382C2212yes—
22#20259281—21no—
23#20259255C2212yes—
24#20259253C2212yes—
25#20259246C2212yes—
26#20259245C2212yes—
27#20259245C2212yes—
28#20259243C2212yes—
29#20259233C2212yes—
30#20259233C2211yescharacteristic
31#20259233C2211yescharacteristic
32#126×C191—21no—
33#125×C191C2211yescharacteristic
34#124×C191C2211yescharacteristic
35#123×C191—21no—
36C22⋊Q8×C191C2211yescharacteristic
37#121×C191C2211yescharacteristic
38C4×Q8×C191C2211yescharacteristic
39C4×D4×C191—21no—
40C4×D4×C191C2211yescharacteristic
41#117×C191—21no—
42C2×C42×C191C2211yescharacteristic
43C2×Dic382—21no—
44C2×Dic382C2312yes—
45#15690277—24no—
46#15690275—24no—
47C2×C191⋊C8—24no—
48C2×C191⋊C8C2312yes—
49C2×C191⋊C8C2312yes—
50C4⋊Dic191—21no—
51C4⋊Dic191—21no—
52C4⋊Dic191—21no—
53C4⋊Dic191C2312yes—
54Dic191⋊C4—22no—
55Dic191⋊C4—22no—
56C22⋊Dic191—22no—
57C4×Dic191—22no—
58C2×Q8×C191C2311yescharacteristic
59C2×D4×C191—21no—
60C2×D4×C191C2311yescharacteristic
61C22×C4×C191—21no—
62C22×C4×C191D411yescharacteristic
63C22×C4×C191D411yescharacteristic
64C22×C4×C191C2311yescharacteristic
65C4⋊C4×C191—21no—
66C4⋊C4×C191—21no—
67C4⋊C4×C191—21no—
68C4⋊C4×C191—21no—
69C4⋊C4×C191C2311yescharacteristic
70C4⋊C4×C191C2311yescharacteristic
71C22⋊C4×C191—21no—
72C22⋊C4×C191—21no—
73C22⋊C4×C191—21no—
74C22⋊C4×C191—21no—
75C22⋊C4×C191—21no—
76C42×C191—21no—
77C42×C191D411yescharacteristic
78C42×C191D411yescharacteristic
79C42×C191C2311yescharacteristic
80C42×C191C2311yescharacteristic
81Dic382—22no—
82Dic382—22no—
83Dic382—42no—
84C2×Dic191—22no—
85C2×Dic191—22no—
86C191⋊C8—22no—
87C191⋊C8#4114yes—
88C23×C191—21no—
89C23×C191C2×D411yescharacteristic
90Q8×C191—22no—
91D4×C191—22no—
92D4×C191—42no—
93C2×C4×C191—22no—
94C2×C4×C191—22no—
95C2×C4×C191—21no—
96C2×C4×C191—21no—
97C2×C4×C191—21no—
98C2×C4×C191—41no—
99C2×C4×C191—42no—
100C2×C4×C191C2×D411yescharacteristic
101C2×C4×C191C2×D411yescharacteristic
102C2×C4×C191C2×D411yescharacteristic
103C2×C4×C191C2×D411yescharacteristic
104C2×C4×C191C2×D411yescharacteristic
105C2×C4×C191C2411yescharacteristic
106C22×C191—21no—
107C22×C191—21no—
108C22×C191—41no—
109C22×C191—41no—
110C22×C191—41no—
111C22×C191#13712yes—
112C22×C191C22×D411yescharacteristic
113C4×C191—21no—
114C4×C191—21no—
115C4×C191—21no—
116C4×C191—21no—
117C4×C191—21no—
118C4×C191—41no—
119C4×C191—41no—
120C4×C191—41no—
121C4×C191C2×#4112yes—
122Dic191—42no—
123Dic191—42no—
124C2×C191—22no—
125C2×C191—81no—
126C2×C191#53211yescharacteristic
127C2×C191C2×#13711yescharacteristic
128C2×C191#57511yescharacteristic
129C191#318111yescharacteristic, complemented, Sylow
130#3181—1911nomaximal, Sylow
131#533—1911no—
132#529D19111yescharacteristic
133#501—1911no—
134#497—1911no—
135#491—1911no—
136#489—1911no—
137#467—1912no—
138#465—1912no—
139#441—1912no—
140#440—1912no—
141C2×#97—1911no—
142C2×Q16—1912no—
143C2×SD16—1912no—
144C22×C8—1912no—
145C4⋊Q8—1911no—
146C4⋊Q8—1911no—
147#126—21no—
148#125—3821no—
149#125C2×D19111yescharacteristic
150#124C2×D19111yescharacteristic
151#123—21no—
152C22⋊Q8—3821no—
153C22⋊Q8—1912no—
154C22⋊Q8C2×D19111yescharacteristic
155#121C2×D19111yescharacteristic
156C4×Q8—3821no—
157C4×Q8—1912no—
158C4×Q8C2×D19111yescharacteristic
159C4×D4—21no—
160C4×D4C2×D19111yescharacteristic
161#117—21no—
162#117—3821no—
163C2×C42C2×D19111yescharacteristic
164#107—1912no—
165#106—1912no—
166#103—1912no—
167#103—1912no—
168#103—1912no—
169#102—1912no—
170#97—1912no—
171#97—1911no—
172#97—1911no—
173C2×Q8—3821no—
174C2×Q8—1912no—
175C2×Q8C22×D19111yescharacteristic
176C2×D4—21no—
177C2×D4C22×D19111yescharacteristic
178C22×C4—21no—
179C22×C4C22⋊D19111yescharacteristic
180C22×C4C22⋊D19111yescharacteristic
181C22×C4C22×D19111yescharacteristic
182Q16—3824no—
183SD16—3824no—
184C2×C8—3824no—
185C2×C8—1912no—
186C2×C8—1912no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—3822no—
192C4⋊C4—3822no—
193C4⋊C4—3821no—
194C4⋊C4—3821no—
195C4⋊C4—3821no—
196C4⋊C4—1912no—
197C4⋊C4C22×D19111yescharacteristic
198C4⋊C4C22×D19111yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—3822no—
205C42—21no—
206C42—3822no—
207C42C22⋊D19111yescharacteristic
208C42C22⋊D19111yescharacteristic
209C42C22×D19111yescharacteristic
210C42C22×D19111yescharacteristic
211C23—21no—
212C23C2×C22⋊D19111yescharacteristic
213Q8—22no—
214Q8—3822no—
215Q8—3822no—
216Q8—7642no—
217D4—22no—
218D4—42no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—3822no—
227C2×C4—3822no—
228C2×C4C2×C22⋊D19111yescharacteristic
229C2×C4C2×C22⋊D19111yescharacteristic
230C2×C4C2×C22⋊D19111yescharacteristic
231C2×C4C2×C22⋊D19111yescharacteristic
232C2×C4C2×C22⋊D19111yescharacteristic
233C2×C4C23×D19111yescharacteristic
234C8—3822no—
235C8—1914no—
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22#2025935212yes—
242C22C22×C22⋊D19111yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—41no—
251C4—7642no—
252C4—7642no—
253C4#41×D19112yes—
254C2—22no—
255C2—81no—
256C2#2713320911yescharacteristic
257C2C2×#2025935211yescharacteristic
258C2#2713346111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#529×C191C2
socleC22×C191C22×D4
prime radical#121×C191C22
hypercenter#529D191
hypocenterC191#3181
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #27133220C2semidirect
2. #27133073C2semidirect
3. #27133058C2semidirect
4. #27133033C2semidirect
5. #27133022C2semidirect
6. #27132902C2semidirect
8. #27132768C2semidirect
9. #27132763C2semidirect
10. C2×#20259233C2semidirect
129. C191#3181semidirect

Maximal subgroups

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

subgroupindex
#271332202
#271330732
#271330582
#271330332
#271330222
#271329022
#271328902
#271327682
#271327632
C2×#202592332
#529×C1912
#3181191