Finite Group Databasebeta

← #37572967

Subgroups — #37572967

Order 24448.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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#27133175C211yesmaximal, characteristic
2C2×#20259279C211yesmaximal, characteristic, complemented
3C2×#20259250C211yesmaximal, characteristic, complemented
4C2×#20259248C211yesmaximal, characteristic, complemented
5#27132626C211yesmaximal, characteristic, complemented
6#27132422C211yesmaximal, characteristic, complemented
7#339×C191C211yesmaximal, characteristic, complemented
8C2×#15690287C2211yescharacteristic
9C22×C4×D191C2211yescharacteristic
10C2×#15690271C2211yescharacteristic
11#20259306—21no—
12#20259302C412yescomplemented
13#20259297C412yescomplemented
14#20259292—21no—
15C4×D764—22no—
16#20259279—22no—
17#20259279—22no—
18C2×Dic191⋊C4—21no—
19C2×(C2×D191)⋊C4—21no—
20C2×C4×Dic191C2211yescharacteristic
21#20259250—22no—
22#20259248—22no—
23(C2×D191)⋊C8—22no—
24C2×M16×C191C2211yescharacteristic
25#117×C191C2211yescharacteristic
26C2×C42×C191C2211yescharacteristic
27C23×D191D411yescharacteristic
28#15690287—21no—
29#15690287—21no—
30#15690287—21no—
31#15690287—21no—
32C2×Dic382C2×C411yescharacteristic
33C2×D764—22no—
34C2×D764C2×C411yescharacteristic
35C2×C22⋊D191—21no—
36C22×Dic191—21no—
37C22×Dic191D411yescharacteristic
38C2×C4×D191—21no—
39C2×C4×D191—21no—
40C2×C4×D191—21no—
41C2×C4×D191—21no—
42C2×C4×D191—21no—
43C2×C4×D191C2×C412yes—
44C2×C4×D191D412yes—
45#15690271—21no—
46#15690271—21no—
47C2×C191⋊C8—21no—
48C4⋊Dic191—22no—
49Dic191⋊C4—24no—
50(C2×D191)⋊C4—24no—
51(C2×D191)⋊C4—22no—
52(C2×D191)⋊C4—22no—
53C4×Dic191—21no—
54C4×Dic191D412yescomplemented
55C22×C4×C191—21no—
56C22×C4×C191C2311yescharacteristic
57M16×C191—21no—
58M16×C191—21no—
59C2×C8×C191—21no—
60C4⋊C4×C191C2×C412yes—
61C22⋊C4×C191—21no—
62C42×C191—21no—
63C42×C191—21no—
64C42×C191D412yescomplemented
65C22×D191—21no—
66C22×D191—21no—
67C22×D191—21no—
68C22×D191—42no—
69C22×D191C22⋊C412yes—
70Dic382—21no—
71Dic382—21no—
72D764—21no—
73D764—21no—
74D764—44no—
75C22⋊D191—41no—
76C22⋊D191—41no—
77C2×Dic191—21no—
78C2×Dic191—21no—
79C2×Dic191—21no—
80C2×Dic191—22no—
81C2×Dic191—42no—
82C2×Dic191C22⋊C412yes—
83C4×D191—22no—
84C4×D191—22no—
85C4×D191—22no—
86C4×D191—22no—
87C4×D191—41no—
88C4×D191—41no—
89C191⋊C8—41no—
90C23×C191C2×D411yescharacteristic
91C2×C4×C191—22no—
92C2×C4×C191—22no—
93C2×C4×C191—42no—
94C2×C4×C191C22×C411yescharacteristic
95C2×C4×C191C2×D411yescharacteristic
96C2×C4×C191C2×D411yescharacteristic
97C2×C4×C191C2×D411yescharacteristic
98C2×C4×C191#4111yescharacteristic
99C2×C4×C191#4111yescharacteristic
100C8×C191—41no—
101C22×C191—22no—
102C22×C191C2×C22⋊C411yescharacteristic
103C22×C191#12011yescharacteristic
104C22×C191#12311yescharacteristic
105C2×D191—22no—
106C2×D191—22no—
107C2×D191—22no—
108C2×D191—42no—
109C2×D191—42no—
110C2×D191—41no—
111C2×D191—42no—
112C4×C191—42no—
113C4×C191—42no—
114C4×C191C4×D411yescharacteristic
115C4×C191C4×D411yescharacteristic
116C4×C191#12111yescharacteristic
117C4×C191#12111yescharacteristic
118Dic191—22no—
119Dic191—42no—
120Dic191—41no—
121C2×C191—21no—
122C2×C191—21no—
123C2×C191#38611yescharacteristic
124C2×C191#42112yes—
125D191—42no—
126D191—81no—
127C191#189711yescharacteristic, complemented, Sylow
128#1897—1911nomaximal, Sylow
129#525—1911no—
130C2×#117—1911no—
131C2×#104—1911no—
132C2×#104—1911no—
133#407—1911no—
134#339—1911no—
135#339D19111yescharacteristic, complemented
136C2×#41—1911no—
137C23×C4—1911no—
138C2×M16—1911no—
139C2×M16C2×D19111yescharacteristic
140#123—3821no—
141C22⋊Q8—1912no—
142#121—1912no—
143#120—3821no—
144C4×D4—3822no—
145#117—3822no—
146#117—3822no—
147#117C2×D19111yescharacteristic
148C2×C4⋊C4—3821no—
149C2×C22⋊C4—3821no—
150C2×C42—1911no—
151C2×C42C2×D19111yescharacteristic
152#104—3822no—
153#104—3822no—
154C22⋊C8—3822no—
155C24—1911no—
156#41—3821no—
157#41—3821no—
158#41—3821no—
159#41—3821no—
160C2×Q8—1911no—
161C2×D4—3822no—
162C2×D4—3821no—
163C2×D4—1911no—
164C22×C4—21no—
165C22×C4—3821no—
166C22×C4—3821no—
167C22×C4—3821no—
168C22×C4—3821no—
169C22×C4—3821no—
170C22×C4—3821no—
171C22×C4—1911no—
172C22×C4—1912no—
173C22×C4—1912no—
174C22×C4C22×D19111yescharacteristic
175M16—21no—
176M16—21no—
177M16—3821no—
178M16—3821no—
179C2×C8—21no—
180C2×C8—3821no—
181C4⋊C4—3824no—
182C4⋊C4—3822no—
183C4⋊C4C4×D19112yes—
184C22⋊C4—21no—
185C22⋊C4—3824no—
186C22⋊C4—3822no—
187C22⋊C4—3822no—
188C42—21no—
189C42—21no—
190C42—3821no—
191C42—1912no—
192C42D76412yescomplemented
193C23—3821no—
194C23—3821no—
195C23—3821no—
196C23—1912no—
197C23—7642no—
198C23D4×D19111yescharacteristic
199Q8—3821no—
200Q8—3821no—
201D4—3821no—
202D4—3821no—
203D4—7644no—
204D4—7641no—
205D4—7641no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—3821no—
210C2×C4—3821no—
211C2×C4—3822no—
212C2×C4—3822no—
213C2×C4—3822no—
214C2×C4—3822no—
215C2×C4—3821no—
216C2×C4—3822no—
217C2×C4—1912no—
218C2×C4—7642no—
219C2×C4—7641no—
220C2×C4—7641no—
221C2×C4C2×C4×D19111yescharacteristic
222C2×C4D4×D19111yescharacteristic
223C2×C4D4×D19111yescharacteristic
224C2×C4C2×D76411yescharacteristic
225C2×C4#1569028711yescharacteristic
226C2×C4#1569028811yescharacteristic
227C8—41no—
228C8—7641no—
229C22—22no—
230C22—3822no—
231C22—3822no—
232C22—3822no—
233C22—7642no—
234C22—7642no—
235C22—7642no—
236C22—7641no—
237C22C22⋊C4×D19111yescharacteristic
238C22#2025929211yescharacteristic
239C22#2025930711yescharacteristic
240C4—42no—
241C4—42no—
242C4—3822no—
243C4—7642no—
244C4—7641no—
245C4#2025928411yescharacteristic
246C4C4×D76411yescharacteristic
247C4#2025929511yescharacteristic
248C4#2025929711yescharacteristic
249C2—21no—
250C2—21no—
251C2—15281no—
252C2—7642no—
253C2#2713254111yescharacteristic
254C2#2713267812yes—
subgroupG/H
centerC2×C4D4×D191
commutatorC2×C4×C191C22×C4
FrattiniC22×C4C22×D191
Fitting#339×C191C2
socleC22×C191C2×C22⋊C4
prime radical#27133175C2
cosocleC22×C4×C191C23
hypercenter#339D191
hypocenterC191#1897
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×#20259279C2semidirect
3. C2×#20259250C2semidirect
4. C2×#20259248C2semidirect
5. #27132626C2semidirect
6. #27132422C2semidirect
7. #339×C191C2semidirect
12. #20259302C4semidirect
13. #20259297C4semidirect
54. C4×Dic191D4semidirect
64. C42×C191D4semidirect
127. C191#1897semidirect
135. #339D191semidirect
192. C42D764semidirect

Maximal subgroups

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

subgroupindex
#271331752
C2×#202592792
C2×#202592502
C2×#202592482
#271326262
#271324222
#339×C1912
#1897191