Finite Group Databasebeta

← #37571499

Subgroups — #37571499

Order 24448.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#27133253C211yesmaximal, characteristic
2#27132802C211yesmaximal, characteristic, complemented
3#27132574C211yesmaximal, characteristic, complemented
4#27132400C211yesmaximal, characteristic, complemented
5#27132398C211yesmaximal, characteristic, complemented
6#27132386C211yesmaximal, characteristic, complemented
7#451×C191C211yesmaximal, characteristic, complemented
8C2×#15690274—21no—
9#20259317—21no—
10#20259311—21no—
11#20259307—21no—
12#20259297—21no—
13#20259296C2211yescharacteristic
14#20259292—21no—
15(C22⋊D191)⋊C4—21no—
16C2×(C2×D191)⋊C4—21no—
17C2×(C2×D191)⋊C4C2211yescharacteristic
18#20259246—21no—
19C22⋊(C191⋊C8)C2211yescharacteristic
20C191⋊#95—21no—
21C191⋊#95C2211yescharacteristic
22C22×Q8×C191C2211yescharacteristic
23C2×Q16×C191—21no—
24C22⋊Q8×C191C2211yescharacteristic
25#103×C191—21no—
26C22⋊C8×C191C2211yescharacteristic
27C23×D191D411yescharacteristic
28C2×D764—21no—
29C2×D764—21no—
30C2×D764D411yescharacteristic
31C2×C22⋊D191—21no—
32C22×Dic191—21no—
33C22×Dic191D411yescharacteristic
34C2×C4×D191—21no—
35#15690274—42no—
36C2×C191⋊C8—21no—
37C4⋊Dic191D411yescharacteristic
38Dic191⋊C4—21no—
39(C2×D191)⋊C4—21no—
40(C2×D191)⋊C4—21no—
41(C2×D191)⋊C4—21no—
42(C2×D191)⋊C4—21no—
43(C2×D191)⋊C4—21no—
44(C2×D191)⋊C4—21no—
45(C2×D191)⋊C4—42no—
46C4×Dic191—21no—
47C2×Q8×C191—21no—
48C2×Q8×C191—21no—
49C2×Q8×C191—42no—
50C2×Q8×C191D411yescharacteristic
51C22×C4×C191—21no—
52C22×C4×C191C2311yescharacteristic
53Q16×C191—42no—
54C2×C8×C191—21no—
55C4⋊C4×C191—21no—
56C4⋊C4×C191D411yescharacteristic
57C22⋊C4×C191—21no—
58C22×D191—21no—
59C22×D191—21no—
60C22×D191—21no—
61C22×D191—21no—
62C22×D191—41no—
63C22×D191—41no—
64D764—41no—
65D764—81no—
66D764—81no—
67C22⋊D191—41no—
68C22⋊D191—41no—
69C2×Dic191—21no—
70C2×Dic191—21no—
71C2×Dic191—41no—
72C2×Dic191—41no—
73C2×Dic191—41no—
74C2×Dic191—41no—
75C4×D191—81no—
76C191⋊C8—41no—
77C23×C191C2×D411yescharacteristic
78Q8×C191—22no—
79Q8×C191—22no—
80Q8×C191—41no—
81Q8×C191—41no—
82Q8×C191—41no—
83C2×C4×C191—22no—
84C2×C4×C191—21no—
85C2×C4×C191—21no—
86C2×C4×C191—41no—
87C2×C4×C191—41no—
88C2×C4×C191—41no—
89C2×C4×C191C2×D411yescharacteristic
90C2×C4×C191C2×D411yescharacteristic
91C8×C191—41no—
92C22×C191—21no—
93C22×C191—21no—
94C22×C191#12011yescharacteristic
95C22×C191#13511yescharacteristic
96C22×C191#13711yescharacteristic
97C2×D191—42no—
98C2×D191—41no—
99C2×D191—41no—
100C2×D191—41no—
101C2×D191—41no—
102C2×D191—41no—
103C2×D191—41no—
104C2×D191—81no—
105C4×C191—21no—
106C4×C191—21no—
107C4×C191—41no—
108C4×C191—42no—
109C4×C191—41no—
110Dic191—41no—
111Dic191—41no—
112Dic191—81no—
113C2×C191—21no—
114C2×C191—21no—
115C2×C191#45211yescharacteristic
116C2×C191#45511yescharacteristic
117C2×C191B(4,𝔽2)11yescharacteristic
118D191—81no—
119D191—81no—
120C191#160611yescharacteristic, complemented, Sylow
121#1606—1911nomaximal, Sylow
122#538—1911no—
123#451D19111yescharacteristic, complemented
124#448—1911no—
125#393—1911no—
126#328—1911no—
127#327—1911no—
128#323—1911no—
129C22×Q8C2×D19111yescharacteristic
130C2×Q16—21no—
131C2×SD16—3821no—
132#126—3821no—
133#124—3821no—
134#123—3821no—
135C22⋊Q8C2×D19111yescharacteristic
136#121—3821no—
137#121—1911no—
138#120—3821no—
139C4×D4—3821no—
140C2×C22⋊C4—3821no—
141C2×C22⋊C4—1911no—
142#103—21no—
143#103—3821no—
144C22⋊C8—1911no—
145C22⋊C8C2×D19111yescharacteristic
146#95—3821no—
147#95—1911no—
148C24—1911no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8C22⋊D19111yescharacteristic
153C2×D4—3821no—
154C2×D4—3821no—
155C2×D4—3821no—
156C2×D4—1911no—
157C22×C4—21no—
158C22×C4—3821no—
159C22×C4—3821no—
160C22×C4—1911no—
161C22×C4C22×D19111yescharacteristic
162Q16—42no—
163SD16—7642no—
164C2×C8—21no—
165C2×C8—3821no—
166C4⋊C4—21no—
167C4⋊C4—3821no—
168C4⋊C4—1911no—
169C4⋊C4C22⋊D19111yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—3821no—
172C22⋊C4—3821no—
173C22⋊C4—3821no—
174C22⋊C4—3821no—
175C22⋊C4—3821no—
176C22⋊C4—3821no—
177C22⋊C4—7642no—
178C42—3821no—
179C23—3821no—
180C23—3821no—
181C23—3821no—
182C23—3821no—
183C23—7641no—
184C23—7641no—
185C23D4×D19111yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191D4—15281no—
192D4—15281no—
193D4—7641no—
194D4—7641no—
195D4—7641no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—3821no—
203C2×C4—3821no—
204C2×C4—15281no—
205C2×C4—7641no—
206C2×C4—7641no—
207C2×C4—7641no—
208C2×C4—7641no—
209C2×C4D4×D19111yescharacteristic
210C2×C4C2×C22⋊D19111yescharacteristic
211C8—41no—
212C8—7641no—
213C22—21no—
214C22—21no—
215C22—15281no—
216C22—7642no—
217C22—7641no—
218C22—7641no—
219C22—7641no—
220C22—7641no—
221C22—7641no—
222C22—7641no—
223C22#2025929011yescharacteristic
224C22#2025934511yescharacteristic
225C22#2025935411yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—42no—
230C4—41no—
231C4—15281no—
232C4—7641no—
233C4—7641no—
234C2—21no—
235C2—21no—
236C2—15281no—
237C2—15281no—
238C2#2713282411yescharacteristic
239C2#2713283911yescharacteristic
240C2#2713284911yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#451×C191C2
socleC22×C191#120
prime radical#27133253C2
hypercenter#451D191
hypocenterC191#1606
LCSG ▷ C22×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #27132802C2semidirect
3. #27132574C2semidirect
4. #27132400C2semidirect
5. #27132398C2semidirect
6. #27132386C2semidirect
7. #451×C191C2semidirect
120. C191#1606semidirect
123. #451D191semidirect

Maximal subgroups

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

subgroupindex
#271332532
#271328022
#271325742
#271324002
#271323982
#271323862
#451×C1912
#1606191