Finite Group Databasebeta

← #37574162

Subgroups — #37574162

Order 24448.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#27133170C211yesmaximal, characteristic
2#27132637C211yesmaximal, characteristic, complemented
3#27132454C212yesmaximal, complemented
4#27132451C212yesmaximal, complemented
5#413×C191C211yesmaximal, characteristic, complemented
6C22×C4×D191C411yescharacteristic, complemented
7#20259303C411yescharacteristic, complemented
8#20259303C2211yescharacteristic
9#20259298—21no—
10#20259293—21no—
11#20259290C411yescharacteristic, complemented
12#20259290C2211yescharacteristic
13C4×C22⋊D191—21no—
14C4×C22⋊D191—21no—
15#20259281—21no—
16#20259281C411yescharacteristic, complemented
17#20259237—21no—
18#20259237C2212yes—
19C2×#41×C191C2211yescharacteristic
20C2×M16×C191—21no—
21#101×C191—21no—
22#100×C191C2212yes—
23C23×D191C2×C411yescharacteristic
24C2×C22⋊D191—21no—
25C2×C22⋊D191—21no—
26C22×Dic191C2×C411yescharacteristic
27C2×C4×D191—21no—
28C2×C4×D191—21no—
29C2×C4×D191—41no—
30C2×C4×D191—41no—
31C4⋊Dic191—21no—
32C4⋊Dic191—41no—
33Dic191⋊C4—21no—
34Dic191⋊C4—21no—
35C22⋊Dic191—21no—
36C22⋊Dic191—21no—
37C22⋊Dic191—22no—
38C22⋊Dic191C2×C411yescharacteristic
39C22⋊Dic191C2×C411yescharacteristic
40(C2×D191)⋊C4—21no—
41(C2×D191)⋊C4—21no—
42C4×Dic191—21no—
43C4×Dic191—41no—
44#41×C191—41no—
45#41×C191—41no—
46C2×Q8×C191D411yescharacteristic
47C2×D4×C191C2×C411yescharacteristic
48C2×D4×C191D411yescharacteristic
49C2×D4×C191C2311yescharacteristic
50C22×C4×C191C2×C411yescharacteristic
51C22×C4×C191D411yescharacteristic
52C22×C4×C191D411yescharacteristic
53M16×C191—22no—
54M16×C191—41no—
55C2×C8×C191—41no—
56C22×D191—21no—
57C22×D191—42no—
58C22×D191—41no—
59C22⋊D191—42no—
60C22⋊D191—42no—
61C2×Dic191—21no—
62C2×Dic191—21no—
63C2×Dic191—21no—
64C2×Dic191—41no—
65C2×Dic191—42no—
66C4×D191—42no—
67C4×D191—41no—
68C4×D191—41no—
69C23×C191C22⋊C411yescharacteristic
70C23×C191C22×C411yescharacteristic
71C23×C191C2×D411yescharacteristic
72Q8×C191—41no—
73D4×C191—41no—
74D4×C191—41no—
75D4×C191—41no—
76C2×C4×C191—21no—
77C2×C4×C191—21no—
78C2×C4×C191—21no—
79C2×C4×C191—21no—
80C2×C4×C191—21no—
81C2×C4×C191—21no—
82C2×C4×C191C22⋊C411yescharacteristic
83C2×C4×C191C22⋊C411yescharacteristic
84C2×C4×C191C22⋊C411yescharacteristic
85C2×C4×C191C2×D411yescharacteristic
86C8×C191—42no—
87C22×C191—21no—
88C22×C191—21no—
89C22×C191—21no—
90C22×C191—41no—
91C22×C191—41no—
92C22×C191—41no—
93C22×C191C2×C22⋊C411yescharacteristic
94C2×D191—22no—
95C2×D191—41no—
96C2×D191—42no—
97C2×D191—41no—
98C2×D191—42no—
99C4×C191—21no—
100C4×C191—21no—
101C4×C191—21no—
102C4×C191#9911yescharacteristic
103C4×C191#9911yescharacteristic
104Dic191—41no—
105Dic191—41no—
106Dic191—41no—
107Dic191—82no—
108C2×C191—21no—
109C2×C191—41no—
110C2×C191—41no—
111C2×C191—41no—
112C2×C191C2×#9911yescharacteristic
113D191—42no—
114C191#212011yescharacteristic, complemented, Sylow
115#2120—1911nomaximal, Sylow
116#525—1911no—
117#413D19111yescharacteristic, complemented
118#410—1911no—
119#352—1912no—
120AΣL(1,𝔽16)—1912no—
121C2×#41C2×D19111yescharacteristic
122C23×C4—1911no—
123C2×M16—21no—
124#123—1911no—
125#123—1911no—
126C22⋊Q8—3821no—
127#121—3821no—
128#120—1911no—
129#120—1911no—
130C4×D4—3821no—
131C4×D4—3821no—
132#117—3821no—
133#117—1911no—
134#101—21no—
135#100C2×D19112yes—
136#99—3821no—
137#99—1912no—
138C24—1911no—
139#41—41no—
140#41—41no—
141C2×Q8C22⋊D19111yescharacteristic
142C2×D4—3821no—
143C2×D4—3821no—
144C2×D4C4×D19111yescharacteristic
145C2×D4D76411yescharacteristic
146C2×D4C22×D19111yescharacteristic
147C22×C4—3821no—
148C22×C4—3821no—
149C22×C4—1911no—
150C22×C4—7641no—
151C22×C4—7641no—
152C22×C4C4×D19111yescharacteristic
153C22×C4C22⋊D19111yescharacteristic
154C22×C4D76411yescharacteristic
155M16—22no—
156M16—41no—
157C2×C8—41no—
158C4⋊C4—3821no—
159C4⋊C4—3821no—
160C4⋊C4—3821no—
161C4⋊C4—7641no—
162C22⋊C4—3821no—
163C22⋊C4—3821no—
164C22⋊C4—3821no—
165C22⋊C4—3821no—
166C22⋊C4—3822no—
167C22⋊C4—1911no—
168C22⋊C4—1911no—
169C42—3821no—
170C42—7641no—
171C23—3821no—
172C23—7641no—
173C23—7642no—
174C23(C2×D191)⋊C411yescharacteristic
175C23C2×C4×D19111yescharacteristic
176C23C2×D76411yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—7642no—
182D4—7642no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—3821no—
190C2×C4—3821no—
191C2×C4—3821no—
192C2×C4—7642no—
193C2×C4—7642no—
194C2×C4—7641no—
195C2×C4—7641no—
196C2×C4—7641no—
197C2×C4(C2×D191)⋊C411yescharacteristic
198C2×C4(C2×D191)⋊C411yescharacteristic
199C2×C4(C2×D191)⋊C411yescharacteristic
200C2×C4C2×C22⋊D19111yescharacteristic
201C8—42no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—3822no—
209C22—7641no—
210C22—7642no—
211C22—7642no—
212C22—7641no—
213C22C2×(C2×D191)⋊C411yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—15282no—
218C4—7641no—
219C4—7641no—
220C4—7641no—
221C4#2025923611yescharacteristic
222C4#2025923611yescharacteristic
223C2—21no—
224C2—41no—
225C2—41no—
226C2—41no—
227C2—7642no—
228C2C2×#2025923611yescharacteristic
subgroupG/H
centerC4#20259236
commutatorC23×C191C22×C4
FrattiniC2×D4C22×D191
Fitting#413×C191C2
socleC2×C191C2×#99
prime radical#27133170C2
cosocleC2×D4×C191C23
hypercenter#413D191
hypocenterC191#2120
LCSG ▷ C23×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #27132637C2semidirect
3. #27132454C2semidirect
4. #27132451C2semidirect
5. #413×C191C2semidirect
6. C22×C4×D191C4semidirect
7. #20259303C4semidirect
11. #20259290C4semidirect
16. #20259281C4semidirect
114. C191#2120semidirect
117. #413D191semidirect

Maximal subgroups

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

subgroupindex
#271331702
#271326372
#271324542
#271324512
#413×C1912
#2120191