Finite Group Databasebeta

← #37583292

Subgroups — #37583292

Order 24448.

416 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#20259301C211yesmaximal, characteristic
2#27132998C212yesmaximal, complemented
3#27132996C212yesmaximal, complemented
4#27132902C214yesmaximal, complemented
5#27132819C212yesmaximal, complemented
6#27132818C212yesmaximal, complemented
7C2×C22⋊(C191⋊C8)C211yesmaximal, characteristic
8C2×C22⋊Q8×C191C211yesmaximal, characteristic
9C22×Dic382C2211yescharacteristic
10C2×#15690277C2214yescomplemented
11C22×C191⋊C8C2212yes—
12#20259301—21no—
13#20259301—21no—
14#20259301C2212yes—
15#20259301C2212yes—
16C2×C4⋊Dic191C2211yescharacteristic
17C2×Dic191⋊C4—21no—
18C2×C22⋊Dic191—21no—
19#20259255C2214yescomplemented
20#20259246C2214yescomplemented
21#20259245C2214yescomplemented
22C22⋊(C191⋊C8)C2212yes—
23C22⋊(C191⋊C8)C2212yes—
24C22×Q8×C191C2211yescharacteristic
25C23×C4×C191C2211yescharacteristic
26C22⋊Q8×C191—21no—
27C22⋊Q8×C191—21no—
28C22⋊Q8×C191C2212yes—
29C22⋊Q8×C191C2212yes—
30C2×C4⋊C4×C191—21no—
31C2×C4⋊C4×C191C2211yescharacteristic
32C2×C22⋊C4×C191—21no—
33C2×Dic382—21no—
34C2×Dic382—42no—
35C2×Dic382C2312yes—
36C22×Dic191—21no—
37C22×Dic191—21no—
38#15690277—44no—
39C2×C191⋊C8—42no—
40C2×C191⋊C8C2314yes—
41C4⋊Dic191—21no—
42C4⋊Dic191C2312yes—
43Dic191⋊C4—22no—
44Dic191⋊C4—22no—
45C22⋊Dic191—22no—
46C22⋊Dic191—22no—
47C24×C191D411yescharacteristic
48C2×Q8×C191—21no—
49C2×Q8×C191—42no—
50C2×Q8×C191C2312yes—
51C22×C4×C191—21no—
52C22×C4×C191—21no—
53C22×C4×C191—42no—
54C22×C4×C191D411yescharacteristic
55C22×C4×C191D411yescharacteristic
56C22×C4×C191D411yescharacteristic
57C22×C4×C191C2311yescharacteristic
58C22×C4×C191C2311yescharacteristic
59C22×C4×C191C2311yescharacteristic
60C4⋊C4×C191—22no—
61C4⋊C4×C191—22no—
62C4⋊C4×C191—21no—
63C4⋊C4×C191C2312yes—
64C22⋊C4×C191—22no—
65C22⋊C4×C191—22no—
66Dic382—24no—
67Dic382—42no—
68C2×Dic191—22no—
69C2×Dic191—22no—
70C2×Dic191—42no—
71C2×Dic191—42no—
72C191⋊C8—24no—
73C23×C191—41no—
74C23×C191—41no—
75C23×C191Q1614yescomplemented
76C23×C191C2×D411yescharacteristic
77C23×C191C2×D411yescharacteristic
78C23×C191C2×D411yescharacteristic
79Q8×C191—24no—
80Q8×C191—42no—
81C2×C4×C191—22no—
82C2×C4×C191—22no—
83C2×C4×C191—22no—
84C2×C4×C191—22no—
85C2×C4×C191—22no—
86C2×C4×C191—41no—
87C2×C4×C191—41no—
88C2×C4×C191—41no—
89C2×C4×C191—42no—
90C2×C4×C191—42no—
91C2×C4×C191C2×D411yescharacteristic
92C2×C4×C191C2×D411yescharacteristic
93C2×C4×C191C2×D411yescharacteristic
94C2×C4×C191C2411yescharacteristic
95C22×C191—21no—
96C22×C191—21no—
97C22×C191—21no—
98C22×C191—21no—
99C22×C191—21no—
100C22×C191—21no—
101C22×C191—44no—
102C22×C191C2×Q1612yes—
103C22×C191C2×Q1612yes—
104C22×C191C2×Q1612yes—
105C22×C191C22×D411yescharacteristic
106C4×C191—21no—
107C4×C191—21no—
108C4×C191—21no—
109C4×C191—42no—
110C4×C191—42no—
111C4×C191H(5,𝔽2)12yes—
112Dic191—42no—
113Dic191—42no—
114C2×C191—22no—
115C2×C191—22no—
116C2×C191—22no—
117C2×C191#53411yescharacteristic
118C2×C191C22×Q1611yescharacteristic
119C2×C191#57511yescharacteristic
120C191#318911yescharacteristic, complemented, Sylow
121#3189—1911nomaximal, Sylow
122C2×C22⋊Q8—1911no—
123C2×C22⋊Q8D19111yescharacteristic
124#484—1912no—
125#484—1912no—
126#467—1914no—
127#451—1912no—
128#451—1912no—
129C2×C22⋊C8—1911no—
130C22×Q8—1911no—
131C22×Q8C2×D19111yescharacteristic
132C23×C4C2×D19111yescharacteristic
133C2×Q16—1914no—
134C22×C8—1912no—
135C22⋊Q8—21no—
136C22⋊Q8—21no—
137C22⋊Q8—3821no—
138C22⋊Q8—3821no—
139C22⋊Q8—1912no—
140C22⋊Q8—1912no—
141C22⋊Q8C2×D19112yes—
142C22⋊Q8C2×D19112yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—3821no—
145C2×C4⋊C4—1911no—
146C2×C4⋊C4C2×D19111yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—3821no—
149#107—1914no—
150#103—1914no—
151#103—1914no—
152C22⋊C8—1912no—
153C22⋊C8—1912no—
154C24C22⋊D19111yescharacteristic
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—3821no—
158C2×Q8—1912no—
159C2×Q8—7642no—
160C2×Q8C22×D19112yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—42no—
164C22×C4—3821no—
165C22×C4—3821no—
166C22×C4C22⋊D19111yescharacteristic
167C22×C4C22⋊D19111yescharacteristic
168C22×C4C22⋊D19111yescharacteristic
169C22×C4C22×D19111yescharacteristic
170C22×C4C22×D19111yescharacteristic
171C22×C4C22×D19111yescharacteristic
172Q16—7644no—
173C2×C8—1914no—
174C2×C8—7642no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—3822no—
179C4⋊C4—3822no—
180C4⋊C4—3821no—
181C4⋊C4—1912no—
182C4⋊C4C22×D19112yes—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—3822no—
186C22⋊C4—3822no—
187C23—41no—
188C23—41no—
189C23#1569027714yescomplemented
190C23C2×C22⋊D19111yescharacteristic
191C23C2×C22⋊D19111yescharacteristic
192C23C2×C22⋊D19111yescharacteristic
193Q8—24no—
194Q8—42no—
195Q8—3824no—
196Q8—7642no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—3822no—
208C2×C4—3822no—
209C2×C4—7642no—
210C2×C4—7642no—
211C2×C4C2×C22⋊D19111yescharacteristic
212C2×C4C2×C22⋊D19111yescharacteristic
213C2×C4C2×C22⋊D19111yescharacteristic
214C2×C4C23×D19111yescharacteristic
215C8—3824no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—44no—
223C22C2×#1569027712yes—
224C22C2×#1569027712yes—
225C22C2×#1569027712yes—
226C22C22×C22⋊D19111yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—42no—
231C4—42no—
232C4—7642no—
233C4—7642no—
234C4#2025936712yes—
235C2—22no—
236C2—22no—
237C2—22no—
238C2#2713322211yescharacteristic
239C2C22×#1569027711yescharacteristic
240C2#2713346111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×C22⋊Q8×C191C2
socleC22×C191C22×D4
prime radicalC24×C191D4
hypercenterC2×C22⋊Q8D191
hypocenterC191#3189
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #27132998C2semidirect
3. #27132996C2semidirect
4. #27132902C2semidirect
5. #27132819C2semidirect
6. #27132818C2semidirect
10. C2×#15690277C22semidirect
19. #20259255C22semidirect
20. #20259246C22semidirect
21. #20259245C22semidirect
75. C23×C191Q16semidirect
120. C191#3189semidirect
189. C23#15690277semidirect

Maximal subgroups

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

subgroupindex
C2×#202593012
#271329982
#271329962
#271329022
#271328192
#271328182
C2×C22⋊(C191⋊C8)2
C2×C22⋊Q8×C1912
#3189191