Finite Group Databasebeta

← #37583908

Subgroups — #37583908

Order 24448.

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

normal subgroups only

This group has 386 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#27133213C211yesmaximal, characteristic
3#27132959C212yesmaximal, complemented
4#27132944C212yesmaximal, complemented
5#27132909C214yesmaximal, complemented
6#27132869C212yesmaximal, complemented
7#27132864C212yesmaximal, complemented
8#423×C191C211yesmaximal, characteristic, complemented
9C2×#15690287—21no—
10C2×#15690276C2214yes—
11#20259322C2211yescharacteristic
12#20259322C2211yescharacteristic
13#20259322C2211yescharacteristic
14#20259319C2211yescharacteristic
15#20259314—21no—
16#20259310—21no—
17#20259301—21no—
18#20259301C2212yes—
19#20259296—21no—
20#20259296C2212yes—
21C4×Dic382—21no—
22C4×Dic382C2212yes—
23C4×D764—21no—
24C4×D764C2212yes—
25#20259281—21no—
26#20259256C2214yes—
27#20259247C2214yes—
28#20259244C2214yes—
29C2×M16×C191C2212yes—
30C2×C42×C191C2211yescharacteristic
31C4⋊C8×C191C2212yes—
32C4⋊C8×C191C2212yes—
33#15690287—44no—
34C2×Dic382—21no—
35C2×Dic382—21no—
36C2×Dic382C2312yes—
37C2×D764—21no—
38C2×D764—21no—
39C2×D764C2312yes—
40C2×C22⋊D191—22no—
41C2×C4×D191—22no—
42#15690276—44no—
43C4⋊Dic191—21no—
44C4⋊Dic191—21no—
45C4⋊Dic191—21no—
46C4⋊Dic191—21no—
47C4⋊Dic191C2312yes—
48C4⋊Dic191C2312yes—
49Dic191⋊C4—22no—
50Dic191⋊C4—22no—
51C22⋊Dic191—22no—
52(C2×D191)⋊C4—22no—
53(C2×D191)⋊C4—22no—
54C4×Dic191—22no—
55C22×C4×C191D411yescharacteristic
56C22×C4×C191D411yescharacteristic
57C22×C4×C191C2311yescharacteristic
58M16×C191—42no—
59C2×C8×C191C2314yes—
60C42×C191D411yescharacteristic
61C42×C191D411yescharacteristic
62C42×C191C2311yescharacteristic
63C42×C191C2311yescharacteristic
64C22×D191—22no—
65Dic382—22no—
66Dic382—22no—
67Dic382—42no—
68Dic382—42no—
69D764—22no—
70D764—22no—
71D764—42no—
72D764—42no—
73C22⋊D191—44no—
74C2×Dic191—22no—
75C2×Dic191—22no—
76C2×Dic191—22no—
77C4×D191—44no—
78C23×C191C2×D411yescharacteristic
79C2×C4×C191—22no—
80C2×C4×C191—22no—
81C2×C4×C191—21no—
82C2×C4×C191—21no—
83C2×C4×C191C2×D411yescharacteristic
84C2×C4×C191C2×D411yescharacteristic
85C2×C4×C191C2×D411yescharacteristic
86C2×C4×C191C2×D411yescharacteristic
87C2×C4×C191C2×D411yescharacteristic
88C2×C4×C191C2411yescharacteristic
89C8×C191—24no—
90C22×C191—21no—
91C22×C191—21no—
92C22×C191—21no—
93C22×C191C22×D411yescharacteristic
94C2×D191—42no—
95C2×D191—42no—
96C2×D191—42no—
97C4×C191—21no—
98C4×C191—21no—
99C4×C191—21no—
100C4×C191AGL(1,ℤ/8)12yes—
101C4×C191#13712yes—
102C4×C191H(5,𝔽2)11yescharacteristic
103C4×C1912−1+411yescharacteristic
104Dic191—42no—
105Dic191—42no—
106Dic191—42no—
107C2×C191—41no—
108C2×C191#53711yescharacteristic
109C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C191C2×#13711yescharacteristic
111D191—82no—
112C191#325411yescharacteristic, complemented, Sylow
113#3254—1911nomaximal, Sylow
114#533—1911no—
115#532—1911no—
116#477—1912no—
117#474—1912no—
118#468—1914no—
119#461—1912no—
120#460—1912no—
121#423D19111yescharacteristic, complemented
122C2×#41—3821no—
123C2×SD16—1914no—
124C2×M16C2×D19112yes—
125C4⋊Q8—1911no—
126C4⋊Q8—1911no—
127C4⋊Q8—1911no—
128#127—1911no—
129#125—3821no—
130#124—3821no—
131C22⋊Q8—3821no—
132C22⋊Q8—1912no—
133#121—3821no—
134#121—1912no—
135C4×Q8—3821no—
136C4×Q8—1912no—
137C4×D4—3821no—
138C4×D4—1912no—
139#117—3821no—
140C2×C42C2×D19111yescharacteristic
141#107—1914no—
142C4⋊C8C2×D19112yes—
143C4⋊C8C2×D19112yes—
144#103—1914no—
145#102—1914no—
146#41—7644no—
147C2×Q8—3821no—
148C2×Q8—3821no—
149C2×Q8—1912no—
150C2×D4—3822no—
151C2×D4—3821no—
152C2×D4—3821no—
153C2×D4—1912no—
154C22×C4—3822no—
155C22×C4D76411yescharacteristic
156C22×C4D76411yescharacteristic
157C22×C4C22×D19111yescharacteristic
158SD16—7644no—
159M16—42no—
160C2×C8C22×D19114yes—
161C4⋊C4—3822no—
162C4⋊C4—3822no—
163C4⋊C4—3821no—
164C4⋊C4—3821no—
165C4⋊C4—3821no—
166C4⋊C4—3821no—
167C4⋊C4—1912no—
168C4⋊C4—1912no—
169C22⋊C4—3822no—
170C22⋊C4—3822no—
171C22⋊C4—3822no—
172C42—3822no—
173C42D76411yescharacteristic
174C42D76411yescharacteristic
175C42C22×D19111yescharacteristic
176C42C22×D19111yescharacteristic
177C23—3822no—
178C23C2×D76411yescharacteristic
179Q8—3822no—
180Q8—3822no—
181Q8—7642no—
182Q8—7642no—
183D4—3822no—
184D4—3822no—
185D4—7642no—
186D4—7644no—
187D4—7642no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—3822no—
193C2×C4—3822no—
194C2×C4—3822no—
195C2×C4—7644no—
196C2×C4C2×D76411yescharacteristic
197C2×C4C2×D76411yescharacteristic
198C2×C4C2×D76411yescharacteristic
199C2×C4C2×D76411yescharacteristic
200C2×C4C2×D76411yescharacteristic
201C2×C4C23×D19111yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—7642no—
207C22—7642no—
208C22—7642no—
209C22C22×D76411yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—7642no—
214C4—7642no—
215C4—7642no—
216C4#2025935012yes—
217C4#2025935512yes—
218C4#2025936711yescharacteristic
219C4#2025937011yescharacteristic
220C2—41no—
221C2—15282no—
222C2#2713324411yescharacteristic
223C2C2×#2025935011yescharacteristic
224C2C2×#2025935511yescharacteristic
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#423×C191C2
socleC22×C191C22×D4
prime radical#27133213C2
hypercenter#423D191
hypocenterC191#3254
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #27133220C2semidirect
3. #27132959C2semidirect
4. #27132944C2semidirect
5. #27132909C2semidirect
6. #27132869C2semidirect
7. #27132864C2semidirect
8. #423×C191C2semidirect
112. C191#3254semidirect
121. #423D191semidirect

Maximal subgroups

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

subgroupindex
#271332202
#271332132
#271329592
#271329442
#271329092
#271328692
#271328642
#423×C1912
#3254191