Finite Group Databasebeta

← #37582986

Subgroups — #37582986

Order 24448.

572 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 165 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×#20259355C212yesmaximal, complemented
2C2×#20259350C212yesmaximal, complemented
3#27133213C211yesmaximal, characteristic, complemented
4C2×#20259310C211yesmaximal, characteristic, complemented
5#27133057C212yesmaximal, complemented
6#27133052C212yesmaximal, complemented
7#27133047C214yesmaximal, complemented
8C4×M16×C191C211yesmaximal, characteristic, complemented
9C2×#15690287C2212yes—
10C22×Dic382C2211yescharacteristic
11C22×D764C2211yescharacteristic
12#20259355—28no—
13#20259350—28no—
14C2×Dic764C2214yescomplemented
15C2×#15690276C2214yes—
16C2×#15690276C2214yescomplemented
17C2×D1528C2214yes—
18#20259322C2211yescharacteristic, complemented
19#20259319C2211yescharacteristic, complemented
20#20259310—22no—
21#20259310C2212yes—
22#20259310C2212yes—
23#20259310C2212yescomplemented
24#20259296—22no—
25C4×D764—22no—
26C2×(C2×D191)⋊C4—22no—
27C2×M16×C191C2212yes—
28C2×C42×C191C2211yescharacteristic
29#97×C191C2212yescomplemented
30C4×C8×C191C2212yescomplemented
31C23×D191—21no—
32#15690287—24no—
33#15690287—24no—
34C2×Dic382—21no—
35C2×Dic382—24no—
36C2×Dic382C2312yes—
37C2×Dic382C2312yes—
38C2×D764—24no—
39C2×D764—21no—
40C2×D764—22no—
41C2×D764C2312yes—
42C2×D764C2312yes—
43C2×C22⋊D191—22no—
44C22×Dic191—21no—
45C2×C4×D191—22no—
46Dic764—28no—
47#15690276—28no—
48#15690276—28no—
49D1528—28no—
50C4⋊Dic191—22no—
51(C2×D191)⋊C4—24no—
52(C2×D191)⋊C4—24no—
53(C2×D191)⋊C4—24no—
54C22×C4×C191D412yes—
55C22×C4×C191C2311yescharacteristic
56M16×C191D418yescomplemented
57C2×C8×C191C2314yes—
58C42×C191D412yes—
59C42×C191C2311yescharacteristic
60C42×C191C2311yescharacteristic
61C22×D191—22no—
62C22×D191—22no—
63C22×D191—44no—
64C22×D191—42no—
65Dic382—22no—
66Dic382—24no—
67Dic382—22no—
68Dic382—42no—
69D764—22no—
70D764—24no—
71D764—22no—
72D764—44no—
73D764—42no—
74C22⋊D191—44no—
75C2×Dic191—22no—
76C2×Dic191—22no—
77C2×Dic191—42no—
78C4×D191—44no—
79C23×C191C2×D411yescharacteristic
80C2×C4×C191—42no—
81C2×C4×C191C2×D412yes—
82C2×C4×C191C2×D412yes—
83C2×C4×C191C2×D411yescharacteristic
84C2×C4×C191C2×D414yes—
85C2×C4×C191C2411yescharacteristic
86C8×C191C2×D418yes—
87C22×C191—22no—
88C22×C191#12712yes—
89C22×C191C22×D411yescharacteristic
90C2×D191—42no—
91C2×D191—42no—
92C2×D191—44no—
93C2×D191—44no—
94C2×D191—84no—
95C4×C191—22no—
96C4×C191—22no—
97C4×C191#12712yes—
98C4×C191C22×D412yes—
99Dic191—42no—
100Dic191—42no—
101C2×C191—22no—
102C2×C191C2×#12711yescharacteristic
103C2×C191#57512yes—
104D191—82no—
105D191—82no—
106C191#315311yescharacteristic, complemented, Sylow
107#3153—1911nomaximal, Sylow
108C2×#137—1912no—
109C2×AGL(1,ℤ/8)—1912no—
110#532—1911no—
111C2×#124—1911no—
112#497—1912no—
113#496—1912no—
114#495—1914no—
115C4×M16D19111yescharacteristic, complemented
116C2×#41—1912no—
117C22×Q8—1911no—
118C22×D4—1911no—
119#137—3828no—
120AGL(1,ℤ/8)—3828no—
121C2×Q16—1914no—
122C2×SD16—1914no—
123C2×SD16—1914no—
124C2×D8—1914no—
125C2×M16C2×D19112yes—
126C4⋊Q8—1911no—
127#127—1911no—
128#124—3822no—
129#124—1912no—
130#124—1912no—
131#124—1912no—
132#121—3822no—
133C4×D4—3822no—
134C2×C22⋊C4—3822no—
135C2×C42C2×D19111yescharacteristic
136#97C2×D19112yescomplemented
137C4×C8C2×D19112yescomplemented
138C24—3821no—
139#41—3824no—
140#41—3824no—
141C2×Q8—3821no—
142C2×Q8—3824no—
143C2×Q8—1912no—
144C2×Q8—1912no—
145C2×D4—3822no—
146C2×D4—3824no—
147C2×D4—3822no—
148C2×D4—3821no—
149C2×D4—1912no—
150C2×D4—1912no—
151C22×C4—3821no—
152C22×C4—3822no—
153C22×C4D76412yes—
154C22×C4C22×D19111yescharacteristic
155Q16—3828no—
156SD16—3828no—
157SD16—3828no—
158D8—3828no—
159M16D76418yescomplemented
160C2×C8C22×D19114yes—
161C4⋊C4—3822no—
162C22⋊C4—3824no—
163C22⋊C4—3824no—
164C22⋊C4—3824no—
165C42D76412yes—
166C42C22×D19111yescharacteristic
167C42C22×D19111yescharacteristic
168C23—3822no—
169C23—3822no—
170C23—7644no—
171C23—7642no—
172C23C2×D76411yescharacteristic
173Q8—3822no—
174Q8—3824no—
175Q8—3822no—
176Q8—7642no—
177D4—3822no—
178D4—3824no—
179D4—3822no—
180D4—7644no—
181D4—7644no—
182D4—7642no—
183C2×C4—42no—
184C2×C4—3822no—
185C2×C4—3822no—
186C2×C4—7642no—
187C2×C4—7644no—
188C2×C4C2×D76412yes—
189C2×C4C2×D76412yes—
190C2×C4C2×D76414yes—
191C2×C4C2×D76411yescharacteristic
192C2×C4C23×D19111yescharacteristic
193C8C2×D76418yes—
194C22—22no—
195C22—15284no—
196C22—7642no—
197C22—7642no—
198C22—7644no—
199C22—7644no—
200C22#2025931912yes—
201C22C22×D76411yescharacteristic
202C4—22no—
203C4—22no—
204C4—7642no—
205C4—7642no—
206C4#2025931912yes—
207C4C22×D76412yes—
208C2—22no—
209C2—15282no—
210C2—15282no—
211C2C2×#2025931911yescharacteristic
212C2#2713346312yes—
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC4×M16×C191C2
socleC22×C191C22×D4
hypercenterC4×M16D191
hypocenterC191#3153
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#20259355C2semidirect
2. C2×#20259350C2semidirect
3. #27133213C2semidirect
4. C2×#20259310C2semidirect
5. #27133057C2semidirect
6. #27133052C2semidirect
7. #27133047C2semidirect
8. C4×M16×C191C2semidirect
14. C2×Dic764C22semidirect
16. C2×#15690276C22semidirect
18. #20259322C22semidirect
19. #20259319C22semidirect
23. #20259310C22semidirect
29. #97×C191C22semidirect
30. C4×C8×C191C22semidirect
56. M16×C191D4semidirect
106. C191#3153semidirect
115. C4×M16D191semidirect
136. #97C2×D191semidirect
137. C4×C8C2×D191semidirect
159. M16D764semidirect

Maximal subgroups

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

subgroupindex
C2×#202593552
C2×#202593502
#271332132
C2×#202593102
#271330572
#271330522
#271330472
C4×M16×C1912
#3153191