Finite Group Databasebeta

← #37582767

Subgroups — #37582767

Order 24448.

394 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 121 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#27133137C211yesmaximal, characteristic
2#27133001C212yesmaximal, complemented
3#27132960C212yesmaximal, complemented
4#27132865C212yesmaximal, complemented
5#27132813C212yesmaximal
6#27132761C212yesmaximal, complemented
7C4×#15690274C212yesmaximal, complemented
8#27132724C211yesmaximal, characteristic, complemented
9#541×C191C211yesmaximal, characteristic, complemented
10C22×D764C2211yescharacteristic
11C2×#15690274C2212yes—
12C2×#15690274C2212yes—
13C4×D764C2212yes—
14C4×D764C2212yes—
15#20259284—22no—
16#20259281C2211yescharacteristic
17C22⋊C4×D191—22no—
18#20259255C2212yescomplemented
19#20259253C2212yescomplemented
20C4⋊(C191⋊C8)C2212yescomplemented
21#20259246C2212yescomplemented
22#20259246C2212yescomplemented
23#20259242C2212yes—
24#20259242C2212yes—
25C22⋊(C191⋊C8)C2212yes—
26#20259233C2211yescharacteristic, complemented
27C4×C191⋊C8C2211yescharacteristic, complemented
28C4⋊Q8×C191C2211yescharacteristic, complemented
29#126×C191—22no—
30#125×C191—22no—
31#125×C191C2211yescharacteristic, complemented
32C22⋊Q8×C191C2212yes—
33C4×Q8×C191C2212yescomplemented
34#117×C191C2211yescharacteristic
35C23×D191—21no—
36C2×D764—24no—
37C2×D764—21no—
38C2×D764C2312yes—
39C2×C4×D191—22no—
40C2×C4×D191—22no—
41#15690274—24no—
42#15690274—24no—
43C2×C191⋊C8C2312yes—
44C2×C191⋊C8C2312yes—
45C4⋊Dic191C2312yes—
46C22⋊Dic191—21no—
47(C2×D191)⋊C4—22no—
48(C2×D191)⋊C4—22no—
49C4×Dic191—21no—
50C2×Q8×C191C2312yes—
51C22×C4×C191C2311yescharacteristic
52C4⋊C4×C191—22no—
53C4⋊C4×C191—22no—
54C4⋊C4×C191—22no—
55C4⋊C4×C191—22no—
56C4⋊C4×C191D412yes—
57C4⋊C4×C191C2312yes—
58C22⋊C4×C191—22no—
59C22⋊C4×C191D412yes—
60C42×C191—22no—
61C42×C191C2311yescharacteristic
62C42×C191C2311yescharacteristic
63C22×D191—22no—
64C22×D191—24no—
65C22×D191—82no—
66D764—24no—
67D764—22no—
68D764#4114yes—
69C2×Dic191—22no—
70C4×D191—82no—
71C4×D191—82no—
72C191⋊C8—22no—
73C191⋊C8—22no—
74C23×C191C2×D411yescharacteristic
75Q8×C191—22no—
76Q8×C191—22no—
77C2×C4×C191—22no—
78C2×C4×C191—22no—
79C2×C4×C191—22no—
80C2×C4×C191C2×D412yes—
81C2×C4×C191C2×D412yes—
82C2×C4×C191C2×D411yescharacteristic
83C2×C4×C191C2411yescharacteristic
84C22×C191—21no—
85C22×C191—22no—
86C22×C191C22×D411yescharacteristic
87C2×D191—24no—
88C2×D191—42no—
89C2×D191—42no—
90C2×D191—84no—
91C4×C191—21no—
92C4×C191—22no—
93C4×C191—22no—
94C4×C191—42no—
95C4×C191—42no—
96C4×C191C2×#4112yes—
97Dic191—42no—
98C2×C191—41no—
99C2×C191#52511yescharacteristic
100C2×C191#57712yes—
101D191—44no—
102C191#312311yescharacteristic, complemented, Sylow
103#3123—1911nomaximal, Sylow
104#541D19111yescharacteristic, complemented
105#518—1911no—
106#485—1912no—
107#478—1912no—
108#461—1912no—
109#450—1912no—
110#440—1912no—
111C4×SD16—1912no—
112#433—1911no—
113C22×D4—1911no—
114C2×SD16—1912no—
115C2×SD16—1912no—
116C4⋊Q8C2×D19111yescharacteristic, complemented
117#126—22no—
118#125—22no—
119#125C2×D19111yescharacteristic, complemented
120C22⋊Q8C2×D19112yes—
121C4×Q8C2×D19112yescomplemented
122C4×D4—3822no—
123C4×D4—1912no—
124C4×D4—1912no—
125#117—1911no—
126#117C2×D19111yescharacteristic
127C2×C22⋊C4—3822no—
128#107—1912no—
129#106—1912no—
130C4⋊C8—1912no—
131#103—1912no—
132#103—1912no—
133#102—1912no—
134#102—1912no—
135C22⋊C8—1912no—
136#97—1911no—
137C4×C8—1911no—
138C24—3821no—
139C2×Q8C22×D19112yes—
140C2×D4—3824no—
141C2×D4—3821no—
142C2×D4—1912no—
143C22×C4—3822no—
144C22×C4—3822no—
145C22×C4C22×D19111yescharacteristic
146SD16—3824no—
147SD16—3824no—
148C2×C8—1912no—
149C2×C8—1912no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—1912no—
155C4⋊C4C22⋊D19112yes—
156C4⋊C4C22×D19112yes—
157C22⋊C4—22no—
158C22⋊C4—3821no—
159C22⋊C4—3822no—
160C22⋊C4—3822no—
161C22⋊C4C22⋊D19112yes—
162C42—22no—
163C42—3821no—
164C42C22×D19111yescharacteristic
165C42C22×D19111yescharacteristic
166C23—3824no—
167C23—3822no—
168C23—15282no—
169C23C2×C22⋊D19111yescharacteristic
170Q8—22no—
171Q8—22no—
172D4—3824no—
173D4—3822no—
174D4—1914no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—3822no—
179C2×C4—15282no—
180C2×C4—15282no—
181C2×C4C2×C22⋊D19112yes—
182C2×C4C2×C22⋊D19111yescharacteristic
183C2×C4C2×C22⋊D19112yes—
184C2×C4C23×D19111yescharacteristic
185C8—3822no—
186C8—3822no—
187C22—22no—
188C22—21no—
189C22—3824no—
190C22—15284no—
191C22—7642no—
192C22—7642no—
193C22C22×C22⋊D19111yescharacteristic
194C4—22no—
195C4—22no—
196C4—21no—
197C4—42no—
198C4—42no—
199C4—7642no—
200C4#41×D19112yes—
201C2—41no—
202C2—7644no—
203C2#2713317011yescharacteristic
204C2#2713347112yes—
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#541×C191C2
socleC22×C191C22×D4
prime radicalC22×D764C22
hypercenter#541D191
hypocenterC191#3123
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #27133001C2semidirect
3. #27132960C2semidirect
4. #27132865C2semidirect
6. #27132761C2semidirect
7. C4×#15690274C2semidirect
8. #27132724C2semidirect
9. #541×C191C2semidirect
18. #20259255C22semidirect
19. #20259253C22semidirect
20. C4⋊(C191⋊C8)C22semidirect
21. #20259246C22semidirect
22. #20259246C22semidirect
26. #20259233C22semidirect
27. C4×C191⋊C8C22semidirect
28. C4⋊Q8×C191C22semidirect
31. #125×C191C22semidirect
33. C4×Q8×C191C22semidirect
102. C191#3123semidirect
104. #541D191semidirect
116. C4⋊Q8C2×D191semidirect
119. #125C2×D191semidirect
121. C4×Q8C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331372
#271330012
#271329602
#271328652
#271328132
#271327612
C4×#156902742
#271327242
#541×C1912
#3123191