Finite Group Databasebeta

← #37582763

Subgroups — #37582763

Order 24448.

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

normal subgroups only

This group has 418 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#27133268C211yesmaximal, characteristic, complemented
2#27132987C212yesmaximal, complemented
3#27132963C212yesmaximal, complemented
4#27132858C212yesmaximal, complemented
5#27132797C212yesmaximal, complemented
6#27132773C212yesmaximal, complemented
7C4×#15690272C212yesmaximal, complemented
8#27132724C211yesmaximal, characteristic, complemented
9#518×C191C211yesmaximal, characteristic, complemented
10C2×#15690272C2212yes—
11C2×#15690272C2212yes—
12#20259319C2211yescharacteristic, complemented
13#20259314C2211yescharacteristic, complemented
14#20259307—22no—
15#20259296C2212yes—
16#20259295—22no—
17C4×D764C2212yescomplemented
18#20259255C2212yescomplemented
19#20259253C2212yescomplemented
20C4⋊(C191⋊C8)C2212yescomplemented
21#20259243C2212yes—
22#20259243C2212yes—
23#20259242C2212yescomplemented
24#20259242C2212yescomplemented
25C22⋊(C191⋊C8)C2212yes—
26#20259233C2211yescharacteristic, complemented
27C4×C191⋊C8C2211yescharacteristic, complemented
28C22×D4×C191C2211yescharacteristic
29C4×D4×C191—22no—
30C4×D4×C191C2212yes—
31C4×D4×C191C2212yes—
32#117×C191C2211yescharacteristic
33#117×C191C2211yescharacteristic
34C2×C22⋊C4×C191—22no—
35C2×D764—22no—
36C2×D764C2312yes—
37C2×C22⋊D191—22no—
38C2×C4×D191—22no—
39#15690272—24no—
40#15690272—24no—
41C2×C191⋊C8C2312yes—
42C2×C191⋊C8C2312yes—
43C4⋊Dic191C2312yes—
44Dic191⋊C4—22no—
45(C2×D191)⋊C4—22no—
46(C2×D191)⋊C4—22no—
47C24×C191—21no—
48C2×D4×C191—24no—
49C2×D4×C191—21no—
50C2×D4×C191C2312yes—
51C22×C4×C191—22no—
52C22×C4×C191—22no—
53C22×C4×C191C2311yescharacteristic
54C4⋊C4×C191D412yes—
55C4⋊C4×C191C2312yes—
56C22⋊C4×C191—21no—
57C22⋊C4×C191—22no—
58C22⋊C4×C191—22no—
59C22⋊C4×C191D412yes—
60C42×C191—21no—
61C42×C191C2311yescharacteristic
62C42×C191C2311yescharacteristic
63C22×D191—22no—
64D764—22no—
65D764—22no—
66D764—82no—
67C22⋊D191—82no—
68C2×Dic191—22no—
69C4×D191—82no—
70C191⋊C8—22no—
71C191⋊C8—22no—
72C23×C191—24no—
73C23×C191—22no—
74C23×C191—82no—
75C23×C191C2×D411yescharacteristic
76D4×C191—24no—
77D4×C191—22no—
78D4×C191#4114yes—
79C2×C4×C191—22no—
80C2×C4×C191—22no—
81C2×C4×C191—82no—
82C2×C4×C191—82no—
83C2×C4×C191C2×D411yescharacteristic
84C2×C4×C191C2×D412yes—
85C2×C4×C191C2×D412yes—
86C2×C4×C191C2411yescharacteristic
87C22×C191—21no—
88C22×C191—24no—
89C22×C191—22no—
90C22×C191—42no—
91C22×C191—42no—
92C22×C191—84no—
93C22×C191C22×D411yescharacteristic
94C2×D191—42no—
95C2×D191—42no—
96C2×D191—42no—
97C4×C191—21no—
98C4×C191—22no—
99C4×C191—22no—
100C4×C191—42no—
101C4×C191C2×#4112yes—
102Dic191—42no—
103C2×C191—41no—
104C2×C191—44no—
105C2×C191#52511yescharacteristic
106C2×C191#57612yes—
107D191—82no—
108C191#312211yescharacteristic, complemented, Sylow
109#3122—1911nomaximal, Sylow
110#540—1911no—
111#518D19111yescharacteristic, complemented
112#482—1912no—
113#478—1912no—
114#459—1912no—
115#447—1912no—
116#442—1912no—
117C4×D8—1912no—
118#433—1911no—
119C22×D4C2×D19111yescharacteristic
120C2×D8—1912no—
121C2×D8—1912no—
122#127—1911no—
123#125—1911no—
124#123—3822no—
125#121—3822no—
126#121—1912no—
127C4×D4—22no—
128C4×D4—1912no—
129C4×D4C2×D19112yes—
130C4×D4C2×D19112yes—
131#117C2×D19111yescharacteristic
132#117C2×D19111yescharacteristic
133C2×C22⋊C4—22no—
134#107—1912no—
135#106—1912no—
136C4⋊C8—1912no—
137#102—1912no—
138#102—1912no—
139#102—1912no—
140#102—1912no—
141C22⋊C8—1912no—
142#97—1911no—
143C4×C8—1911no—
144C24—21no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—3822no—
148C2×D4—3822no—
149C2×D4—1912no—
150C2×D4C22×D19112yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—3822no—
154C22×C4C22×D19111yescharacteristic
155D8—3824no—
156D8—3824no—
157C2×C8—1912no—
158C2×C8—1912no—
159C4⋊C4—3822no—
160C4⋊C4—1912no—
161C4⋊C4C22⋊D19112yes—
162C4⋊C4C22×D19112yes—
163C22⋊C4—21no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—3822no—
167C22⋊C4—3822no—
168C22⋊C4C22⋊D19112yes—
169C42—21no—
170C42C22×D19111yescharacteristic
171C42C22×D19111yescharacteristic
172C23—24no—
173C23—22no—
174C23—82no—
175C23—3822no—
176C23C2×C22⋊D19111yescharacteristic
177D4—24no—
178D4—22no—
179D4—3822no—
180D4—3822no—
181D4—15282no—
182D4—15282no—
183D4#1569028714yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—3822no—
189C2×C4—15282no—
190C2×C4C2×C22⋊D19111yescharacteristic
191C2×C4C2×C22⋊D19112yes—
192C2×C4C2×C22⋊D19112yes—
193C2×C4C23×D19111yescharacteristic
194C8—3822no—
195C8—3822no—
196C22—22no—
197C22—21no—
198C22—24no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—7642no—
203C22—7642no—
204C22—7642no—
205C22C22×C22⋊D19111yescharacteristic
206C4—22no—
207C4—22no—
208C4—21no—
209C4—42no—
210C4—7642no—
211C4C2×#1569028712yes—
212C2—41no—
213C2—44no—
214C2—15282no—
215C2#2713317411yescharacteristic
216C2#2713346612yes—
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#518×C191C2
socleC22×C191C22×D4
hypercenter#518D191
hypocenterC191#3122
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #27133268C2semidirect
2. #27132987C2semidirect
3. #27132963C2semidirect
4. #27132858C2semidirect
5. #27132797C2semidirect
6. #27132773C2semidirect
7. C4×#15690272C2semidirect
8. #27132724C2semidirect
9. #518×C191C2semidirect
12. #20259319C22semidirect
13. #20259314C22semidirect
17. C4×D764C22semidirect
18. #20259255C22semidirect
19. #20259253C22semidirect
20. C4⋊(C191⋊C8)C22semidirect
23. #20259242C22semidirect
24. #20259242C22semidirect
26. #20259233C22semidirect
27. C4×C191⋊C8C22semidirect
108. C191#3122semidirect
111. #518D191semidirect

Maximal subgroups

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

subgroupindex
#271332682
#271329872
#271329632
#271328582
#271327972
#271327732
C4×#156902722
#271327242
#518×C1912
#3122191