Finite Group Databasebeta

← #37582772

Subgroups — #37582772

Order 24448.

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

normal subgroups only

This group has 394 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#27133275C211yesmaximal, characteristic, complemented
2#27133004C212yesmaximal, complemented
3#27132963C212yesmaximal, complemented
4#27132868C212yesmaximal, complemented
5#27132814C212yesmaximal
6#27132763C212yesmaximal, complemented
7C4×#15690275C212yesmaximal, complemented
8#27132724C211yesmaximal, characteristic, complemented
9#518×C191C211yesmaximal, characteristic
10C2×#15690275C2212yes—
11C2×#15690275C2212yes—
12#20259322C2211yescharacteristic, complemented
13#20259315—22no—
14#20259314C2211yescharacteristic, complemented
15#20259312—22no—
16#20259301C2212yes—
17C4×Dic382C2212yescomplemented
18#20259255C2212yescomplemented
19#20259253C2212yescomplemented
20C4⋊(C191⋊C8)C2212yescomplemented
21#20259245C2212yescomplemented
22#20259245C2212yescomplemented
23#20259243C2212yes—
24#20259243C2212yes—
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×Dic382C2312yes—
36#15690275—24no—
37#15690275—24no—
38C2×C191⋊C8C2312yes—
39C2×C191⋊C8C2312yes—
40C4⋊Dic191—22no—
41C4⋊Dic191C2312yes—
42Dic191⋊C4—22no—
43Dic191⋊C4—22no—
44Dic191⋊C4—22no—
45C22⋊Dic191—22no—
46C4×Dic191—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
63Dic382—22no—
64Dic382—22no—
65C2×Dic191—22no—
66C2×Dic191—22no—
67C191⋊C8—22no—
68C191⋊C8—22no—
69C23×C191—24no—
70C23×C191—22no—
71C23×C191—82no—
72C23×C191C2×D411yescharacteristic
73D4×C191—24no—
74D4×C191—22no—
75D4×C191#4114yes—
76C2×C4×C191—22no—
77C2×C4×C191—22no—
78C2×C4×C191—82no—
79C2×C4×C191—82no—
80C2×C4×C191C2×D411yescharacteristic
81C2×C4×C191C2×D412yes—
82C2×C4×C191C2×D412yes—
83C2×C4×C191C2411yescharacteristic
84C22×C191—21no—
85C22×C191—24no—
86C22×C191—22no—
87C22×C191—42no—
88C22×C191—42no—
89C22×C191—84no—
90C22×C191C22×D411yescharacteristic
91C4×C191—21no—
92C4×C191—22no—
93C4×C191—22no—
94C4×C191—42no—
95C4×C191C2×#4112yes—
96Dic191—42no—
97Dic191—42no—
98C2×C191—41no—
99C2×C191—44no—
100C2×C191#52511yescharacteristic
101C2×C191#57712yes—
102C191#312311yescharacteristic, complemented, Sylow
103#3123—1911nomaximal, Sylow
104#541—1911no—
105#518D19111yescharacteristic
106#485—1912no—
107#478—1912no—
108#461—1912no—
109#450—1912no—
110#440—1912no—
111C4×SD16—1912no—
112#433—1911no—
113C22×D4C2×D19111yescharacteristic
114C2×SD16—1912no—
115C2×SD16—1912no—
116C4⋊Q8—1911no—
117#126—3822no—
118#125—3822no—
119#125—1911no—
120C22⋊Q8—1912no—
121C4×Q8—1912no—
122C4×D4—22no—
123C4×D4C2×D19112yes—
124C4×D4C2×D19112yes—
125#117C2×D19111yescharacteristic
126#117C2×D19111yescharacteristic
127C2×C22⋊C4—22no—
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—21no—
139C2×Q8—1912no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4C22×D19112yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4C22×D19111yescharacteristic
146SD16—3824no—
147SD16—3824no—
148C2×C8—1912no—
149C2×C8—1912no—
150C4⋊C4—3822no—
151C4⋊C4—3822no—
152C4⋊C4—3822no—
153C4⋊C4—3822no—
154C4⋊C4—1912no—
155C4⋊C4C22⋊D19112yes—
156C4⋊C4C22×D19112yes—
157C22⋊C4—21no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—3822no—
161C22⋊C4C22⋊D19112yes—
162C42—21no—
163C42—3822no—
164C42C22×D19111yescharacteristic
165C42C22×D19111yescharacteristic
166C23—24no—
167C23—22no—
168C23—82no—
169C23C2×C22⋊D19111yescharacteristic
170Q8—3822no—
171Q8—3822no—
172D4—24no—
173D4—22no—
174D4#1569028714yes—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—3822no—
180C2×C4—3822no—
181C2×C4C2×C22⋊D19111yescharacteristic
182C2×C4C2×C22⋊D19112yes—
183C2×C4C2×C22⋊D19112yes—
184C2×C4C23×D19111yescharacteristic
185C8—3822no—
186C8—3822no—
187C22—22no—
188C22—21no—
189C22—24no—
190C22—42no—
191C22—42no—
192C22—84no—
193C22C22×C22⋊D19111yescharacteristic
194C4—22no—
195C4—22no—
196C4—21no—
197C4—42no—
198C4—7642no—
199C4—7642no—
200C4C2×#1569028712yes—
201C2—41no—
202C2—44no—
203C2#2713317411yescharacteristic
204C2#2713347412yes—
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#518×C191C2
socleC22×C191C22×D4
prime radicalC22×D4×C191C22
hypercenter#518D191
hypocenterC191#3123
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. #27133275C2semidirect
2. #27133004C2semidirect
3. #27132963C2semidirect
4. #27132868C2semidirect
6. #27132763C2semidirect
7. C4×#15690275C2semidirect
8. #27132724C2semidirect
12. #20259322C22semidirect
14. #20259314C22semidirect
17. C4×Dic382C22semidirect
18. #20259255C22semidirect
19. #20259253C22semidirect
20. C4⋊(C191⋊C8)C22semidirect
21. #20259245C22semidirect
22. #20259245C22semidirect
26. #20259233C22semidirect
27. C4×C191⋊C8C22semidirect
102. C191#3123semidirect

Maximal subgroups

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

subgroupindex
#271332752
#271330042
#271329632
#271328682
#271328142
#271327632
C4×#156902752
#271327242
#518×C1912
#3123191