Finite Group Databasebeta

← #37573744

Subgroups — #37573744

Order 24448.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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#27133244C211yesmaximal, characteristic
2C2×#20259250C212yesmaximal, complemented
3#27132662C212yesmaximal, complemented
4#27132432C211yesmaximal, characteristic, complemented
5#384×C191C211yesmaximal, characteristic, complemented
6C2×#15690287C2212yes—
7C2×#15690271C2212yes—
8#20259302—22no—
9#20259297—22no—
10#20259296—22no—
11#20259250—24no—
12#20259245—22no—
13#20259242—22no—
14C2×C4⋊C4×C191—22no—
15C2×C4⋊C4×C191C2211yescharacteristic
16C2×C4⋊C4×C191C2211yescharacteristic
17C2×C42×C191C2211yescharacteristic
18#15690287—22no—
19#15690287—22no—
20#15690287—22no—
21#15690287—22no—
22C2×Dic382D412yes—
23C2×D764—22no—
24C2×D764D412yes—
25C2×C22⋊D191—22no—
26C2×C4×D191—22no—
27#15690271—22no—
28#15690271—22no—
29C2×C191⋊C8—22no—
30C4⋊Dic191—22no—
31(C2×D191)⋊C4—24no—
32C22×C4×C191—21no—
33C22×C4×C191—21no—
34C22×C4×C191D412yes—
35C22×C4×C191C2311yescharacteristic
36C4⋊C4×C191—22no—
37C4⋊C4×C191—22no—
38C4⋊C4×C191—44no—
39C42×C191—21no—
40C42×C191D412yescomplemented
41C22×D191—22no—
42Dic382—22no—
43Dic382—22no—
44D764—22no—
45D764—22no—
46D764—82no—
47C22⋊D191—42no—
48C22⋊D191—42no—
49C2×Dic191—22no—
50C4×D191—42no—
51C4×D191—42no—
52C191⋊C8—42no—
53C23×C191C2×D411yescharacteristic
54C2×C4×C191—22no—
55C2×C4×C191—22no—
56C2×C4×C191—22no—
57C2×C4×C191—22no—
58C2×C4×C191—42no—
59C2×C4×C191—42no—
60C2×C4×C191—41no—
61C2×C4×C191C2×D412yes—
62C2×C4×C191C2×D411yescharacteristic
63C2×C4×C191#4111yescharacteristic
64C2×C4×C191#4112yes—
65C22×C191—21no—
66C22×C191—21no—
67C22×C191#12011yescharacteristic
68C22×C191#12111yescharacteristic
69C22×C191#12411yescharacteristic
70C2×D191—42no—
71C2×D191—42no—
72C2×D191—42no—
73C4×C191—24no—
74C4×C191—42no—
75C4×C191—81no—
76C4×C191#12112yes—
77C4×C191#12312yes—
78Dic191—42no—
79C2×C191—21no—
80C2×C191—21no—
81C2×C191#39411yescharacteristic
82C2×C191#45311yescharacteristic
83C2×C191#45611yescharacteristic
84D191—82no—
85C191#203911yescharacteristic, complemented, Sylow
86#2039—1911nomaximal, Sylow
87#537—1911no—
88C2×#104—1912no—
89#417—1912no—
90#384D19111yescharacteristic, complemented
91#343—1911no—
92C2×#41—1912no—
93C2×M16—1912no—
94C22⋊Q8—3822no—
95#121—3822no—
96#121—3822no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4C2×D19111yescharacteristic
99C2×C4⋊C4C2×D19111yescharacteristic
100C2×C42C2×D19111yescharacteristic
101#104—3824no—
102#103—3822no—
103#102—3822no—
104#41—3822no—
105#41—3822no—
106#41—3822no—
107#41—3822no—
108C2×Q8—1912no—
109C2×D4—3822no—
110C2×D4—3822no—
111C2×D4—1912no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—3822no—
115C22×C4C22⋊D19112yes—
116C22×C4C22×D19111yescharacteristic
117M16—3822no—
118M16—3822no—
119C2×C8—3822no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—44no—
123C4⋊C4—3822no—
124C22⋊C4—3824no—
125C42—21no—
126C42D76412yescomplemented
127C23—3822no—
128C23C2×C22⋊D19111yescharacteristic
129Q8—3822no—
130Q8—3822no—
131D4—3822no—
132D4—3822no—
133D4—15282no—
134D4—7642no—
135D4—7642no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—3822no—
144C2×C4—7642no—
145C2×C4—7642no—
146C2×C4D4×D19112yes—
147C2×C4C2×D76411yescharacteristic
148C2×C4#1569028711yescharacteristic
149C2×C4#1569028812yes—
150C8—7642no—
151C22—21no—
152C22—21no—
153C22—7642no—
154C22—7642no—
155C22—7642no—
156C22#2025929011yescharacteristic
157C22#2025929611yescharacteristic
158C22#2025931111yescharacteristic
159C4—24no—
160C4—42no—
161C4—81no—
162C4—7642no—
163C4#2025929712yes—
164C4#2025930712yes—
165C2—21no—
166C2—21no—
167C2—15282no—
168C2#2713258011yescharacteristic
169C2#2713283011yescharacteristic
170C2#2713284511yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#384×C191C2
socleC22×C191#120
prime radical#27133244C2
hypercenter#384D191
hypocenterC191#2039
LCSG ▷ C22×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C2×#20259250C2semidirect
3. #27132662C2semidirect
4. #27132432C2semidirect
5. #384×C191C2semidirect
40. C42×C191D4semidirect
85. C191#2039semidirect
90. #384D191semidirect
126. C42D764semidirect

Maximal subgroups

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

subgroupindex
#271332442
C2×#202592502
#271326622
#271324322
#384×C1912
#2039191