Finite Group Databasebeta

← #37573067

Subgroups — #37573067

Order 24448.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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#27133227C211yesmaximal, characteristic
2#27132647C211yesmaximal, characteristic, complemented
3C2×#20259238C212yesmaximal, complemented
4#27132435C212yesmaximal, complemented
5#568×C191C211yesmaximal, characteristic, complemented
6C22×D764C2211yescharacteristic
7C2×#15690271C2212yes—
8#20259318—21no—
9#20259311—21no—
10#20259296—22no—
11#20259290C414yes—
12#20259281C2211yescharacteristic
13#20259241—21no—
14#20259239—21no—
15#20259238—24no—
16C2×#41×C191C2211yescharacteristic
17#131×C191—24no—
18C2×M16×C191—22no—
19C2×M16×C191C2212yes—
20C23×D191C2×C412yes—
21C2×D764—22no—
22C2×D764—21no—
23C2×D764—21no—
24C2×D764—22no—
25C2×C22⋊D191—24no—
26#15690271—22no—
27#15690271—22no—
28C2×C191⋊C8—22no—
29C4⋊Dic191—21no—
30C22⋊Dic191C2×C412yes—
31(C2×D191)⋊C4—24no—
32C4×Dic191—21no—
33#41×C191—22no—
34#41×C191—22no—
35C2×Q8×C191D411yescharacteristic
36C2×D4×C191C2×C412yes—
37C2×D4×C191D411yescharacteristic
38C22×C4×C191D412yes—
39C22×C4×C191C2311yescharacteristic
40M16×C191—24no—
41M16×C191—22no—
42M16×C191D414yescomplemented
43C2×C8×C191—24no—
44C2×C8×C191—22no—
45C22×D191—22no—
46C22×D191—24no—
47C22×D191—82no—
48D764—41no—
49D764—42no—
50D764—41no—
51C22⋊D191—84no—
52C2×Dic191—22no—
53C191⋊C8—42no—
54C23×C191C22⋊C412yes—
55C23×C191C22×C411yescharacteristic
56Q8×C191—22no—
57D4×C191—24no—
58D4×C191—22no—
59C2×C4×C191—22no—
60C2×C4×C191—22no—
61C2×C4×C191C22⋊C412yes—
62C2×C4×C191C2×D412yes—
63C2×C4×C191C2×D411yescharacteristic
64C2×C4×C191C2×D411yescharacteristic
65C2×C4×C191#4112yes—
66C8×C191—24no—
67C22×C191—22no—
68C22×C191—41no—
69C22×C191—42no—
70C22×C191C2×C22⋊C411yescharacteristic
71C22×C191C4×D412yes—
72C2×D191—44no—
73C2×D191—42no—
74C2×D191—84no—
75C4×C191—22no—
76C4×C191#12011yescharacteristic
77C4×C191#12112yes—
78C4×C191#12311yescharacteristic
79Dic191—42no—
80C2×C191—21no—
81C2×C191—22no—
82C2×C191—42no—
83C2×C191#38611yescharacteristic
84D191—82no—
85C191#191411yescharacteristic, complemented, Sylow
86#1914—1911nomaximal, Sylow
87#568D19111yescharacteristic, complemented
88#535—1911no—
89#413—1911no—
90C2×#100—1912no—
91#344—1912no—
92C2×#41C2×D19111yescharacteristic
93C22×D4—1911no—
94#131—24no—
95C2×M16—22no—
96C2×M16—1912no—
97C2×M16C2×D19112yes—
98#127—3821no—
99#124—3821no—
100#121—3822no—
101#120—1914no—
102#117—1911no—
103#101—3821no—
104#100—3821no—
105#100—3824no—
106C24—1912no—
107#41—22no—
108#41—22no—
109C2×Q8D76411yescharacteristic
110C2×D4—3824no—
111C2×D4—3822no—
112C2×D4—3821no—
113C2×D4—3821no—
114C2×D4—3822no—
115C2×D4C4×D19112yes—
116C2×D4D76411yescharacteristic
117C22×C4C22⋊D19112yes—
118C22×C4C22×D19111yescharacteristic
119M16—24no—
120M16—22no—
121M16—3822no—
122M16—3822no—
123M16C22⋊D19114yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—3822no—
127C4⋊C4—3821no—
128C22⋊C4—3824no—
129C22⋊C4—1912no—
130C42—3821no—
131C23—3824no—
132C23—3822no—
133C23—15282no—
134C23(C2×D191)⋊C412yes—
135C23C2×C4×D19111yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—15284no—
140D4—7641no—
141D4—7642no—
142D4—7641no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—3822no—
146C2×C4(C2×D191)⋊C412yes—
147C2×C4C2×C22⋊D19112yes—
148C2×C4C2×C22⋊D19111yescharacteristic
149C2×C4C2×D76411yescharacteristic
150C2×C4#1569028712yes—
151C8—24no—
152C8—7642no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—15284no—
157C22—7644no—
158C22—7642no—
159C22C2×(C2×D191)⋊C411yescharacteristic
160C22C4×C22⋊D19112yes—
161C4—22no—
162C4—7642no—
163C4#2025929111yescharacteristic
164C4#2025929612yes—
165C4#2025930511yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—15282no—
170C2#2713254211yescharacteristic
subgroupG/H
centerC2#27132542
commutatorC23×C191C22×C4
FrattiniC22×C4C22×D191
Fitting#568×C191C2
socleC2×C191#386
prime radical#27133227C2
cosocleC22×C4×C191C23
hypercenter#568D191
hypocenterC191#1914
LCSG ▷ C23×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #27132647C2semidirect
3. C2×#20259238C2semidirect
4. #27132435C2semidirect
5. #568×C191C2semidirect
42. M16×C191D4semidirect
85. C191#1914semidirect
87. #568D191semidirect
123. M16C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
#271332272
#271326472
C2×#202592382
#271324352
#568×C1912
#1914191