Finite Group Databasebeta

← #37571110

Subgroups — #37571110

Order 24448.

270 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 63 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#27133212C211yesmaximal, characteristic
2#27132615C212yesmaximal, complemented
3#27132403C211yesmaximal, characteristic, complemented
4#27132402C212yesmaximal, complemented
5#329×C191C211yesmaximal, characteristic, complemented
6C2×#15690288C411yescharacteristic
7C2×#15690286C411yescharacteristic
8C2×#15690270—22no—
9#20259320C411yescharacteristic
10#20259318C411yescharacteristic
11#20259310C2211yescharacteristic
12#20259295—22no—
13(C22⋊D191)⋊C4—22no—
14C42×D191C2211yescharacteristic
15#20259233C2212yes—
16C8×Dic191—22no—
17#124×C191C2211yescharacteristic
18#97×C191C2212yes—
19#15690288—42no—
20#15690286—42no—
21C2×Dic382C2×C411yescharacteristic
22C2×D764—21no—
23C2×D764C2×C411yescharacteristic
24C2×C22⋊D191—21no—
25C2×C22⋊D191—22no—
26C22×Dic191—21no—
27C2×C4×D191—21no—
28C2×C4×D191C2×C411yescharacteristic
29C2×C4×D191D412yes—
30#15690270—28no—
31C2×C191⋊C8—22no—
32Dic191⋊C4—22no—
33(C2×D191)⋊C4—22no—
34C4×Dic191C2×C411yescharacteristic
35C4×Dic191D412yes—
36C2×Q8×C191C2×C411yescharacteristic
37C2×D4×C191C2×C411yescharacteristic
38C2×C8×C191—22no—
39C22⋊C4×C191—22no—
40C42×C191C2311yescharacteristic
41C22×D191—21no—
42C22×D191C22⋊C411yescharacteristic
43Dic382—41no—
44D764—41no—
45D764—42no—
46C22⋊D191—42no—
47C22⋊D191—44no—
48C2×Dic191—21no—
49C2×Dic191—42no—
50C2×Dic191C22⋊C411yescharacteristic
51C2×Dic191C22⋊C412yes—
52C4×D191—22no—
53C4×D191—24no—
54C4×D191—42no—
55C191⋊C8—24no—
56C23×C191—21no—
57Q8×C191—41no—
58D4×C191—41no—
59C2×C4×C191—21no—
60C2×C4×C191C22×C411yescharacteristic
61C2×C4×C191C2×D412yes—
62C8×C191—24no—
63C22×C191—41no—
64C22×C191—42no—
65C22×C191C2×C22⋊C411yescharacteristic
66C2×D191—21no—
67C2×D191—22no—
68C2×D191—41no—
69C2×D191—42no—
70C4×C191—21no—
71C4×C191—22no—
72C4×C191—41no—
73Dic191—41no—
74Dic191#10012yes—
75Dic191#10414yes—
76C2×C191—81no—
77C2×C191C2×#10011yescharacteristic
78C2×C191C2×#10412yes—
79D191—41no—
80D191—81no—
81C191#152511yescharacteristic, complemented, Sylow
82#1525—1911nomaximal, Sylow
83#532—1911no—
84C4×M16—1912no—
85#329—1911no—
86#329—1912no—
87#329D19111yescharacteristic, complemented
88C2×#41—1911no—
89C2×#41—1911no—
90C2×M16—3822no—
91C4⋊Q8—1911no—
92#127—1911no—
93#124—1911no—
94#124C2×D19111yescharacteristic
95#121—3822no—
96C4×D4—3822no—
97C2×C42—1911no—
98#97—1912no—
99#97C2×D19112yes—
100C4×C8—3822no—
101#41—7642no—
102#41—7642no—
103C2×Q8—1911no—
104C2×Q8C4×D19111yescharacteristic
105C2×D4—3821no—
106C2×D4—3821no—
107C2×D4—3822no—
108C2×D4—1911no—
109C2×D4C4×D19111yescharacteristic
110C22×C4—3821no—
111C22×C4—3821no—
112C22×C4—1911no—
113C22×C4—1912no—
114M16—3828no—
115C2×C8—22no—
116C2×C8—3822no—
117C4⋊C4—3822no—
118C22⋊C4—22no—
119C22⋊C4—3822no—
120C42—1911no—
121C42—1912no—
122C42C22×D19111yescharacteristic
123C23—21no—
124C23—3821no—
125C23—1911no—
126Q8—41no—
127Q8—7641no—
128D4—41no—
129D4—7641no—
130D4—7642no—
131D4—7644no—
132D4—7642no—
133C2×C4—21no—
134C2×C4—3824no—
135C2×C4—3822no—
136C2×C4—3821no—
137C2×C4—1912no—
138C2×C4—1911no—
139C2×C4—7642no—
140C2×C4—7642no—
141C2×C4C2×C4×D19111yescharacteristic
142C2×C4D4×D19112yes—
143C8—24no—
144C8—3824no—
145C22—42no—
146C22—41no—
147C22—3822no—
148C22—3821no—
149C22—7642no—
150C22—7641no—
151C22C22⋊C4×D19111yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—1914no—
156C4—1912no—
157C4—7641no—
158C2—81no—
159C2—15281no—
160C2—7641no—
161C2#100×D19111yescharacteristic
162C2#104×D19112yes—
subgroupG/H
centerC22C22⋊C4×D191
commutatorC2×C4×C191C22×C4
FrattiniC42C22×D191
Fitting#329×C191C2
socleC22×C191C2×C22⋊C4
prime radical#27133212C2
cosocleC42×C191C23
hypercenter#329D191
hypocenterC191#1525
LCSG ▷ C2×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #27132615C2semidirect
3. #27132403C2semidirect
4. #27132402C2semidirect
5. #329×C191C2semidirect
81. C191#1525semidirect
87. #329D191semidirect

Maximal subgroups

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

subgroupindex
#271332122
#271326152
#271324032
#271324022
#329×C1912
#1525191