Finite Group Databasebeta

← #37571445

Subgroups — #37571445

Order 24448.

316 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 53 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#27133289C211yesmaximal, characteristic, complemented
2#27132816C212yesmaximal, complemented
3#27132572C211yesmaximal, characteristic
4#27132401C212yesmaximal, complemented
5#323×C191C211yesmaximal, characteristic, complemented
6C22×D764C2211yescharacteristic
7C22×C22⋊D191—21no—
8C2×#15690276—22no—
9#20259301—21no—
10#20259301C2212yes—
11#20259299—22no—
12#20259299—22no—
13C2×C22⋊Dic191—21no—
14C2×(C2×D191)⋊C4—21no—
15#20259244—22no—
16C191⋊#95C2211yescharacteristic
17C2×C22⋊C4×C191C2211yescharacteristic
18C22⋊C8×C191C2212yes—
19C23×D191—21no—
20C2×Dic382—21no—
21C2×Dic382D412yescomplemented
22C2×D764—22no—
23C2×D764—42no—
24C2×C22⋊D191—42no—
25C2×C22⋊D191—42no—
26C2×C22⋊D191—42no—
27C22×Dic191—21no—
28C22×Dic191—21no—
29C22×Dic191—21no—
30#15690276—44no—
31C4⋊Dic191—21no—
32C4⋊Dic191D412yescomplemented
33Dic191⋊C4—22no—
34Dic191⋊C4—22no—
35C22⋊Dic191—22no—
36C22⋊Dic191—22no—
37(C2×D191)⋊C4—42no—
38C24×C191D411yescharacteristic
39C22×C4×C191D411yescharacteristic
40C22×C4×C191C2311yescharacteristic
41C2×C8×C191—22no—
42C22⋊C4×C191—22no—
43C22×D191—22no—
44C22×D191—41no—
45C22×D191—41no—
46C22×D191—41no—
47C22×D191—41no—
48C22×D191—41no—
49C22×D191—41no—
50Dic382—42no—
51Dic382—81no—
52D764—24no—
53D764—42no—
54C22⋊D191—48no—
55C2×Dic191—22no—
56C2×Dic191—22no—
57C2×Dic191—22no—
58C2×Dic191—41no—
59C2×Dic191—41no—
60C2×Dic191—81no—
61C2×Dic191—81no—
62C23×C191—21no—
63C23×C191—21no—
64C23×C191—41no—
65C23×C191—41no—
66C23×C191SD1612yescomplemented
67C23×C191C2×D411yescharacteristic
68C2×C4×C191—21no—
69C2×C4×C191—41no—
70C2×C4×C191—41no—
71C2×C4×C191C2×D412yes—
72C8×C191—42no—
73C22×C191—21no—
74C22×C191—21no—
75C22×C191—22no—
76C22×C191—42no—
77C22×C191—41no—
78C22×C191—41no—
79C22×C191—81no—
80C22×C191#12011yescharacteristic
81C22×C191C2×SD1611yescharacteristic
82C22×C191#13711yescharacteristic
83C2×D191—42no—
84C2×D191—42no—
85C2×D191—42no—
86C2×D191—81no—
87C2×D191—81no—
88C2×D191—81no—
89C2×D191—81no—
90C4×C191—22no—
91C4×C191—41no—
92Dic191—42no—
93Dic191—42no—
94Dic191—42no—
95C2×C191—21no—
96C2×C191—21no—
97C2×C191—42no—
98C2×C191#44811yescharacteristic
99C2×C191#45311yescharacteristic
100C2×C191B(4,𝔽2)11yescharacteristic
101D191—82no—
102C191#159611yescharacteristic, complemented, Sylow
103#1596—1911nomaximal, Sylow
104#543—1911no—
105#450—1912no—
106#392—1911no—
107#328—1912no—
108#323D19111yescharacteristic, complemented
109C22×D4—3821no—
110C22×D4—1911no—
111C2×SD16—3822no—
112C22⋊Q8—3822no—
113C22⋊Q8—3822no—
114C22⋊Q8—3821no—
115C22⋊Q8—1912no—
116C2×C22⋊C4—3821no—
117C2×C22⋊C4—3821no—
118C2×C22⋊C4C2×D19111yescharacteristic
119#102—3822no—
120C22⋊C8C2×D19112yes—
121#95—1911no—
122C24—3821no—
123C24D76411yescharacteristic
124C2×Q8—3821no—
125C2×Q8—1912no—
126C2×D4—3822no—
127C2×D4—7642no—
128C2×D4—7642no—
129C2×D4—7642no—
130C2×D4—7642no—
131C22×C4—3821no—
132C22×C4—3821no—
133C22×C4—3821no—
134C22×C4D76411yescharacteristic
135C22×C4C22×D19111yescharacteristic
136SD16—7644no—
137C2×C8—22no—
138C4⋊C4—3822no—
139C4⋊C4—3821no—
140C4⋊C4—3822no—
141C4⋊C4—1912no—
142C22⋊C4—22no—
143C22⋊C4—3822no—
144C22⋊C4—3822no—
145C22⋊C4—7642no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—3822no—
151C23—7641no—
152C23—7641no—
153C23—7641no—
154C23—7641no—
155C23—7641no—
156C23—7641no—
157C23#1569027612yescomplemented
158C23C2×D76411yescharacteristic
159Q8—15281no—
160Q8—7642no—
161D4—3824no—
162D4—7642no—
163D4—7648no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—3822no—
168C2×C4—3822no—
169C2×C4—3822no—
170C2×C4—15281no—
171C2×C4—15281no—
172C2×C4—7641no—
173C2×C4—7641no—
174C2×C4D4×D19112yes—
175C8—42no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—15281no—
184C22—15281no—
185C22—15281no—
186C22—15281no—
187C22—7642no—
188C22—7642no—
189C22—7642no—
190C22#2025929211yescharacteristic
191C22C2×#1569027611yescharacteristic
192C22#2025935511yescharacteristic
193C4—22no—
194C4—41no—
195C4—7642no—
196C4—7642no—
197C4—7642no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—15282no—
202C2#2713280411yescharacteristic
203C2#2713283311yescharacteristic
204C2#2713285111yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#323×C191C2
socleC22×C191#120
prime radical#27132572C2
hypercenter#323D191
hypocenterC191#1596
LCSG ▷ C22×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #27133289C2semidirect
2. #27132816C2semidirect
4. #27132401C2semidirect
5. #323×C191C2semidirect
21. C2×Dic382D4semidirect
32. C4⋊Dic191D4semidirect
66. C23×C191SD16semidirect
102. C191#1596semidirect
108. #323D191semidirect
157. C23#15690276semidirect

Maximal subgroups

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

subgroupindex
#271332892
#271328162
#271325722
#271324012
#323×C1912
#1596191