Finite Group Databasebeta

← #37574534

Subgroups — #37574534

Order 24448.

226 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 41 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#27133416C211yesmaximal, characteristic, complemented
2#27132854C211yesmaximal, characteristic
3#27132852C211yesmaximal, characteristic
4#27132843C211yesmaximal, characteristic
5#27132457C211yesmaximal, characteristic, complemented
6#27132454C211yesmaximal, characteristic, complemented
7#352×C191C211yesmaximal, characteristic, complemented
8#20259367C2211yescharacteristic
9#20259355—21no—
10#20259321—21no—
11#20259315—21no—
12#20259315—21no—
13#20259312—21no—
14#20259309C2211yescharacteristic
15#20259306—21no—
16#20259305—21no—
17#20259303C2211yescharacteristic
18#20259301—21no—
19#20259299—21no—
20#20259248—21no—
21#20259237C2211yescharacteristic
22#20259236—21no—
23#20259236—21no—
24#123×C191C2211yescharacteristic
25#100×C191C2211yescharacteristic
26#99×C191C2211yescharacteristic
27#15690287—21no—
28#15690286—41no—
29C2×Dic382—21no—
30C2×Dic382D411yescharacteristic
31C2×C22⋊D191—21no—
32C2×C22⋊D191—21no—
33D4×D191—41no—
34C22×Dic191D411yescharacteristic
35Dic764—41no—
36#15690276—41no—
37C4⋊Dic191—21no—
38C4⋊Dic191—21no—
39C4⋊Dic191—41no—
40Dic191⋊C4—21no—
41Dic191⋊C4—21no—
42Dic191⋊C4—21no—
43Dic191⋊C4—21no—
44Dic191⋊C4—21no—
45Dic191⋊C4—41no—
46C22⋊Dic191—21no—
47C22⋊Dic191—21no—
48C22⋊Dic191—21no—
49C22⋊Dic191—21no—
50C22⋊Dic191D411yescharacteristic
51(C2×D191)⋊C4—41no—
52(C2×D191)⋊C4—41no—
53C4×Dic191—21no—
54C4×Dic191D411yescharacteristic
55C2×D4×C191C2311yescharacteristic
56C22×C4×C191D411yescharacteristic
57M16×C191—21no—
58C4⋊C4×C191—21no—
59C22⋊C4×C191—21no—
60C22⋊C4×C191—21no—
61C22⋊C4×C191D411yescharacteristic
62C22×D191—21no—
63C22×D191—21no—
64Dic382—21no—
65Dic382—41no—
66Dic382—81no—
67D764—21no—
68C22⋊D191—41no—
69C22⋊D191—41no—
70C22⋊D191—41no—
71C2×Dic191—21no—
72C2×Dic191—21no—
73C2×Dic191—21no—
74C2×Dic191—21no—
75C2×Dic191—21no—
76C2×Dic191—21no—
77C2×Dic191—41no—
78C2×Dic191—41no—
79C4×D191—41no—
80C23×C191C2×D411yescharacteristic
81C23×C191C2×D411yescharacteristic
82D4×C191—41no—
83C2×C4×C191—21no—
84C2×C4×C191—21no—
85C2×C4×C191—41no—
86C2×C4×C191—41no—
87C2×C4×C191C2×D411yescharacteristic
88C8×C191—41no—
89C22×C191—21no—
90C22×C191—21no—
91C22×C191—41no—
92C22×C191—41no—
93C22×C191#12011yescharacteristic
94C2×D191—41no—
95C2×D191—81no—
96C2×D191—81no—
97C4×C191—21no—
98C4×C191—41no—
99C4×C191—41no—
100C4×C191—81no—
101Dic191—41no—
102Dic191—41no—
103Dic191—41no—
104Dic191—41no—
105Dic191—41no—
106Dic191—81no—
107C2×C191—21no—
108C2×C191—41no—
109C2×C191—41no—
110C2×C191B(4,𝔽2)11yescharacteristic
111D191—81no—
112C191#220111yescharacteristic, complemented, Sylow
113#2201—1911nomaximal, Sylow
114#563—1911no—
115#458—1911no—
116#458—1911no—
117#455—1911no—
118#354—1911no—
119#352—1911no—
120#352D19111yescharacteristic, complemented
121H(5,𝔽2)—1911no—
122#137—3821no—
123C4⋊Q8—3821no—
124#126—3821no—
125#126—3821no—
126#125—3821no—
127#124—1911no—
128#123—3821no—
129#123—3821no—
130#123—1911no—
131#123C2×D19111yescharacteristic
132C22⋊Q8—3821no—
133C22⋊Q8—3821no—
134#104—3821no—
135#100C2×D19111yescharacteristic
136#99—3821no—
137#99—3821no—
138#99—1911no—
139#99C2×D19111yescharacteristic
140#41—3821no—
141#41—7641no—
142C2×Q8—3821no—
143C2×Q8—1911no—
144C2×D4—3821no—
145C2×D4—3821no—
146C2×D4—7641no—
147C2×D4C22×D19111yescharacteristic
148C22×C4—1911no—
149C22×C4D76411yescharacteristic
150Q16—7641no—
151SD16—7641no—
152M16—21no—
153C4⋊C4—21no—
154C4⋊C4—3821no—
155C4⋊C4—3821no—
156C4⋊C4—3821no—
157C4⋊C4—3821no—
158C4⋊C4—3821no—
159C4⋊C4—3821no—
160C4⋊C4—3821no—
161C4⋊C4—7641no—
162C4⋊C4—7641no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—3821no—
166C22⋊C4—3821no—
167C22⋊C4—3821no—
168C22⋊C4—3821no—
169C22⋊C4—1911no—
170C22⋊C4—7641no—
171C22⋊C4—7641no—
172C22⋊C4D76411yescharacteristic
173C42—3821no—
174C42—1911no—
175C23—3821no—
176C23—3821no—
177C23D4×D19111yescharacteristic
178C23C2×D76411yescharacteristic
179Q8—3821no—
180Q8—15281no—
181Q8—7641no—
182D4—41no—
183D4—3821no—
184D4—7641no—
185D4—7641no—
186D4—7641no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—3821no—
192C2×C4—3821no—
193C2×C4—3821no—
194C2×C4—3821no—
195C2×C4—3821no—
196C2×C4—3821no—
197C2×C4—7641no—
198C2×C4—7641no—
199C2×C4—7641no—
200C2×C4D4×D19111yescharacteristic
201C8—41no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—15281no—
207C22—15281no—
208C22—7641no—
209C22#2025929211yescharacteristic
210C4—21no—
211C4—41no—
212C4—41no—
213C4—81no—
214C4—15281no—
215C4—7641no—
216C4—7641no—
217C4—7641no—
218C4—7641no—
219C4—7641no—
220C2—21no—
221C2—41no—
222C2—41no—
223C2—15281no—
224C2#2713285111yescharacteristic
subgroupG/H
centerC2#27132851
commutator, cosocleC2×D4×C191C23
FrattiniC2×D4C22×D191
Fitting#352×C191C2
socleC2×C191B(4,𝔽2)
prime radical#20259367C22
hypercenter#352D191
hypocenterC191#2201
LCSG ▷ C2×D4×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
1. #27133416C2semidirect
5. #27132457C2semidirect
6. #27132454C2semidirect
7. #352×C191C2semidirect
112. C191#2201semidirect
120. #352D191semidirect

Maximal subgroups

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

subgroupindex
#271334162
#271328542
#271328522
#271328432
#271324572
#271324542
#352×C1912
#2201191