Finite Group Databasebeta

← #37572491

Subgroups — #37572491

Order 24448.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#20259368C211yesmaximal, characteristic
2#27132679C212yesmaximal, complemented
3#27132669C212yesmaximal, complemented
4#27132637C211yesmaximal, characteristic, complemented
5#413×C191C211yesmaximal, characteristic, complemented
6#20259368—24no—
7#20259368—24no—
8C2×#15690288—21no—
9C2×#15690287C412yescomplemented
10C2×#15690287C2211yescharacteristic
11C22×D764C412yescomplemented
12C22×D764C2211yescharacteristic
13C2×D4×D191—21no—
14C2×D4×D191—22no—
15#20259281C2212yescomplemented
16#20259248—24no—
17#20259244—24no—
18#20259237—22no—
19C2×#41×C191C2211yescharacteristic
20C2×M16×C191C2212yescomplemented
21#101×C191—21no—
22#100×C191—21no—
23C23×D191—22no—
24C23×D191—21no—
25#15690288—28no—
26#15690287—24no—
27#15690287—24no—
28#15690287—22no—
29#15690287D414yes—
30C2×Dic382C2×C411yescharacteristic
31C2×D764—24no—
32C2×D764—24no—
33C2×D764—22no—
34C2×D764—21no—
35C2×D764—22no—
36C2×D764C2×C411yescharacteristic
37C2×D764C2×C412yes—
38C2×D764D414yes—
39C2×C22⋊D191—22no—
40C2×C22⋊D191—21no—
41D4×D191—216no—
42D4×D191—28no—
43C2×C4×D191—22no—
44C2×C4×D191—21no—
45C4⋊Dic191—22no—
46C22⋊Dic191—22no—
47C4×Dic191—22no—
48#41×C191—22no—
49#41×C191—22no—
50C2×Q8×C191D411yescharacteristic
51C2×D4×C191D411yescharacteristic
52C2×D4×C191D412yes—
53C22×C4×C191C2×C412yes—
54C22×C4×C191C2311yescharacteristic
55M16×C191—22no—
56M16×C191—22no—
57C2×C8×C191—22no—
58C22×D191—22no—
59C22×D191—21no—
60C22×D191—28no—
61C22×D191—24no—
62C22×D191—82no—
63C22×D191—84no—
64Dic382—21no—
65Dic382C22⋊C412yes—
66D764—28no—
67D764—22no—
68D764—21no—
69D764—24no—
70D764C22⋊C412yes—
71D764C22⋊C414yes—
72C22⋊D191—28no—
73C22⋊D191—24no—
74C2×Dic191—21no—
75C2×Dic191—42no—
76C4×D191—28no—
77C4×D191—24no—
78C23×C191C22⋊C412yes—
79C23×C191C2×D411yescharacteristic
80Q8×C191—22no—
81D4×C191—24no—
82D4×C191—22no—
83C2×C4×C191—22no—
84C2×C4×C191—22no—
85C2×C4×C191C22⋊C412yes—
86C2×C4×C191C22×C411yescharacteristic
87C2×C4×C191C2×D412yes—
88C2×C4×C191C2×D412yes—
89C2×C4×C191C2×D411yescharacteristic
90C8×C191—42no—
91C22×C191—22no—
92C22×C191—41no—
93C22×C191—42no—
94C22×C191C2×C22⋊C411yescharacteristic
95C22×C191#12012yes—
96C2×D191—24no—
97C2×D191—22no—
98C2×D191—44no—
99C2×D191—42no—
100C2×D191—84no—
101C2×D191—88no—
102C4×C191—22no—
103C4×C191C2×C22⋊C411yescharacteristic
104C4×C191C2×C22⋊C411yescharacteristic
105C4×C191#12012yes—
106Dic191—22no—
107Dic191—82no—
108C2×C191—22no—
109C2×C191—21no—
110C2×C191—42no—
111C2×C191#37911yescharacteristic
112D191—44no—
113D191—42no—
114C191#179511yescharacteristic, complemented, Sylow
115#1795—1911nomaximal, Sylow
116C2×H(5,𝔽2)—1911no—
117#421—1912no—
118#418—1912no—
119#413D19111yescharacteristic, complemented
120#410—1911no—
121H(5,𝔽2)—3824no—
122H(5,𝔽2)—3824no—
123C2×#41—3821no—
124C2×#41—1912no—
125C2×#41—1911no—
126C2×#41C2×D19111yescharacteristic
127C22×D4—3821no—
128C22×D4—3822no—
129C22×D4—1912no—
130C22×D4—1911no—
131C2×M16C2×D19112yescomplemented
132#117—1912no—
133#104—3824no—
134#102—3824no—
135#101—21no—
136#100—21no—
137#99—3822no—
138C24—3822no—
139C24—3821no—
140#41—22no—
141#41—22no—
142#41—3824no—
143#41—3824no—
144#41—3828no—
145#41—3822no—
146#41—1914no—
147C2×Q8—1911no—
148C2×Q8C22⋊D19111yescharacteristic
149C2×D4—3824no—
150C2×D4—3824no—
151C2×D4—38216no—
152C2×D4—3828no—
153C2×D4—3822no—
154C2×D4—3821no—
155C2×D4—3822no—
156C2×D4—3822no—
157C2×D4—3821no—
158C2×D4—1911no—
159C2×D4—1912no—
160C2×D4—1914no—
161C2×D4C22⋊D19111yescharacteristic
162C2×D4D76412yes—
163C22×C4—3822no—
164C22×C4—3821no—
165C22×C4C4×D19112yes—
166C22×C4C22×D19111yescharacteristic
167M16—22no—
168M16—22no—
169C2×C8—22no—
170C4⋊C4—3822no—
171C22⋊C4—3822no—
172C42—3822no—
173C23—3828no—
174C23—3824no—
175C23—3822no—
176C23—3821no—
177C23—15282no—
178C23—15284no—
179C23(C2×D191)⋊C412yes—
180C23C2×D76411yescharacteristic
181Q8—22no—
182Q8—3821no—
183Q8—1912no—
184D4—24no—
185D4—22no—
186D4—3828no—
187D4—3828no—
188D4—3824no—
189D4—3822no—
190D4—3821no—
191D4—3824no—
192D4—1912no—
193D4—1914no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—3828no—
197C2×C4—3821no—
198C2×C4—3824no—
199C2×C4—7642no—
200C2×C4(C2×D191)⋊C412yes—
201C2×C4C2×C4×D19111yescharacteristic
202C2×C4D4×D19112yes—
203C2×C4D4×D19112yes—
204C2×C4C2×C22⋊D19111yescharacteristic
205C8—42no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22—3824no—
210C22—3822no—
211C22—15284no—
212C22—15288no—
213C22—7644no—
214C22—7642no—
215C22C2×(C2×D191)⋊C411yescharacteristic
216C22#2025929212yes—
217C4—22no—
218C4—3822no—
219C4—15282no—
220C4C22⋊C4×D19111yescharacteristic
221C4C22⋊C4×D19111yescharacteristic
222C4#2025929012yes—
223C2—22no—
224C2—21no—
225C2—42no—
226C2—7644no—
227C2—7642no—
228C2#2713251811yescharacteristic
subgroupG/H
centerC2#27132518
commutatorC2×C4×C191C22×C4
FrattiniC22×C4C22×D191
Fitting#413×C191C2
socleC2×C191#379
prime radicalC2×#20259368C2
cosocleC22×C4×C191C23
hypercenter#413D191
hypocenterC191#1795
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #27132679C2semidirect
3. #27132669C2semidirect
4. #27132637C2semidirect
5. #413×C191C2semidirect
9. C2×#15690287C4semidirect
11. C22×D764C4semidirect
15. #20259281C22semidirect
20. C2×M16×C191C22semidirect
114. C191#1795semidirect
119. #413D191semidirect
131. C2×M16C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593682
#271326792
#271326692
#271326372
#413×C1912
#1795191