Finite Group Databasebeta

← #37583991

Subgroups — #37583991

Order 24448.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#27133284C212yesmaximal, complemented
2#27132954C212yesmaximal, complemented
3#27132949C212yesmaximal, complemented
4#27132921C211yesmaximal, characteristic, complemented
5#27132916C211yesmaximal, characteristic
6#27132892C212yesmaximal, complemented
7#27132886C212yesmaximal, complemented
8#27132879C212yesmaximal, complemented
9#424×C191C211yesmaximal, characteristic, complemented
10C2×Dic764C2211yescharacteristic
11C2×#15690276C2212yes—
12C2×D1528C2211yescharacteristic
13#20259322C2212yes—
14#20259317—22no—
15#20259310C2212yes—
16#20259308—22no—
17#20259304—22no—
18#20259301C2212yes—
19#20259300—22no—
20#20259296C2212yes—
21C4×Dic382C2212yes—
22C4×D764C2212yes—
23#20259254C2211yescharacteristic
24#20259254C2212yes—
25#20259254C2211yescharacteristic
26#20259247C2212yes—
27#20259247C2212yes—
28#20259244C2212yes—
29#20259244C2212yes—
30C2×M16×C191C2211yescharacteristic
31C22×C8×C191C2211yescharacteristic
32#117×C191C2211yescharacteristic
33C4⋊C8×C191C2212yes—
34C4⋊C8×C191C2212yes—
35C2×Dic382—22no—
36C2×Dic382C2312yes—
37C2×D764C2312yes—
38C2×C22⋊D191—22no—
39C2×C4×D191—22no—
40Dic764—41no—
41#15690276—42no—
42D1528—41no—
43C4⋊Dic191—22no—
44C4⋊Dic191C2312yes—
45C4⋊Dic191C2312yes—
46Dic191⋊C4—22no—
47Dic191⋊C4—22no—
48C22⋊Dic191—22no—
49(C2×D191)⋊C4—22no—
50(C2×D191)⋊C4—22no—
51(C2×D191)⋊C4—22no—
52(C2×D191)⋊C4—22no—
53C4×Dic191—22no—
54C22×C4×C191C2311yescharacteristic
55M16×C191—41no—
56C2×C8×C191—41no—
57C2×C8×C191C2311yescharacteristic
58C2×C8×C191C2312yes—
59C2×C8×C191C2311yescharacteristic
60C4⋊C4×C191D412yes—
61C22⋊C4×C191D411yescharacteristic
62C22⋊C4×C191D411yescharacteristic
63C42×C191C2312yes—
64C22×D191—22no—
65Dic382—22no—
66Dic382—22no—
67Dic382—82no—
68D764—22no—
69D764—22no—
70C22⋊D191—82no—
71C2×Dic191—22no—
72C2×Dic191—22no—
73C2×Dic191—22no—
74C4×D191—82no—
75C23×C191C2×D411yescharacteristic
76C2×C4×C191—21no—
77C2×C4×C191—21no—
78C2×C4×C191C2×D412yes—
79C2×C4×C191C2×D412yes—
80C2×C4×C191C2×D411yescharacteristic
81C2×C4×C191C2411yescharacteristic
82C8×C191—22no—
83C8×C191—21no—
84C8×C191—21no—
85C22×C191—21no—
86C22×C191—22no—
87C22×C191C22×D411yescharacteristic
88C2×D191—42no—
89C2×D191—42no—
90C2×D191—42no—
91C4×C191—21no—
92C4×C191—22no—
93C4×C191—22no—
94C4×C191H(5,𝔽2)11yescharacteristic
95C4×C1912−1+411yescharacteristic
96Dic191—42no—
97Dic191—42no—
98Dic191—42no—
99C2×C191—41no—
100C2×C191#53711yescharacteristic
101C2×C191#57712yes—
102D191—82no—
103C191#326311yescharacteristic, complemented, Sylow
104#3263—1911nomaximal, Sylow
105#542—1912no—
106#476—1912no—
107#475—1912no—
108#470—1911no—
109#469—1911no—
110#465—1912no—
111#464—1912no—
112#463—1912no—
113#424D19111yescharacteristic, complemented
114C2×Q16—1911no—
115C2×SD16—1912no—
116C2×D8—1911no—
117C2×M16C2×D19111yescharacteristic
118C22×C8C2×D19111yescharacteristic
119C4⋊Q8—1912no—
120#126—3822no—
121#124—3822no—
122#124—1912no—
123#123—3822no—
124C22⋊Q8—3822no—
125C22⋊Q8—1912no—
126#121—1912no—
127C4×Q8—1912no—
128C4×D4—1912no—
129#117C2×D19111yescharacteristic
130#106—1912no—
131#106—1911no—
132#106—1911no—
133C4⋊C8C2×D19112yes—
134C4⋊C8C2×D19112yes—
135#103—1912no—
136#103—1912no—
137#102—1912no—
138#102—1912no—
139C2×Q8—3822no—
140C2×Q8—1912no—
141C2×D4—3822no—
142C2×D4—1912no—
143C22×C4—3822no—
144C22×C4C22×D19111yescharacteristic
145Q16—7641no—
146SD16—7642no—
147D8—7641no—
148M16—41no—
149C2×C8—41no—
150C2×C8C22×D19111yescharacteristic
151C2×C8C22×D19112yes—
152C2×C8C22×D19111yescharacteristic
153C4⋊C4—3822no—
154C4⋊C4—3822no—
155C4⋊C4—3822no—
156C4⋊C4—1912no—
157C4⋊C4—1912no—
158C4⋊C4D76412yes—
159C22⋊C4—3822no—
160C22⋊C4—3822no—
161C22⋊C4—3822no—
162C22⋊C4—3822no—
163C22⋊C4—3822no—
164C22⋊C4D76411yescharacteristic
165C22⋊C4D76411yescharacteristic
166C42—3822no—
167C42C22×D19112yes—
168C23—3822no—
169C23C2×D76411yescharacteristic
170Q8—3822no—
171Q8—3822no—
172Q8—15282no—
173D4—3822no—
174D4—3822no—
175D4—15282no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—3822no—
179C2×C4—3822no—
180C2×C4—3822no—
181C2×C4—15282no—
182C2×C4C2×D76412yes—
183C2×C4C2×D76412yes—
184C2×C4C2×D76411yescharacteristic
185C2×C4C23×D19111yescharacteristic
186C8—22no—
187C8—21no—
188C8—21no—
189C22—22no—
190C22—21no—
191C22—7642no—
192C22—7642no—
193C22—7642no—
194C22C22×D76411yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—7642no—
199C4—7642no—
200C4—7642no—
201C4#2025936711yescharacteristic
202C4#2025937011yescharacteristic
203C2—41no—
204C2—15282no—
205C2#2713324411yescharacteristic
206C2#2713347512yes—
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#424×C191C2
socleC22×C191C22×D4
prime radical#27132916C2
hypercenter#424D191
hypocenterC191#3263
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #27133284C2semidirect
2. #27132954C2semidirect
3. #27132949C2semidirect
4. #27132921C2semidirect
6. #27132892C2semidirect
7. #27132886C2semidirect
8. #27132879C2semidirect
9. #424×C191C2semidirect
103. C191#3263semidirect
113. #424D191semidirect

Maximal subgroups

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

subgroupindex
#271332842
#271329542
#271329492
#271329212
#271329162
#271328922
#271328862
#271328792
#424×C1912
#3263191