Finite Group Databasebeta

← #37572483

Subgroups — #37572483

Order 24448.

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

normal subgroups only

This group has 548 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
1#27133499C211yesmaximal, characteristic
2#27132679C212yesmaximal, complemented
3#27132660C212yesmaximal, complemented
4#27132637C211yesmaximal, characteristic, complemented
5#408×C191C211yesmaximal, characteristic, complemented
6#20259370C412yescomplemented
7#20259369—22no—
8#20259368—22no—
9#20259367—22no—
10#20259367C412yescomplemented
11C2×#15690287—22no—
12C2×#15690287C2212yes—
13#41×D191—24no—
14#20259281C2212yescomplemented
15#20259248—24no—
16#20259247—22no—
17#20259244—22no—
18#20259237—22no—
19C2×#41×C191C2211yescharacteristic
20C2×M16×C191C2211yescharacteristic, complemented
21C22×C8×C191C2211yescharacteristic, complemented
22C22⋊C8×C191—22no—
23#15690288—24no—
24#15690287—24no—
25#15690287—24no—
26#15690287—24no—
27#15690287—22no—
28#15690287—22no—
29#15690287C2×C414yes—
30#15690287D414yes—
31#15690286—28no—
32#15690286—24no—
33C2×Dic382—22no—
34C2×Dic382D412yes—
35Q8×D191—24no—
36C2×D764—22no—
37C2×D764D412yes—
38C2×C22⋊D191—24no—
39C2×C22⋊D191—22no—
40D4×D191—28no—
41D4×D191—24no—
42C2×C4×D191—24no—
43C2×C4×D191—22no—
44C4⋊Dic191—22no—
45C22⋊Dic191—22no—
46C4×Dic191—22no—
47#41×C191—22no—
48#41×C191—22no—
49C2×Q8×C191C2×C411yescharacteristic
50C2×D4×C191C2×C411yescharacteristic
51C2×D4×C191D412yes—
52C22×C4×C191D412yes—
53C22×C4×C191C2311yescharacteristic
54M16×C191—22no—
55C2×C8×C191—21no—
56C2×C8×C191—21no—
57C2×C8×C191—22no—
58C22×D191—24no—
59C22×D191—22no—
60Dic382—24no—
61Dic382—22no—
62Dic382C22⋊C414yes—
63D764—24no—
64D764—22no—
65D764C22⋊C414yes—
66C22⋊D191—28no—
67C22⋊D191—24no—
68C22⋊D191—24no—
69C2×Dic191—24no—
70C2×Dic191—22no—
71C2×Dic191—42no—
72C4×D191—28no—
73C4×D191—24no—
74C4×D191—24no—
75C23×C191C22⋊C412yes—
76C23×C191C2×D411yescharacteristic
77Q8×C191—22no—
78D4×C191—24no—
79D4×C191—22no—
80C2×C4×C191—22no—
81C2×C4×C191—22no—
82C2×C4×C191C22⋊C412yes—
83C2×C4×C191C22×C411yescharacteristic
84C2×C4×C191C2×D412yes—
85C2×C4×C191C2×D412yes—
86C2×C4×C191C2×D411yescharacteristic
87C8×C191—41no—
88C8×C191—41no—
89C22×C191—22no—
90C22×C191—41no—
91C22×C191—42no—
92C22×C191C2×C22⋊C411yescharacteristic
93C22×C191#12012yes—
94C2×D191—24no—
95C2×D191—82no—
96C2×D191—84no—
97C4×C191—22no—
98C4×C191C2×C22⋊C412yes—
99C4×C191#12011yescharacteristic
100C4×C191#12011yescharacteristic
101Dic191—24no—
102Dic191—82no—
103C2×C191—22no—
104C2×C191—21no—
105C2×C191—42no—
106C2×C191#37911yescharacteristic
107D191—44no—
108C191#179411yescharacteristic, complemented, Sylow
109#1794—1911nomaximal, Sylow
110#585—1911no—
111#421—1912no—
112#416—1912no—
113#410—1911no—
114#408D19111yescharacteristic, complemented
1152−1+4—3822no—
1162−1+4—1912no—
117H(5,𝔽2)—3822no—
118H(5,𝔽2)—3822no—
119H(5,𝔽2)—1912no—
120C2×#41—3824no—
121C2×#41—3822no—
122C2×#41—1912no—
123C2×#41C2×D19111yescharacteristic
124C2×M16C2×D19111yescharacteristic, complemented
125C22×C8C2×D19111yescharacteristic, complemented
126#117—1912no—
127#104—3824no—
128#103—3822no—
129#102—3822no—
130#99—3822no—
131C22⋊C8—22no—
132#41—22no—
133#41—22no—
134#41—3824no—
135#41—3824no—
136#41—3828no—
137#41—3824no—
138#41—3824no—
139#41—3824no—
140#41—3822no—
141#41—3822no—
142#41—1914no—
143#41—1914no—
144C2×Q8—3822no—
145C2×Q8—3824no—
146C2×Q8—1912no—
147C2×Q8C4×D19111yescharacteristic
148C2×D4—3822no—
149C2×D4—3828no—
150C2×D4—3824no—
151C2×D4—3824no—
152C2×D4—3822no—
153C2×D4—1912no—
154C2×D4C4×D19111yescharacteristic
155C2×D4D76412yes—
156C22×C4—3824no—
157C22×C4—3822no—
158C22×C4C22⋊D19112yes—
159C22×C4C22×D19111yescharacteristic
160M16—22no—
161C2×C8—22no—
162C2×C8—21no—
163C2×C8—21no—
164C4⋊C4—3822no—
165C22⋊C4—3822no—
166C42—3822no—
167C23—3824no—
168C23—3822no—
169C23(C2×D191)⋊C412yes—
170C23C2×D76411yescharacteristic
171Q8—22no—
172Q8—3824no—
173Q8—3822no—
174Q8—1914no—
175D4—24no—
176D4—22no—
177D4—3824no—
178D4—3828no—
179D4—3824no—
180D4—3824no—
181D4—3822no—
182D4—1914no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—3828no—
186C2×C4—3824no—
187C2×C4—3822no—
188C2×C4—3824no—
189C2×C4—3824no—
190C2×C4—7642no—
191C2×C4(C2×D191)⋊C412yes—
192C2×C4C2×C4×D19111yescharacteristic
193C2×C4D4×D19112yes—
194C2×C4D4×D19112yes—
195C2×C4C2×C22⋊D19111yescharacteristic
196C8—41no—
197C8—41no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—3824no—
202C22—15282no—
203C22—15284no—
204C22C2×(C2×D191)⋊C411yescharacteristic
205C22#2025929212yes—
206C4—22no—
207C4—3824no—
208C4—15282no—
209C4C22⋊C4×D19112yes—
210C4#2025929011yescharacteristic
211C4#2025929011yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—7644no—
216C2#2713251811yescharacteristic
subgroupG/H
centerC4#20259290
commutatorC2×C4×C191C22×C4
FrattiniC22×C4C22×D191
Fitting#408×C191C2
socleC2×C191#379
prime radical#27133499C2
cosocleC22×C4×C191C23
hypercenter#408D191
hypocenterC191#1794
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #27132679C2semidirect
3. #27132660C2semidirect
4. #27132637C2semidirect
5. #408×C191C2semidirect
6. #20259370C4semidirect
10. #20259367C4semidirect
14. #20259281C22semidirect
20. C2×M16×C191C22semidirect
21. C22×C8×C191C22semidirect
108. C191#1794semidirect
114. #408D191semidirect
124. C2×M16C2×D191semidirect
125. C22×C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271334992
#271326792
#271326602
#271326372
#408×C1912
#1794191