Finite Group Databasebeta

← #38742682

Subgroups — #38742682

Order 25216.

588 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 121 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#27161096C212yesmaximal, complemented
2#27161084C212yesmaximal, complemented
3#27161063C211yesmaximal, characteristic, complemented
4#27161053C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C197C211yesmaximal, characteristic
6C2×#15691342C2212yescomplemented
7#20263285C2212yescomplemented
8#20263253—24no—
9#20263247—24no—
10#20263245—21no—
11#20263243—21no—
12#20263241—22no—
13H(5,𝔽2)×C197—24no—
14H(5,𝔽2)×C197—24no—
15C2×#41×C197—21no—
16C2×#41×C197C412yescomplemented
17C2×#41×C197C2211yescharacteristic
18C2×#41×C197C2211yescharacteristic
19C22×D4×C197—22no—
20C22×D4×C197—21no—
21C22×D4×C197C412yescomplemented
22C22×D4×C197C2211yescharacteristic
23#15691342—22no—
24#15691342—22no—
25C2×#13096802—22no—
26C4⋊Dic197—22no—
27C22⋊Dic197—22no—
28C4×Dic197—22no—
29C24×C197—22no—
30C24×C197—21no—
31#41×C197—28no—
32#41×C197—24no—
33#41×C197—24no—
34#41×C197—22no—
35#41×C197—22no—
36#41×C197—22no—
37#41×C197D414yes—
38C2×Q8×C197C2×C411yescharacteristic
39C2×Q8×C197D411yescharacteristic
40C2×D4×C197—216no—
41C2×D4×C197—28no—
42C2×D4×C197—22no—
43C2×D4×C197—21no—
44C2×D4×C197—24no—
45C2×D4×C197—24no—
46C2×D4×C197—22no—
47C2×D4×C197—21no—
48C2×D4×C197—22no—
49C2×D4×C197C2×C411yescharacteristic
50C2×D4×C197C2×C412yes—
51C2×D4×C197D412yes—
52C2×D4×C197D411yescharacteristic
53C2×D4×C197D414yes—
54C22×C4×C197—22no—
55C22×C4×C197—21no—
56C22×C4×C197C2×C412yes—
57C22×C4×C197C2311yescharacteristic
58C2×Dic197—42no—
59#13096802—42no—
60C23×C197—28no—
61C23×C197—22no—
62C23×C197—21no—
63C23×C197—24no—
64C23×C197—82no—
65C23×C197—84no—
66C23×C197C22⋊C412yes—
67C23×C197C2×D411yescharacteristic
68Q8×C197—21no—
69Q8×C197—22no—
70Q8×C197C22⋊C412yes—
71D4×C197—28no—
72D4×C197—28no—
73D4×C197—24no—
74D4×C197—24no—
75D4×C197—22no—
76D4×C197—21no—
77D4×C197—22no—
78D4×C197—24no—
79D4×C197C22⋊C412yes—
80D4×C197C22⋊C414yes—
81C2×C4×C197—28no—
82C2×C4×C197—21no—
83C2×C4×C197—22no—
84C2×C4×C197—22no—
85C2×C4×C197—24no—
86C2×C4×C197C22⋊C412yes—
87C2×C4×C197C22×C411yescharacteristic
88C2×C4×C197C2×D412yes—
89C2×C4×C197C2×D412yes—
90C2×C4×C197C2×D411yescharacteristic
91C22×C197—22no—
92C22×C197—24no—
93C22×C197—22no—
94C22×C197—41no—
95C22×C197—42no—
96C22×C197—44no—
97C22×C197—42no—
98C22×C197—84no—
99C22×C197—88no—
100C22×C197C2×C22⋊C411yescharacteristic
101C22×C197#12012yes—
102C4×C197—22no—
103C4×C197—22no—
104C4×C197C2×C22⋊C411yescharacteristic
105C4×C197C2×C22⋊C411yescharacteristic
106C4×C197#12012yes—
107Dic197—82no—
108C2×C197—22no—
109C2×C197—21no—
110C2×C197—44no—
111C2×C197—42no—
112C2×C197—42no—
113C2×C197#37911yescharacteristic
114C197#179511yescharacteristic, complemented, Sylow
115#1795—1971nomaximal, Sylow
116C2×H(5,𝔽2)D19711yescharacteristic
117#421—1972no—
118#418—1972no—
119#413—1971no—
120#410—1971no—
121H(5,𝔽2)—24no—
122H(5,𝔽2)—24no—
123C2×#41—21no—
124C2×#41Dic19712yescomplemented
125C2×#41C2×D19711yescharacteristic
126C2×#41C2×D19711yescharacteristic
127C22×D4—22no—
128C22×D4—21no—
129C22×D4Dic19712yescomplemented
130C22×D4C2×D19711yescharacteristic
131C2×M16—1972no—
132#117—1972no—
133#104—3944no—
134#102—3944no—
135#101—3941no—
136#100—3941no—
137#99—3942no—
138C24—22no—
139C24—21no—
140#41—28no—
141#41—24no—
142#41—24no—
143#41—22no—
144#41—22no—
145#41—22no—
146#41C22⋊D19714yes—
147C2×Q8C2×Dic19711yescharacteristic
148C2×Q8C22⋊D19711yescharacteristic
149C2×D4—216no—
150C2×D4—28no—
151C2×D4—22no—
152C2×D4—21no—
153C2×D4—24no—
154C2×D4—24no—
155C2×D4—22no—
156C2×D4—21no—
157C2×D4—22no—
158C2×D4C2×Dic19711yescharacteristic
159C2×D4C2×Dic19712yes—
160C2×D4C22⋊D19712yes—
161C2×D4C22⋊D19711yescharacteristic
162C2×D4C22⋊D19714yes—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4C2×Dic19712yes—
166C22×C4C22×D19711yescharacteristic
167M16—3942no—
168M16—3942no—
169C2×C8—3942no—
170C4⋊C4—3942no—
171C22⋊C4—3942no—
172C42—3942no—
173C23—28no—
174C23—22no—
175C23—21no—
176C23—24no—
177C23—82no—
178C23—84no—
179C23C22⋊Dic19712yes—
180C23C2×C22⋊D19711yescharacteristic
181Q8—21no—
182Q8—22no—
183Q8C22⋊Dic19712yes—
184D4—28no—
185D4—28no—
186D4—24no—
187D4—24no—
188D4—22no—
189D4—21no—
190D4—22no—
191D4—24no—
192D4C22⋊Dic19712yes—
193D4C22⋊Dic19714yes—
194C2×C4—28no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—7882no—
200C2×C4C22⋊Dic19712yes—
201C2×C4C22×Dic19711yescharacteristic
202C2×C4C2×C22⋊D19712yes—
203C2×C4C2×C22⋊D19712yes—
204C2×C4C2×C22⋊D19711yescharacteristic
205C8—7882no—
206C22—22no—
207C22—24no—
208C22—22no—
209C22—41no—
210C22—42no—
211C22—44no—
212C22—42no—
213C22—84no—
214C22—88no—
215C22C2×C22⋊Dic19711yescharacteristic
216C22#2026329512yes—
217C4—22no—
218C4—22no—
219C4—15762no—
220C4C2×C22⋊Dic19711yescharacteristic
221C4C2×C22⋊Dic19711yescharacteristic
222C4#2026329512yes—
223C2—22no—
224C2—21no—
225C2—44no—
226C2—42no—
227C2—42no—
228C2#2716093511yescharacteristic
subgroupG/H
centerC2#27160935
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting, prime radicalC2×H(5,𝔽2)×C197C2
socleC2×C197#379
cosocleC22×C4×C197C23
hypercenterC2×H(5,𝔽2)D197
hypocenterC197#1795
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #27161096C2semidirect
2. #27161084C2semidirect
3. #27161063C2semidirect
4. #27161053C2semidirect
6. C2×#15691342C22semidirect
7. #20263285C22semidirect
16. C2×#41×C197C4semidirect
21. C22×D4×C197C4semidirect
114. C197#1795semidirect
124. C2×#41Dic197semidirect
129. C22×D4Dic197semidirect

Maximal subgroups

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

subgroupindex
#271610962
#271610842
#271610632
#271610532
C2×H(5,𝔽2)×C1972
#1795197