Finite Group Databasebeta

← #38752428

Subgroups — #38752428

Order 25216.

452 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 137 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#27161877C212yesmaximal, complemented
2C2×#20263332C211yesmaximal, characteristic, complemented
3#27161351C212yesmaximal, complemented
4#27161346C212yesmaximal, complemented
5#27161339C214yesmaximal, complemented
6C2×#20263262C211yesmaximal, characteristic, complemented
7#27161063C212yesmaximal, complemented
8C2×#135×C197C211yesmaximal, characteristic, complemented
9C2×#15691358C2211yescharacteristic
10#20263357C2214yes—
11#20263356—22no—
12#20263352C2214yes—
13#20263351—22no—
14#20263346—24no—
15#20263332—21no—
16#20263332—21no—
17#20263332C2212yes—
18#20263332C2212yes—
19C2×#15691342C2212yes—
20C2×#15691342C2211yescharacteristic
21C2×#15691341—21no—
22C2×C8×D197—21no—
23#20263262C2212yescomplemented
24#20263262C2212yescomplemented
25#20263245C2214yescomplemented
26#20263243C2214yescomplemented
27C2×#41×C197C2212yes—
28#135×C197—21no—
29#135×C197—21no—
30#135×C197—21no—
31#135×C197—21no—
32C2×Q16×C197C2211yescharacteristic, complemented
33C2×SD16×C197C2212yescomplemented
34C2×D8×C197C2211yescharacteristic, complemented
35C22×C8×C197C2211yescharacteristic
36#15691358—21no—
37#15691358—21no—
38#15691358D412yes—
39#15691358C2312yes—
40C2×Dic394D411yescharacteristic
41C2×D788D411yescharacteristic
42C2×C22⋊D197—21no—
43C2×C4×D197—21no—
44#15691348—24no—
45#15691346—24no—
46#15691345—24no—
47#15691343—24no—
48#15691342—21no—
49#15691342—22no—
50#15691342C2312yes—
51#15691342C2314yes—
52#15691341—22no—
53#15691341—22no—
54C2×#13096802—21no—
55C2×#13096802—22no—
56C8×D197—22no—
57C8×D197—22no—
58#41×C197—22no—
59#41×C197—22no—
60#41×C197—22no—
61#41×C197—22no—
62C2×Q8×C197C2312yes—
63C2×D4×C197—22no—
64C2×D4×C197C2312yes—
65C22×C4×C197—22no—
66C22×C4×C197C2311yescharacteristic
67Q16×C197—21no—
68Q16×C197—21no—
69SD16×C197—22no—
70SD16×C197—22no—
71D8×C197—21no—
72D8×C197—21no—
73C2×C8×C197D411yescharacteristic
74C2×C8×C197D411yescharacteristic
75C2×C8×C197D411yescharacteristic
76C2×C8×C197D411yescharacteristic
77C2×C8×C197C2311yescharacteristic
78C2×C8×C197C2311yescharacteristic
79C22×D197—21no—
80Dic394—21no—
81Dic394C2×D412yes—
82D788—21no—
83D788C2×D412yes—
84C22⋊D197—22no—
85C22⋊D197—22no—
86C2×Dic197—21no—
87C4×D197—22no—
88C4×D197—22no—
89#13096802—22no—
90#13096802—24no—
91C23×C197—22no—
92C23×C197#4111yescharacteristic
93Q8×C197—22no—
94Q8×C197—22no—
95D4×C197—22no—
96D4×C197—22no—
97D4×C197—42no—
98D4×C197—42no—
99C2×C4×C197—22no—
100C2×C4×C197—42no—
101C2×C4×C197—42no—
102C2×C4×C197C2×D411yescharacteristic
103C2×C4×C197C2×D411yescharacteristic
104C2×C4×C197C2×D411yescharacteristic
105C2×C4×C197C2×D411yescharacteristic
106C2×C4×C197#4111yescharacteristic
107C2×C4×C197C2411yescharacteristic
108C8×C197C2×D411yescharacteristic
109C8×C197C2×D411yescharacteristic
110C8×C197C2×D411yescharacteristic
111C8×C197C2×D411yescharacteristic
112C22×C197—41no—
113C22×C197—42no—
114C22×C197—82no—
115C22×C197#12111yescharacteristic
116C22×C197#12111yescharacteristic
117C22×C197C2×#4111yescharacteristic
118C2×D197—22no—
119C2×D197—42no—
120C4×C197—42no—
121C4×C197#12111yescharacteristic
122C4×C197#12111yescharacteristic
123C4×C197C22×D411yescharacteristic
124C4×C197C22×D411yescharacteristic
125Dic197—22no—
126C2×C197—21no—
127C2×C197—21no—
128C2×C197—21no—
129C2×C197—82no—
130C2×C197C2×#12111yescharacteristic
131D197—42no—
132C197#306711yescharacteristic, complemented, Sylow
133#3067—1971nomaximal, Sylow
134#575—1972no—
135C2×#135D19711yescharacteristic, complemented
136C2×#131—1971no—
137#473—1972no—
138#472—1972no—
139#471—1974no—
140C2×#108—1971no—
141#413—1972no—
142C2×#41—1971no—
143C2×#41C2×D19712yes—
144#137—3942no—
145#137—1974no—
146AGL(1,ℤ/8)—3942no—
147AGL(1,ℤ/8)—1974no—
148#135—21no—
149#135—21no—
150#135—21no—
151#135—21no—
152#135—3944no—
153C2×Q16C2×D19711yescharacteristic, complemented
154C2×SD16C2×D19712yescomplemented
155C2×D8C2×D19711yescharacteristic, complemented
156#131—3941no—
157#131—3941no—
158#131—1972no—
159#131—1972no—
160C2×M16—3941no—
161C2×M16—1971no—
162C2×M16—1972no—
163C22×C8—3941no—
164C22×C8C2×D19711yescharacteristic
165#108—1972no—
166#108—1972no—
167#101—1974no—
168#100—1974no—
169#41—22no—
170#41—22no—
171#41—22no—
172#41—22no—
173#41—3941no—
174#41—3941no—
175#41—1972no—
176#41—1972no—
177C2×Q8—1971no—
178C2×Q8C22×D19712yes—
179C2×D4—22no—
180C2×D4—3941no—
181C2×D4—1971no—
182C2×D4C22×D19712yes—
183C22×C4—22no—
184C22×C4—3941no—
185C22×C4C22×D19711yescharacteristic
186Q16—21no—
187Q16—21no—
188Q16—3944no—
189SD16—22no—
190SD16—22no—
191SD16—3944no—
192SD16—3944no—
193D8—21no—
194D8—21no—
195D8—3944no—
196M16—3942no—
197M16—3941no—
198M16—3942no—
199M16—3942no—
200M16—1972no—
201M16—1974no—
202C2×C8—3942no—
203C2×C8—3941no—
204C2×C8—3942no—
205C2×C8—3942no—
206C2×C8C22⋊D19711yescharacteristic
207C2×C8C22⋊D19711yescharacteristic
208C2×C8C22⋊D19711yescharacteristic
209C2×C8C22⋊D19711yescharacteristic
210C2×C8C22×D19711yescharacteristic
211C2×C8C22×D19711yescharacteristic
212C23—22no—
213C23—3941no—
214C23#1569135711yescharacteristic
215Q8—22no—
216Q8—22no—
217Q8—3941no—
218Q8—1972no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—42no—
223D4—3942no—
224D4—3942no—
225D4—3941no—
226D4—1972no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—3941no—
231C2×C4—3942no—
232C2×C4—3942no—
233C2×C4D4×D19711yescharacteristic
234C2×C4D4×D19711yescharacteristic
235C2×C4C2×C22⋊D19711yescharacteristic
236C2×C4C2×C22⋊D19711yescharacteristic
237C2×C4#1569135711yescharacteristic
238C2×C4C23×D19711yescharacteristic
239C8—3944no—
240C8—3942no—
241C8C2×C22⋊D19711yescharacteristic
242C8C2×C22⋊D19711yescharacteristic
243C8C2×C22⋊D19711yescharacteristic
244C8C2×C22⋊D19711yescharacteristic
245C22—41no—
246C22—42no—
247C22—82no—
248C22—7882no—
249C22—3942no—
250C22#2026329711yescharacteristic
251C22#2026329711yescharacteristic
252C22C2×#1569135711yescharacteristic
253C4—42no—
254C4—3942no—
255C4#2026329711yescharacteristic
256C4#2026329711yescharacteristic
257C4C2×D4×D19711yescharacteristic
258C4C22×C22⋊D19711yescharacteristic
259C2—21no—
260C2—21no—
261C2—21no—
262C2—82no—
263C2—7882no—
264C2C2×#2026329711yescharacteristic
subgroupG/H
centerC4#20263297
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#135×C197C2
socleC2×C197C2×#121
hypercenterC2×#135D197
hypocenterC197#3067
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #27161877C2semidirect
2. C2×#20263332C2semidirect
3. #27161351C2semidirect
4. #27161346C2semidirect
5. #27161339C2semidirect
6. C2×#20263262C2semidirect
7. #27161063C2semidirect
8. C2×#135×C197C2semidirect
23. #20263262C22semidirect
24. #20263262C22semidirect
25. #20263245C22semidirect
26. #20263243C22semidirect
32. C2×Q16×C197C22semidirect
33. C2×SD16×C197C22semidirect
34. C2×D8×C197C22semidirect
132. C197#3067semidirect
135. C2×#135D197semidirect
153. C2×Q16C2×D197semidirect
154. C2×SD16C2×D197semidirect
155. C2×D8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618772
C2×#202633322
#271613512
#271613462
#271613392
C2×#202632622
#271610632
C2×#135×C1972
#3067197