Finite Group Databasebeta

← #38752190

Subgroups — #38752190

Order 25216.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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×#20263351C211yesmaximal, characteristic, complemented
2C2×#20263322C211yesmaximal, characteristic, complemented
3#41×Dic197C211yesmaximal, characteristic, complemented
4#27161277C214yesmaximal, complemented
5#27161272C214yesmaximal, complemented
6#27161102C211yesmaximal, characteristic, complemented
7#27161085C211yesmaximal, characteristic, complemented
8#27161084C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C197C211yesmaximal, characteristic, complemented
10C22×D788C2211yescharacteristic
11C22×C22⋊D197—22no—
12#20263351—24no—
13C2×#15691345C2212yescomplemented
14C2×#15691343C2212yescomplemented
15C2×#15691342C2211yescharacteristic
16#20263322—21no—
17#20263322—21no—
18#20263322C2214yes—
19Q8×Dic197C2212yescomplemented
20D4×Dic197—22no—
21D4×Dic197C2212yescomplemented
22#20263285—21no—
23#20263285C2211yescharacteristic
24C2×C4×Dic197—21no—
25C2×C4×Dic197C2211yescharacteristic
26#20263255C2214yescomplemented
27#20263248C2214yescomplemented
28#20263247C2214yescomplemented
29C2×#41×C197C2211yescharacteristic
30C22×D4×C197C2211yescharacteristic
31AGL(1,ℤ/8)×C197—24no—
32C2×SD16×C197C2212yescomplemented
33C2×D8×C197C2212yescomplemented
34C2×M16×C197C2211yescharacteristic
35C23×D197—21no—
36C2×D788—22no—
37C2×D788—21no—
38C2×D788—22no—
39C2×D788C2312yes—
40C2×C22⋊D197—24no—
41C2×C22⋊D197—24no—
42C2×C22⋊D197—48no—
43C22×Dic197—22no—
44C22×Dic197D412yes—
45#15691345—24no—
46#15691343—24no—
47#15691342—22no—
48C2×#13096802C2312yes—
49C4⋊Dic197—22no—
50C4⋊Dic197C2312yes—
51C22⋊Dic197—22no—
52C22⋊Dic197—21no—
53C4×Dic197—22no—
54C4×Dic197—21no—
55C4×Dic197D411yescharacteristic
56C4×Dic197D411yescharacteristic
57C4×Dic197C2312yes—
58C24×C197—21no—
59#41×C197—22no—
60#41×C197D414yescomplemented
61C2×Q8×C197C2311yescharacteristic
62C2×D4×C197—21no—
63C2×D4×C197—22no—
64C2×D4×C197—21no—
65C2×D4×C197—22no—
66C2×D4×C197C2312yes—
67C2×D4×C197C2311yescharacteristic
68C22×C4×C197—21no—
69C22×C4×C197C2311yescharacteristic
70SD16×C197—24no—
71D8×C197—24no—
72M16×C197—22no—
73C2×C8×C197C2312yes—
74C22×D197—22no—
75C22×D197—44no—
76C22×D197—42no—
77D788—22no—
78D788—22no—
79D788—42no—
80C22⋊D197—48no—
81C22⋊D197—48no—
82C2×Dic197—24no—
83C2×Dic197—22no—
84C2×Dic197—44no—
85C2×Dic197C2×D412yes—
86C2×Dic197C2×D412yes—
87#13096802—22no—
88C23×C197—21no—
89C23×C197—22no—
90C23×C197—44no—
91C23×C197—42no—
92C23×C197C2×D411yescharacteristic
93Q8×C197—21no—
94Q8×C197C2×D412yescomplemented
95D4×C197—24no—
96D4×C197—22no—
97D4×C197—21no—
98D4×C197—22no—
99D4×C197—42no—
100D4×C197C2×D412yescomplemented
101C2×C4×C197—21no—
102C2×C4×C197—24no—
103C2×C4×C197C2×D411yescharacteristic
104C2×C4×C197C2×D412yes—
105C2×C4×C197#4112yes—
106C2×C4×C197C2411yescharacteristic
107C8×C197—22no—
108C22×C197—22no—
109C22×C197—22no—
110C22×C197—42no—
111C22×C197—42no—
112C22×C197—44no—
113C22×C197—84no—
114C22×C197#12112yes—
115C22×C197C22×D411yescharacteristic
116C2×D197—42no—
117C2×D197—44no—
118C2×D197—84no—
119C4×C197—22no—
120C4×C197#12112yes—
121C4×C197C22×D411yescharacteristic
122C4×C197C2×#4111yescharacteristic
123Dic197—22no—
124Dic197—42no—
125Dic197AGL(1,ℤ/8)14yescomplemented
126C2×C197—22no—
127C2×C197—42no—
128C2×C197—82no—
129C2×C197C2×#12111yescharacteristic
130C2×C197C2×AGL(1,ℤ/8)12yes—
131D197—82no—
132C197#303811yescharacteristic, complemented, Sylow
133#3038—1971nomaximal, Sylow
134C2×AGL(1,ℤ/8)—1971no—
135C2×AGL(1,ℤ/8)D19711yescharacteristic, complemented
136C2×#127—1971no—
137C4×#41—1971no—
138#460—1974no—
139#459—1974no—
140#423—1971no—
141#418—1971no—
142#418—1971no—
143C2×#41C2×D19711yescharacteristic
144C22×D4—3942no—
145C22×D4—1971no—
146C22×D4C2×D19711yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—3944no—
149C2×SD16—1972no—
150C2×SD16C2×D19712yescomplemented
151C2×D8—1972no—
152C2×D8C2×D19712yescomplemented
153C2×M16—1971no—
154C2×M16C2×D19711yescharacteristic
155#127—3941no—
156#127—3941no—
157#127—1974no—
158C4×Q8—1972no—
159C4×D4—3942no—
160C4×D4—1972no—
161#117—3941no—
162#117—1971no—
163C2×C42—3941no—
164C2×C42—1971no—
165C4⋊C8—1974no—
166#102—1974no—
167#102—1974no—
168C24—21no—
169C24—3941no—
170#41—22no—
171#41C22⋊D19714yescomplemented
172C2×Q8C22×D19711yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—7888no—
178C2×D4—3944no—
179C2×D4—3944no—
180C2×D4—3941no—
181C2×D4—3942no—
182C2×D4—3942no—
183C2×D4—1972no—
184C2×D4C22×D19712yes—
185C2×D4C22×D19711yescharacteristic
186C22×C4—21no—
187C22×C4—3942no—
188C22×C4—1972no—
189C22×C4C22×D19711yescharacteristic
190SD16—24no—
191SD16—3944no—
192D8—24no—
193D8—3944no—
194M16—22no—
195M16—3942no—
196C2×C8—1972no—
197C2×C8C22×D19712yes—
198C4⋊C4—3942no—
199C4⋊C4—1972no—
200C22⋊C4—3941no—
201C22⋊C4—3942no—
202C42—3942no—
203C42—3941no—
204C42—1971no—
205C42—1971no—
206C42—1972no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—7884no—
212C23—7882no—
213C23—3942no—
214C23D4×D19711yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D19712yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—7888no—
223D4—7888no—
224D4—7882no—
225D4—3942no—
226D4—3942no—
227D4C2×C22⋊D19712yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—7884no—
231C2×C4—3942no—
232C2×C4—3944no—
233C2×C4—1972no—
234C2×C4—1972no—
235C2×C4D4×D19711yescharacteristic
236C2×C4C2×C22⋊D19712yes—
237C2×C4#1569135712yes—
238C2×C4C23×D19711yescharacteristic
239C8—22no—
240C8—3942no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—44no—
245C22—42no—
246C22—84no—
247C22—7884no—
248C22—7882no—
249C22—15764no—
250C22#2026329812yes—
251C22C2×D4×D19711yescharacteristic
252C4—22no—
253C4—7882no—
254C4—3942no—
255C4—1974no—
256C4#2026329812yes—
257C4C22×C22⋊D19711yescharacteristic
258C4C2×#1569135711yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—15762no—
263C2C2×#2026329811yescharacteristic
264C2AGL(1,ℤ/8)×D19712yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×AGL(1,ℤ/8)×C197C2
socleC22×C197C22×D4
hypercenterC2×AGL(1,ℤ/8)D197
hypocenterC197#3038
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20263351C2semidirect
2. C2×#20263322C2semidirect
3. #41×Dic197C2semidirect
4. #27161277C2semidirect
5. #27161272C2semidirect
6. #27161102C2semidirect
7. #27161085C2semidirect
8. #27161084C2semidirect
9. C2×AGL(1,ℤ/8)×C197C2semidirect
13. C2×#15691345C22semidirect
14. C2×#15691343C22semidirect
19. Q8×Dic197C22semidirect
21. D4×Dic197C22semidirect
26. #20263255C22semidirect
27. #20263248C22semidirect
28. #20263247C22semidirect
32. C2×SD16×C197C22semidirect
33. C2×D8×C197C22semidirect
60. #41×C197D4semidirect
94. Q8×C197C2×D4semidirect
100. D4×C197C2×D4semidirect
125. Dic197AGL(1,ℤ/8)semidirect
132. C197#3038semidirect
135. C2×AGL(1,ℤ/8)D197semidirect
150. C2×SD16C2×D197semidirect
152. C2×D8C2×D197semidirect
171. #41C22⋊D197semidirect
216. Q8C2×C22⋊D197semidirect
227. D4C2×C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633512
C2×#202633222
#41×Dic1972
#271612772
#271612722
#271611022
#271610852
#271610842
C2×AGL(1,ℤ/8)×C1972
#3038197