Finite Group Databasebeta

← #27206951

Subgroups — #27206951

Order 13184.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D412C211yesmaximal, characteristic, complemented
2#20268139C211yesmaximal, characteristic, complemented
3#20268104C212yesmaximal, complemented
4#20268068C211yesmaximal, characteristic, complemented
5#20268025C212yesmaximal, complemented
6#20267986C211yesmaximal, characteristic, complemented
7#20267924C212yesmaximal, complemented
8C4×#13110090C211yesmaximal, characteristic, complemented
9D4×C103⋊C8C211yesmaximal, characteristic, complemented
10C2×#15692941C212yesmaximal, complemented
11#547×C103C211yesmaximal, characteristic, complemented
12C22×D412C2212yes—
13C2×D4×D103—21no—
14C2×#13110090C2211yescharacteristic
15C2×#13110090C2212yescomplemented
16C22×C103⋊C8C2212yescomplemented
17#15693018C2211yescharacteristic
18#15692996—21no—
19#15692995C2212yes—
20#15692991—22no—
21C4×D412C2211yescharacteristic
22#15692954C2212yescomplemented
23#15692954C2211yescharacteristic, complemented
24C4⋊(C103⋊C8)C2211yescharacteristic, complemented
25#15692942C2211yescharacteristic, complemented
26#15692941C2211yescharacteristic
27#15692941C2212yes—
28#15692941C2212yescomplemented
29#15692941C2212yescomplemented
30#15692941C2212yescomplemented
31C22⋊(C103⋊C8)C2212yescomplemented
32C4×C103⋊C8C2211yescharacteristic, complemented
33C2×#41×C103—21no—
34C4⋊Q8×C103C2211yescharacteristic, complemented
35#123×C103—22no—
36C22⋊Q8×C103—21no—
37#121×C103C2212yescomplemented
38C4×D4×C103C2211yescharacteristic
39C4×D4×C103C2211yescharacteristic, complemented
40C2×C4⋊C4×C103C2212yescomplemented
41C23×D103—22no—
42C2×D412—24no—
43C2×D412—21no—
44C2×D412—22no—
45C2×D412—21no—
46C2×D412—42no—
47C2×D412C2312yes—
48C2×D412C2311yescharacteristic
49C2×C22⋊D103—22no—
50D4×D103—44no—
51C2×C4×D103—21no—
52#13110090—22no—
53#13110090—42no—
54C2×C103⋊C8—42no—
55C2×C103⋊C8C2311yescharacteristic
56C2×C103⋊C8C2311yescharacteristic
57C2×C103⋊C8C2312yes—
58C4⋊Dic103C2311yescharacteristic
59(C2×D103)⋊C4—21no—
60(C2×D103)⋊C4—22no—
61#41×C103—44no—
62C2×Q8×C103—21no—
63C2×D4×C103—22no—
64C2×D4×C103D411yescharacteristic
65C2×D4×C103C2311yescharacteristic
66C22×C4×C103—22no—
67C22×C4×C103—21no—
68C22×C4×C103C2312yes—
69C4⋊C4×C103—22no—
70C4⋊C4×C103—21no—
71C4⋊C4×C103D411yescharacteristic
72C4⋊C4×C103C2311yescharacteristic
73C4⋊C4×C103C2312yes—
74C22⋊C4×C103—21no—
75C22⋊C4×C103—22no—
76C22⋊C4×C103D412yes—
77C42×C103C2311yescharacteristic
78C23×C103—21no—
79C23×C103C2×D412yes—
80C22×D103—21no—
81C22×D103—22no—
82C22×D103—44no—
83C22×D103—44no—
84C22×D103—82no—
85Q8×C103—42no—
86D4×C103—21no—
87D4×C103—21no—
88D4×C103—44no—
89D4×C103D814yescomplemented
90C2×C4×C103—22no—
91C2×C4×C103—22no—
92C2×C4×C103—21no—
93C2×C4×C103—22no—
94C2×C4×C103—44no—
95C2×C4×C103—42no—
96C2×C4×C103C2×D411yescharacteristic
97C2×C4×C103C2×D412yes—
98C2×C4×C103C2×D411yescharacteristic
99C2×C4×C103C2411yescharacteristic
100C22⋊D103—44no—
101C2×Dic103—21no—
102D412—22no—
103D412—21no—
104D412—22no—
105D412—44no—
106D412—42no—
107D412—81no—
108D412#4112yescomplemented
109C4×D103—42no—
110C103⋊C8—21no—
111C103⋊C8—21no—
112C103⋊C8—22no—
113C22×C103—22no—
114C22×C103—22no—
115C22×C103—41no—
116C22×C103—41no—
117C22×C103—41no—
118C22×C103C2×D814yes—
119C22×C103C22×D411yescharacteristic
120C2×D103—22no—
121C2×D103—41no—
122C2×D103—41no—
123C2×D103—42no—
124C2×D103—42no—
125C2×D103—42no—
126C2×D103—88no—
127C4×C103—22no—
128C4×C103—21no—
129C4×C103—41no—
130C4×C103—42no—
131C4×C103C2×D812yes—
132C4×C103C2×#4111yescharacteristic
133C4×C103H(5,𝔽2)11yescharacteristic
134Dic103—41no—
135C2×C103—24no—
136C2×C103—81no—
137C2×C103#54611yescharacteristic
138C2×C103C22×D811yescharacteristic
139C2×C103#57711yescharacteristic
140D103—42no—
141D103—82no—
142#3294—1031nomaximal, Sylow
143C103#329411yescharacteristic, complemented, Sylow
144#547D10311yescharacteristic, complemented
145D42—1031no—
146#486—1031no—
147#480—1032no—
148#474—1031no—
149#466—1032no—
150#459—1031no—
151#447—1032no—
152C4×D8—1031no—
153C8×D4—1031no—
154C2×#102—1032no—
155C2×#41—21no—
156C22×D4—1032no—
157C22×D4—2061no—
158C2×D8—1032no—
159C2×D8—1031no—
160C22×C8—1032no—
161C4⋊Q8C2×D10311yescharacteristic, complemented
162#127—1031no—
163#123—22no—
164C22⋊Q8—21no—
165#121—1032no—
166#121—2061no—
167#121C2×D10312yescomplemented
168#120—2062no—
169C4×D4—1031no—
170C4×D4C2×D10311yescharacteristic
171C4×D4C2×D10311yescharacteristic, complemented
172C2×C4⋊C4C2×D10312yescomplemented
173#107—1031no—
174#107—1032no—
175C4⋊C8—1031no—
176#102—1031no—
177#102—1031no—
178#102—1032no—
179#102—1032no—
180#102—1032no—
181#102—1032no—
182C22⋊C8—1032no—
183C4×C8—1031no—
184C24—2062no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—1031no—
189C2×D4—1032no—
190C2×D4—4122no—
191C2×D4—4124no—
192C2×D4—2064no—
193C2×D4—2062no—
194C2×D4—2061no—
195C2×D4—2062no—
196C2×D4—2061no—
197C2×D4C22⋊D10311yescharacteristic
198C2×D4C22×D10311yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—2061no—
202C22×C4C22×D10312yes—
203D8—4122no—
204D8—2062no—
205C2×C8—1031no—
206C2×C8—1031no—
207C2×C8—1032no—
208C2×C8—4122no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—1031no—
212C4⋊C4C22⋊D10311yescharacteristic
213C4⋊C4C22×D10311yescharacteristic
214C4⋊C4C22×D10312yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—2061no—
218C22⋊C4—2062no—
219C22⋊C4C22⋊D10312yes—
220C42C22×D10311yescharacteristic
221C23—21no—
222C23—8242no—
223C23—4124no—
224C23—4124no—
225C23—2062no—
226C23—2061no—
227C23C2×C22⋊D10312yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—8241no—
233D4—1032no—
234D4—4124no—
235D4—4124no—
236D4—4122no—
237D4—2062no—
238D4—2062no—
239D4—2061no—
240D4#1311009014yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—4122no—
248C2×C4—2061no—
249C2×C4C2×C22⋊D10311yescharacteristic
250C2×C4C2×C22⋊D10312yes—
251C2×C4C2×C22⋊D10311yescharacteristic
252C2×C4C23×D10311yescharacteristic
253C8—2061no—
254C8—2061no—
255C8—2062no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—8248no—
262C22—4122no—
263C22—4122no—
264C22—4122no—
265C22—4121no—
266C22—4121no—
267C22—2062no—
268C22C2×#1311009014yes—
269C22C22×C22⋊D10311yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—4121no—
275C4C2×#1311009012yes—
276C4#41×D10311yescharacteristic
277C4#1569306711yescharacteristic
278C2—24no—
279C2—81no—
280C2—8242no—
281C2—4122no—
282C2#2026842911yescharacteristic
283C2C22×#1311009011yescharacteristic
284C2#2026859911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#547×C103C2
socleC22×C103C22×D4
hypercenter#547D103
hypocenterC103#3294
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D412C2semidirect
2. #20268139C2semidirect
3. #20268104C2semidirect
4. #20268068C2semidirect
5. #20268025C2semidirect
6. #20267986C2semidirect
7. #20267924C2semidirect
8. C4×#13110090C2semidirect
9. D4×C103⋊C8C2semidirect
10. C2×#15692941C2semidirect
11. #547×C103C2semidirect
15. C2×#13110090C22semidirect
16. C22×C103⋊C8C22semidirect
22. #15692954C22semidirect
23. #15692954C22semidirect
24. C4⋊(C103⋊C8)C22semidirect
25. #15692942C22semidirect
28. #15692941C22semidirect
29. #15692941C22semidirect
30. #15692941C22semidirect
31. C22⋊(C103⋊C8)C22semidirect
32. C4×C103⋊C8C22semidirect
34. C4⋊Q8×C103C22semidirect
37. #121×C103C22semidirect
39. C4×D4×C103C22semidirect
40. C2×C4⋊C4×C103C22semidirect
89. D4×C103D8semidirect
108. D412#41semidirect
143. C103#3294semidirect
144. #547D103semidirect
161. C4⋊Q8C2×D103semidirect
167. #121C2×D103semidirect
171. C4×D4C2×D103semidirect
172. C2×C4⋊C4C2×D103semidirect
240. D4#13110090semidirect

Maximal subgroups

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

subgroupindex
D4×D4122
#202681392
#202681042
#202680682
#202680252
#202679862
#202679242
C4×#131100902
D4×C103⋊C82
C2×#156929412
#547×C1032
#3294103