Finite Group Databasebeta

← #36335829

Subgroups — #36335829

Order 21376.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×SD16×D167C212yesmaximal, complemented
2Q8×D668C211yesmaximal, characteristic, complemented
3#24923904C211yesmaximal, characteristic
4#24923641C211yesmaximal, characteristic, complemented
5#24923497C212yesmaximal, complemented
6#24923475C211yesmaximal, characteristic, complemented
7#24923421C212yesmaximal, complemented
8#24923406C212yesmaximal, complemented
9C4×#14581937C211yesmaximal, characteristic, complemented
10#24923334C211yesmaximal, characteristic, complemented
11#460×C167C211yesmaximal, characteristic, complemented
12C2×Q8×D167C2212yescomplemented
13C2×D4×D167—21no—
14C2×D4×D167C2212yes—
15C2×#14581939C2212yes—
16C2×#14581938C2212yescomplemented
17C2×#14581938C2212yescomplemented
18C2×#14581937C2211yescharacteristic
19SD16×D167—28no—
20C2×C8×D167C2212yescomplemented
21#19128832—21no—
22#19128832C2211yescharacteristic, complemented
23#19128812—22no—
24#19128812C2212yescomplemented
25#19128803—21no—
26#19128803C2212yes—
27#19128800—22no—
28C4×D668—21no—
29C4×D668C2211yescharacteristic
30C4×D668C2211yescharacteristic
31#19128763C2211yescharacteristic, complemented
32#19128757C2212yescomplemented
33#19128756C2211yescharacteristic, complemented
34#19128752C2211yescharacteristic
35(C2×D167)⋊C8C2212yescomplemented
36C4×C167⋊C8C2211yescharacteristic, complemented
37C2×SD16×C167C2212yescomplemented
38#127×C167C2211yescharacteristic
39C4×Q8×C167C2211yescharacteristic, complemented
40C4⋊C8×C167C2211yescharacteristic, complemented
41#102×C167C2212yescomplemented
42C23×D167—22no—
43C2×Dic334—22no—
44C2×Dic334C2312yes—
45Q8×D167—24no—
46Q8×D167—42no—
47C2×D668D411yescharacteristic
48C2×D668C2311yescharacteristic
49C2×C22⋊D167—22no—
50C2×C22⋊D167—22no—
51D4×D167—24no—
52D4×D167—42no—
53D4×D167—44no—
54C2×C4×D167—22no—
55C2×C4×D167—21no—
56C2×C4×D167C2312yes—
57#14581939—24no—
58#14581938—24no—
59#14581938—42no—
60#14581937D414yescomplemented
61C2×C167⋊C8C2311yescharacteristic
62C2×C167⋊C8C2311yescharacteristic
63C8×D167—24no—
64C4⋊Dic167—22no—
65C4⋊Dic167—21no—
66C4⋊Dic167—21no—
67C4⋊Dic167D411yescharacteristic
68C4⋊Dic167C2311yescharacteristic
69C22⋊Dic167—22no—
70(C2×D167)⋊C4—22no—
71(C2×D167)⋊C4—21no—
72(C2×D167)⋊C4D412yes—
73C2×Q8×C167C2311yescharacteristic
74C2×D4×C167—21no—
75C2×D4×C167—21no—
76C2×D4×C167C2312yes—
77SD16×C167—24no—
78C2×C8×C167C2312yes—
79C4⋊C4×C167—21no—
80C4⋊C4×C167C2311yescharacteristic
81C42×C167—21no—
82C42×C167C2311yescharacteristic
83C22×D167—21no—
84C22×D167—44no—
85C22×D167—44no—
86C22×D167—82no—
87C22×D167C2×D412yes—
88Dic334—22no—
89Dic334—22no—
90Dic334—44no—
91D668—21no—
92D668SD1614yescomplemented
93D668C2×D412yes—
94C22⋊D167—44no—
95C22⋊D167—44no—
96C2×Dic167—21no—
97C2×Dic167—22no—
98C2×Dic167C2×D412yes—
99C4×D167—22no—
100C4×D167—22no—
101C4×D167—44no—
102C4×D167—42no—
103C167⋊C8—21no—
104C167⋊C8C2×D412yescomplemented
105C23×C167—22no—
106Q8×C167—21no—
107Q8×C167C2×D412yescomplemented
108D4×C167—22no—
109D4×C167—22no—
110D4×C167—42no—
111D4×C167—81no—
112C2×C4×C167—21no—
113C2×C4×C167—21no—
114C2×C4×C167C2×D411yescharacteristic
115C2×C4×C167C2×D411yescharacteristic
116C2×C4×C167C2411yescharacteristic
117C8×C167—22no—
118C22×C167—42no—
119C22×C167—42no—
120C22×C167—42no—
121C22×C167C22×D411yescharacteristic
122C2×D167—22no—
123C2×D167—22no—
124C2×D167—22no—
125C2×D167—41no—
126C2×D167—41no—
127C2×D167—88no—
128C2×D167C2×SD1614yescomplemented
129C4×C167—22no—
130C4×C167—21no—
131C4×C167—41no—
132C4×C167C2×SD1612yescomplemented
133C4×C167C22×D411yescharacteristic
134C4×C167H(5,𝔽2)11yescharacteristic
135Dic167—22no—
136Dic167—42no—
137Dic167—41no—
138C2×C167—82no—
139C2×C167D4211yescharacteristic
140C2×C167C22×SD1611yescharacteristic
141C2×C167#57711yescharacteristic
142D167—24no—
143D167—42no—
144C167D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1671nomaximal, Sylow
146C22×SD16—1672no—
147D4×Q8—1671no—
148D42—1671no—
149#492—1671no—
150#465—1672no—
151#461—1671no—
152#460D16711yescharacteristic, complemented
153#450—1672no—
154#448—1672no—
155C4×SD16—1671no—
156C8×D4—1671no—
157C22×Q8—1672no—
158C22×D4—3341no—
159C22×D4—1672no—
160C2×SD16—3348no—
161C2×SD16—1672no—
162C2×SD16—1672no—
163C2×SD16—1671no—
164C2×SD16—1672no—
165C2×SD16C2×D16712yescomplemented
166C22×C8—1672no—
167C4⋊Q8—3341no—
168C4⋊Q8—1671no—
169#127C2×D16711yescharacteristic
170C22⋊Q8—3342no—
171C22⋊Q8—1672no—
172#121—3341no—
173#121—1672no—
174#120—3342no—
175C4×Q8C2×D16711yescharacteristic, complemented
176C4×D4—3341no—
177C4×D4—1671no—
178C4×D4—1671no—
179#106—1671no—
180C4⋊C8C2×D16711yescharacteristic, complemented
181#103—1671no—
182#103—1672no—
183#102—1671no—
184#102C2×D16712yescomplemented
185C22⋊C8—1672no—
186C4×C8—1671no—
187C24—3342no—
188C2×Q8—3342no—
189C2×Q8—3344no—
190C2×Q8—6682no—
191C2×Q8—1672no—
192C2×Q8C22×D16711yescharacteristic
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—3344no—
196C2×D4—3342no—
197C2×D4—3342no—
198C2×D4—6682no—
199C2×D4—6684no—
200C2×D4—1671no—
201C2×D4—1671no—
202C2×D4C22×D16712yes—
203C22×C4—3342no—
204C22×C4—3341no—
205C22×C4—1672no—
206SD16—24no—
207SD16—3344no—
208SD16—3344no—
209SD16—6682no—
210SD16—1674no—
211C2×C8—3344no—
212C2×C8—1671no—
213C2×C8—1671no—
214C2×C8C22×D16712yes—
215C4⋊C4—21no—
216C4⋊C4—3342no—
217C4⋊C4—3341no—
218C4⋊C4—3341no—
219C4⋊C4—1671no—
220C4⋊C4—1671no—
221C4⋊C4C22×D16711yescharacteristic
222C22⋊C4—3342no—
223C22⋊C4—3342no—
224C22⋊C4—3341no—
225C22⋊C4—1672no—
226C42—21no—
227C42C22×D16711yescharacteristic
228C23—22no—
229C23—13362no—
230C23—3341no—
231C23—6684no—
232C23—6684no—
233C23—1672no—
234Q8—21no—
235Q8—3342no—
236Q8—3342no—
237Q8—6684no—
238Q8D4×D16712yescomplemented
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—3341no—
244D4—6684no—
245D4—6684no—
246D4—1674no—
247D4—1672no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—3342no—
251C2×C4—3342no—
252C2×C4—3341no—
253C2×C4—3342no—
254C2×C4—6684no—
255C2×C4—6682no—
256C2×C4—1672no—
257C2×C4D4×D16711yescharacteristic
258C2×C4D4×D16711yescharacteristic
259C2×C4C23×D16711yescharacteristic
260C8—22no—
261C8—3341no—
262C8—1672no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—13368no—
267C22—3342no—
268C22—3342no—
269C22—3342no—
270C22—6681no—
271C22—6681no—
272C22—1674no—
273C22C2×D4×D16711yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—3342no—
278C4—6681no—
279C4—6682no—
280C4SD16×D16712yescomplemented
281C4C2×D4×D16711yescharacteristic
282C4#1912887811yescharacteristic
283C2—82no—
284C2—3344no—
285C2—6682no—
286C2#2492390311yescharacteristic
287C2C2×SD16×D16711yescharacteristic
288C2#2492407911yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#460×C167C2
socleC22×C167C22×D4
prime radical#24923904C2
hypercenter#460D167
hypocenterC167D4×SD16
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D167C2semidirect
2. Q8×D668C2semidirect
4. #24923641C2semidirect
5. #24923497C2semidirect
6. #24923475C2semidirect
7. #24923421C2semidirect
8. #24923406C2semidirect
9. C4×#14581937C2semidirect
10. #24923334C2semidirect
11. #460×C167C2semidirect
12. C2×Q8×D167C22semidirect
16. C2×#14581938C22semidirect
17. C2×#14581938C22semidirect
20. C2×C8×D167C22semidirect
22. #19128832C22semidirect
24. #19128812C22semidirect
31. #19128763C22semidirect
32. #19128757C22semidirect
33. #19128756C22semidirect
35. (C2×D167)⋊C8C22semidirect
36. C4×C167⋊C8C22semidirect
37. C2×SD16×C167C22semidirect
39. C4×Q8×C167C22semidirect
40. C4⋊C8×C167C22semidirect
41. #102×C167C22semidirect
60. #14581937D4semidirect
92. D668SD16semidirect
104. C167⋊C8C2×D4semidirect
107. Q8×C167C2×D4semidirect
128. C2×D167C2×SD16semidirect
132. C4×C167C2×SD16semidirect
144. C167D4×SD16semidirect
152. #460D167semidirect
165. C2×SD16C2×D167semidirect
175. C4×Q8C2×D167semidirect
180. C4⋊C8C2×D167semidirect
184. #102C2×D167semidirect
238. Q8D4×D167semidirect
280. C4SD16×D167semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1672
Q8×D6682
#249239042
#249236412
#249234972
#249234752
#249234212
#249234062
C4×#145819372
#249233342
#460×C1672
D4×SD16167