Finite Group Databasebeta

← #36333769

Subgroups — #36333769

Order 21376.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#19128865C211yesmaximal, characteristic, complemented
2C2×M16×D167C211yesmaximal, characteristic, complemented
3#24923781C211yesmaximal, characteristic
4C2×#19128812C211yesmaximal, characteristic, complemented
5#24923528C212yesmaximal, complemented
6#24923516C212yesmaximal, complemented
7#24923511C212yesmaximal, complemented
8#24923499C212yesmaximal, complemented
9#24923277C211yesmaximal, characteristic, complemented
10#24923268C211yesmaximal, characteristic, complemented
11#428×C167C211yesmaximal, characteristic, complemented
12C2×#14581950C2211yescharacteristic
13C22×Dic334C2211yescharacteristic
14C22×C4×D167C2211yescharacteristic
15#19128865—24no—
16C2×Dic668C2212yescomplemented
17C2×#14581939C2212yescomplemented
18C2×#14581934C2211yescharacteristic
19C2×#14581933C2212yes—
20M16×D167—24no—
21C2×C8×D167C2212yes—
22#19128816—21no—
23#19128812—21no—
24#19128812—21no—
25#19128812C2212yes—
26#19128812C2211yescharacteristic
27#19128812C2212yes—
28#19128812C2211yescharacteristic
29#19128807C2211yescharacteristic
30#19128807C2211yescharacteristic
31#19128802—21no—
32C4×D668—21no—
33C4×D668—21no—
34C2×C4⋊Dic167—21no—
35C2×(C2×D167)⋊C4—21no—
36#19128755C2212yescomplemented
37#19128755C2212yescomplemented
38#19128755C2212yescomplemented
39#19128752C2212yescomplemented
40C2×M16×C167C2211yescharacteristic
41#117×C167C2211yescharacteristic
42C2×C4⋊C4×C167C2211yescharacteristic
43#107×C167C2212yescomplemented
44#106×C167C2212yescomplemented
45C23×D167D411yescharacteristic
46#14581950—22no—
47#14581950—22no—
48C2×Dic334—21no—
49C2×Dic334—22no—
50C2×Dic334—22no—
51C2×Dic334C2312yes—
52C2×Dic334C2311yescharacteristic
53C2×D668—21no—
54C2×D668—21no—
55C2×D668C2311yescharacteristic
56C2×C22⋊D167—21no—
57C22×Dic167—21no—
58C22×Dic167D411yescharacteristic
59C2×C4×D167—22no—
60C2×C4×D167—21no—
61C2×C4×D167—22no—
62C2×C4×D167D411yescharacteristic
63C2×C4×D167D411yescharacteristic
64C2×C4×D167C2311yescharacteristic
65C2×C4×D167C2311yescharacteristic
66Dic668—24no—
67#14581939—24no—
68#14581934—22no—
69#14581933—24no—
70C2×C167⋊C8C2312yes—
71C8×D167—24no—
72C4⋊Dic167—22no—
73C4⋊Dic167—22no—
74C4⋊Dic167—21no—
75C4⋊Dic167—21no—
76(C2×D167)⋊C4—22no—
77(C2×D167)⋊C4—22no—
78(C2×D167)⋊C4—21no—
79(C2×D167)⋊C4—21no—
80(C2×D167)⋊C4—21no—
81(C2×D167)⋊C4—21no—
82C22×C4×C167—21no—
83C22×C4×C167C2311yescharacteristic
84M16×C167D414yescomplemented
85C2×C8×C167C2312yes—
86C4⋊C4×C167—21no—
87C4⋊C4×C167C2311yescharacteristic
88C4⋊C4×C167C2312yes—
89C4⋊C4×C167C2311yescharacteristic
90C22⋊C4×C167—21no—
91C42×C167—21no—
92C22×D167—22no—
93C22×D167—22no—
94C22×D167—21no—
95C22×D167—41no—
96C22×D167C2×D411yescharacteristic
97C22×D167C2×D411yescharacteristic
98Dic334—22no—
99Dic334—21no—
100Dic334—22no—
101Dic334—21no—
102Dic334—42no—
103D668—21no—
104D668—21no—
105D668—42no—
106D668—42no—
107C22⋊D167—42no—
108C2×Dic167—22no—
109C2×Dic167—21no—
110C2×Dic167—22no—
111C2×Dic167—42no—
112C2×Dic167C2×D411yescharacteristic
113C2×Dic167C2×D411yescharacteristic
114C4×D167—22no—
115C4×D167—22no—
116C4×D167—21no—
117C4×D167—21no—
118C4×D167—21no—
119C4×D167—21no—
120C4×D167—42no—
121C167⋊C8—22no—
122C23×C167C2×D411yescharacteristic
123C2×C4×C167—21no—
124C2×C4×C167—21no—
125C2×C4×C167—22no—
126C2×C4×C167—42no—
127C2×C4×C167C2×D411yescharacteristic
128C2×C4×C167C2×D412yes—
129C2×C4×C167#4112yes—
130C2×C4×C167C2411yescharacteristic
131C8×C167C2×D414yes—
132C22×C167—21no—
133C22×C167—21no—
134C22×C167#12112yes—
135C22×C167C22×D411yescharacteristic
136C2×D167—21no—
137C2×D167—21no—
138C2×D167—21no—
139C2×D167—21no—
140C2×D167—21no—
141C2×D167—44no—
142C2×D167—41no—
143C2×D167—41no—
144C2×D167—41no—
145C2×D167#13712yescomplemented
146C4×C167—41no—
147C4×C167—42no—
148C4×C167—41no—
149C4×C167#12112yes—
150C4×C167C22×D411yescharacteristic
151C4×C167C2×#4111yescharacteristic
152Dic167—21no—
153Dic167—21no—
154Dic167—41no—
155Dic167—42no—
156C2×C167—22no—
157C2×C167C2×#12111yescharacteristic
158C2×C167C2×#13711yescharacteristic
159C2×C167#57511yescharacteristic
160D167—22no—
161D167—41no—
162D167—81no—
163C167#305611yescharacteristic, complemented, Sylow
164#3056—1671nomaximal, Sylow
165C2×#137—1671no—
166C22×M16—1671no—
167#525—1671no—
168C2×C22⋊Q8—1671no—
169#470—1672no—
170#468—1672no—
171#467—1672no—
172#465—1672no—
173#428D16711yescharacteristic, complemented
174#419—1671no—
175#417—1671no—
176C2×#41—1671no—
177C22×Q8—1671no—
178C23×C4—1671no—
179#137—3344no—
180C2×Q16—1672no—
181C2×SD16—1672no—
182C2×M16—3344no—
183C2×M16—1671no—
184C2×M16—1672no—
185C2×M16C2×D16711yescharacteristic
186C22×C8—1672no—
187#123—3341no—
188C22⋊Q8—3341no—
189C22⋊Q8—3341no—
190C22⋊Q8—1672no—
191C22⋊Q8—1672no—
192C22⋊Q8—1671no—
193C22⋊Q8—1671no—
194#121—1671no—
195#121—1671no—
196#120—3341no—
197C4×D4—3341no—
198C4×D4—3341no—
199#117C2×D16711yescharacteristic
200C2×C4⋊C4—3341no—
201C2×C4⋊C4C2×D16711yescharacteristic
202C2×C22⋊C4—3341no—
203#107C2×D16712yescomplemented
204#106C2×D16712yescomplemented
205#103—1672no—
206#103—1672no—
207#103—1672no—
208#102—1672no—
209C24—1671no—
210#41—3342no—
211#41—3342no—
212C2×Q8—3341no—
213C2×Q8—3342no—
214C2×Q8—3342no—
215C2×Q8—1672no—
216C2×Q8—1671no—
217C2×D4—3341no—
218C2×D4—3341no—
219C2×D4—3341no—
220C2×D4—1671no—
221C22×C4—21no—
222C22×C4—3342no—
223C22×C4—3342no—
224C22×C4—3341no—
225C22×C4—3341no—
226C22×C4—1671no—
227C22×C4—1671no—
228C22×C4—1671no—
229C22×C4—1671no—
230C22×C4—1671no—
231C22×C4C22×D16711yescharacteristic
232Q16—3344no—
233SD16—3344no—
234M16—3342no—
235M16—3344no—
236M16D66814yescomplemented
237C2×C8—3344no—
238C2×C8—1672no—
239C2×C8C22×D16712yes—
240C4⋊C4—21no—
241C4⋊C4—3342no—
242C4⋊C4—3342no—
243C4⋊C4—3341no—
244C4⋊C4—3341no—
245C4⋊C4C22×D16712yes—
246C4⋊C4C22×D16711yescharacteristic
247C4⋊C4C22×D16711yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—3342no—
250C22⋊C4—3342no—
251C22⋊C4—3341no—
252C22⋊C4—3341no—
253C22⋊C4—3341no—
254C22⋊C4—3341no—
255C42—21no—
256C23—3342no—
257C23—3342no—
258C23—3341no—
259C23—6681no—
260C23—1671no—
261C23—1671no—
262C23D4×D16711yescharacteristic
263Q8—3342no—
264Q8—3341no—
265Q8—3342no—
266Q8—3341no—
267Q8—6682no—
268D4—3341no—
269D4—3341no—
270D4—6682no—
271D4—6682no—
272D4—6682no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—3342no—
278C2×C4—3341no—
279C2×C4—3341no—
280C2×C4—3341no—
281C2×C4—3341no—
282C2×C4—3342no—
283C2×C4—3342no—
284C2×C4—3341no—
285C2×C4—3342no—
286C2×C4—6682no—
287C2×C4—6682no—
288C2×C4—1671no—
289C2×C4—1671no—
290C2×C4D4×D16711yescharacteristic
291C2×C4C2×D66812yes—
292C2×C4#1458195112yes—
293C2×C4C23×D16711yescharacteristic
294C8—3342no—
295C8C2×D66814yes—
296C22—21no—
297C22—21no—
298C22—3341no—
299C22—3341no—
300C22—3341no—
301C22—3341no—
302C22—3341no—
303C22—6684no—
304C22—6681no—
305C22—6681no—
306C22—6681no—
307C22—1672no—
308C22#1912880712yes—
309C22C2×D4×D16711yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—3341no—
314C4—3341no—
315C4—6681no—
316C4—6682no—
317C4#1912880712yes—
318C4C22×D66811yescharacteristic
319C4C2×#1458195111yescharacteristic
320C2—22no—
321C2—13361no—
322C2—3342no—
323C2—6681no—
324C2C2×#1912880711yescharacteristic
325C2#137×D16711yescharacteristic
326C2#2492406511yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#428×C167C2
socleC22×C167C22×D4
prime radical#24923781C2
hypercenter#428D167
hypocenterC167#3056
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#19128865C2semidirect
2. C2×M16×D167C2semidirect
4. C2×#19128812C2semidirect
5. #24923528C2semidirect
6. #24923516C2semidirect
7. #24923511C2semidirect
8. #24923499C2semidirect
9. #24923277C2semidirect
10. #24923268C2semidirect
11. #428×C167C2semidirect
16. C2×Dic668C22semidirect
17. C2×#14581939C22semidirect
36. #19128755C22semidirect
37. #19128755C22semidirect
38. #19128755C22semidirect
39. #19128752C22semidirect
43. #107×C167C22semidirect
44. #106×C167C22semidirect
84. M16×C167D4semidirect
145. C2×D167#137semidirect
163. C167#3056semidirect
173. #428D167semidirect
203. #107C2×D167semidirect
204. #106C2×D167semidirect
236. M16D668semidirect

Maximal subgroups

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

subgroupindex
C2×#191288652
C2×M16×D1672
#249237812
C2×#191288122
#249235282
#249235162
#249235112
#249234992
#249232772
#249232682
#428×C1672
#3056167