Finite Group Databasebeta

← #36334605

Subgroups — #36334605

Order 21376.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19128863C211yesmaximal, characteristic, complemented
2C2×#19128860C211yesmaximal, characteristic, complemented
3C2×#19128857C211yesmaximal, characteristic, complemented
4#24923815C211yesmaximal, characteristic, complemented
5C2×#19128821C211yesmaximal, characteristic, complemented
6#24923662C212yesmaximal, complemented
7#24923655C212yesmaximal, complemented
8#24923654C212yesmaximal, complemented
9#24923650C212yesmaximal, complemented
10M16×Dic167C211yesmaximal, characteristic, complemented
11C2×#137×C167C211yesmaximal, characteristic, complemented
12C2×#14581950C2211yescharacteristic
13C22×D668C2211yescharacteristic
14#19128863—24no—
15#19128860—24no—
16#19128857—24no—
17C2×#14581940C2212yescomplemented
18C2×#14581939C2212yes—
19C2×#14581937C2212yes—
20C2×#14581937C2212yescomplemented
21C2×D1336C2212yes—
22C2×#14581935C2212yes—
23C2×#14581934C2211yescharacteristic
24#19128830C2211yescharacteristic, complemented
25#19128828C2211yescharacteristic, complemented
26#19128821—22no—
27#19128821C2212yes—
28#19128821C2212yes—
29#19128821C2211yescharacteristic, complemented
30#19128819C2211yescharacteristic, complemented
31#19128804—22no—
32C4×C22⋊D167—22no—
33C2×(C2×D167)⋊C4—22no—
34C2×C4×Dic167C2211yescharacteristic
35#19128742C2212yescomplemented
36C8×Dic167C2212yescomplemented
37C2×#41×C167C2211yescharacteristic
38C22×Q8×C167C2211yescharacteristic
39#137×C167—24no—
40C2×Q16×C167C2212yescomplemented
41C2×SD16×C167C2212yescomplemented
42C2×M16×C167C2211yescharacteristic
43C23×D167—21no—
44#14581950—22no—
45#14581950—22no—
46C2×Dic334C2311yescharacteristic
47C2×D668—22no—
48C2×D668—22no—
49C2×D668—21no—
50C2×D668C2312yes—
51C2×D668C2311yescharacteristic
52C2×C22⋊D167—22no—
53C2×C22⋊D167—21no—
54C22×Dic167D412yes—
55C2×C4×D167—21no—
56#14581940—24no—
57#14581939—24no—
58#14581937—24no—
59#14581937—24no—
60D1336—24no—
61#14581935—24no—
62#14581934D414yescomplemented
63C2×C167⋊C8C2312yes—
64Dic167⋊C4—22no—
65C22⋊Dic167—22no—
66(C2×D167)⋊C4—24no—
67(C2×D167)⋊C4—24no—
68(C2×D167)⋊C4—22no—
69C4×Dic167D412yes—
70C4×Dic167C2311yescharacteristic
71C4×Dic167C2311yescharacteristic
72#41×C167—22no—
73#41×C167—22no—
74C2×Q8×C167—21no—
75C2×Q8×C167—22no—
76C2×Q8×C167—22no—
77C2×Q8×C167C2311yescharacteristic
78C2×Q8×C167C2312yes—
79C2×D4×C167—21no—
80C2×D4×C167C2311yescharacteristic
81C22×C4×C167—21no—
82C22×C4×C167—21no—
83C22×C4×C167C2311yescharacteristic
84Q16×C167—24no—
85SD16×C167—24no—
86M16×C167D414yescomplemented
87C2×C8×C167C2312yes—
88C22×D167—22no—
89C22×D167—21no—
90C22×D167—44no—
91C22×D167—42no—
92Dic334—21no—
93Dic334—21no—
94D668—22no—
95D668—21no—
96D668—22no—
97D668—21no—
98D668—42no—
99C22⋊D167—44no—
100C22⋊D167—42no—
101C2×Dic167—21no—
102C2×Dic167—42no—
103C2×Dic167C2×D412yes—
104C2×Dic167C2×D412yes—
105C4×D167—42no—
106C167⋊C8C2×D414yes—
107C23×C167—21no—
108C23×C167C2×D411yescharacteristic
109Q8×C167—22no—
110Q8×C167—21no—
111Q8×C167—22no—
112Q8×C167—21no—
113Q8×C167—42no—
114D4×C167—21no—
115D4×C167—21no—
116D4×C167—42no—
117C2×C4×C167—21no—
118C2×C4×C167—22no—
119C2×C4×C167—42no—
120C2×C4×C167—42no—
121C2×C4×C167C2×D411yescharacteristic
122C2×C4×C167C2×D412yes—
123C2×C4×C167C2×D412yes—
124C2×C4×C167C2411yescharacteristic
125C8×C167C2×D414yes—
126C22×C167—22no—
127C22×C167—41no—
128C22×C167—42no—
129C22×C167#12712yes—
130C22×C167C22×D411yescharacteristic
131C2×D167—41no—
132C2×D167—42no—
133C2×D167—42no—
134C2×D167—44no—
135C2×D167—84no—
136C4×C167—41no—
137C4×C167—42no—
138C4×C167#12712yes—
139C4×C167C22×D411yescharacteristic
140C4×C167C22×D411yescharacteristic
141Dic167—22no—
142Dic167—22no—
143Dic167—41no—
144C2×C167—22no—
145C2×C167—81no—
146C2×C167C2×#12711yescharacteristic
147C2×C167#57512yes—
148D167—81no—
149D167—82no—
150C167#315311yescharacteristic, complemented, Sylow
151#3153—1671nomaximal, Sylow
152C2×#137—1671no—
153C2×#137D16711yescharacteristic, complemented
154C2×AGL(1,ℤ/8)—1671no—
155C2×AGL(1,ℤ/8)—1671no—
156#532—1671no—
157C2×#124—1671no—
158#497—1672no—
159#496—1672no—
160#495—1672no—
161#495—1672no—
162C4×M16—1671no—
163C2×#41—1671no—
164C2×#41C2×D16711yescharacteristic
165C22×Q8C2×D16711yescharacteristic
166C22×D4—1671no—
167#137—24no—
168#137—3344no—
169AGL(1,ℤ/8)—3344no—
170AGL(1,ℤ/8)—3344no—
171C2×Q16—1672no—
172C2×Q16C2×D16712yescomplemented
173C2×SD16—1672no—
174C2×SD16—1672no—
175C2×SD16—1672no—
176C2×SD16C2×D16712yescomplemented
177C2×D8—1672no—
178C2×D8—1672no—
179C2×M16—1671no—
180C2×M16C2×D16711yescharacteristic
181C4⋊Q8—1671no—
182#127—1671no—
183#124—3342no—
184#124—1672no—
185#124—1672no—
186#124—1671no—
187#124—1671no—
188#121—3342no—
189C4×D4—3342no—
190C2×C22⋊C4—3342no—
191C2×C42—1671no—
192#97—1672no—
193C4×C8—1672no—
194C24—3341no—
195#41—22no—
196#41—22no—
197#41—3342no—
198#41—3342no—
199C2×Q8—21no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—1671no—
203C2×Q8C22×D16712yes—
204C2×Q8C22×D16711yescharacteristic
205C2×D4—21no—
206C2×D4—3341no—
207C2×D4—3342no—
208C2×D4—3342no—
209C2×D4—3341no—
210C2×D4—3342no—
211C2×D4—1672no—
212C2×D4—1671no—
213C2×D4C22×D16711yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—3341no—
217C22×C4—1672no—
218C22×C4C22×D16711yescharacteristic
219Q16—24no—
220Q16—3344no—
221SD16—24no—
222SD16—3344no—
223SD16—3344no—
224SD16—3344no—
225D8—3344no—
226D8—3344no—
227M16—1674no—
228M16C22⋊D16714yescomplemented
229C2×C8—1672no—
230C2×C8C22×D16712yes—
231C4⋊C4—3342no—
232C22⋊C4—3344no—
233C22⋊C4—3344no—
234C22⋊C4—3342no—
235C22⋊C4—3342no—
236C42—1672no—
237C42—1671no—
238C42—1671no—
239C23—21no—
240C23—3342no—
241C23—3341no—
242C23—6684no—
243C23—6682no—
244C23D4×D16711yescharacteristic
245Q8—21no—
246Q8—22no—
247Q8—22no—
248Q8—21no—
249Q8—42no—
250Q8—3341no—
251Q8—3341no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—3342no—
256D4—3341no—
257D4—3341no—
258D4—3342no—
259D4—6684no—
260D4—6682no—
261D4—6682no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—3341no—
267C2×C4—6682no—
268C2×C4—6682no—
269C2×C4—1672no—
270C2×C4—1672no—
271C2×C4D4×D16712yes—
272C2×C4D4×D16711yescharacteristic
273C2×C4C2×C22⋊D16712yes—
274C2×C4C23×D16711yescharacteristic
275C8—1674no—
276C8C2×C22⋊D16714yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—13364no—
281C22—6681no—
282C22—6682no—
283C22—6682no—
284C22—6684no—
285C22#1912882812yes—
286C22C2×D4×D16711yescharacteristic
287C4—41no—
288C4—42no—
289C4—3342no—
290C4—3342no—
291C4—6681no—
292C4#1912882812yes—
293C4C2×D4×D16711yescharacteristic
294C4C22×C22⋊D16711yescharacteristic
295C2—22no—
296C2—81no—
297C2—13362no—
298C2—13361no—
299C2C2×#1912882811yescharacteristic
300C2#2492406812yes—
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC2×#137×C167C2
socleC22×C167C22×D4
hypercenterC2×#137D167
hypocenterC167#3153
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#19128863C2semidirect
2. C2×#19128860C2semidirect
3. C2×#19128857C2semidirect
4. #24923815C2semidirect
5. C2×#19128821C2semidirect
6. #24923662C2semidirect
7. #24923655C2semidirect
8. #24923654C2semidirect
9. #24923650C2semidirect
10. M16×Dic167C2semidirect
11. C2×#137×C167C2semidirect
17. C2×#14581940C22semidirect
20. C2×#14581937C22semidirect
24. #19128830C22semidirect
25. #19128828C22semidirect
29. #19128821C22semidirect
30. #19128819C22semidirect
35. #19128742C22semidirect
36. C8×Dic167C22semidirect
40. C2×Q16×C167C22semidirect
41. C2×SD16×C167C22semidirect
62. #14581934D4semidirect
86. M16×C167D4semidirect
150. C167#3153semidirect
153. C2×#137D167semidirect
172. C2×Q16C2×D167semidirect
176. C2×SD16C2×D167semidirect
228. M16C22⋊D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191288632
C2×#191288602
C2×#191288572
#249238152
C2×#191288212
#249236622
#249236552
#249236542
#249236502
M16×Dic1672
C2×#137×C1672
#3153167