Finite Group Databasebeta

← #36337273

Subgroups — #36337273

Order 21376.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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×#19128855C211yesmaximal, characteristic, complemented
2#135×D167C211yesmaximal, characteristic, complemented
3#24923711C212yesmaximal, complemented
4#24923710C211yesmaximal, characteristic, complemented
5#24923707C211yesmaximal, characteristic, complemented
6#24923706C212yesmaximal, complemented
7#24923704C211yesmaximal, characteristic, complemented
8#24923702C212yesmaximal, complemented
9#24923700C212yesmaximal, complemented
10M32×D167C211yesmaximal, characteristic, complemented
11#510×C167C211yesmaximal, characteristic, complemented
12C2×#14581951—21no—
13#41×D167—21no—
14#19128855—21no—
15#19128855—21no—
16#19128855C2212yes—
17#19128855C2212yescomplemented
18#19128855C2211yescharacteristic, complemented
19#19128854C2211yescharacteristic
20#19128853—21no—
21#19128852—21no—
22#19128851C2211yescharacteristic, complemented
23Q16×D167C2211yescharacteristic, complemented
24C2×#14581937—21no—
25SD16×D167—21no—
26C2×D1336C2211yescharacteristic, complemented
27D8×D167C2211yescharacteristic, complemented
28C2×C8×D167C2211yescharacteristic
29#19128778C2212yescomplemented
30#19128777C2212yescomplemented
31#19128775C2212yescomplemented
32#19128775C2212yescomplemented
33D2672C2212yescomplemented
34#19128773C2212yescomplemented
35#19128772C2211yescharacteristic, complemented
36#19128771C2212yescomplemented
37C16×D167C2212yescomplemented
38#135×C167C2211yescharacteristic
39C2×Q16×C167C2211yescharacteristic, complemented
40Q32×C167C2212yescomplemented
41SD32×C167C2212yescomplemented
42M32×C167C2211yescharacteristic, complemented
43#14581951—21no—
44#14581951—22no—
45#14581951—22no—
46#14581951—41no—
47#14581951—41no—
48#14581950—21no—
49#14581950—41no—
50#14581949—21no—
51#14581949—41no—
52Q8×D167—21no—
53C2×D668—21no—
54C2×D668—41no—
55D4×D167—21no—
56D4×D167—41no—
57C2×C4×D167—41no—
58C2×C4×D167—41no—
59C2×C4×D167D411yescharacteristic
60Dic668C2311yescharacteristic
61#14581940—21no—
62#14581939—21no—
63#14581938—21no—
64#14581937—21no—
65#14581937—22no—
66#14581937—22no—
67D1336—21no—
68D1336C2311yescharacteristic
69D1336C2312yes—
70#14581935—21no—
71C2×C167⋊C8D411yescharacteristic
72C8×D167D411yescharacteristic
73C8×D167D411yescharacteristic
74C8×D167C2311yescharacteristic
75C8×D167C2311yescharacteristic
76C167⋊C16C2312yes—
77#41×C167—21no—
78C2×Q8×C167—21no—
79Q16×C167—21no—
80Q16×C167C2311yescharacteristic
81Q16×C167C2312yes—
82SD16×C167—21no—
83D8×C167C2311yescharacteristic
84C2×C8×C167C2311yescharacteristic
85C16×C167C2312yes—
86C22×D167—41no—
87C22×D167—41no—
88C22×D167D811yescharacteristic
89Dic334—21no—
90Dic334—41no—
91D668—21no—
92D668—22no—
93D668—41no—
94D668—42no—
95D668—42no—
96D668—41no—
97C22⋊D167—41no—
98C22⋊D167—41no—
99C22⋊D167—41no—
100C2×Dic167—41no—
101C2×Dic167D811yescharacteristic
102C4×D167—41no—
103C4×D167—41no—
104C4×D167—41no—
105C4×D167—42no—
106C4×D167—42no—
107C4×D167D811yescharacteristic
108C4×D167D811yescharacteristic
109C4×D167C2×D411yescharacteristic
110C4×D167C2×D411yescharacteristic
111C167⋊C8C2×D411yescharacteristic
112C167⋊C8C2×D411yescharacteristic
113Q8×C167—21no—
114Q8×C167—22no—
115Q8×C167—41no—
116D4×C167—21no—
117D4×C167—41no—
118C2×C4×C167—41no—
119C2×C4×C167—41no—
120C2×C4×C167C2×D411yescharacteristic
121C8×C167C2×D411yescharacteristic
122C8×C167C2411yescharacteristic
123C22×C167—41no—
124C22×C167C2×D811yescharacteristic
125C2×D167—41no—
126C2×D167—41no—
127C2×D167—42no—
128C2×D167—82no—
129C2×D167—82no—
130C2×D167C2×D811yescharacteristic
131C2×D167C2×D811yescharacteristic
132C4×C167—41no—
133C4×C167—42no—
134C4×C167C2×D811yescharacteristic
135C4×C167C22×D411yescharacteristic
136Dic167—41no—
137Dic167C2×D811yescharacteristic
138Dic167C2×D811yescharacteristic
139C2×C167—21no—
140C2×C167—81no—
141C2×C167C22×D811yescharacteristic
142D167—21no—
143D167—21no—
144D167—81no—
145D167—82no—
146C167#341411yescharacteristic, complemented, Sylow
147#3414—1671nomaximal, Sylow
148C2×#135—1671no—
149C2×#135—1671no—
150#510—1671no—
151#510—1672no—
152#510D16711yescharacteristic, complemented
153#509—1671no—
154#509—1672no—
155#509—1671no—
156#508—1672no—
157#508—1672no—
158C2×M32—1671no—
159C2×#41—3341no—
160C2×#41—3341no—
161#135—3341no—
162#135—3341no—
163#135—3341no—
164#135—3341no—
165#135—1672no—
166#135—1671no—
167#135—1672no—
168#135—1671no—
169#135—1671no—
170#135C2×D16711yescharacteristic
171C2×Q16—1671no—
172C2×Q16C2×D16711yescharacteristic, complemented
173C2×SD16—3341no—
174C2×SD16—3341no—
175C2×D8—1671no—
176C2×D8—1671no—
177C22×C8—1671no—
178Q32—1672no—
179Q32C2×D16712yescomplemented
180SD32—1672no—
181SD32—1672no—
182SD32—1672no—
183SD32C2×D16712yescomplemented
184D16—1672no—
185D16—1672no—
186M32—1671no—
187M32—1672no—
188M32C2×D16711yescharacteristic, complemented
189C2×C16—1672no—
190#41—21no—
191#41—3341no—
192#41—3342no—
193#41—3341no—
194#41—3342no—
195#41—3341no—
196#41—6681no—
197#41—6681no—
198#41—6681no—
199#41—6681no—
200C2×Q8—21no—
201C2×Q8—3341no—
202C2×D4—3341no—
203C2×D4—3341no—
204C2×D4—6681no—
205C2×D4—6681no—
206C22×C4—6681no—
207C22×C4—6681no—
208C22×C4—1671no—
209Q16—21no—
210Q16—3341no—
211Q16—1671no—
212Q16C22×D16711yescharacteristic
213Q16C22×D16712yes—
214SD16—21no—
215SD16—3341no—
216SD16—3342no—
217SD16—3341no—
218SD16—3342no—
219SD16—3341no—
220D8—3341no—
221D8—3341no—
222D8—1672no—
223D8—1671no—
224D8C22×D16711yescharacteristic
225C2×C8—1671no—
226C2×C8—1671no—
227C2×C8—1671no—
228C2×C8—1671no—
229C2×C8—1671no—
230C2×C8C22×D16711yescharacteristic
231C16—1672no—
232C16C22×D16712yes—
233C23—6681no—
234C23—6681no—
235C23—1671no—
236Q8—21no—
237Q8—22no—
238Q8—41no—
239Q8—3341no—
240Q8—6681no—
241D4—21no—
242D4—41no—
243D4—3341no—
244D4—3342no—
245D4—6681no—
246D4—6682no—
247D4—6682no—
248D4—6681no—
249D4—6681no—
250D4—6681no—
251D4—6681no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—6681no—
255C2×C4—6681no—
256C2×C4—6682no—
257C2×C4—6682no—
258C2×C4—6681no—
259C2×C4—6681no—
260C2×C4—1671no—
261C2×C4—1671no—
262C2×C4—1671no—
263C2×C4—1671no—
264C2×C4—1671no—
265C2×C4D4×D16711yescharacteristic
266C8—1671no—
267C8—1671no—
268C8D4×D16711yescharacteristic
269C8C23×D16711yescharacteristic
270C22—41no—
271C22—13362no—
272C22—13362no—
273C22—6681no—
274C22—6682no—
275C22—6681no—
276C22—1671no—
277C22—1671no—
278C22D8×D16711yescharacteristic
279C4—41no—
280C4—42no—
281C4—6681no—
282C4—1671no—
283C4—1671no—
284C4D8×D16711yescharacteristic
285C4C2×D4×D16711yescharacteristic
286C2—21no—
287C2—81no—
288C2—13362no—
289C2—13361no—
290C2—3341no—
291C2—3341no—
292C2C2×D8×D16711yescharacteristic
subgroupG/H
centerC2C2×D8×D167
commutator, cosocleC8×C167C24
FrattiniC8C23×D167
Fitting#510×C167C2
socleC2×C167C22×D8
hypercenter#510D167
hypocenterC167#3414
LCSG ▷ C8×C167 ▷ C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. C2×#19128855C2semidirect
2. #135×D167C2semidirect
3. #24923711C2semidirect
4. #24923710C2semidirect
5. #24923707C2semidirect
6. #24923706C2semidirect
7. #24923704C2semidirect
8. #24923702C2semidirect
9. #24923700C2semidirect
10. M32×D167C2semidirect
11. #510×C167C2semidirect
17. #19128855C22semidirect
18. #19128855C22semidirect
22. #19128851C22semidirect
23. Q16×D167C22semidirect
26. C2×D1336C22semidirect
27. D8×D167C22semidirect
29. #19128778C22semidirect
30. #19128777C22semidirect
31. #19128775C22semidirect
32. #19128775C22semidirect
33. D2672C22semidirect
34. #19128773C22semidirect
35. #19128772C22semidirect
36. #19128771C22semidirect
37. C16×D167C22semidirect
39. C2×Q16×C167C22semidirect
40. Q32×C167C22semidirect
41. SD32×C167C22semidirect
42. M32×C167C22semidirect
146. C167#3414semidirect
152. #510D167semidirect
172. C2×Q16C2×D167semidirect
179. Q32C2×D167semidirect
183. SD32C2×D167semidirect
188. M32C2×D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191288552
#135×D1672
#249237112
#249237102
#249237072
#249237062
#249237042
#249237022
#249237002
M32×D1672
#510×C1672
#3414167