Finite Group Databasebeta

← #36327252

Subgroups — #36327252

Order 21376.

496 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128827C211yesmaximal, characteristic
2#24923167C211yesmaximal, characteristic, complemented
3#24923161C211yesmaximal, characteristic
4#24923160C211yesmaximal, characteristic
5#24923160C211yesmaximal, characteristic
6#24923155C211yesmaximal, characteristic, complemented
7#24923152C211yesmaximal, characteristic, complemented
8#24923150C211yesmaximal, characteristic, complemented
9#24923140C211yesmaximal, characteristic
10C4×Dic167⋊C4C211yesmaximal, characteristic, complemented
11C4×Dic167⋊C4C211yesmaximal, characteristic, complemented
12C4×(C2×D167)⋊C4C211yesmaximal, characteristic
13#24923116C211yesmaximal, characteristic
14#24923114C211yesmaximal, characteristic, complemented
15C4×C4⋊C4×C167C211yesmaximal, characteristic, complemented
16C22×C4×D167C2211yescharacteristic
17#19128827C418yescomplemented
18C2×C4⋊Dic167C2211yescharacteristic
19C2×C4⋊Dic167C2211yescharacteristic
20C2×Dic167⋊C4C2211yescharacteristic
21C2×Dic167⋊C4C2211yescharacteristic
22C2×Dic167⋊C4C2211yescharacteristic
23C2×Dic167⋊C4C2211yescharacteristic
24C2×(C2×D167)⋊C4C2211yescharacteristic
25C2×(C2×D167)⋊C4C2211yescharacteristic
26C2×(C2×D167)⋊C4C2211yescharacteristic
27C2×(C2×D167)⋊C4C2211yescharacteristic
28C2×(C2×D167)⋊C4C2211yescharacteristic
29C2×(C2×D167)⋊C4C2211yescharacteristic
30C2×C4×Dic167C2211yescharacteristic
31C2×C4×Dic167C2211yescharacteristic
32C2×C4×Dic167C2211yescharacteristic
33C2×C4×Dic167C2211yescharacteristic
34C2×C4×Dic167C2211yescharacteristic
35C167⋊#95C2211yescharacteristic
36C167⋊#95C2211yescharacteristic
37C167⋊#95C2211yescharacteristic
38C167⋊#95C2211yescharacteristic
39C167⋊#95C2211yescharacteristic
40C167⋊#95C2211yescharacteristic
41C167⋊#95C2211yescharacteristic
42C167⋊#95C2211yescharacteristic
43C167⋊#95C2211yescharacteristic
44C167⋊#95C2211yescharacteristic
45C2×C4⋊C4×C167C2211yescharacteristic
46C2×C4⋊C4×C167C2211yescharacteristic
47C2×C42×C167C2211yescharacteristic
48C2×C42×C167C2211yescharacteristic
49C2×C42×C167C2211yescharacteristic
50#95×C167C2211yescharacteristic
51#95×C167C2211yescharacteristic
52C23×D167C2311yescharacteristic
53C22×Dic167C2311yescharacteristic
54C22×Dic167C2311yescharacteristic
55C22×Dic167C2311yescharacteristic
56C22×Dic167C2311yescharacteristic
57C22×Dic167C2311yescharacteristic
58C22×Dic167C2311yescharacteristic
59C22×Dic167C2311yescharacteristic
60C2×C4×D167—22no—
61C2×C4×D167—22no—
62C2×C4×D167—41no—
63C4⋊Dic167—41no—
64C4⋊Dic167C2×C414yescomplemented
65Dic167⋊C4—22no—
66Dic167⋊C4—22no—
67Dic167⋊C4—22no—
68Dic167⋊C4C2×C414yescomplemented
69(C2×D167)⋊C4—22no—
70(C2×D167)⋊C4—22no—
71(C2×D167)⋊C4—41no—
72(C2×D167)⋊C4C2×C414yes—
73(C2×D167)⋊C4C2×C414yes—
74(C2×D167)⋊C4C2×C414yes—
75C4×Dic167—22no—
76C4×Dic167—22no—
77C4×Dic167—22no—
78C4×Dic167—41no—
79C4×Dic167C2×C414yescomplemented
80C22×C4×C167C2311yescharacteristic
81C22×C4×C167C2311yescharacteristic
82C22×C4×C167C2311yescharacteristic
83C22×C4×C167C2311yescharacteristic
84C22×C4×C167C2311yescharacteristic
85C22×C4×C167C2311yescharacteristic
86C22×C4×C167C2311yescharacteristic
87C4⋊C4×C167—22no—
88C4⋊C4×C167C2×C414yescomplemented
89C42×C167—22no—
90C42×C167—41no—
91C42×C167—41no—
92C22×D167—21no—
93C22×D167—21no—
94C22×D167—21no—
95C22×D167—21no—
96C22×D167—21no—
97C22×D167—21no—
98C22×D167C22×C412yes—
99C2×Dic167—21no—
100C2×Dic167—21no—
101C2×Dic167—21no—
102C2×Dic167—21no—
103C2×Dic167—21no—
104C2×Dic167—21no—
105C2×Dic167—21no—
106C2×Dic167—21no—
107C2×Dic167—21no—
108C2×Dic167—21no—
109C2×Dic167—21no—
110C2×Dic167—21no—
111C2×Dic167—21no—
112C2×Dic167—21no—
113C2×Dic167—21no—
114C2×Dic167—21no—
115C2×Dic167—21no—
116C2×Dic167C22×C412yes—
117C2×Dic167C22×C412yes—
118C2×Dic167C22×C412yes—
119C2×Dic167#4112yes—
120C4×D167—44no—
121C23×C167C2411yescharacteristic
122C2×C4×C167—21no—
123C2×C4×C167—21no—
124C2×C4×C167—21no—
125C2×C4×C167—21no—
126C2×C4×C167—21no—
127C2×C4×C167—21no—
128C2×C4×C167—21no—
129C2×C4×C167—21no—
130C2×C4×C167—21no—
131C2×C4×C167—21no—
132C2×C4×C167—21no—
133C2×C4×C167—21no—
134C2×C4×C167—21no—
135C2×C4×C167—21no—
136C2×C4×C167—21no—
137C2×C4×C167C22×C412yes—
138C2×C4×C167C22×C412yes—
139C2×C4×C167C22×C412yes—
140C2×C4×C167#4112yes—
141C2×C4×C167#4112yes—
142C2×C4×C167#4112yes—
143C22×C167C23×C411yescharacteristic
144C22×C167C2×#4111yescharacteristic
145C22×C167C2×#4111yescharacteristic
146C22×C167C2×#4111yescharacteristic
147C22×C167C2×#4111yescharacteristic
148C22×C167H(5,𝔽2)11yescharacteristic
149C22×C1672−1+411yescharacteristic
150C2×D167—22no—
151C2×D167—22no—
152C2×D167—22no—
153C2×D167—41no—
154C2×D167—41no—
155C2×D167—41no—
156C2×D167—41no—
157C4×C167—22no—
158C4×C167—22no—
159C4×C167—22no—
160C4×C167—22no—
161C4×C167—22no—
162C4×C167—22no—
163C4×C167—41no—
164Dic167—22no—
165Dic167—22no—
166Dic167—22no—
167Dic167—22no—
168Dic167—41no—
169Dic167—41no—
170Dic167—41no—
171C2×C167C4×#4111yescharacteristic
172C2×C167C4×#4111yescharacteristic
173C2×C167#52011yescharacteristic
174C2×C167#55111yescharacteristic
175C2×C167#55211yescharacteristic
176C2×C167#55311yescharacteristic
177C2×C167#55611yescharacteristic
178D167—42no—
179C167#237311yescharacteristic, complemented, Sylow
180#2373—1671nomaximal, Sylow
181C2×#126—1671no—
182#389—1671no—
183#388—1671no—
184#388—1671no—
185#388—1671no—
186#387—1671no—
187#387—1671no—
188#387—1671no—
189#385—1671no—
190C4×C4⋊C4—1671no—
191C4×C4⋊C4—1671no—
192C4×C4⋊C4D16711yescharacteristic, complemented
193C4×C22⋊C4—1671no—
194C4×C22⋊C4—1671no—
195#376—1671no—
196C23×C4—1671no—
197#126—1678no—
198C2×C4⋊C4—1671no—
199C2×C4⋊C4—1671no—
200C2×C4⋊C4—1671no—
201C2×C4⋊C4—1671no—
202C2×C4⋊C4—1671no—
203C2×C4⋊C4—1671no—
204C2×C4⋊C4C2×D16711yescharacteristic
205C2×C4⋊C4C2×D16711yescharacteristic
206C2×C22⋊C4—1671no—
207C2×C22⋊C4—1671no—
208C2×C22⋊C4—1671no—
209C2×C22⋊C4—1671no—
210C2×C22⋊C4—1671no—
211C2×C22⋊C4—1671no—
212C2×C42—1671no—
213C2×C42—1671no—
214C2×C42—1671no—
215C2×C42—1671no—
216C2×C42—1671no—
217C2×C42C2×D16711yescharacteristic
218C2×C42C2×D16711yescharacteristic
219C2×C42C2×D16711yescharacteristic
220#95—1671no—
221#95—1671no—
222#95—1671no—
223#95—1671no—
224#95—1671no—
225#95—1671no—
226#95—1671no—
227#95—1671no—
228#95—1671no—
229#95—1671no—
230#95C2×D16711yescharacteristic
231#95C2×D16711yescharacteristic
232C24—1671no—
233C22×C4—3342no—
234C22×C4—3342no—
235C22×C4—6681no—
236C22×C4—1671no—
237C22×C4—1671no—
238C22×C4—1671no—
239C22×C4—1671no—
240C22×C4—1671no—
241C22×C4—1671no—
242C22×C4—1671no—
243C22×C4C22×D16711yescharacteristic
244C22×C4C22×D16711yescharacteristic
245C22×C4C22×D16711yescharacteristic
246C22×C4C22×D16711yescharacteristic
247C22×C4C22×D16711yescharacteristic
248C22×C4C22×D16711yescharacteristic
249C22×C4C22×D16711yescharacteristic
250C4⋊C4—22no—
251C4⋊C4—3342no—
252C4⋊C4—3342no—
253C4⋊C4—3342no—
254C4⋊C4—6681no—
255C4⋊C4—1674no—
256C4⋊C4—1674no—
257C4⋊C4C4×D16714yescomplemented
258C22⋊C4—3342no—
259C22⋊C4—3342no—
260C22⋊C4—6681no—
261C22⋊C4—1674no—
262C22⋊C4—1674no—
263C22⋊C4—1674no—
264C42—22no—
265C42—41no—
266C42—41no—
267C42—3342no—
268C42—3342no—
269C42—3342no—
270C42—6681no—
271C42—1674no—
272C23—3341no—
273C23—3341no—
274C23—3341no—
275C23—3341no—
276C23—3341no—
277C23—3341no—
278C23—1672no—
279C23C23×D16711yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—3341no—
296C2×C4—3341no—
297C2×C4—3341no—
298C2×C4—3341no—
299C2×C4—3341no—
300C2×C4—3341no—
301C2×C4—3341no—
302C2×C4—3341no—
303C2×C4—3341no—
304C2×C4—3341no—
305C2×C4—3341no—
306C2×C4—3341no—
307C2×C4—3341no—
308C2×C4—3341no—
309C2×C4—3341no—
310C2×C4—3341no—
311C2×C4—3341no—
312C2×C4—6684no—
313C2×C4—1672no—
314C2×C4—1672no—
315C2×C4—1672no—
316C2×C4—1672no—
317C2×C4C2×C4×D16712yes—
318C2×C4C2×C4×D16712yes—
319C2×C4C2×C4×D16712yes—
320C2×C4#1458195012yes—
321C2×C4#1458195012yes—
322C2×C4#1458195112yes—
323C22—3342no—
324C22—3342no—
325C22—3342no—
326C22—6681no—
327C22—6681no—
328C22—6681no—
329C22—6681no—
330C22C22×C4×D16711yescharacteristic
331C22#41×D16711yescharacteristic
332C22C2×#1458195011yescharacteristic
333C22C2×#1458195011yescharacteristic
334C22C2×#1458195111yescharacteristic
335C22#1912887711yescharacteristic
336C22#1912887911yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—3342no—
345C4—3342no—
346C4—3342no—
347C4—3342no—
348C4—6681no—
349C4—6681no—
350C4—6681no—
351C2—6682no—
352C2C4×#1458195011yescharacteristic
353C2C4×#1458195111yescharacteristic
354C2#2492374811yescharacteristic
355C2#2492394811yescharacteristic
356C2#2492395211yescharacteristic
357C2#2492396711yescharacteristic
358C2#2492399211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D167
commutatorC22×C167C23×C4
FittingC4×C4⋊C4×C167C2
socle, cosocleC23×C167C24
prime radicalC23×D167C23
hypercenterC4×C4⋊C4D167
hypocenterC167#2373
LCSG ▷ C22×C167 ▷ C167
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #24923167C2semidirect
6. #24923155C2semidirect
7. #24923152C2semidirect
8. #24923150C2semidirect
10. C4×Dic167⋊C4C2semidirect
11. C4×Dic167⋊C4C2semidirect
14. #24923114C2semidirect
15. C4×C4⋊C4×C167C2semidirect
17. #19128827C4semidirect
64. C4⋊Dic167C2×C4semidirect
68. Dic167⋊C4C2×C4semidirect
79. C4×Dic167C2×C4semidirect
88. C4⋊C4×C167C2×C4semidirect
179. C167#2373semidirect
192. C4×C4⋊C4D167semidirect
257. C4⋊C4C4×D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191288272
#249231672
#249231612
#249231602
#249231552
#249231522
#249231502
#249231402
C4×Dic167⋊C42
C4×(C2×D167)⋊C42
#249231162
#249231142
C4×C4⋊C4×C1672
#2373167