Finite Group Databasebeta

← #36326997

Subgroups — #36326997

Order 21376.

532 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128790C211yesmaximal, characteristic
2#24923155C211yesmaximal, characteristic, complemented
3#24923154C211yesmaximal, characteristic, complemented
4#24923152C211yesmaximal, characteristic, complemented
5#24923143C211yesmaximal, characteristic
6#24923140C211yesmaximal, characteristic
7#24923136C211yesmaximal, characteristic, complemented
8#24923134C211yesmaximal, characteristic, complemented
9#24923131C211yesmaximal, characteristic
10C4×Dic167⋊C4C211yesmaximal, characteristic, complemented
11C4×(C2×D167)⋊C4C211yesmaximal, characteristic
12#24923116C211yesmaximal, characteristic
13#24923114C211yesmaximal, characteristic, complemented
14#95×D167C211yesmaximal, characteristic
15#387×C167C211yesmaximal, characteristic, complemented
16C22×C4×D167C2211yescharacteristic
17C22×C4×D167C2211yescharacteristic
18C22×C4×D167C2211yescharacteristic
19#19128790C418yescomplemented
20C2×C4⋊Dic167C2211yescharacteristic
21C2×C4⋊Dic167C2211yescharacteristic
22C2×Dic167⋊C4C2211yescharacteristic
23C2×Dic167⋊C4C2211yescharacteristic
24C2×(C2×D167)⋊C4C2211yescharacteristic
25C2×(C2×D167)⋊C4C2211yescharacteristic
26C2×(C2×D167)⋊C4C2211yescharacteristic
27C2×(C2×D167)⋊C4C2211yescharacteristic
28C2×C4×Dic167C2211yescharacteristic
29C2×C4×Dic167C2211yescharacteristic
30C2×C4×Dic167C2211yescharacteristic
31C2×C4×Dic167C2211yescharacteristic
32C2×C4×Dic167C2211yescharacteristic
33C167⋊#95C2211yescharacteristic
34C167⋊#95C2211yescharacteristic
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
48#95×C167C2211yescharacteristic
49#95×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—22no—
63C2×C4×D167—22no—
64C2×C4×D167—22no—
65C2×C4×D167—22no—
66C2×C4×D167—41no—
67C2×C4×D167—41no—
68C2×C4×D167C2×C414yes—
69C4⋊Dic167—41no—
70C4⋊Dic167C2×C414yescomplemented
71Dic167⋊C4—22no—
72Dic167⋊C4—22no—
73(C2×D167)⋊C4—22no—
74(C2×D167)⋊C4—22no—
75(C2×D167)⋊C4C2×C414yes—
76(C2×D167)⋊C4C2×C414yes—
77C4×Dic167—22no—
78C4×Dic167—22no—
79C4×Dic167—41no—
80C4×Dic167C2×C414yescomplemented
81C4×Dic167C2×C414yescomplemented
82C22×C4×C167C2311yescharacteristic
83C22×C4×C167C2311yescharacteristic
84C22×C4×C167C2311yescharacteristic
85C22×C4×C167C2311yescharacteristic
86C22×C4×C167C2311yescharacteristic
87C22×C4×C167C2311yescharacteristic
88C22×C4×C167C2311yescharacteristic
89C4⋊C4×C167—41no—
90C4⋊C4×C167C2×C414yescomplemented
91C42×C167—41no—
92C22×D167—21no—
93C22×D167—21no—
94C22×D167—21no—
95C22×D167—21no—
96C22×D167C22×C412yes—
97C22×D167#4112yes—
98C22×D167#4112yes—
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—28no—
121C4×D167—44no—
122C4×D167—44no—
123C23×C167C2411yescharacteristic
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×C167—21no—
138C2×C4×C167—21no—
139C2×C4×C167—21no—
140C2×C4×C167—21no—
141C2×C4×C167C22×C412yes—
142C2×C4×C167C22×C412yes—
143C2×C4×C167C22×C412yes—
144C2×C4×C167#4112yes—
145C22×C167C23×C411yescharacteristic
146C22×C167C2×#4111yescharacteristic
147C22×C167C2×#4111yescharacteristic
148C22×C167C2×#4111yescharacteristic
149C22×C167C2×#4111yescharacteristic
150C22×C167H(5,𝔽2)11yescharacteristic
151C22×C1672−1+411yescharacteristic
152C2×D167—22no—
153C2×D167—22no—
154C2×D167—22no—
155C2×D167—22no—
156C2×D167—22no—
157C2×D167—22no—
158C2×D167#11714yes—
159C4×C167—22no—
160C4×C167—22no—
161C4×C167—22no—
162C4×C167—22no—
163C4×C167—41no—
164C4×C167—41no—
165C4×C167—41no—
166Dic167—22no—
167Dic167—22no—
168Dic167—22no—
169Dic167—22no—
170Dic167—41no—
171Dic167—41no—
172Dic167—41no—
173C2×C167C2×#11711yescharacteristic
174C2×C167C4×#4111yescharacteristic
175C2×C167#52011yescharacteristic
176C2×C167#55111yescharacteristic
177C2×C167#55111yescharacteristic
178C2×C167#55211yescharacteristic
179C2×C167#55211yescharacteristic
180D167—24no—
181C167#234311yescharacteristic, complemented, Sylow
182#2343—1671nomaximal, Sylow
183C2×#117—1671no—
184#387—1671no—
185#387—1671no—
186#387—1671no—
187#387D16711yescharacteristic, complemented
188#385—1671no—
189#385—1671no—
190#383—1671no—
191#382—1671no—
192#381—1671no—
193C4×C4⋊C4—1671no—
194C4×C22⋊C4—1671no—
195C4×C22⋊C4—1671no—
196#376—1671no—
197C2×#95—1671no—
198C23×C4—1671no—
199C23×C4—1671no—
200C23×C4—1671no—
201#117—1678no—
202C2×C4⋊C4—1671no—
203C2×C4⋊C4—1671no—
204C2×C4⋊C4—1671no—
205C2×C4⋊C4—1671no—
206C2×C4⋊C4C2×D16711yescharacteristic
207C2×C4⋊C4C2×D16711yescharacteristic
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
218#95—1671no—
219#95—1671no—
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
232#95C2×D16711yescharacteristic
233#95C2×D16711yescharacteristic
234C24—1671no—
235C22×C4—3342no—
236C22×C4—3342no—
237C22×C4—3342no—
238C22×C4—3342no—
239C22×C4—3342no—
240C22×C4—3342no—
241C22×C4—6681no—
242C22×C4—6681no—
243C22×C4—1671no—
244C22×C4—1671no—
245C22×C4—1671no—
246C22×C4—1671no—
247C22×C4—1671no—
248C22×C4—1671no—
249C22×C4—1671no—
250C22×C4—1674no—
251C22×C4C22×D16711yescharacteristic
252C22×C4C22×D16711yescharacteristic
253C22×C4C22×D16711yescharacteristic
254C22×C4C22×D16711yescharacteristic
255C22×C4C22×D16711yescharacteristic
256C22×C4C22×D16711yescharacteristic
257C22×C4C22×D16711yescharacteristic
258C4⋊C4—41no—
259C4⋊C4—3342no—
260C4⋊C4—3342no—
261C4⋊C4—6681no—
262C4⋊C4—1674no—
263C4⋊C4C4×D16714yescomplemented
264C22⋊C4—3342no—
265C22⋊C4—3342no—
266C22⋊C4—1674no—
267C22⋊C4—1674no—
268C42—41no—
269C42—3342no—
270C42—3342no—
271C42—6681no—
272C42—1674no—
273C42—1674no—
274C23—3341no—
275C23—3341no—
276C23—3341no—
277C23—3341no—
278C23—1672no—
279C23—1672no—
280C23—1672no—
281C23C23×D16711yescharacteristic
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—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—3348no—
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—3341no—
313C2×C4—3341no—
314C2×C4—3341no—
315C2×C4—3341no—
316C2×C4—3341no—
317C2×C4—6684no—
318C2×C4—6684no—
319C2×C4—1672no—
320C2×C4—1672no—
321C2×C4—1672no—
322C2×C4—1672no—
323C2×C4C2×C4×D16712yes—
324C2×C4C2×C4×D16712yes—
325C2×C4C2×C4×D16712yes—
326C2×C4#1458195112yes—
327C22—3342no—
328C22—3342no—
329C22—3342no—
330C22—3342no—
331C22—3342no—
332C22—3342no—
333C22—1674no—
334C22C22×C4×D16711yescharacteristic
335C22#41×D16711yescharacteristic
336C22#41×D16711yescharacteristic
337C22#41×D16711yescharacteristic
338C22C2×#1458195111yescharacteristic
339C22#1912887711yescharacteristic
340C22#1912887911yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—3342no—
349C4—3342no—
350C4—3342no—
351C4—3342no—
352C4—6681no—
353C4—6681no—
354C4—6681no—
355C2—3344no—
356C2#117×D16711yescharacteristic
357C2C4×#1458195111yescharacteristic
358C2#2492374811yescharacteristic
359C2#2492394611yescharacteristic
360C2#2492394711yescharacteristic
361C2#2492395511yescharacteristic
362C2#2492395711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D167
commutatorC22×C167C23×C4
Fitting#387×C167C2
socle, cosocleC23×C167C24
prime radicalC23×D167C23
hypercenter#387D167
hypocenterC167#2343
LCSG ▷ C22×C167 ▷ C167
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #24923155C2semidirect
3. #24923154C2semidirect
4. #24923152C2semidirect
7. #24923136C2semidirect
8. #24923134C2semidirect
10. C4×Dic167⋊C4C2semidirect
13. #24923114C2semidirect
15. #387×C167C2semidirect
19. #19128790C4semidirect
70. C4⋊Dic167C2×C4semidirect
80. C4×Dic167C2×C4semidirect
81. C4×Dic167C2×C4semidirect
90. C4⋊C4×C167C2×C4semidirect
181. C167#2343semidirect
187. #387D167semidirect
263. C4⋊C4C4×D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191287902
#249231552
#249231542
#249231522
#249231432
#249231402
#249231362
#249231342
#249231312
C4×Dic167⋊C42
C4×(C2×D167)⋊C42
#249231162
#249231142
#95×D1672
#387×C1672
#2343167