Finite Group Databasebeta

← #36327077

Subgroups — #36327077

Order 21376.

588 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128794C211yesmaximal, characteristic
2#24923161C211yesmaximal, characteristic, complemented
3#24923159C211yesmaximal, characteristic, complemented
4#24923155C211yesmaximal, characteristic, complemented
5#24923147C211yesmaximal, characteristic
6#24923147C211yesmaximal, characteristic
7#24923136C211yesmaximal, characteristic, complemented
8#24923131C211yesmaximal, characteristic, complemented
9#24923126C211yesmaximal, characteristic, complemented
10C4×(C2×D167)⋊C4C211yesmaximal, characteristic, complemented
11C4×(C2×D167)⋊C4C211yesmaximal, characteristic, complemented
12#24923116C211yesmaximal, characteristic, complemented
13#24923114C211yesmaximal, characteristic, complemented
14#95×D167C211yesmaximal, characteristic, complemented
15#387×C167C211yesmaximal, characteristic, complemented
16C22×D668C2211yescharacteristic
17C22×C4×D167C2211yescharacteristic
18C22×C4×D167C2211yescharacteristic
19C22×C4×D167C2211yescharacteristic
20C22×C4×D167C2211yescharacteristic
21#19128794C418yescomplemented
22C2×C4⋊Dic167C2211yescharacteristic
23C2×(C2×D167)⋊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×(C2×D167)⋊C4C2211yescharacteristic
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
48#95×C167C2211yescharacteristic
49#95×C167C2211yescharacteristic
50#95×C167C2211yescharacteristic
51#95×C167C2211yescharacteristic
52C23×D167C2311yescharacteristic
53C23×D167C2311yescharacteristic
54C2×D668—22no—
55C2×D668—22no—
56C2×D668C2×C414yes—
57C22×Dic167C2311yescharacteristic
58C22×Dic167C2311yescharacteristic
59C22×Dic167C2311yescharacteristic
60C22×Dic167C2311yescharacteristic
61C22×Dic167C2311yescharacteristic
62C22×Dic167C2311yescharacteristic
63C2×C4×D167—22no—
64C2×C4×D167—22no—
65C2×C4×D167—22no—
66C2×C4×D167—22no—
67C2×C4×D167—41no—
68C2×C4×D167—41no—
69C2×C4×D167—41no—
70C2×C4×D167—41no—
71C2×C4×D167—41no—
72C2×C4×D167—41no—
73C2×C4×D167C2×C414yes—
74C2×C4×D167C2×C414yescomplemented
75C4⋊Dic167—41no—
76(C2×D167)⋊C4—22no—
77(C2×D167)⋊C4—22no—
78(C2×D167)⋊C4—22no—
79(C2×D167)⋊C4—22no—
80(C2×D167)⋊C4—41no—
81(C2×D167)⋊C4—41no—
82(C2×D167)⋊C4C2×C414yes—
83(C2×D167)⋊C4C2×C414yescomplemented
84C4×Dic167—22no—
85C4×Dic167—22no—
86C4×Dic167—41no—
87C4×Dic167C2×C414yescomplemented
88C22×C4×C167C2311yescharacteristic
89C22×C4×C167C2311yescharacteristic
90C22×C4×C167C2311yescharacteristic
91C22×C4×C167C2311yescharacteristic
92C22×C4×C167C2311yescharacteristic
93C22×C4×C167C2311yescharacteristic
94C22×C4×C167C2311yescharacteristic
95C4⋊C4×C167—41no—
96C4⋊C4×C167C2×C414yescomplemented
97C42×C167—41no—
98C22×D167—21no—
99C22×D167—21no—
100C22×D167—21no—
101C22×D167—21no—
102C22×D167—21no—
103C22×D167—21no—
104C22×D167—21no—
105C22×D167—21no—
106C22×D167—21no—
107C22×D167—21no—
108C22×D167C22×C412yes—
109C22×D167C22×C412yes—
110C22×D167#4112yes—
111C22×D167#4112yes—
112D668—28no—
113C2×Dic167—21no—
114C2×Dic167—21no—
115C2×Dic167—21no—
116C2×Dic167—21no—
117C2×Dic167—21no—
118C2×Dic167—21no—
119C2×Dic167—21no—
120C2×Dic167—21no—
121C2×Dic167—21no—
122C2×Dic167—21no—
123C2×Dic167—21no—
124C2×Dic167—21no—
125C2×Dic167—21no—
126C2×Dic167—21no—
127C2×Dic167—21no—
128C2×Dic167C22×C412yes—
129C2×Dic167C22×C412yes—
130C2×Dic167#4112yes—
131C4×D167—44no—
132C4×D167—44no—
133C4×D167—44no—
134C4×D167—44no—
135C23×C167C2411yescharacteristic
136C2×C4×C167—21no—
137C2×C4×C167—21no—
138C2×C4×C167—21no—
139C2×C4×C167—21no—
140C2×C4×C167—21no—
141C2×C4×C167—21no—
142C2×C4×C167—21no—
143C2×C4×C167—21no—
144C2×C4×C167—21no—
145C2×C4×C167—21no—
146C2×C4×C167—21no—
147C2×C4×C167—21no—
148C2×C4×C167—21no—
149C2×C4×C167—21no—
150C2×C4×C167—21no—
151C2×C4×C167—21no—
152C2×C4×C167—21no—
153C2×C4×C167C22×C412yes—
154C2×C4×C167C22×C412yes—
155C2×C4×C167C22×C412yes—
156C2×C4×C167#4112yes—
157C22×C167C23×C411yescharacteristic
158C22×C167C2×#4111yescharacteristic
159C22×C167C2×#4111yescharacteristic
160C22×C167C2×#4111yescharacteristic
161C22×C167C2×#4111yescharacteristic
162C22×C167H(5,𝔽2)11yescharacteristic
163C22×C167H(5,𝔽2)11yescharacteristic
164C2×D167—22no—
165C2×D167—22no—
166C2×D167—22no—
167C2×D167—22no—
168C2×D167—22no—
169C2×D167—22no—
170C2×D167—22no—
171C2×D167—22no—
172C2×D167—22no—
173C2×D167—41no—
174C2×D167—41no—
175C2×D167—41no—
176C2×D167—41no—
177C2×D167#11714yes—
178C4×C167—22no—
179C4×C167—22no—
180C4×C167—22no—
181C4×C167—22no—
182C4×C167—41no—
183C4×C167—41no—
184C4×C167—41no—
185Dic167—22no—
186Dic167—22no—
187Dic167—22no—
188Dic167—41no—
189Dic167—41no—
190Dic167—41no—
191C2×C167C2×#11711yescharacteristic
192C2×C167C4×#4111yescharacteristic
193C2×C167#51811yescharacteristic
194C2×C167#55111yescharacteristic
195C2×C167#55111yescharacteristic
196C2×C167#55311yescharacteristic
197C2×C167#55311yescharacteristic
198D167—24no—
199D167—42no—
200C167#235311yescharacteristic, complemented, Sylow
201#2353—1671nomaximal, Sylow
202C2×C4×D4—1671no—
203#388—1671no—
204#388—1671no—
205#387—1671no—
206#387D16711yescharacteristic, complemented
207#386—1671no—
208#386—1671no—
209#383—1671no—
210#381—1671no—
211#380—1671no—
212C4×C22⋊C4—1671no—
213C4×C22⋊C4—1671no—
214C4×C22⋊C4—1671no—
215#376—1671no—
216C2×#95—1671no—
217C22×D4—1671no—
218C23×C4—1671no—
219C23×C4—1671no—
220C23×C4—1671no—
221C23×C4—1671no—
222C4×D4—1678no—
223C2×C4⋊C4—1671no—
224C2×C4⋊C4C2×D16711yescharacteristic
225C2×C4⋊C4C2×D16711yescharacteristic
226C2×C22⋊C4—1671no—
227C2×C22⋊C4—1671no—
228C2×C22⋊C4—1671no—
229C2×C22⋊C4—1671no—
230C2×C22⋊C4—1671no—
231C2×C22⋊C4—1671no—
232C2×C22⋊C4—1671no—
233C2×C22⋊C4—1671no—
234C2×C42—1671no—
235C2×C42—1671no—
236C2×C42—1671no—
237C2×C42—1671no—
238C2×C42C2×D16711yescharacteristic
239#95—1671no—
240#95—1671no—
241#95—1671no—
242#95—1671no—
243#95—1671no—
244#95—1671no—
245#95—1671no—
246#95—1671no—
247#95—1671no—
248#95—1671no—
249#95C2×D16711yescharacteristic
250#95C2×D16711yescharacteristic
251#95C2×D16711yescharacteristic
252#95C2×D16711yescharacteristic
253C24—1671no—
254C24—1671no—
255C2×D4—3342no—
256C2×D4—3342no—
257C2×D4—1674no—
258C22×C4—3342no—
259C22×C4—3342no—
260C22×C4—3342no—
261C22×C4—3342no—
262C22×C4—6681no—
263C22×C4—6681no—
264C22×C4—6681no—
265C22×C4—6681no—
266C22×C4—6681no—
267C22×C4—6681no—
268C22×C4—1671no—
269C22×C4—1671no—
270C22×C4—1671no—
271C22×C4—1671no—
272C22×C4—1671no—
273C22×C4—1671no—
274C22×C4—1674no—
275C22×C4—1674no—
276C22×C4C22×D16711yescharacteristic
277C22×C4C22×D16711yescharacteristic
278C22×C4C22×D16711yescharacteristic
279C22×C4C22×D16711yescharacteristic
280C22×C4C22×D16711yescharacteristic
281C22×C4C22×D16711yescharacteristic
282C22×C4C22×D16711yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—6681no—
285C4⋊C4C4×D16714yescomplemented
286C22⋊C4—3342no—
287C22⋊C4—3342no—
288C22⋊C4—3342no—
289C22⋊C4—3342no—
290C22⋊C4—6681no—
291C22⋊C4—6681no—
292C22⋊C4—1674no—
293C22⋊C4—1674no—
294C42—41no—
295C42—3342no—
296C42—3342no—
297C42—6681no—
298C42—1674no—
299C23—3341no—
300C23—3341no—
301C23—3341no—
302C23—3341no—
303C23—3341no—
304C23—3341no—
305C23—3341no—
306C23—3341no—
307C23—3341no—
308C23—3341no—
309C23—1672no—
310C23—1672no—
311C23—1672no—
312C23—1672no—
313C23C23×D16711yescharacteristic
314D4—3348no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—3341no—
333C2×C4—3341no—
334C2×C4—3341no—
335C2×C4—3341no—
336C2×C4—3341no—
337C2×C4—3341no—
338C2×C4—3341no—
339C2×C4—3341no—
340C2×C4—3341no—
341C2×C4—3341no—
342C2×C4—3341no—
343C2×C4—3341no—
344C2×C4—3341no—
345C2×C4—3341no—
346C2×C4—3341no—
347C2×C4—6684no—
348C2×C4—6684no—
349C2×C4—6684no—
350C2×C4—6684no—
351C2×C4—1672no—
352C2×C4—1672no—
353C2×C4—1672no—
354C2×C4C2×C4×D16712yes—
355C2×C4C2×C4×D16712yes—
356C2×C4C2×C4×D16712yes—
357C2×C4#1458195112yes—
358C22—3342no—
359C22—3342no—
360C22—3342no—
361C22—3342no—
362C22—3342no—
363C22—3342no—
364C22—3342no—
365C22—3342no—
366C22—3342no—
367C22—6681no—
368C22—6681no—
369C22—6681no—
370C22—6681no—
371C22—1674no—
372C22C22×C4×D16711yescharacteristic
373C22#41×D16711yescharacteristic
374C22#41×D16711yescharacteristic
375C22#41×D16711yescharacteristic
376C22C2×#1458195111yescharacteristic
377C22#1912887711yescharacteristic
378C22#1912887811yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—3342no—
387C4—3342no—
388C4—3342no—
389C4—6681no—
390C4—6681no—
391C4—6681no—
392C2—3344no—
393C2—6682no—
394C2#117×D16711yescharacteristic
395C2C4×#1458195111yescharacteristic
396C2#2492374111yescharacteristic
397C2#2492394411yescharacteristic
398C2#2492394611yescharacteristic
399C2#2492396411yescharacteristic
400C2#2492397211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D167
commutatorC22×C167C23×C4
Fitting#387×C167C2
socle, cosocleC23×C167C24
prime radicalC22×D668C22
hypercenter#387D167
hypocenterC167#2353
LCSG ▷ C22×C167 ▷ C167
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #24923161C2semidirect
3. #24923159C2semidirect
4. #24923155C2semidirect
7. #24923136C2semidirect
8. #24923131C2semidirect
9. #24923126C2semidirect
10. C4×(C2×D167)⋊C4C2semidirect
11. C4×(C2×D167)⋊C4C2semidirect
12. #24923116C2semidirect
13. #24923114C2semidirect
14. #95×D167C2semidirect
15. #387×C167C2semidirect
21. #19128794C4semidirect
74. C2×C4×D167C2×C4semidirect
83. (C2×D167)⋊C4C2×C4semidirect
87. C4×Dic167C2×C4semidirect
96. C4⋊C4×C167C2×C4semidirect
200. C167#2353semidirect
206. #387D167semidirect
285. C4⋊C4C4×D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191287942
#249231612
#249231592
#249231552
#249231472
#249231362
#249231312
#249231262
C4×(C2×D167)⋊C42
#249231162
#249231142
#95×D1672
#387×C1672
#2353167