Finite Group Databasebeta

← #36327072

Subgroups — #36327072

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×(C22⋊D167)⋊C4C211yesmaximal, characteristic
2#24923160C211yesmaximal, characteristic, complemented
3#24923158C211yesmaximal, characteristic, complemented
4#24923153C211yesmaximal, characteristic, complemented
5#24923151C211yesmaximal, characteristic, complemented
6#24923148C211yesmaximal, characteristic
7#24923146C211yesmaximal, characteristic
8#24923136C211yesmaximal, characteristic, complemented
9#24923131C211yesmaximal, characteristic, complemented
10#24923127C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic167C211yesmaximal, characteristic, complemented
12C4×(C2×D167)⋊C4C211yesmaximal, characteristic, complemented
13#24923115C211yesmaximal, characteristic, complemented
14C2×C167⋊#95C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C167C211yesmaximal, characteristic, complemented
16C22×C22⋊D167C2211yescharacteristic
17C23×Dic167C2211yescharacteristic
18C23×Dic167C2211yescharacteristic
19C22×C4×D167C2211yescharacteristic
20(C22⋊D167)⋊C4C418yescomplemented
21C2×Dic167⋊C4C2211yescharacteristic
22C2×Dic167⋊C4C2211yescharacteristic
23C2×Dic167⋊C4C2211yescharacteristic
24C2×C22⋊Dic167C2211yescharacteristic
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
33C167⋊#95C2211yescharacteristic
34C167⋊#95C2211yescharacteristic
35C167⋊#95C2211yescharacteristic
36C167⋊#95C2211yescharacteristic
37C167⋊#95C2211yescharacteristic
38C167⋊#95C2211yescharacteristic
39C167⋊#95C2211yescharacteristic
40C167⋊#95C2211yescharacteristic
41C167⋊#95C2211yescharacteristic
42C167⋊#95C2211yescharacteristic
43C167⋊#95C2211yescharacteristic
44C167⋊#95C2211yescharacteristic
45C23×C4×C167C2211yescharacteristic
46C2×C22⋊C4×C167C2211yescharacteristic
47C2×C22⋊C4×C167C2211yescharacteristic
48C2×C42×C167C2211yescharacteristic
49C2×C42×C167C2211yescharacteristic
50#95×C167C2211yescharacteristic
51#95×C167C2211yescharacteristic
52C23×D167C2311yescharacteristic
53C2×C22⋊D167—22no—
54C2×C22⋊D167—22no—
55C2×C22⋊D167C2×C414yes—
56C22×Dic167—22no—
57C22×Dic167—22no—
58C22×Dic167—41no—
59C22×Dic167—41no—
60C22×Dic167—41no—
61C22×Dic167C2×C414yes—
62C22×Dic167C2311yescharacteristic
63C22×Dic167C2311yescharacteristic
64C22×Dic167C2311yescharacteristic
65C22×Dic167C2311yescharacteristic
66C22×Dic167C2311yescharacteristic
67C22×Dic167C2311yescharacteristic
68C22×Dic167C2311yescharacteristic
69C2×C4×D167—41no—
70C2×C4×D167—41no—
71C2×C4×D167C2×C414yescomplemented
72Dic167⋊C4—41no—
73Dic167⋊C4—41no—
74Dic167⋊C4C2×C414yescomplemented
75C22⋊Dic167—22no—
76(C2×D167)⋊C4—22no—
77(C2×D167)⋊C4—22no—
78(C2×D167)⋊C4—41no—
79(C2×D167)⋊C4—41no—
80(C2×D167)⋊C4C2×C414yescomplemented
81C4×Dic167—22no—
82C4×Dic167—41no—
83C4×Dic167C2×C414yescomplemented
84C24×C167C2311yescharacteristic
85C22×C4×C167—22no—
86C22×C4×C167—22no—
87C22×C4×C167—41no—
88C22×C4×C167C2311yescharacteristic
89C22×C4×C167C2311yescharacteristic
90C22×C4×C167C2311yescharacteristic
91C22×C4×C167C2311yescharacteristic
92C22×C4×C167C2311yescharacteristic
93C22×C4×C167C2311yescharacteristic
94C22⋊C4×C167—22no—
95C22⋊C4×C167C2×C414yes—
96C42×C167—22no—
97C42×C167—41no—
98C22×D167—21no—
99C22×D167—21no—
100C22×D167—21no—
101C22×D167—21no—
102C22×D167—21no—
103C22×D167—21no—
104C22×D167C22×C412yes—
105C22⋊D167—28no—
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×Dic167—21no—
117C2×Dic167—21no—
118C2×Dic167—21no—
119C2×Dic167—21no—
120C2×Dic167—21no—
121C2×Dic167—21no—
122C2×Dic167—21no—
123C2×Dic167—44no—
124C2×Dic167—44no—
125C2×Dic167C22×C412yes—
126C2×Dic167C22×C412yes—
127C2×Dic167C22×C412yes—
128C2×Dic167#4112yes—
129C4×D167—44no—
130C23×C167—21no—
131C23×C167—21no—
132C23×C167—21no—
133C23×C167—21no—
134C23×C167C22×C412yes—
135C23×C167#4112yes—
136C23×C167#4112yes—
137C23×C167C2411yescharacteristic
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×C167—44no—
154C2×C4×C167C22×C412yes—
155C2×C4×C167C22×C412yes—
156C2×C4×C167#4112yes—
157C22×C167—22no—
158C22×C167—22no—
159C22×C167—22no—
160C22×C167—22no—
161C22×C167—22no—
162C22×C167—22no—
163C22×C167#11714yes—
164C22×C167C23×C411yescharacteristic
165C22×C167C2×#4111yescharacteristic
166C22×C167C2×#4111yescharacteristic
167C22×C167C2×#4111yescharacteristic
168C22×C167C2×#4111yescharacteristic
169C22×C167H(5,𝔽2)11yescharacteristic
170C22×C167H(5,𝔽2)11yescharacteristic
171C2×D167—22no—
172C2×D167—22no—
173C2×D167—22no—
174C2×D167—41no—
175C2×D167—41no—
176C2×D167—41no—
177C2×D167—41no—
178C4×C167—22no—
179C4×C167—22no—
180C4×C167—22no—
181C4×C167—41no—
182C4×C167—41no—
183C4×C167—41no—
184Dic167—22no—
185Dic167—22no—
186Dic167—22no—
187Dic167—22no—
188Dic167—41no—
189Dic167—41no—
190Dic167—41no—
191C2×C167—24no—
192C2×C167C2×#11711yescharacteristic
193C2×C167C4×#4111yescharacteristic
194C2×C167#51811yescharacteristic
195C2×C167#55111yescharacteristic
196C2×C167#55111yescharacteristic
197C2×C167#55311yescharacteristic
198C2×C167#55311yescharacteristic
199D167—42no—
200C167#235311yescharacteristic, complemented, Sylow
201#2353—1671nomaximal, Sylow
202C2×C4×D4—1671no—
203#388—1671no—
204#388—1671no—
205#387—1671no—
206#387—1671no—
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⋊C4D16711yescharacteristic, complemented
215#376—1671no—
216C2×#95—1671no—
217C22×D4—1671no—
218C23×C4—1671no—
219C23×C4—1671no—
220C23×C4—1671no—
221C23×C4C2×D16711yescharacteristic
222C4×D4—1678no—
223C2×C4⋊C4—1671no—
224C2×C4⋊C4—1671no—
225C2×C4⋊C4—1671no—
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⋊C4C2×D16711yescharacteristic
233C2×C22⋊C4C2×D16711yescharacteristic
234C2×C42—1671no—
235C2×C42—1671no—
236C2×C42—1671no—
237C2×C42C2×D16711yescharacteristic
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#95—1671no—
250#95—1671no—
251#95C2×D16711yescharacteristic
252#95C2×D16711yescharacteristic
253C24—1671no—
254C24C22×D16711yescharacteristic
255C2×D4—3342no—
256C2×D4—3342no—
257C2×D4—1674no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—3342no—
262C22×C4—3342no—
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—1671no—
275C22×C4—1674no—
276C22×C4—1674no—
277C22×C4C22×D16711yescharacteristic
278C22×C4C22×D16711yescharacteristic
279C22×C4C22×D16711yescharacteristic
280C22×C4C22×D16711yescharacteristic
281C22×C4C22×D16711yescharacteristic
282C22×C4C22×D16711yescharacteristic
283C4⋊C4—6681no—
284C4⋊C4—6681no—
285C4⋊C4—1674no—
286C22⋊C4—22no—
287C22⋊C4—3342no—
288C22⋊C4—3342no—
289C22⋊C4—3342no—
290C22⋊C4—6681no—
291C22⋊C4—6681no—
292C22⋊C4—1674no—
293C22⋊C4C4×D16714yes—
294C42—22no—
295C42—41no—
296C42—3342no—
297C42—6681no—
298C42—1674no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—3341no—
304C23—3341no—
305C23—3341no—
306C23—3341no—
307C23—3341no—
308C23—3341no—
309C23—1672no—
310C23C2×C4×D16712yes—
311C23#1458195012yes—
312C23#1458195012yes—
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—44no—
331C2×C4—3341no—
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—3341no—
348C2×C4—6684no—
349C2×C4—6684no—
350C2×C4—6684no—
351C2×C4—1672no—
352C2×C4—1672no—
353C2×C4—1672no—
354C2×C4—1672no—
355C2×C4C2×C4×D16712yes—
356C2×C4C2×C4×D16712yes—
357C2×C4#1458194912yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—3342no—
365C22—3342no—
366C22—3342no—
367C22—6681no—
368C22—6681no—
369C22—6681no—
370C22—6681no—
371C22#1912878914yes—
372C22C22×C4×D16711yescharacteristic
373C22#41×D16711yescharacteristic
374C22C2×#1458194911yescharacteristic
375C22C2×#1458195011yescharacteristic
376C22C2×#1458195011yescharacteristic
377C22#1912887711yescharacteristic
378C22#1912887811yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—3342no—
386C4—3342no—
387C4—3342no—
388C4—3342no—
389C4—6681no—
390C4—6681no—
391C4—6681no—
392C2—24no—
393C2—6682no—
394C2C2×#1912878911yescharacteristic
395C2C4×#1458194911yescharacteristic
396C2#2492374111yescharacteristic
397C2#2492394311yescharacteristic
398C2#2492394511yescharacteristic
399C2#2492396511yescharacteristic
400C2#2492396811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D167
commutatorC22×C167C23×C4
FittingC4×C22⋊C4×C167C2
socle, cosocleC23×C167C24
prime radicalC22×C22⋊D167C22
hypercenterC4×C22⋊C4D167
hypocenterC167#2353
LCSG ▷ C22×C167 ▷ C167
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #24923160C2semidirect
3. #24923158C2semidirect
4. #24923153C2semidirect
5. #24923151C2semidirect
8. #24923136C2semidirect
9. #24923131C2semidirect
10. #24923127C2semidirect
11. C22⋊C4×Dic167C2semidirect
12. C4×(C2×D167)⋊C4C2semidirect
13. #24923115C2semidirect
14. C2×C167⋊#95C2semidirect
15. C4×C22⋊C4×C167C2semidirect
20. (C22⋊D167)⋊C4C4semidirect
71. C2×C4×D167C2×C4semidirect
74. Dic167⋊C4C2×C4semidirect
80. (C2×D167)⋊C4C2×C4semidirect
83. C4×Dic167C2×C4semidirect
200. C167#2353semidirect
214. C4×C22⋊C4D167semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D167)⋊C42
#249231602
#249231582
#249231532
#249231512
#249231482
#249231462
#249231362
#249231312
#249231272
C22⋊C4×Dic1672
C4×(C2×D167)⋊C42
#249231152
C2×C167⋊#952
C4×C22⋊C4×C1672
#2353167