Finite Group Databasebeta

← #36337198

Subgroups — #36337198

Order 21376.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923992C211yesmaximal, characteristic, complemented
2#24923664C211yesmaximal, characteristic, complemented
3#24923656C211yesmaximal, characteristic
4#24923651C211yesmaximal, characteristic, complemented
5#24923491C211yesmaximal, characteristic, complemented
6#24923488C211yesmaximal, characteristic, complemented
7#24923484C211yesmaximal, characteristic, complemented
8#24923481C211yesmaximal, characteristic, complemented
9#24923474C211yesmaximal, characteristic, complemented
10#24923466C211yesmaximal, characteristic, complemented
11#24923400C211yesmaximal, characteristic, complemented
12#24923379C211yesmaximal, characteristic, complemented
13#24923374C211yesmaximal, characteristic, complemented
14C4×#14581938C211yesmaximal, characteristic, complemented
15#559×C167C211yesmaximal, characteristic, complemented
16C2×#14581940C2211yescharacteristic
17C2×#14581940C2211yescharacteristic
18C2×#14581938C2211yescharacteristic
19C2×#14581938C2211yescharacteristic
20C2×#14581938C2211yescharacteristic
21C2×#14581938C2211yescharacteristic
22C2×#14581937C2211yescharacteristic
23C2×#14581935C2211yescharacteristic
24C2×#14581935C2211yescharacteristic
25#19128832C2211yescharacteristic
26#19128827—21no—
27#19128827—21no—
28#19128820C2211yescharacteristic
29#19128820C2211yescharacteristic
30#19128808—21no—
31C4×Dic334C2211yescharacteristic
32C4×Dic334C2211yescharacteristic
33C4×D668C2211yescharacteristic
34#19128790—21no—
35#19128763C2211yescharacteristic
36C4⋊(C167⋊C8)C2211yescharacteristic
37C4⋊(C167⋊C8)C2211yescharacteristic
38C4⋊(C167⋊C8)C2211yescharacteristic
39#19128756C2211yescharacteristic
40#19128755C2211yescharacteristic
41#19128755C2211yescharacteristic
42#19128753C2211yescharacteristic
43#19128753C2211yescharacteristic
44#19128752C2211yescharacteristic
45#19128743C2211yescharacteristic
46#19128743C2211yescharacteristic
47C4×C167⋊C8C2211yescharacteristic
48#127×C167C2211yescharacteristic
49#124×C167—21no—
50#124×C167C2211yescharacteristic
51#124×C167C2211yescharacteristic
52#123×C167—21no—
53#123×C167—21no—
54C4×Q8×C167C2211yescharacteristic
55C4×Q8×C167C2211yescharacteristic
56C4×D4×C167—21no—
57C4×D4×C167C2211yescharacteristic
58C4×D4×C167C2211yescharacteristic
59C2×Dic334C2311yescharacteristic
60C2×Dic334C2311yescharacteristic
61C2×D668C2311yescharacteristic
62C2×C4×D167—21no—
63#14581940—22no—
64#14581940—41no—
65#14581938—22no—
66#14581938—41no—
67#14581938—41no—
68#14581938—41no—
69#14581937—41no—
70#14581935—22no—
71#14581935—41no—
72C2×C167⋊C8C2311yescharacteristic
73C2×C167⋊C8C2311yescharacteristic
74C2×C167⋊C8C2311yescharacteristic
75C2×C167⋊C8C2311yescharacteristic
76C4⋊Dic167—21no—
77C4⋊Dic167—21no—
78C4⋊Dic167C2311yescharacteristic
79Dic167⋊C4—21no—
80Dic167⋊C4—21no—
81(C2×D167)⋊C4—21no—
82(C2×D167)⋊C4—21no—
83(C2×D167)⋊C4—21no—
84(C2×D167)⋊C4—21no—
85(C2×D167)⋊C4—21no—
86C4×Dic167—21no—
87C4×Dic167—21no—
88C2×Q8×C167D411yescharacteristic
89C2×Q8×C167C2311yescharacteristic
90C2×D4×C167—21no—
91C2×D4×C167—21no—
92C2×D4×C167C2311yescharacteristic
93C2×D4×C167C2311yescharacteristic
94C22×C4×C167—21no—
95C22×C4×C167—21no—
96C4⋊C4×C167—21no—
97C4⋊C4×C167D411yescharacteristic
98C4⋊C4×C167D411yescharacteristic
99C4⋊C4×C167D411yescharacteristic
100C4⋊C4×C167C2311yescharacteristic
101C22⋊C4×C167—21no—
102C22⋊C4×C167—21no—
103C22⋊C4×C167—21no—
104C22⋊C4×C167—21no—
105C22⋊C4×C167—21no—
106C22⋊C4×C167—21no—
107C42×C167—21no—
108C42×C167C2311yescharacteristic
109C42×C167C2311yescharacteristic
110C42×C167C2311yescharacteristic
111C22×D167—21no—
112Dic334—21no—
113Dic334—21no—
114Dic334—21no—
115Dic334—21no—
116D668—21no—
117D668—21no—
118C2×Dic167—21no—
119C2×Dic167—21no—
120C2×Dic167—21no—
121C4×D167—42no—
122C167⋊C8—21no—
123C167⋊C8—21no—
124C167⋊C8—21no—
125C167⋊C8#4112yes—
126C23×C167—21no—
127C23×C167—21no—
128Q8×C167—21no—
129Q8×C167—21no—
130Q8×C167—41no—
131D4×C167—21no—
132D4×C167—21no—
133D4×C167—21no—
134D4×C167—21no—
135D4×C167—42no—
136D4×C167—81no—
137C2×C4×C167—21no—
138C2×C4×C167—21no—
139C2×C4×C167—81no—
140C2×C4×C167—81no—
141C2×C4×C167C2×D411yescharacteristic
142C2×C4×C167C2×D411yescharacteristic
143C2×C4×C167C2×D411yescharacteristic
144C2×C4×C167C2×D411yescharacteristic
145C2×C4×C167C2×D411yescharacteristic
146C2×C4×C167C2×D411yescharacteristic
147C2×C4×C167C2411yescharacteristic
148C22×C167—41no—
149C22×C167—41no—
150C22×C167—41no—
151C22×C167—41no—
152C22×C167—41no—
153C22×C167—41no—
154C22×C167C22×D411yescharacteristic
155C2×D167—41no—
156C2×D167—41no—
157C2×D167—41no—
158C4×C167—21no—
159C4×C167—21no—
160C4×C167—21no—
161C4×C167—21no—
162C4×C167—21no—
163C4×C167—21no—
164C4×C167—41no—
165C4×C167—41no—
166C4×C167C2×#4111yescharacteristic
167C4×C167H(5,𝔽2)11yescharacteristic
168Dic167—41no—
169Dic167—41no—
170Dic167—41no—
171C2×C167—81no—
172C2×C167—81no—
173C2×C167#55011yescharacteristic
174C2×C167#57511yescharacteristic
175C2×C167#57711yescharacteristic
176D167—81no—
177C167#340111yescharacteristic, complemented, Sylow
178#3401—1671nomaximal, Sylow
179#559D16711yescharacteristic, complemented
180#556—1671no—
181#497—1671no—
182#496—1671no—
183#495—1671no—
184#464—1671no—
185#464—1671no—
186#463—1671no—
187#463—1671no—
188#461—1671no—
189#460—1671no—
190#446—1671no—
191#442—1671no—
192#441—1671no—
193C4×SD16—1671no—
194C2×Q16—1671no—
195C2×Q16—1671no—
196C2×SD16—1671no—
197C2×SD16—1671no—
198C2×SD16—1671no—
199C2×SD16—1671no—
200C2×SD16—1671no—
201C2×D8—1671no—
202C2×D8—1671no—
203C4⋊Q8—1671no—
204#127C2×D16711yescharacteristic
205#126—3341no—
206#126—3341no—
207#124—21no—
208#124—1671no—
209#124—1671no—
210#124C2×D16711yescharacteristic
211#124C2×D16711yescharacteristic
212#123—21no—
213#123—21no—
214C22⋊Q8—3341no—
215C4×Q8—1671no—
216C4×Q8—1671no—
217C4×Q8C2×D16711yescharacteristic
218C4×Q8C2×D16711yescharacteristic
219C4×D4—21no—
220C4×D4—1671no—
221C4×D4C2×D16711yescharacteristic
222C4×D4C2×D16711yescharacteristic
223#117—3341no—
224#106—1671no—
225C4⋊C8—1671no—
226C4⋊C8—1671no—
227C4⋊C8—1671no—
228#103—1671no—
229#103—1671no—
230#103—1671no—
231#102—1671no—
232#102—1671no—
233#102—1671no—
234#97—1671no—
235#97—1671no—
236C4×C8—1671no—
237C2×Q8—1671no—
238C2×Q8—1671no—
239C2×Q8C22⋊D16711yescharacteristic
240C2×Q8C22×D16711yescharacteristic
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—1671no—
244C2×D4C22×D16711yescharacteristic
245C2×D4C22×D16711yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—3341no—
249Q16—3342no—
250Q16—6681no—
251SD16—3342no—
252SD16—6681no—
253SD16—6681no—
254SD16—6681no—
255SD16—6681no—
256D8—3342no—
257D8—6681no—
258C2×C8—1671no—
259C2×C8—1671no—
260C2×C8—1671no—
261C2×C8—1671no—
262C4⋊C4—21no—
263C4⋊C4—3341no—
264C4⋊C4—3341no—
265C4⋊C4—3341no—
266C4⋊C4—3341no—
267C4⋊C4—1671no—
268C4⋊C4C22⋊D16711yescharacteristic
269C4⋊C4C22⋊D16711yescharacteristic
270C4⋊C4C22⋊D16711yescharacteristic
271C4⋊C4C22×D16711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—3341no—
279C22⋊C4—3341no—
280C22⋊C4—3341no—
281C22⋊C4—3341no—
282C22⋊C4—3341no—
283C42—21no—
284C42—3341no—
285C42—3341no—
286C42C22×D16711yescharacteristic
287C42C22×D16711yescharacteristic
288C42C22×D16711yescharacteristic
289C23—21no—
290C23—21no—
291C23—3341no—
292Q8—21no—
293Q8—21no—
294Q8—41no—
295Q8—3341no—
296Q8—3341no—
297Q8—3341no—
298Q8—3341no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—42no—
304D4—81no—
305D4—3341no—
306D4—3341no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—3341no—
312C2×C4—3341no—
313C2×C4—3341no—
314C2×C4—6682no—
315C2×C4C2×C22⋊D16711yescharacteristic
316C2×C4C2×C22⋊D16711yescharacteristic
317C2×C4C2×C22⋊D16711yescharacteristic
318C2×C4C2×C22⋊D16711yescharacteristic
319C2×C4C2×C22⋊D16711yescharacteristic
320C2×C4C2×C22⋊D16711yescharacteristic
321C2×C4C23×D16711yescharacteristic
322C8—3341no—
323C8—3341no—
324C8—3341no—
325C8—1672no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—6681no—
333C22—6681no—
334C22—6681no—
335C22C22×C22⋊D16711yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—6681no—
345C4—6681no—
346C4—6681no—
347C4#41×D16711yescharacteristic
348C4#1912887811yescharacteristic
349C2—81no—
350C2—81no—
351C2—13361no—
352C2#2492393811yescharacteristic
353C2#2492406711yescharacteristic
354C2#2492408011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#559×C167C2
socleC22×C167C22×D4
prime radical#24923656C2
hypercenter#559D167
hypocenterC167#3401
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #24923992C2semidirect
2. #24923664C2semidirect
4. #24923651C2semidirect
5. #24923491C2semidirect
6. #24923488C2semidirect
7. #24923484C2semidirect
8. #24923481C2semidirect
9. #24923474C2semidirect
10. #24923466C2semidirect
11. #24923400C2semidirect
12. #24923379C2semidirect
13. #24923374C2semidirect
14. C4×#14581938C2semidirect
15. #559×C167C2semidirect
177. C167#3401semidirect
179. #559D167semidirect

Maximal subgroups

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

subgroupindex
#249239922
#249236642
#249236562
#249236512
#249234912
#249234882
#249234842
#249234812
#249234742
#249234662
#249234002
#249233792
#249233742
C4×#145819382
#559×C1672
#3401167