Finite Group Databasebeta

← #36337213

Subgroups — #36337213

Order 21376.

356 conjugacy classes of subgroups.

The subgroup lattice has 344 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#24923661C211yesmaximal, characteristic, complemented
4#24923651C211yesmaximal, characteristic
5#24923491C211yesmaximal, characteristic, complemented
6#24923488C211yesmaximal, characteristic, complemented
7#24923484C211yesmaximal, characteristic
8#24923481C211yesmaximal, characteristic
9#24923479C211yesmaximal, characteristic, complemented
10#24923476C211yesmaximal, characteristic, complemented
11#24923400C211yesmaximal, characteristic, complemented
12#24923369C211yesmaximal, characteristic, complemented
13#24923367C211yesmaximal, characteristic, complemented
14C4×#14581940C211yesmaximal, characteristic, complemented
15#556×C167C211yesmaximal, characteristic, complemented
16C2×#14581940C2211yescharacteristic
17C2×#14581940C2211yescharacteristic
18C2×#14581940C2211yescharacteristic
19C2×#14581940C2211yescharacteristic
20C2×#14581938C2211yescharacteristic
21C2×#14581938C2211yescharacteristic
22C2×#14581937C2211yescharacteristic
23C2×#14581937C2211yescharacteristic
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#19128765C2211yescharacteristic
36C4⋊(C167⋊C8)C2211yescharacteristic
37C4⋊(C167⋊C8)C2211yescharacteristic
38C4⋊(C167⋊C8)C2211yescharacteristic
39#19128756C2211yescharacteristic
40#19128756C2211yescharacteristic
41#19128755C2211yescharacteristic
42#19128755C2211yescharacteristic
43#19128753C2211yescharacteristic
44#19128752C2211yescharacteristic
45#19128743C2211yescharacteristic
46#19128743C2211yescharacteristic
47C4×C167⋊C8C2211yescharacteristic
48C4⋊Q8×C167C2211yescharacteristic
49#126×C167—21no—
50#126×C167—21no—
51#124×C167C2211yescharacteristic
52#124×C167C2211yescharacteristic
53C22⋊Q8×C167—21no—
54C4×Q8×C167C2211yescharacteristic
55C4×Q8×C167C2211yescharacteristic
56C4×Q8×C167C2211yescharacteristic
57C4×D4×C167C2211yescharacteristic
58#117×C167—21no—
59C2×Dic334C2311yescharacteristic
60C2×Dic334C2311yescharacteristic
61C2×D668C2311yescharacteristic
62C2×C4×D167—21no—
63#14581940—22no—
64#14581940—41no—
65#14581940—41no—
66#14581940—41no—
67#14581938—22no—
68#14581938—41no—
69#14581937—22no—
70#14581937—41no—
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×Q8×C167C2311yescharacteristic
91C2×D4×C167C2311yescharacteristic
92C22×C4×C167—21no—
93C4⋊C4×C167—21no—
94C4⋊C4×C167—21no—
95C4⋊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—
106C42×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—
127Q8×C167—21no—
128Q8×C167—21no—
129Q8×C167—21no—
130Q8×C167—21no—
131Q8×C167—41no—
132D4×C167—21no—
133D4×C167—21no—
134C2×C4×C167—21no—
135C2×C4×C167—21no—
136C2×C4×C167—21no—
137C2×C4×C167—42no—
138C2×C4×C167C2×D411yescharacteristic
139C2×C4×C167C2×D411yescharacteristic
140C2×C4×C167C2×D411yescharacteristic
141C2×C4×C167C2×D411yescharacteristic
142C2×C4×C167C2×D411yescharacteristic
143C2×C4×C167C2×D411yescharacteristic
144C2×C4×C167C2411yescharacteristic
145C22×C167—41no—
146C22×C167—41no—
147C22×C167—41no—
148C22×C167C22×D411yescharacteristic
149C2×D167—41no—
150C2×D167—41no—
151C2×D167—41no—
152C4×C167—21no—
153C4×C167—21no—
154C4×C167—21no—
155C4×C167—21no—
156C4×C167—21no—
157C4×C167—21no—
158C4×C167—41no—
159C4×C167—41no—
160C4×C167—41no—
161C4×C167C2×#4111yescharacteristic
162C4×C167H(5,𝔽2)11yescharacteristic
163Dic167—41no—
164Dic167—41no—
165Dic167—41no—
166C2×C167—81no—
167C2×C167#55011yescharacteristic
168C2×C167#57511yescharacteristic
169C2×C167#57811yescharacteristic
170D167—81no—
171C167#340211yescharacteristic, complemented, Sylow
172#3402—1671nomaximal, Sylow
173#556—1671no—
174#556D16711yescharacteristic, complemented
175#497—1671no—
176#497—1671no—
177#495—1671no—
178#464—1671no—
179#464—1671no—
180#463—1671no—
181#463—1671no—
182#462—1671no—
183#462—1671no—
184#446—1671no—
185#440—1671no—
186#440—1671no—
187C4×Q16—1671no—
188C2×Q16—1671no—
189C2×Q16—1671no—
190C2×Q16—1671no—
191C2×Q16—1671no—
192C2×SD16—1671no—
193C2×SD16—1671no—
194C2×SD16—1671no—
195C2×SD16—1671no—
196C2×D8—1671no—
197C4⋊Q8—1671no—
198C4⋊Q8C2×D16711yescharacteristic
199#126—21no—
200#126—21no—
201#126—3341no—
202#126—3341no—
203#124—1671no—
204#124—1671no—
205#124C2×D16711yescharacteristic
206#124C2×D16711yescharacteristic
207C22⋊Q8—21no—
208C22⋊Q8—3341no—
209C4×Q8—1671no—
210C4×Q8—1671no—
211C4×Q8C2×D16711yescharacteristic
212C4×Q8C2×D16711yescharacteristic
213C4×Q8C2×D16711yescharacteristic
214C4×D4—1671no—
215C4×D4C2×D16711yescharacteristic
216#117—21no—
217#117—3341no—
218#107—1671no—
219C4⋊C8—1671no—
220C4⋊C8—1671no—
221C4⋊C8—1671no—
222#103—1671no—
223#103—1671no—
224#103—1671no—
225#103—1671no—
226#102—1671no—
227#102—1671no—
228#97—1671no—
229#97—1671no—
230C4×C8—1671no—
231C2×Q8—1671no—
232C2×Q8—1671no—
233C2×Q8C22⋊D16711yescharacteristic
234C2×Q8C22×D16711yescharacteristic
235C2×Q8C22×D16711yescharacteristic
236C2×D4—1671no—
237C2×D4C22×D16711yescharacteristic
238C22×C4—21no—
239C22×C4—3341no—
240Q16—3342no—
241Q16—6681no—
242Q16—6681no—
243Q16—6681no—
244SD16—3342no—
245SD16—3342no—
246SD16—6681no—
247SD16—6681no—
248D8—6681no—
249C2×C8—1671no—
250C2×C8—1671no—
251C2×C8—1671no—
252C2×C8—1671no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—3341no—
258C4⋊C4—3341no—
259C4⋊C4—3341no—
260C4⋊C4—3341no—
261C4⋊C4—1671no—
262C4⋊C4C22⋊D16711yescharacteristic
263C4⋊C4C22⋊D16711yescharacteristic
264C4⋊C4C22⋊D16711yescharacteristic
265C4⋊C4C22×D16711yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—3341no—
272C22⋊C4—3341no—
273C22⋊C4—3341no—
274C22⋊C4—3341no—
275C22⋊C4—3341no—
276C42—21no—
277C42—21no—
278C42—3341no—
279C42—3341no—
280C42C22×D16711yescharacteristic
281C42C22×D16711yescharacteristic
282C42C22×D16711yescharacteristic
283C23—21no—
284C23—3341no—
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—41no—
290Q8—3341no—
291Q8—3341no—
292Q8—3341no—
293Q8—3341no—
294D4—21no—
295D4—21no—
296D4—3341no—
297D4—3341no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—3341no—
303C2×C4—3341no—
304C2×C4—3341no—
305C2×C4—6682no—
306C2×C4C2×C22⋊D16711yescharacteristic
307C2×C4C2×C22⋊D16711yescharacteristic
308C2×C4C2×C22⋊D16711yescharacteristic
309C2×C4C2×C22⋊D16711yescharacteristic
310C2×C4C2×C22⋊D16711yescharacteristic
311C2×C4C2×C22⋊D16711yescharacteristic
312C2×C4C23×D16711yescharacteristic
313C8—3341no—
314C8—3341no—
315C8—3341no—
316C8—1672no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—6681no—
321C22—6681no—
322C22—6681no—
323C22C22×C22⋊D16711yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—6681no—
334C4—6681no—
335C4—6681no—
336C4#41×D16711yescharacteristic
337C4#1912887811yescharacteristic
338C2—81no—
339C2—13361no—
340C2#2492393811yescharacteristic
341C2#2492406711yescharacteristic
342C2#2492408411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#556×C167C2
socleC22×C167C22×D4
prime radicalC2×#14581935C22
hypercenter#556D167
hypocenterC167#3402
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #24923992C2semidirect
2. #24923664C2semidirect
3. #24923661C2semidirect
5. #24923491C2semidirect
6. #24923488C2semidirect
9. #24923479C2semidirect
10. #24923476C2semidirect
11. #24923400C2semidirect
12. #24923369C2semidirect
13. #24923367C2semidirect
14. C4×#14581940C2semidirect
15. #556×C167C2semidirect
171. C167#3402semidirect
174. #556D167semidirect

Maximal subgroups

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

subgroupindex
#249239922
#249236642
#249236612
#249236512
#249234912
#249234882
#249234842
#249234812
#249234792
#249234762
#249234002
#249233692
#249233672
C4×#145819402
#556×C1672
#3402167