Finite Group Databasebeta

← #36363691

Subgroups — #36363691

Order 22144.

380 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012045C211yesmaximal, characteristic
2C4×Q8×D173C211yesmaximal, characteristic
3#26011823C211yesmaximal, characteristic
4#26011820C211yesmaximal, characteristic
5#26011775C211yesmaximal, characteristic, complemented
6#26011774C211yesmaximal, characteristic, complemented
7#26011700C211yesmaximal, characteristic, complemented
8#26011699C211yesmaximal, characteristic, complemented
9#26011648C211yesmaximal, characteristic
10#26011645C211yesmaximal, characteristic
11#26011599C211yesmaximal, characteristic, complemented
12#26011596C211yesmaximal, characteristic, complemented
13#26011596C211yesmaximal, characteristic, complemented
14#26011546C211yesmaximal, characteristic
15#441×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×Dic692C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20#19149678C2211yescharacteristic
21#19149677C2211yescharacteristic
22#19149669—21no—
23#19149659C2211yescharacteristic
24#19149657—21no—
25#19149655C2211yescharacteristic
26Q8×Dic173C2211yescharacteristic
27C4×Dic346—21no—
28C4×Dic346—21no—
29C4×Dic346C2211yescharacteristic
30C4×Dic346C2211yescharacteristic
31#19149644—21no—
32#19149644C2211yescharacteristic
33#19149636—21no—
34C4⋊C4×D173—21no—
35C4⋊C4×D173C2211yescharacteristic
36C42×D173—21no—
37C42×D173C2211yescharacteristic
38#19149611C2211yescharacteristic
39#19149610C2211yescharacteristic
40#19149610C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149602C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149590C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54C4×Q8×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346—21no—
61C2×Dic346C2311yescharacteristic
62C2×Dic346C2311yescharacteristic
63Q8×D173—22no—
64Q8×D173—22no—
65C2×C4×D173—21no—
66C2×C4×D173—21no—
67C2×C4×D173D411yescharacteristic
68C2×C4×D173D411yescharacteristic
69C2×C4×D173C2311yescharacteristic
70Dic692—22no—
71#14583816—22no—
72#14583816—22no—
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C4⋊Dic173—21no—
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583803—21no—
88#14583803—21no—
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173D411yescharacteristic
94C4×Dic173D411yescharacteristic
95C4×Dic173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C2×Q8×C173C2311yescharacteristic
98Q16×C173—22no—
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173—21no—
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C42×C173—21no—
106C42×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173C2×D411yescharacteristic
109Dic346—24no—
110Dic346—21no—
111Dic346—21no—
112Dic346—21no—
113Dic346—42no—
114Dic346#4112yes—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173C2×D411yescharacteristic
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C4×D173—24no—
123C4×D173—22no—
124C4×D173—21no—
125C4×D173—21no—
126C4×D173—22no—
127C4×D173—42no—
128#13063436—21no—
129#13063436—21no—
130Q8×C173—21no—
131Q8×C173—21no—
132Q8×C173—21no—
133Q8×C173#4112yes—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2411yescharacteristic
141C8×C173—21no—
142C8×C173—21no—
143C22×C173C22×D411yescharacteristic
144C2×D173—21no—
145C2×D173—21no—
146C2×D173#13712yes—
147C4×C173—22no—
148C4×C173—21no—
149C4×C173—21no—
150C4×C173—41no—
151C4×C173—41no—
152C4×C173—41no—
153C4×C173C2×#4111yescharacteristic
154C4×C173C2×#4111yescharacteristic
155Dic173—22no—
156Dic173—21no—
157Dic173—21no—
158Dic173—41no—
159Dic173—41no—
160Dic173—41no—
161Dic173#13712yes—
162C2×C173#52511yescharacteristic
163C2×C173C2×#13711yescharacteristic
164C2×C173C2×#13711yescharacteristic
165D173—22no—
166C173#311311yescharacteristic, complemented, Sylow
167#3113—1731nomaximal, Sylow
168#533—1731no—
169C2×C4×Q8—1731no—
170#483—1731no—
171#483—1731no—
172#475—1731no—
173#475—1731no—
174#462—1731no—
175#462—1731no—
176#451—1731no—
177#451—1731no—
178#441—1731no—
179#441—1731no—
180#441—1731no—
181#441D17311yescharacteristic, complemented
182#432—1731no—
183C22×Q8—1731no—
184C2×Q16—1731no—
185C2×Q16—1731no—
186C2×Q16—1731no—
187C2×Q16C2×D17311yescharacteristic
188C4⋊Q8—1731no—
189C4⋊Q8—1731no—
190#125—3461no—
191C22⋊Q8—3461no—
192C22⋊Q8—1731no—
193C22⋊Q8—1731no—
194C4×Q8—3461no—
195C4×Q8—3461no—
196C4×Q8—3461no—
197C4×Q8—1731no—
198C4×Q8—1731no—
199C4×Q8—1731no—
200C4×Q8—1731no—
201C4×Q8C2×D17311yescharacteristic
202C4×Q8C2×D17311yescharacteristic
203#117—3461no—
204C2×C4⋊C4—3461no—
205C2×C4⋊C4—1731no—
206C2×C42—3461no—
207C2×C42—1731no—
208#106—1731no—
209#106—1731no—
210#106—1731no—
211#106C2×D17311yescharacteristic
212C4⋊C8—1731no—
213C4⋊C8—1731no—
214#103—1731no—
215#103—1731no—
216#103—1731no—
217#103—1731no—
218#103—1731no—
219#103—1731no—
220#103C2×D17311yescharacteristic
221#103C2×D17311yescharacteristic
222C22⋊C8—1731no—
223C22⋊C8—1731no—
224#97—1731no—
225#97C2×D17311yescharacteristic
226C2×Q8—3461no—
227C2×Q8—3461no—
228C2×Q8—3462no—
229C2×Q8—3462no—
230C2×Q8—1731no—
231C2×Q8—1731no—
232C2×Q8C22×D17311yescharacteristic
233C2×Q8C22×D17311yescharacteristic
234C22×C4—3461no—
235C22×C4—3461no—
236C22×C4—1731no—
237C22×C4—1731no—
238C22×C4—1731no—
239Q16—22no—
240Q16—3462no—
241Q16—3462no—
242Q16—3462no—
243C2×C8—1731no—
244C2×C8—1731no—
245C2×C8C22×D17311yescharacteristic
246C2×C8C22×D17311yescharacteristic
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—3461no—
250C4⋊C4—3461no—
251C4⋊C4—3461no—
252C4⋊C4—3461no—
253C4⋊C4—3461no—
254C4⋊C4—3461no—
255C4⋊C4—3461no—
256C4⋊C4—3461no—
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—1731no—
260C4⋊C4—1731no—
261C4⋊C4C22×D17311yescharacteristic
262C4⋊C4C22×D17311yescharacteristic
263C22⋊C4—3461no—
264C22⋊C4—3461no—
265C42—21no—
266C42—21no—
267C42—3461no—
268C42—3461no—
269C42—3461no—
270C42—3461no—
271C42—1731no—
272C42—1731no—
273C42—1731no—
274C42C22×D17311yescharacteristic
275C23—1731no—
276Q8—21no—
277Q8—21no—
278Q8—21no—
279Q8—3464no—
280Q8—3461no—
281Q8—3461no—
282Q8—3461no—
283Q8—1732no—
284Q8—6922no—
285Q8#1458382612yes—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—3464no—
291C2×C4—3462no—
292C2×C4—3462no—
293C2×C4—3461no—
294C2×C4—3461no—
295C2×C4—3461no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—1731no—
300C2×C4—1731no—
301C2×C4—1731no—
302C2×C4—6922no—
303C2×C4D4×D17311yescharacteristic
304C2×C4D4×D17311yescharacteristic
305C2×C4C23×D17311yescharacteristic
306C8—21no—
307C8—21no—
308C8—3461no—
309C8—3461no—
310C22—3461no—
311C22—3461no—
312C22—1732no—
313C22C2×D4×D17311yescharacteristic
314C4—22no—
315C4—21no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—3462no—
321C4—3461no—
322C4—3461no—
323C4—1732no—
324C4—6921no—
325C4—6921no—
326C4—6921no—
327C4#41×D17311yescharacteristic
328C4C2×#1458382611yescharacteristic
329C2—3462no—
330C2#2601200011yescharacteristic
331C2#137×D17311yescharacteristic
332C2#137×D17311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#441×C173C2
socleC22×C173C22×D4
prime radicalC22×D173C2×D4
hypercenter#441D173
hypocenterC173#3113
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
5. #26011775C2semidirect
6. #26011774C2semidirect
7. #26011700C2semidirect
8. #26011699C2semidirect
11. #26011599C2semidirect
12. #26011596C2semidirect
13. #26011596C2semidirect
15. #441×C173C2semidirect
166. C173#3113semidirect
181. #441D173semidirect

Maximal subgroups

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

subgroupindex
#260120452
C4×Q8×D1732
#260118232
#260118202
#260117752
#260117742
#260117002
#260116992
#260116482
#260116452
#260115992
#260115962
#260115462
#441×C1732
#3113173