Finite Group Databasebeta

← #36365556

Subgroups — #36365556

Order 22144.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012150C211yesmaximal, characteristic, complemented
2#26012145C211yesmaximal, characteristic, complemented
3#26011851C211yesmaximal, characteristic, complemented
4#26011827C211yesmaximal, characteristic, complemented
5#26011805C211yesmaximal, characteristic, complemented
6#26011787C211yesmaximal, characteristic, complemented
7#26011737C211yesmaximal, characteristic, complemented
8#26011727C211yesmaximal, characteristic, complemented
9#26011707C211yesmaximal, characteristic, complemented
10#26011638C211yesmaximal, characteristic, complemented
11#26011632C211yesmaximal, characteristic, complemented
12#26011593C211yesmaximal, characteristic, complemented
13#26011493C211yesmaximal, characteristic, complemented
14C2×#19149604C211yesmaximal, characteristic, complemented
15#435×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×#14583825—21no—
18C22×Dic346C2211yescharacteristic
19C2×#14583815C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21C2×D1384C2211yescharacteristic
22#19149679C2211yescharacteristic
23#19149667C2211yescharacteristic
24#19149665—21no—
25#19149663—21no—
26#19149663—21no—
27#19149659—21no—
28#19149658C2211yescharacteristic
29#19149657—21no—
30#19149653C2211yescharacteristic
31#19149653C2211yescharacteristic
32#19149653C2211yescharacteristic
33#19149651—21no—
34C4×Dic346C2211yescharacteristic
35C4×D692C2211yescharacteristic
36#19149639—21no—
37C2×C4⋊Dic173C2211yescharacteristic
38C2×C4⋊Dic173C2211yescharacteristic
39#19149613C2211yescharacteristic
40#19149613C2211yescharacteristic
41#19149613C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149604C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149604C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149601C2211yescharacteristic
50#19149601C2211yescharacteristic
51#19149601C2211yescharacteristic
52C2×M16×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55C4⋊C8×C173C2211yescharacteristic
56C22⋊C8×C173C2211yescharacteristic
57C22⋊C8×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59#14583826—22no—
60#14583826—41no—
61#14583825—44no—
62C2×Dic346—22no—
63C2×Dic346—21no—
64C2×Dic346—21no—
65C2×Dic346—41no—
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C22×Dic173—21no—
75C22×Dic173—21no—
76C22×Dic173—21no—
77C2×C4×D173—21no—
78C2×C4×D173—21no—
79#14583815—22no—
80#14583815—41no—
81D1384—41no—
82C4⋊Dic173—21no—
83C4⋊Dic173—21no—
84C4⋊Dic173—21no—
85C4⋊Dic173C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87C4⋊Dic173C2311yescharacteristic
88C4⋊Dic173C2311yescharacteristic
89#14583805—21no—
90#14583805—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98C4×Dic173—21no—
99C2×D4×C173D411yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C22×C4×C173C2311yescharacteristic
102M16×C173—41no—
103C2×C8×C173—41no—
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173D411yescharacteristic
109C22⋊C4×C173D411yescharacteristic
110C22⋊C4×C173D411yescharacteristic
111C42×C173C2311yescharacteristic
112C22×D173—21no—
113C22×D173—21no—
114Dic346—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—42no—
118Dic346—42no—
119Dic346#4112yescomplemented
120D692—21no—
121D692—21no—
122D692—21no—
123D692—21no—
124C22⋊D173—42no—
125C22⋊D173—42no—
126C22⋊D173—42no—
127C22⋊D173—42no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—42no—
135C2×Dic173—42no—
136C2×Dic173—42no—
137C4×D173—42no—
138C4×D173—42no—
139C23×C173C2×D411yescharacteristic
140C23×C173C2×D411yescharacteristic
141D4×C173—22no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2×D411yescharacteristic
150C2×C4×C173C2411yescharacteristic
151C8×C173—21no—
152C8×C173—21no—
153C8×C173—21no—
154C8×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173AGL(1,ℤ/8)12yes—
161C22×C173C22×D411yescharacteristic
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—41no—
168C4×C173—21no—
169C4×C173—21no—
170C4×C173—21no—
171C4×C173—21no—
172C4×C173C2×#4111yescharacteristic
173C4×C173H(5,𝔽2)11yescharacteristic
174Dic173—22no—
175Dic173—41no—
176Dic173—41no—
177Dic173—41no—
178Dic173—41no—
179Dic173—41no—
180C2×C173—22no—
181C2×C173—41no—
182C2×C173#54611yescharacteristic
183C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C173#57811yescharacteristic
185D173—81no—
186D173—81no—
187C173#330811yescharacteristic, complemented, Sylow
188#3308—1731nomaximal, Sylow
189#548—1731no—
190#547—1731no—
191#489—1731no—
192#484—1731no—
193#480—1731no—
194#477—1731no—
195#468—1731no—
196#466—1731no—
197#463—1731no—
198#449—1731no—
199#448—1731no—
200#440—1731no—
201#435D17311yescharacteristic, complemented
202#417—1731no—
203C2×#103—1731no—
204C2×#41—3461no—
205C2×#41—1731no—
206C22×Q8—1731no—
207C2×SD16—1731no—
208C2×SD16—1731no—
209C2×D8—1731no—
210C2×M16C2×D17311yescharacteristic
211C22×C8C2×D17311yescharacteristic
212C4⋊Q8—1731no—
213#124—3461no—
214#124—1731no—
215#123—3461no—
216#123—3461no—
217C22⋊Q8—3461no—
218C22⋊Q8—3461no—
219C22⋊Q8—1731no—
220#121—3461no—
221#121—1731no—
222#121—1731no—
223#121—1731no—
224C4×Q8—1731no—
225C4×D4—3461no—
226C4×D4—1731no—
227C4×D4C2×D17311yescharacteristic
228C2×C4⋊C4—1731no—
229C2×C4⋊C4—1731no—
230#107—1731no—
231#107—1731no—
232#107—1731no—
233C4⋊C8C2×D17311yescharacteristic
234#103—1731no—
235#103—1731no—
236#103—1731no—
237#103—1731no—
238#103—1731no—
239#103—1731no—
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244C22⋊C8C2×D17311yescharacteristic
245C22⋊C8C2×D17311yescharacteristic
246#97C2×D17311yescharacteristic
247#41—3462no—
248#41—6921no—
249#41—6924no—
250C2×Q8—3462no—
251C2×Q8—3461no—
252C2×Q8—3461no—
253C2×Q8—1731no—
254C2×Q8—1731no—
255C2×Q8—6921no—
256C2×D4—3461no—
257C2×D4—3461no—
258C2×D4—3461no—
259C2×D4—3461no—
260C2×D4—1731no—
261C2×D4—1731no—
262C2×D4D69211yescharacteristic
263C22×C4—3461no—
264C22×C4—3461no—
265C22×C4—3461no—
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4C22×D17311yescharacteristic
269C22×C4C22×D17311yescharacteristic
270SD16—3462no—
271SD16—6921no—
272D8—6921no—
273M16—41no—
274C2×C8—41no—
275C2×C8C22×D17311yescharacteristic
276C2×C8C22×D17311yescharacteristic
277C2×C8C22×D17311yescharacteristic
278C2×C8C22×D17311yescharacteristic
279C4⋊C4—3461no—
280C4⋊C4—3461no—
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—3461no—
284C4⋊C4—1731no—
285C4⋊C4—1731no—
286C4⋊C4—1731no—
287C4⋊C4—1731no—
288C4⋊C4D69211yescharacteristic
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4—3461no—
296C22⋊C4D69211yescharacteristic
297C22⋊C4D69211yescharacteristic
298C42—3461no—
299C42C22×D17311yescharacteristic
300C23—3461no—
301C23—3461no—
302C23C2×D69211yescharacteristic
303C23C2×D69211yescharacteristic
304Q8—3461no—
305Q8—3461no—
306Q8—3461no—
307Q8—1732no—
308Q8—6922no—
309Q8—6922no—
310D4—22no—
311D4—3461no—
312D4—3461no—
313D4—3461no—
314D4—3461no—
315D4—6922no—
316D4—6922no—
317D4—6922no—
318D4—6922no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—6922no—
330C2×C4—6922no—
331C2×C4—6922no—
332C2×C4—6922no—
333C2×C4—6922no—
334C2×C4C2×D69211yescharacteristic
335C2×C4C2×D69211yescharacteristic
336C2×C4C2×D69211yescharacteristic
337C2×C4C2×D69211yescharacteristic
338C2×C4C23×D17311yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—6921no—
349C22—6921no—
350C22—6921no—
351C22—6921no—
352C22—6921no—
353C22—6921no—
354C22#1914970712yes—
355C22C22×D69211yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—3462no—
361C4—6921no—
362C4—6921no—
363C4—6921no—
364C4—6921no—
365C4—6921no—
366C4#41×D17311yescharacteristic
367C4#1914972411yescharacteristic
368C2—22no—
369C2—41no—
370C2—13841no—
371C2—13841no—
372C2#2601213611yescharacteristic
373C2C2×#1914970711yescharacteristic
374C2#2601230811yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#435×C173C2
socleC22×C173C22×D4
hypercenter#435D173
hypocenterC173#3308
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #26012150C2semidirect
2. #26012145C2semidirect
3. #26011851C2semidirect
4. #26011827C2semidirect
5. #26011805C2semidirect
6. #26011787C2semidirect
7. #26011737C2semidirect
8. #26011727C2semidirect
9. #26011707C2semidirect
10. #26011638C2semidirect
11. #26011632C2semidirect
12. #26011593C2semidirect
13. #26011493C2semidirect
14. C2×#19149604C2semidirect
15. #435×C173C2semidirect
119. Dic346#41semidirect
187. C173#3308semidirect
201. #435D173semidirect

Maximal subgroups

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

subgroupindex
#260121502
#260121452
#260118512
#260118272
#260118052
#260117872
#260117372
#260117272
#260117072
#260116382
#260116322
#260115932
#260114932
C2×#191496042
#435×C1732
#3308173