Finite Group Databasebeta

← #36365001

Subgroups — #36365001

Order 22144.

370 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012097C211yesmaximal, characteristic, complemented
2#26012092C211yesmaximal, characteristic, complemented
3#26011794C211yesmaximal, characteristic, complemented
4#26011790C211yesmaximal, characteristic, complemented
5#26011788C211yesmaximal, characteristic, complemented
6#26011745C211yesmaximal, characteristic, complemented
7#26011743C211yesmaximal, characteristic, complemented
8#26011728C211yesmaximal, characteristic, complemented
9#26011726C211yesmaximal, characteristic, complemented
10#26011687C211yesmaximal, characteristic, complemented
11#26011686C211yesmaximal, characteristic, complemented
12#26011685C211yesmaximal, characteristic, complemented
13#26011685C211yesmaximal, characteristic, complemented
14#26011516C211yesmaximal, characteristic, complemented
15#478×C173C211yesmaximal, characteristic, complemented
16C2×D1384C2211yescharacteristic
17C2×D1384C2211yescharacteristic
18C2×#14583811C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×#14583809C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic
22#19149676C2211yescharacteristic
23#19149675C2211yescharacteristic
24#19149670C2211yescharacteristic
25#19149664—21no—
26#19149664—21no—
27#19149662—21no—
28#19149661—21no—
29#19149654—21no—
30#19149654—21no—
31#19149654C2211yescharacteristic
32#19149654C2211yescharacteristic
33#19149654C2211yescharacteristic
34#19149653—21no—
35#19149652—21no—
36#19149650C2211yescharacteristic
37C4×D692C2211yescharacteristic
38#19149641C2211yescharacteristic
39#19149641C2211yescharacteristic
40#19149636C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149610C2211yescharacteristic
43C4⋊#13063436C2211yescharacteristic
44#19149608C2211yescharacteristic
45#19149608C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149599C2211yescharacteristic
51#19149599C2211yescharacteristic
52#125×C173C2211yescharacteristic
53C4×D4×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C2×D692—21no—
60C2×D692—21no—
61C2×D692—21no—
62C2×D692—21no—
63C2×D692—21no—
64C2×D692C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173C2311yescharacteristic
74D1384—41no—
75D1384—41no—
76#14583811—41no—
77#14583811—41no—
78#14583809—41no—
79C2×#13063436C2311yescharacteristic
80C2×#13063436C2311yescharacteristic
81C8×D173—41no—
82C4⋊Dic173—21no—
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805D411yescharacteristic
86#14583805D411yescharacteristic
87C22⋊Dic173—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803D411yescharacteristic
95#14583803D411yescharacteristic
96C4×Dic173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173—21no—
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173—21no—
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—21no—
110C22×D173—21no—
111C22×D173C2×D411yescharacteristic
112D692—21no—
113D692—21no—
114D692—21no—
115D692—21no—
116D692—21no—
117D692—21no—
118D692—81no—
119D692—81no—
120D692—81no—
121D692—81no—
122D692—81no—
123C22⋊D173—81no—
124C22⋊D173—81no—
125C22⋊D173—81no—
126C2×Dic173—21no—
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C4×D173—21no—
131C4×D173—21no—
132C4×D173—81no—
133C4×D173—81no—
134C4×D173—81no—
135#13063436—21no—
136#13063436—21no—
137C23×C173—21no—
138D4×C173—21no—
139D4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—81no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2411yescharacteristic
147C8×C173—21no—
148C8×C173—21no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173C22×D411yescharacteristic
153C2×D173—21no—
154C2×D173—21no—
155C2×D173—21no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C2×D173—41no—
160C2×D173—41no—
161C2×D173—41no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C4×C173—21no—
166C4×C173—21no—
167C4×C173—41no—
168C4×C173—41no—
169C4×C173—41no—
170C4×C173H(5,𝔽2)11yescharacteristic
171C4×C1732−1+411yescharacteristic
172Dic173—21no—
173Dic173—21no—
174Dic173—21no—
175Dic173—41no—
176C2×C173—81no—
177C2×C173#53711yescharacteristic
178C2×C173#57611yescharacteristic
179C2×C173#57611yescharacteristic
180D173—41no—
181D173—81no—
182D173—81no—
183D173—81no—
184C173#325911yescharacteristic, complemented, Sylow
185#3259—1731nomaximal, Sylow
186#540—1731no—
187#540—1731no—
188#478—1731no—
189#478—1731no—
190#478—1731no—
191#478D17311yescharacteristic, complemented
192#469—1731no—
193#469—1731no—
194#466—1731no—
195#466—1731no—
196#459—1731no—
197#459—1731no—
198#459—1731no—
199#459—1731no—
200#424—1731no—
201C2×D8—1731no—
202C2×D8—1731no—
203C2×D8—1731no—
204C2×D8—1731no—
205C2×M16—1731no—
206C22×C8—1731no—
207#127—1731no—
208#127—1731no—
209#125—1731no—
210#125C2×D17311yescharacteristic
211#123—3461no—
212#123—3461no—
213#123—3461no—
214#123—3461no—
215#121—3461no—
216#121—3461no—
217#121—3461no—
218#121—3461no—
219#121—1731no—
220#121—1731no—
221#121—1731no—
222#121—1731no—
223C4×D4—1731no—
224C4×D4—1731no—
225C4×D4—1731no—
226C4×D4C2×D17311yescharacteristic
227#117—1731no—
228#107—1731no—
229#107C2×D17311yescharacteristic
230#106—1731no—
231#106C2×D17311yescharacteristic
232C4⋊C8—1731no—
233C4⋊C8—1731no—
234C4⋊C8—1731no—
235C4⋊C8C2×D17311yescharacteristic
236#102—1731no—
237#102—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241#102—1731no—
242#102C2×D17311yescharacteristic
243#102C2×D17311yescharacteristic
244C2×D4—3461no—
245C2×D4—3461no—
246C2×D4—3461no—
247C2×D4—3461no—
248C2×D4—3461no—
249C2×D4—3461no—
250C2×D4—3461no—
251C2×D4—3461no—
252C2×D4—1731no—
253C2×D4—1731no—
254C2×D4—1731no—
255C2×D4C22×D17311yescharacteristic
256C22×C4—21no—
257C22×C4—3461no—
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4—1731no—
261D8—6921no—
262D8—6921no—
263D8—6921no—
264D8—6921no—
265M16—6921no—
266C2×C8—1731no—
267C2×C8—1731no—
268C2×C8—6921no—
269C2×C8C22×D17311yescharacteristic
270C2×C8C22×D17311yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—1731no—
276C4⋊C4—1731no—
277C4⋊C4—1731no—
278C4⋊C4C22×D17311yescharacteristic
279C4⋊C4C22×D17311yescharacteristic
280C4⋊C4C22×D17311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—1731no—
290C22⋊C4—1731no—
291C42—1731no—
292C42C22×D17311yescharacteristic
293C23—21no—
294C23—3461no—
295C23—3461no—
296C23—3461no—
297C23—1731no—
298D4—21no—
299D4—21no—
300D4—3461no—
301D4—3461no—
302D4—3461no—
303D4—3461no—
304D4—3461no—
305D4—3461no—
306D4—13841no—
307D4—13841no—
308D4—13841no—
309D4—13841no—
310D4—13841no—
311D4—13841no—
312D4—13841no—
313D4—13841no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—81no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—13841no—
322C2×C4—13841no—
323C2×C4—13841no—
324C2×C4—1731no—
325C2×C4—1731no—
326C2×C4—1731no—
327C2×C4D4×D17311yescharacteristic
328C2×C4D4×D17311yescharacteristic
329C2×C4C23×D17311yescharacteristic
330C8—21no—
331C8—21no—
332C8—3461no—
333C8—3461no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—3461no—
338C22—3461no—
339C22—3461no—
340C22—6921no—
341C22—6921no—
342C22—6921no—
343C22—6921no—
344C22—6921no—
345C22—6921no—
346C22—6921no—
347C22—6921no—
348C22—6921no—
349C22C2×D4×D17311yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—3461no—
356C4—3461no—
357C4—3461no—
358C4—6921no—
359C4#1914972511yescharacteristic
360C4#1914972711yescharacteristic
361C2—81no—
362C2—13841no—
363C2—13841no—
364C2—13841no—
365C2—6921no—
366C2#2601207111yescharacteristic
367C2#2601229311yescharacteristic
368C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#478×C173C2
socleC22×C173C22×D4
hypercenter#478D173
hypocenterC173#3259
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #26012097C2semidirect
2. #26012092C2semidirect
3. #26011794C2semidirect
4. #26011790C2semidirect
5. #26011788C2semidirect
6. #26011745C2semidirect
7. #26011743C2semidirect
8. #26011728C2semidirect
9. #26011726C2semidirect
10. #26011687C2semidirect
11. #26011686C2semidirect
12. #26011685C2semidirect
13. #26011685C2semidirect
14. #26011516C2semidirect
15. #478×C173C2semidirect
184. C173#3259semidirect
191. #478D173semidirect

Maximal subgroups

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

subgroupindex
#260120972
#260120922
#260117942
#260117902
#260117882
#260117452
#260117432
#260117282
#260117262
#260116872
#260116862
#260116852
#260115162
#478×C1732
#3259173