Finite Group Databasebeta

← #36364251

Subgroups — #36364251

Order 22144.

374 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012115C211yesmaximal, characteristic, complemented
2#26012097C211yesmaximal, characteristic, complemented
3#26011880C211yesmaximal, characteristic, complemented
4#26011848C211yesmaximal, characteristic, complemented
5#26011839C211yesmaximal, characteristic, complemented
6#26011745C211yesmaximal, characteristic, complemented
7#26011738C211yesmaximal, characteristic, complemented
8#26011728C211yesmaximal, characteristic, complemented
9#26011721C211yesmaximal, characteristic, complemented
10#26011602C211yesmaximal, characteristic, complemented
11#26011590C211yesmaximal, characteristic, complemented
12#26011578C211yesmaximal, characteristic, complemented
13#26011573C211yesmaximal, characteristic, complemented
14#26011448C211yesmaximal, characteristic, complemented
15#501×C173C211yesmaximal, characteristic, complemented
16C2×#14583815C2211yescharacteristic
17C2×#14583815C2211yescharacteristic
18C2×D1384C2211yescharacteristic
19C2×D1384C2211yescharacteristic
20C2×#14583809C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic
22#19149676C2211yescharacteristic
23#19149674—21no—
24#19149668—21no—
25#19149667C2211yescharacteristic
26#19149664—21no—
27#19149664—21no—
28#19149664—21no—
29#19149659C2211yescharacteristic
30#19149655—21no—
31#19149654—21no—
32#19149654—21no—
33#19149654C2211yescharacteristic
34#19149654C2211yescharacteristic
35#19149654C2211yescharacteristic
36#19149644C2211yescharacteristic
37#19149641C2211yescharacteristic
38#19149641C2211yescharacteristic
39#19149641C2211yescharacteristic
40#19149636C2211yescharacteristic
41#19149602C2211yescharacteristic
42#19149602C2211yescharacteristic
43#19149599C2211yescharacteristic
44#19149599C2211yescharacteristic
45#19149599C2211yescharacteristic
46#19149599C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149589C2211yescharacteristic
50C4×#13063436C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C4⋊Q8×C173C2211yescharacteristic
53#125×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#107×C173C2211yescharacteristic
56#106×C173C2211yescharacteristic
57#106×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59C2×Dic346C2311yescharacteristic
60C2×D692—21no—
61C2×D692—21no—
62C2×D692—21no—
63C2×D692—21no—
64C2×D692—21no—
65C2×D692C2311yescharacteristic
66C2×D692C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C4×D173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173C2311yescharacteristic
72#14583815—22no—
73#14583815—22no—
74D1384—22no—
75D1384—22no—
76#14583809—22no—
77C2×#13063436C2311yescharacteristic
78C2×#13063436C2311yescharacteristic
79C8×D173—22no—
80C4⋊Dic173—21no—
81#14583805—21no—
82#14583805D411yescharacteristic
83#14583805D411yescharacteristic
84#14583803—21no—
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803D411yescharacteristic
94#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173—21no—
98C2×C8×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C4⋊C4×C173—21no—
101C4⋊C4×C173—21no—
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C4⋊C4×C173C2311yescharacteristic
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—21no—
110C22×D173—21no—
111C22×D173C2×D411yescharacteristic
112Dic346—21no—
113Dic346—21no—
114D692—21no—
115D692—21no—
116D692—21no—
117D692—21no—
118D692—21no—
119D692—21no—
120D692—81no—
121D692—81no—
122D692—81no—
123D692—81no—
124D692—81no—
125C2×Dic173—21no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—81no—
132C4×D173—81no—
133C4×D173—81no—
134#13063436—21no—
135#13063436—21no—
136Q8×C173—81no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2411yescharacteristic
144C8×C173#4112yes—
145C8×C173#4112yes—
146C22×C173C22×D411yescharacteristic
147C2×D173—21no—
148C2×D173—21no—
149C2×D173—21no—
150C2×D173—41no—
151C2×D173—41no—
152C2×D173—41no—
153C2×D173—41no—
154C2×D173—41no—
155C2×D173—41no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C4×C173—21no—
160C4×C173—21no—
161C4×C173—41no—
162C4×C173—41no—
163C4×C173—41no—
164C4×C173—41no—
165C4×C173C2×#4111yescharacteristic
166C4×C173C2×#4111yescharacteristic
167Dic173—21no—
168Dic173—21no—
169Dic173—21no—
170Dic173—41no—
171C2×C173#53211yescharacteristic
172C2×C173#57611yescharacteristic
173C2×C173#57711yescharacteristic
174D173—41no—
175D173—81no—
176D173—81no—
177D173—81no—
178C173#317911yescharacteristic, complemented, Sylow
179#3179—1731nomaximal, Sylow
180#542—1731no—
181#540—1731no—
182#501D17311yescharacteristic, complemented
183#496—1731no—
184#488—1731no—
185#486—1731no—
186#469—1731no—
187#468—1731no—
188#466—1731no—
189#465—1731no—
190#442—1731no—
191#440—1731no—
192C4×SD16—1731no—
193C4×D8—1731no—
194#405—1731no—
195C2×SD16—1731no—
196C2×SD16—1731no—
197C2×D8—1731no—
198C2×D8—1731no—
199C2×M16—1731no—
200C22×C8—1731no—
201C4⋊Q8C2×D17311yescharacteristic
202#127—1731no—
203#126—3461no—
204#125C2×D17311yescharacteristic
205#124—3461no—
206#124—1731no—
207#123—3461no—
208#123—3461no—
209#123—3461no—
210C22⋊Q8—3461no—
211C22⋊Q8—1731no—
212#121—3461no—
213#121—3461no—
214#121—1731no—
215#121—1731no—
216#121—1731no—
217C4×Q8—1731no—
218C4×D4—1731no—
219C4×D4—1731no—
220C4×D4—1731no—
221#117—1731no—
222#107C2×D17311yescharacteristic
223#107C2×D17311yescharacteristic
224#106C2×D17311yescharacteristic
225#106C2×D17311yescharacteristic
226#103—1731no—
227#103—1731no—
228#102—1731no—
229#102—1731no—
230#102—1731no—
231#102—1731no—
232#102—1731no—
233#102—1731no—
234#97—1731no—
235#97C2×D17311yescharacteristic
236C4×C8—1731no—
237C4×C8—1731no—
238C2×Q8—21no—
239C2×Q8—1731no—
240C2×D4—3461no—
241C2×D4—3461no—
242C2×D4—3461no—
243C2×D4—3461no—
244C2×D4—3461no—
245C2×D4—1731no—
246C2×D4—1731no—
247C2×D4—1731no—
248C22×C4—3461no—
249C22×C4—3461no—
250C22×C4—3461no—
251C22×C4—1731no—
252SD16—3462no—
253SD16—3462no—
254D8—3462no—
255D8—3462no—
256M16—3462no—
257C2×C8—3462no—
258C2×C8—1731no—
259C2×C8—1731no—
260C2×C8C22×D17311yescharacteristic
261C2×C8C22×D17311yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—1731no—
268C4⋊C4—1731no—
269C4⋊C4C22×D17311yescharacteristic
270C4⋊C4C22×D17311yescharacteristic
271C4⋊C4C22×D17311yescharacteristic
272C4⋊C4C22×D17311yescharacteristic
273C22⋊C4—3461no—
274C22⋊C4—3461no—
275C22⋊C4—3461no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—1731no—
283C22⋊C4—1731no—
284C42—3461no—
285C42—1731no—
286C42C22×D17311yescharacteristic
287C23—3461no—
288C23—3461no—
289C23—3461no—
290C23—1731no—
291Q8—81no—
292Q8—3461no—
293Q8—3461no—
294D4—3461no—
295D4—3461no—
296D4—3461no—
297D4—3461no—
298D4—3461no—
299D4—3461no—
300D4—13841no—
301D4—13841no—
302D4—13841no—
303D4—13841no—
304D4—13841no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—13841no—
313C2×C4—13841no—
314C2×C4—13841no—
315C2×C4—1731no—
316C2×C4—1731no—
317C2×C4—1731no—
318C2×C4D4×D17311yescharacteristic
319C2×C4D4×D17311yescharacteristic
320C2×C4C23×D17311yescharacteristic
321C8—3461no—
322C8—3461no—
323C8#1458382712yes—
324C8#1458382712yes—
325C22—3461no—
326C22—3461no—
327C22—3461no—
328C22—6921no—
329C22—6921no—
330C22—6921no—
331C22—6921no—
332C22—6921no—
333C22—6921no—
334C22—6921no—
335C22—6921no—
336C22—6921no—
337C22C2×D4×D17311yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—3461no—
345C4—3461no—
346C4—3461no—
347C4—6921no—
348C4C2×#1458382711yescharacteristic
349C4C2×#1458382711yescharacteristic
350C2—13841no—
351C2—13841no—
352C2—13841no—
353C2—6921no—
354C2#2601204211yescharacteristic
355C2#2601229511yescharacteristic
356C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#501×C173C2
socleC22×C173C22×D4
hypercenter#501D173
hypocenterC173#3179
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. #26012115C2semidirect
2. #26012097C2semidirect
3. #26011880C2semidirect
4. #26011848C2semidirect
5. #26011839C2semidirect
6. #26011745C2semidirect
7. #26011738C2semidirect
8. #26011728C2semidirect
9. #26011721C2semidirect
10. #26011602C2semidirect
11. #26011590C2semidirect
12. #26011578C2semidirect
13. #26011573C2semidirect
14. #26011448C2semidirect
15. #501×C173C2semidirect
178. C173#3179semidirect
182. #501D173semidirect

Maximal subgroups

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

subgroupindex
#260121152
#260120972
#260118802
#260118482
#260118392
#260117452
#260117382
#260117282
#260117212
#260116022
#260115902
#260115782
#260115732
#260114482
#501×C1732
#3179173