Finite Group Databasebeta

← #36364242

Subgroups — #36364242

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#26012113C211yesmaximal, characteristic, complemented
2#26012092C211yesmaximal, characteristic, complemented
3#26011898C211yesmaximal, characteristic, complemented
4#26011877C211yesmaximal, characteristic, complemented
5#26011847C211yesmaximal, characteristic, complemented
6#26011836C211yesmaximal, characteristic, complemented
7#26011745C211yesmaximal, characteristic, complemented
8#26011734C211yesmaximal, characteristic, complemented
9#26011726C211yesmaximal, characteristic, complemented
10#26011723C211yesmaximal, characteristic, complemented
11#26011601C211yesmaximal, characteristic, complemented
12#26011589C211yesmaximal, characteristic, complemented
13C4×D1384C211yesmaximal, characteristic, complemented
14#26011445C211yesmaximal, characteristic, complemented
15C4×SD16×C173C211yesmaximal, characteristic, complemented
16C2×#14583813C2211yescharacteristic
17C2×D1384C2211yescharacteristic
18C2×#14583811C2211yescharacteristic
19C2×#14583809C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21#19149678C2211yescharacteristic
22#19149675C2211yescharacteristic
23#19149673—21no—
24#19149670C2211yescharacteristic
25#19149668C2211yescharacteristic
26#19149667—21no—
27#19149664—21no—
28#19149662—21no—
29#19149661—21no—
30#19149657—21no—
31#19149655C2211yescharacteristic
32#19149654C2211yescharacteristic
33#19149654C2211yescharacteristic
34#19149653—21no—
35#19149652—21no—
36#19149650C2211yescharacteristic
37C4×D692C2211yescharacteristic
38C4×D692C2211yescharacteristic
39#19149636C2211yescharacteristic
40#19149613C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149610C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149590C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×D692—21no—
61C2×D692—21no—
62C2×D692C2311yescharacteristic
63C2×D692C2311yescharacteristic
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C4×D173—21no—
68C2×C4×D173—21no—
69C2×C4×D173C2311yescharacteristic
70#14583813—22no—
71D1384—22no—
72#14583811—22no—
73#14583809—22no—
74C2×#13063436C2311yescharacteristic
75C2×#13063436C2311yescharacteristic
76C8×D173—22no—
77C4⋊Dic173—21no—
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84#14583805D411yescharacteristic
85C22⋊Dic173—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×Dic173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173—21no—
99SD16×C173—22no—
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C42×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—21no—
110C22×D173C2×D411yescharacteristic
111Dic346—81no—
112D692—21no—
113D692—21no—
114D692—21no—
115D692—21no—
116D692—81no—
117D692—81no—
118C22⋊D173—81no—
119C22⋊D173—81no—
120C22⋊D173—81no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C4×D173—21no—
127C4×D173—21no—
128C4×D173—81no—
129C4×D173—81no—
130#13063436—21no—
131#13063436#4112yes—
132C23×C173—21no—
133Q8×C173—21no—
134Q8×C173—21no—
135D4×C173—21no—
136D4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—81no—
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2411yescharacteristic
144C8×C173—21no—
145C8×C173#4112yes—
146C22×C173—41no—
147C22×C173—41no—
148C22×C173—41no—
149C22×C173C22×D411yescharacteristic
150C2×D173—21no—
151C2×D173—21no—
152C2×D173—21no—
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×C173C2×#4111yescharacteristic
165C4×C173C2×#4111yescharacteristic
166Dic173—21no—
167Dic173—21no—
168Dic173—21no—
169Dic173—41no—
170Dic173—41no—
171C2×C173—81no—
172C2×C173#53211yescharacteristic
173C2×C173#57611yescharacteristic
174C2×C173#57711yescharacteristic
175D173—41no—
176D173—81no—
177D173—81no—
178C173#317911yescharacteristic, complemented, Sylow
179#3179—1731nomaximal, Sylow
180#542—1731no—
181#540—1731no—
182#501—1731no—
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×SD16D17311yescharacteristic, complemented
193C4×D8—1731no—
194#405—1731no—
195C2×SD16—1731no—
196C2×SD16C2×D17311yescharacteristic
197C2×D8—1731no—
198C2×D8—1731no—
199C2×M16—1731no—
200C22×C8—1731no—
201C4⋊Q8—1731no—
202#127—1731no—
203#126—3461no—
204#125—1731no—
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×Q8C2×D17311yescharacteristic
218C4×D4—1731no—
219C4×D4—1731no—
220C4×D4C2×D17311yescharacteristic
221#117—1731no—
222#107—1731no—
223#107—1731no—
224#106—1731no—
225#106C2×D17311yescharacteristic
226#103—1731no—
227#103C2×D17311yescharacteristic
228#102—1731no—
229#102—1731no—
230#102—1731no—
231#102—1731no—
232#102—1731no—
233#102C2×D17311yescharacteristic
234#97—1731no—
235#97—1731no—
236C4×C8—1731no—
237C4×C8C2×D17311yescharacteristic
238C2×Q8—3461no—
239C2×Q8C22×D17311yescharacteristic
240C2×D4—3461no—
241C2×D4—3461no—
242C2×D4—3461no—
243C2×D4—3461no—
244C2×D4—3461no—
245C2×D4—1731no—
246C2×D4—1731no—
247C2×D4C22×D17311yescharacteristic
248C22×C4—21no—
249C22×C4—3461no—
250C22×C4—3461no—
251C22×C4—1731no—
252SD16—22no—
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—3461no—
264C4⋊C4—3461no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—1731no—
268C4⋊C4—1731no—
269C4⋊C4—1731no—
270C4⋊C4—1731no—
271C4⋊C4C22×D17311yescharacteristic
272C4⋊C4C22×D17311yescharacteristic
273C22⋊C4—21no—
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—21no—
285C42—1731no—
286C42C22×D17311yescharacteristic
287C23—21no—
288C23—3461no—
289C23—3461no—
290C23—1731no—
291Q8—21no—
292Q8—21no—
293Q8—13841no—
294D4—21no—
295D4—21no—
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—81no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—3461no—
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—21no—
322C8—3461no—
323C8—1732no—
324C8#1458382612yes—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—3461no—
329C22—3461no—
330C22—3461no—
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—3461no—
344C4—3461no—
345C4—3461no—
346C4—6921no—
347C4—6921no—
348C4#41×D17311yescharacteristic
349C4C2×#1458382611yescharacteristic
350C2—81no—
351C2—13841no—
352C2—13841no—
353C2—6921no—
354C2#2601203811yescharacteristic
355C2#2601229311yescharacteristic
356C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×SD16×C173C2
socleC22×C173C22×D4
hypercenterC4×SD16D173
hypocenterC173#3179
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #26012113C2semidirect
2. #26012092C2semidirect
3. #26011898C2semidirect
4. #26011877C2semidirect
5. #26011847C2semidirect
6. #26011836C2semidirect
7. #26011745C2semidirect
8. #26011734C2semidirect
9. #26011726C2semidirect
10. #26011723C2semidirect
11. #26011601C2semidirect
12. #26011589C2semidirect
13. C4×D1384C2semidirect
14. #26011445C2semidirect
15. C4×SD16×C173C2semidirect
178. C173#3179semidirect
192. C4×SD16D173semidirect

Maximal subgroups

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

subgroupindex
#260121132
#260120922
#260118982
#260118772
#260118472
#260118362
#260117452
#260117342
#260117262
#260117232
#260116012
#260115892
C4×D13842
#260114452
C4×SD16×C1732
#3179173