Finite Group Databasebeta

← #36364243

Subgroups — #36364243

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#26012110C211yesmaximal, characteristic, complemented
2#26012095C211yesmaximal, characteristic, complemented
3#26011900C211yesmaximal, characteristic, complemented
4#26011845C211yesmaximal, characteristic, complemented
5#26011837C211yesmaximal, characteristic, complemented
6#26011741C211yesmaximal, characteristic, complemented
7#26011734C211yesmaximal, characteristic, complemented
8#26011724C211yesmaximal, characteristic, complemented
9#26011717C211yesmaximal, characteristic, complemented
10#26011603C211yesmaximal, characteristic, complemented
11#26011592C211yesmaximal, characteristic, complemented
12SD16×Dic173C211yesmaximal, characteristic, complemented
13D8×Dic173C211yesmaximal, characteristic, complemented
14#26011448C211yesmaximal, characteristic, complemented
15#496×C173C211yesmaximal, characteristic, complemented
16C2×#14583809C2211yescharacteristic
17C2×C8×D173C2211yescharacteristic
18#19149679C2211yescharacteristic
19#19149672—21no—
20#19149671C2211yescharacteristic
21#19149665—21no—
22#19149663—21no—
23#19149660—21no—
24#19149660—21no—
25#19149656—21no—
26#19149655C2211yescharacteristic
27#19149651—21no—
28#19149651—21no—
29#19149650C2211yescharacteristic
30#19149650C2211yescharacteristic
31#19149650C2211yescharacteristic
32Q8×Dic173C2211yescharacteristic
33D4×Dic173C2211yescharacteristic
34D4×Dic173C2211yescharacteristic
35D4×Dic173C2211yescharacteristic
36#19149636C2211yescharacteristic
37#19149613C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149603C2211yescharacteristic
42#19149603C2211yescharacteristic
43#19149600C2211yescharacteristic
44#19149600C2211yescharacteristic
45#19149600C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149589C2211yescharacteristic
50C4×#13063436C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54C2×D8×C173C2211yescharacteristic
55C2×D8×C173C2211yescharacteristic
56#127×C173C2211yescharacteristic
57#124×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×C22⋊D173—21no—
61C2×C22⋊D173—21no—
62C2×C22⋊D173—21no—
63C22×Dic173—21no—
64C22×Dic173—21no—
65C22×Dic173—21no—
66C2×C4×D173C2311yescharacteristic
67#14583809—22no—
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
70C8×D173—22no—
71C4⋊Dic173—21no—
72C4⋊Dic173—21no—
73C4⋊Dic173C2311yescharacteristic
74C4⋊Dic173C2311yescharacteristic
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77#14583805—21no—
78#14583805—21no—
79#14583805—21no—
80#14583805D411yescharacteristic
81#14583805D411yescharacteristic
82C22⋊Dic173—21no—
83C22⋊Dic173—21no—
84C22⋊Dic173—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88#14583803—21no—
89#14583803D411yescharacteristic
90#14583803D411yescharacteristic
91C4×Dic173—21no—
92C4×Dic173C2311yescharacteristic
93C2×Q8×C173C2311yescharacteristic
94C2×D4×C173—21no—
95C2×D4×C173—21no—
96C2×D4×C173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98C2×D4×C173C2311yescharacteristic
99SD16×C173—22no—
100SD16×C173—22no—
101D8×C173—22no—
102D8×C173—22no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173C2×D411yescharacteristic
109Dic346—81no—
110C22⋊D173—81no—
111C22⋊D173—81no—
112C22⋊D173—81no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—81no—
118C2×Dic173—81no—
119C2×Dic173—81no—
120C2×Dic173C2×D411yescharacteristic
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C4×D173—21no—
124C4×D173—21no—
125#13063436—21no—
126#13063436—21no—
127C23×C173—21no—
128C23×C173—21no—
129C23×C173—21no—
130Q8×C173—21no—
131Q8×C173—21no—
132D4×C173—21no—
133D4×C173—21no—
134D4×C173—21no—
135D4×C173—21no—
136D4×C173—21no—
137D4×C173—21no—
138D4×C173—81no—
139D4×C173—81no—
140C2×C4×C173—21no—
141C2×C4×C173C2×D411yescharacteristic
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2411yescharacteristic
144C8×C173#4112yes—
145C8×C173#4112yes—
146C22×C173—41no—
147C22×C173—41no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173—41no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173C22×D411yescharacteristic
156C2×D173—21no—
157C2×D173—21no—
158C2×D173—21no—
159C4×C173—21no—
160C4×C173—21no—
161C4×C173—41no—
162C4×C173C2×#4111yescharacteristic
163C4×C173C2×#4111yescharacteristic
164Dic173—21no—
165Dic173—21no—
166Dic173—21no—
167Dic173—41no—
168Dic173—41no—
169Dic173—41no—
170Dic173—41no—
171C2×C173—81no—
172C2×C173—81no—
173C2×C173—81no—
174C2×C173#53211yescharacteristic
175C2×C173#57611yescharacteristic
176C2×C173#57711yescharacteristic
177D173—41no—
178C173#317911yescharacteristic, complemented, Sylow
179#3179—1731nomaximal, Sylow
180#542—1731no—
181#540—1731no—
182#501—1731no—
183#496D17311yescharacteristic, complemented
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×SD16C2×D17311yescharacteristic
196C2×SD16C2×D17311yescharacteristic
197C2×D8C2×D17311yescharacteristic
198C2×D8C2×D17311yescharacteristic
199C2×M16—1731no—
200C22×C8—1731no—
201C4⋊Q8—1731no—
202#127C2×D17311yescharacteristic
203#126—3461no—
204#125—1731no—
205#124—3461no—
206#124C2×D17311yescharacteristic
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#107—1731no—
223#107—1731no—
224#106—1731no—
225#106—1731no—
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—3461no—
239C2×Q8C22×D17311yescharacteristic
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—3461no—
243C2×D4—3461no—
244C2×D4—3461no—
245C2×D4C22×D17311yescharacteristic
246C2×D4C22×D17311yescharacteristic
247C2×D4C22×D17311yescharacteristic
248C22×C4—3461no—
249C22×C4—3461no—
250C22×C4—3461no—
251C22×C4—1731no—
252SD16—22no—
253SD16—22no—
254D8—22no—
255D8—22no—
256M16—3462no—
257C2×C8—3462no—
258C2×C8—1731no—
259C2×C8—1731no—
260C2×C8C22×D17311yescharacteristic
261C2×C8C22×D17311yescharacteristic
262C4⋊C4—3461no—
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⋊C4—1731no—
272C4⋊C4—1731no—
273C22⋊C4—21no—
274C22⋊C4—21no—
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—21no—
288C23—21no—
289C23—21no—
290C23—1731no—
291Q8—21no—
292Q8—21no—
293Q8—13841no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—81no—
301D4—81no—
302D4—13841no—
303D4—13841no—
304D4—13841no—
305C2×C4—21no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—3461no—
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#1458382512yes—
324C8#1458382512yes—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—3461no—
335C22—3461no—
336C22—3461no—
337C22C2×D4×D17311yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—3461no—
342C4—3461no—
343C4—3461no—
344C4—6921no—
345C4—6921no—
346C4—6921no—
347C4—6921no—
348C4C2×#1458382511yescharacteristic
349C4C2×#1458382511yescharacteristic
350C2—81no—
351C2—81no—
352C2—81no—
353C2—6921no—
354C2#2601203911yescharacteristic
355C2#2601229511yescharacteristic
356C2#2601230111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#496×C173C2
socleC22×C173C22×D4
hypercenter#496D173
hypocenterC173#3179
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #26012110C2semidirect
2. #26012095C2semidirect
3. #26011900C2semidirect
4. #26011845C2semidirect
5. #26011837C2semidirect
6. #26011741C2semidirect
7. #26011734C2semidirect
8. #26011724C2semidirect
9. #26011717C2semidirect
10. #26011603C2semidirect
11. #26011592C2semidirect
12. SD16×Dic173C2semidirect
13. D8×Dic173C2semidirect
14. #26011448C2semidirect
15. #496×C173C2semidirect
178. C173#3179semidirect
183. #496D173semidirect

Maximal subgroups

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

subgroupindex
#260121102
#260120952
#260119002
#260118452
#260118372
#260117412
#260117342
#260117242
#260117172
#260116032
#260115922
SD16×Dic1732
D8×Dic1732
#260114482
#496×C1732
#3179173