Finite Group Databasebeta

← #36365432

Subgroups — #36365432

Order 22144.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012199C211yesmaximal, characteristic
2Q8×C22⋊D173C211yesmaximal, characteristic
3#26011840C211yesmaximal, characteristic, complemented
4#26011827C211yesmaximal, characteristic, complemented
5#26011825C211yesmaximal, characteristic, complemented
6#26011784C211yesmaximal, characteristic, complemented
7#26011730C211yesmaximal, characteristic, complemented
8#26011729C211yesmaximal, characteristic, complemented
9#26011699C211yesmaximal, characteristic, complemented
10#26011645C211yesmaximal, characteristic, complemented
11Q16×Dic173C211yesmaximal, characteristic, complemented
12#26011554C211yesmaximal, characteristic
13C2×#19149603C211yesmaximal, characteristic, complemented
14#103×D173C211yesmaximal, characteristic, complemented
15#451×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C22×#13063436C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20#19149678C2211yescharacteristic
21#19149677—21no—
22#19149677C2211yescharacteristic
23#19149669—21no—
24#19149659—21no—
25#19149658—21no—
26#19149658C2211yescharacteristic
27#19149656—21no—
28#19149655—21no—
29#19149655C2211yescharacteristic
30#19149655C2211yescharacteristic
31Q8×Dic173C2211yescharacteristic
32Q8×Dic173C2211yescharacteristic
33C4×C22⋊D173—21no—
34C4×C22⋊D173C2211yescharacteristic
35#19149639—21no—
36C2×C4⋊Dic173C2211yescharacteristic
37C4⋊C4×D173C2211yescharacteristic
38#19149613C2211yescharacteristic
39#19149613C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149603C2211yescharacteristic
48#19149603C2211yescharacteristic
49#19149602C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C22×Q8×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic
54C2×Q16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346—21no—
61C2×Dic346C2311yescharacteristic
62Q8×D173—22no—
63Q8×D173—41no—
64C2×C22⋊D173D411yescharacteristic
65C22×Dic173—21no—
66C2×C4×D173—21no—
67C2×C4×D173—21no—
68C2×C4×D173C2311yescharacteristic
69#14583816—41no—
70C2×#13063436—41no—
71C2×#13063436C2311yescharacteristic
72C2×#13063436C2311yescharacteristic
73C8×D173—41no—
74C4⋊Dic173—21no—
75C4⋊Dic173—21no—
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805D411yescharacteristic
88C22⋊Dic173—21no—
89C22⋊Dic173D411yescharacteristic
90#14583803—21no—
91#14583803—21no—
92#14583803D411yescharacteristic
93C4×Dic173—21no—
94C4×Dic173—21no—
95C4×Dic173C2311yescharacteristic
96C2×Q8×C173—21no—
97C2×Q8×C173—22no—
98C2×Q8×C173—41no—
99C2×Q8×C173C2311yescharacteristic
100C2×Q8×C173C2311yescharacteristic
101C2×Q8×C173C2311yescharacteristic
102C22×C4×C173—21no—
103C22×C4×C173C2311yescharacteristic
104Q16×C173—22no—
105Q16×C173—41no—
106C2×C8×C173C2311yescharacteristic
107C2×C8×C173C2311yescharacteristic
108C4⋊C4×C173—21no—
109C4⋊C4×C173C2311yescharacteristic
110C22⋊C4×C173—21no—
111C22×D173C2×D411yescharacteristic
112Dic346—21no—
113Dic346—21no—
114Dic346—42no—
115Dic346—81no—
116C22⋊D173Q1614yescomplemented
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—81no—
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C4×D173—21no—
126C4×D173—21no—
127C4×D173—42no—
128C4×D173—81no—
129#13063436—21no—
130#13063436—21no—
131C23×C173C2×D411yescharacteristic
132Q8×C173—21no—
133Q8×C173—21no—
134Q8×C173—21no—
135Q8×C173—21no—
136Q8×C173—21no—
137Q8×C173—42no—
138Q8×C173#4112yes—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—42no—
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2411yescharacteristic
148C8×C173—21no—
149C8×C173—21no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C173C2×Q1612yes—
153C22×C173C22×D411yescharacteristic
154C2×D173—21no—
155C2×D173—21no—
156C2×D173C2×Q1612yes—
157C4×C173—21no—
158C4×C173—22no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173C2×#4111yescharacteristic
163C4×C173H(5,𝔽2)11yescharacteristic
164Dic173—21no—
165Dic173—21no—
166Dic173—41no—
167Dic173—41no—
168Dic173—41no—
169Dic173—41no—
170Dic173C2×Q1612yes—
171C2×C173—22no—
172C2×C173#54611yescharacteristic
173C2×C173C22×Q1611yescharacteristic
174C2×C173#57711yescharacteristic
175D173—22no—
176C173#329911yescharacteristic, complemented, Sylow
177#3299—1731nomaximal, Sylow
178#554—1731no—
179D4×Q8—1731no—
180#487—1731no—
181#484—1731no—
182#484—1731no—
183#477—1731no—
184#467—1731no—
185#467—1731no—
186#462—1731no—
187#451—1731no—
188#451D17311yescharacteristic, complemented
189C4×Q16—1731no—
190C8×D4—1731no—
191C2×#103—1731no—
192C2×#103—1731no—
193C22×Q8—1731no—
194C22×Q8C2×D17311yescharacteristic
195C2×Q16—1731no—
196C2×Q16C2×D17311yescharacteristic
197C2×Q16C2×D17311yescharacteristic
198C22×C8—1731no—
199C22×C8—1731no—
200C4⋊Q8—3461no—
201C4⋊Q8—1731no—
202C4⋊Q8—1731no—
203#125—3461no—
204C22⋊Q8—3461no—
205C22⋊Q8—3461no—
206C22⋊Q8—3461no—
207C22⋊Q8—3461no—
208C22⋊Q8—1731no—
209C22⋊Q8—1731no—
210C22⋊Q8—1731no—
211C22⋊Q8C2×D17311yescharacteristic
212C4×Q8—1731no—
213C4×Q8—1731no—
214C4×D4—3461no—
215C4×D4—3461no—
216C4×D4—1731no—
217C2×C4⋊C4—1731no—
218C2×C4⋊C4—1731no—
219#107—1731no—
220#107—1731no—
221#107—1731no—
222C4⋊C8—1731no—
223#103—1731no—
224#103—1731no—
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103—1731no—
230#103—1731no—
231#103C2×D17311yescharacteristic
232#103C2×D17311yescharacteristic
233C22⋊C8—1731no—
234C22⋊C8C2×D17311yescharacteristic
235C4×C8—1731no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—3461no—
240C2×Q8—3461no—
241C2×Q8—3462no—
242C2×Q8—1731no—
243C2×Q8—6921no—
244C2×Q8C22×D17311yescharacteristic
245C2×Q8C22×D17311yescharacteristic
246C2×Q8C22×D17311yescharacteristic
247C2×D4—1731no—
248C22×C4—21no—
249C22×C4—3461no—
250C22×C4—3461no—
251C22×C4—3461no—
252C22×C4—1731no—
253C22×C4C22×D17311yescharacteristic
254Q16—22no—
255Q16—41no—
256Q16—6921no—
257C2×C8—1731no—
258C2×C8—1731no—
259C2×C8—6921no—
260C2×C8—6921no—
261C2×C8C22×D17311yescharacteristic
262C2×C8C22×D17311yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—3461no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—3461no—
271C4⋊C4—3461no—
272C4⋊C4—3461no—
273C4⋊C4—3461no—
274C4⋊C4—1731no—
275C4⋊C4—1731no—
276C4⋊C4—1731no—
277C4⋊C4—1731no—
278C4⋊C4C22×D17311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—1731no—
284C22⋊C4—1731no—
285C42—3461no—
286C42—3461no—
287C42—1731no—
288C23—1731no—
289C23D4×D17311yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—3461no—
297Q8—3461no—
298Q8—13841no—
299Q8—6922no—
300Q8#1458382512yes—
301D4—1734no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—13841no—
316C2×C4—13841no—
317C2×C4—1731no—
318C2×C4—1731no—
319C2×C4—1731no—
320C2×C4—6922no—
321C2×C4D4×D17311yescharacteristic
322C2×C4C23×D17311yescharacteristic
323C8—21no—
324C8—21no—
325C8—3461no—
326C8—3461no—
327C22—21no—
328C22—21no—
329C22—3461no—
330C22—3461no—
331C22—1732no—
332C22Q16×D17312yes—
333C22C2×D4×D17311yescharacteristic
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—3461no—
340C4—3461no—
341C4—1732no—
342C4—6921no—
343C4—6921no—
344C4—6921no—
345C4—6921no—
346C4C2×#1458382511yescharacteristic
347C4#1914972411yescharacteristic
348C2—22no—
349C2—3462no—
350C2#2601212811yescharacteristic
351C2C2×Q16×D17311yescharacteristic
352C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#451×C173C2
socleC22×C173C22×D4
prime radicalC2×C22⋊D173D4
hypercenter#451D173
hypocenterC173#3299
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #26011840C2semidirect
4. #26011827C2semidirect
5. #26011825C2semidirect
6. #26011784C2semidirect
7. #26011730C2semidirect
8. #26011729C2semidirect
9. #26011699C2semidirect
10. #26011645C2semidirect
11. Q16×Dic173C2semidirect
13. C2×#19149603C2semidirect
14. #103×D173C2semidirect
15. #451×C173C2semidirect
116. C22⋊D173Q16semidirect
176. C173#3299semidirect
188. #451D173semidirect

Maximal subgroups

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

subgroupindex
#260121992
Q8×C22⋊D1732
#260118402
#260118272
#260118252
#260117842
#260117302
#260117292
#260116992
#260116452
Q16×Dic1732
#260115542
C2×#191496032
#103×D1732
#451×C1732
#3299173