Finite Group Databasebeta

← #36365434

Subgroups — #36365434

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
2#26012154C211yesmaximal, characteristic
3#26011840C211yesmaximal, characteristic, complemented
4#26011828C211yesmaximal, characteristic, complemented
5#26011824C211yesmaximal, characteristic, complemented
6#26011785C211yesmaximal, characteristic, complemented
7#26011731C211yesmaximal, characteristic, complemented
8#26011699C211yesmaximal, characteristic, complemented
9#26011646C211yesmaximal, characteristic, complemented
10#26011645C211yesmaximal, characteristic, complemented
11#26011583C211yesmaximal, characteristic, complemented
12C8×C22⋊D173C211yesmaximal, characteristic
13C2×#19149604C211yesmaximal, characteristic, complemented
14#103×D173C211yesmaximal, characteristic, complemented
15#467×C173C211yesmaximal, characteristic, complemented
16C22×Dic346C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21#19149678—21no—
22#19149678C2211yescharacteristic
23#19149677C2211yescharacteristic
24#19149669—21no—
25#19149659—21no—
26#19149659C2211yescharacteristic
27#19149657—21no—
28#19149656—21no—
29#19149656—21no—
30#19149655C2211yescharacteristic
31Q8×Dic173C2211yescharacteristic
32#19149644C2211yescharacteristic
33C4×C22⋊D173C2211yescharacteristic
34#19149639—21no—
35#19149639—21no—
36C2×C4⋊Dic173C2211yescharacteristic
37C4⋊C4×D173C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149608C2211yescharacteristic
41#19149604C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149602C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346—21no—
61C2×Dic346—22no—
62C2×Dic346—21no—
63C2×Dic346—41no—
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66Q8×D173—22no—
67Q8×D173—41no—
68C2×C22⋊D173D411yescharacteristic
69C22×Dic173—21no—
70C22×Dic173—21no—
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173C2311yescharacteristic
74#14583816—22no—
75#14583816—41no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C8×D173—41no—
79C4⋊Dic173—21no—
80C4⋊Dic173—21no—
81C4⋊Dic173—21no—
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805—21no—
90#14583805D411yescharacteristic
91C22⋊Dic173D411yescharacteristic
92#14583803—21no—
93#14583803—21no—
94#14583803D411yescharacteristic
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C2×Q8×C173C2311yescharacteristic
99C2×Q8×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101Q16×C173—41no—
102C2×C8×C173—41no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C4⋊C4×C173—21no—
106C4⋊C4×C173—21no—
107C4⋊C4×C173C2311yescharacteristic
108C4⋊C4×C173C2311yescharacteristic
109C22⋊C4×C173—21no—
110C22⋊C4×C173—21no—
111C22×D173C2×D411yescharacteristic
112Dic346—21no—
113Dic346—21no—
114Dic346—21no—
115Dic346—42no—
116Dic346—42no—
117Dic346—81no—
118Dic346#4112yes—
119C22⋊D173Q1614yescomplemented
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—42no—
125C2×Dic173—81no—
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—21no—
130C4×D173—21no—
131C4×D173—42no—
132C4×D173—81no—
133#13063436—21no—
134#13063436—21no—
135C23×C173C2×D411yescharacteristic
136Q8×C173—21no—
137Q8×C173—21no—
138Q8×C173—21no—
139Q8×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—21no—
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—41no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173C2×#4111yescharacteristic
163C4×C173H(5,𝔽2)11yescharacteristic
164Dic173—21no—
165Dic173—22no—
166Dic173—21no—
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#467D17311yescharacteristic, complemented
186#462—1731no—
187#451—1731no—
188#451—1731no—
189C4×Q16—1731no—
190C8×D4—1731no—
191C2×#103—1731no—
192C2×#103—1731no—
193C22×Q8—1731no—
194C22×Q8—1731no—
195C2×Q16—1731no—
196C2×Q16—1731no—
197C2×Q16C2×D17311yescharacteristic
198C22×C8—1731no—
199C22×C8C2×D17311yescharacteristic
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⋊Q8C2×D17311yescharacteristic
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#107C2×D17311yescharacteristic
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⋊C8—1731no—
235C4×C8—1731no—
236C2×Q8—3461no—
237C2×Q8—3461no—
238C2×Q8—3462no—
239C2×Q8—3461no—
240C2×Q8—3462no—
241C2×Q8—1731no—
242C2×Q8—1731no—
243C2×Q8—6921no—
244C2×Q8—6921no—
245C2×Q8C22×D17311yescharacteristic
246C2×Q8C22×D17311yescharacteristic
247C2×D4—1731no—
248C22×C4—3461no—
249C22×C4—3461no—
250C22×C4—3461no—
251C22×C4—3461no—
252C22×C4—1731no—
253C22×C4C22×D17311yescharacteristic
254Q16—41no—
255Q16—3462no—
256Q16—6921no—
257C2×C8—41no—
258C2×C8—1731no—
259C2×C8—1731no—
260C2×C8—6921no—
261C2×C8C22×D17311yescharacteristic
262C2×C8C22×D17311yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—21no—
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⋊C4C22×D17311yescharacteristic
278C4⋊C4C22×D17311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
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—3461no—
295Q8—3461no—
296Q8—3461no—
297Q8—13841no—
298Q8—1732no—
299Q8—6922no—
300Q8—6922no—
301D4—1734no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—13841no—
315C2×C4—13841no—
316C2×C4—1731no—
317C2×C4—1731no—
318C2×C4—1731no—
319C2×C4—6922no—
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—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—3461no—
340C4—3461no—
341C4—3462no—
342C4—1732no—
343C4—6921no—
344C4—6921no—
345C4—6921no—
346C4#41×D17311yescharacteristic
347C4#1914972411yescharacteristic
348C2—22no—
349C2—3462no—
350C2#2601213011yescharacteristic
351C2C2×Q16×D17311yescharacteristic
352C2#2601229811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#467×C173C2
socleC22×C173C22×D4
prime radicalC2×C22⋊D173D4
hypercenter#467D173
hypocenterC173#3299
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
3. #26011840C2semidirect
4. #26011828C2semidirect
5. #26011824C2semidirect
6. #26011785C2semidirect
7. #26011731C2semidirect
8. #26011699C2semidirect
9. #26011646C2semidirect
10. #26011645C2semidirect
11. #26011583C2semidirect
13. C2×#19149604C2semidirect
14. #103×D173C2semidirect
15. #467×C173C2semidirect
119. C22⋊D173Q16semidirect
176. C173#3299semidirect
185. #467D173semidirect

Maximal subgroups

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

subgroupindex
#260121992
#260121542
#260118402
#260118282
#260118242
#260117852
#260117312
#260116992
#260116462
#260116452
#260115832
C8×C22⋊D1732
C2×#191496042
#103×D1732
#467×C1732
#3299173