Finite Group Databasebeta

← #36365659

Subgroups — #36365659

Order 22144.

382 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#26012181C211yesmaximal, characteristic, complemented
2#26011859C211yesmaximal, characteristic, complemented
3#26011829C211yesmaximal, characteristic, complemented
4#26011824C211yesmaximal, characteristic, complemented
5#26011791C211yesmaximal, characteristic, complemented
6#26011747C211yesmaximal, characteristic, complemented
7#26011730C211yesmaximal, characteristic, complemented
8#26011698C211yesmaximal, characteristic, complemented
9#26011653C211yesmaximal, characteristic, complemented
10#26011653C211yesmaximal, characteristic, complemented
11#26011591C211yesmaximal, characteristic, complemented
12#26011563C211yesmaximal, characteristic, complemented
13#26011492C211yesmaximal, characteristic, complemented
14#26011487C211yesmaximal, characteristic, complemented
15#547×C173C211yesmaximal, characteristic
16C2×#14583816C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583810C2211yescharacteristic
20C22×#13063436C2211yescharacteristic
21#19149672—21no—
22#19149671C2211yescharacteristic
23#19149669—21no—
24#19149660—21no—
25#19149658C2211yescharacteristic
26#19149658C2211yescharacteristic
27C4×Dic346C2211yescharacteristic
28#19149638C2211yescharacteristic
29#19149635—21no—
30C2×C4⋊Dic173C2211yescharacteristic
31#19149612C2211yescharacteristic
32#19149612C2211yescharacteristic
33#19149610C2211yescharacteristic
34C4⋊#13063436C2211yescharacteristic
35#19149603C2211yescharacteristic
36#19149603C2211yescharacteristic
37#19149603C2211yescharacteristic
38#19149603C2211yescharacteristic
39#19149603C2211yescharacteristic
40#19149603C2211yescharacteristic
41#19149602C2211yescharacteristic
42#19149600C2211yescharacteristic
43#19149600C2211yescharacteristic
44#19149600C2211yescharacteristic
45C22⋊#13063436C2211yescharacteristic
46C22⋊#13063436C2211yescharacteristic
47#19149590C2211yescharacteristic
48C2×#41×C173C2211yescharacteristic
49C2×#41×C173C2211yescharacteristic
50C4⋊Q8×C173C2211yescharacteristic
51#123×C173—21no—
52#123×C173—21no—
53C22⋊Q8×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173—21no—
56C4×D4×C173C2211yescharacteristic
57C4×D4×C173C2211yescharacteristic
58C2×C4⋊C4×C173—21no—
59C2×Dic346C2311yescharacteristic
60C22×Dic173—21no—
61#14583816—41no—
62#14583816—41no—
63#14583814—22no—
64#14583810—41no—
65C2×#13063436—41no—
66C2×#13063436C2311yescharacteristic
67C2×#13063436C2311yescharacteristic
68C2×#13063436C2311yescharacteristic
69C2×#13063436C2311yescharacteristic
70C4⋊Dic173—21no—
71C4⋊Dic173C2311yescharacteristic
72C4⋊Dic173C2311yescharacteristic
73C4⋊Dic173C2311yescharacteristic
74#14583805—21no—
75#14583805—21no—
76#14583805—21no—
77#14583805—21no—
78#14583805—21no—
79C22⋊Dic173—21no—
80C22⋊Dic173—21no—
81C22⋊Dic173—21no—
82C4×Dic173—21no—
83C4×Dic173—21no—
84#41×C173—22no—
85#41×C173—22no—
86#41×C173—41no—
87#41×C173—41no—
88C2×Q8×C173C2311yescharacteristic
89C2×Q8×C173C2311yescharacteristic
90C2×D4×C173—21no—
91C2×D4×C173—21no—
92C2×D4×C173D411yescharacteristic
93C2×D4×C173C2311yescharacteristic
94C22×C4×C173—21no—
95C22×C4×C173—21no—
96C22×C4×C173—21no—
97C22×C4×C173C2311yescharacteristic
98C22×C4×C173C2311yescharacteristic
99C4⋊C4×C173—21no—
100C4⋊C4×C173—21no—
101C4⋊C4×C173—21no—
102C4⋊C4×C173—21no—
103C4⋊C4×C173D411yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C22⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C22⋊C4×C173D411yescharacteristic
109C22⋊C4×C173D411yescharacteristic
110C42×C173C2311yescharacteristic
111Dic346—21no—
112Dic346—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—42no—
118#13063436—21no—
119#13063436—21no—
120#13063436—21no—
121#13063436—21no—
122C23×C173—21no—
123C23×C173C2×D411yescharacteristic
124C23×C173C2×D411yescharacteristic
125Q8×C173—21no—
126Q8×C173—21no—
127Q8×C173—21no—
128Q8×C173—21no—
129D4×C173—21no—
130D4×C173—41no—
131D4×C173—42no—
132D4×C173—42no—
133D4×C173#4112yes—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—42no—
142C2×C4×C173—42no—
143C2×C4×C173—42no—
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2411yescharacteristic
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—22no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173C22×D411yescharacteristic
159C4×C173—21no—
160C4×C173—21no—
161C4×C173—21no—
162C4×C173—21no—
163C4×C173—41no—
164C4×C173—41no—
165C4×C173—41no—
166C4×C173C2×#4111yescharacteristic
167C4×C173H(5,𝔽2)11yescharacteristic
168Dic173—41no—
169Dic173—41no—
170Dic173—41no—
171Dic173—41no—
172C2×C173—41no—
173C2×C173—41no—
174C2×C173—42no—
175C2×C173#54611yescharacteristic
176C2×C173#57511yescharacteristic
177C2×C173#57811yescharacteristic
178C173#331511yescharacteristic, complemented, Sylow
179#3315—1731nomaximal, Sylow
180#552—1731no—
181#547D17311yescharacteristic
182#491—1731no—
183#485—1731no—
184#484—1731no—
185#478—1731no—
186#470—1731no—
187#467—1731no—
188#462—1731no—
189#452—1731no—
190#452—1731no—
191#440—1731no—
192#435—1731no—
193#417—1731no—
194#416—1731no—
195C2×#41C2×D17311yescharacteristic
196C2×#41C2×D17311yescharacteristic
197C2×Q16—1731no—
198C2×Q16—1731no—
199C2×SD16—1731no—
200C2×M16—1731no—
201C22×C8—1731no—
202C4⋊Q8C2×D17311yescharacteristic
203#126—3461no—
204#125—3461no—
205#125—1731no—
206#123—21no—
207#123—21no—
208#123—3461no—
209C22⋊Q8—1731no—
210C22⋊Q8—1731no—
211C22⋊Q8C2×D17311yescharacteristic
212C22⋊Q8C2×D17311yescharacteristic
213#121—21no—
214C4×Q8—1731no—
215C4×D4C2×D17311yescharacteristic
216C4×D4C2×D17311yescharacteristic
217#117—3461no—
218#117—1731no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—1731no—
221#107—1731no—
222#107—1731no—
223#106—1731no—
224C4⋊C8—1731no—
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103—1731no—
230#103—1731no—
231#103—1731no—
232#102—1731no—
233#102—1731no—
234#102—1731no—
235C22⋊C8—1731no—
236C22⋊C8—1731no—
237#97—1731no—
238#41—22no—
239#41—22no—
240#41—41no—
241#41—41no—
242C2×Q8—1731no—
243C2×Q8C22×D17311yescharacteristic
244C2×Q8C22×D17311yescharacteristic
245C2×D4—21no—
246C2×D4—21no—
247C2×D4C22⋊D17311yescharacteristic
248C2×D4C22×D17311yescharacteristic
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—3461no—
253C22×C4C22×D17311yescharacteristic
254C22×C4C22×D17311yescharacteristic
255Q16—6921no—
256Q16—6921no—
257SD16—3462no—
258M16—6921no—
259C2×C8—1731no—
260C2×C8—1731no—
261C2×C8—1731no—
262C2×C8—1731no—
263C2×C8—6921no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
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—21no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4C22⋊D17311yescharacteristic
286C22⋊C4C22⋊D17311yescharacteristic
287C42—3461no—
288C42—3461no—
289C42C22×D17311yescharacteristic
290C23—21no—
291C23C2×C22⋊D17311yescharacteristic
292C23C2×C22⋊D17311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—3461no—
298Q8—3461no—
299D4—21no—
300D4—41no—
301D4—42no—
302D4—42no—
303D4#1458382512yes—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—6922no—
319C2×C4C2×C22⋊D17311yescharacteristic
320C2×C4C2×C22⋊D17311yescharacteristic
321C2×C4C2×C22⋊D17311yescharacteristic
322C2×C4C2×C22⋊D17311yescharacteristic
323C2×C4C23×D17311yescharacteristic
324C8—3461no—
325C8—3461no—
326C8—3461no—
327C8—3461no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—22no—
335C22—41no—
336C22—41no—
337C22C22×C22⋊D17311yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—6921no—
346C4—6921no—
347C4—6921no—
348C4—6921no—
349C4C2×#1458382511yescharacteristic
350C4#1914972411yescharacteristic
351C2—41no—
352C2—41no—
353C2—42no—
354C2#2601213511yescharacteristic
355C2#2601228911yescharacteristic
356C2#2601230611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting, prime radical#547×C173C2
socleC22×C173C22×D4
hypercenter#547D173
hypocenterC173#3315
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #26012181C2semidirect
2. #26011859C2semidirect
3. #26011829C2semidirect
4. #26011824C2semidirect
5. #26011791C2semidirect
6. #26011747C2semidirect
7. #26011730C2semidirect
8. #26011698C2semidirect
9. #26011653C2semidirect
10. #26011653C2semidirect
11. #26011591C2semidirect
12. #26011563C2semidirect
13. #26011492C2semidirect
14. #26011487C2semidirect
178. C173#3315semidirect

Maximal subgroups

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

subgroupindex
#260121812
#260118592
#260118292
#260118242
#260117912
#260117472
#260117302
#260116982
#260116532
#260115912
#260115632
#260114922
#260114872
#547×C1732
#3315173