Finite Group Databasebeta

← #36364464

Subgroups — #36364464

Order 22144.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012071C211yesmaximal, characteristic, complemented
2#26012069C211yesmaximal, characteristic, complemented
3#26011825C211yesmaximal, characteristic, complemented
4#26011824C211yesmaximal, characteristic, complemented
5#26011805C211yesmaximal, characteristic, complemented
6#26011803C211yesmaximal, characteristic, complemented
7#26011740C211yesmaximal, characteristic, complemented
8#26011734C211yesmaximal, characteristic, complemented
9#26011729C211yesmaximal, characteristic, complemented
10#26011726C211yesmaximal, characteristic, complemented
11#26011652C211yesmaximal, characteristic, complemented
12#26011636C211yesmaximal, characteristic, complemented
13#26011635C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic, complemented
15#452×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19C2×#14583811C2211yescharacteristic
20C2×#14583809C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic
22#19149658—21no—
23#19149657—21no—
24#19149655C2211yescharacteristic
25#19149655C2211yescharacteristic
26#19149655C2211yescharacteristic
27#19149654—21no—
28#19149654C2211yescharacteristic
29#19149653—21no—
30#19149653C2211yescharacteristic
31#19149652—21no—
32#19149652—21no—
33#19149651—21no—
34#19149651—21no—
35#19149650C2211yescharacteristic
36#19149650C2211yescharacteristic
37C2×C4⋊Dic173C2211yescharacteristic
38C4⋊C4×D173C2211yescharacteristic
39#19149613C2211yescharacteristic
40#19149613C2211yescharacteristic
41#19149612C2211yescharacteristic
42#19149612C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583827—41no—
60#14583827—41no—
61#14583826—41no—
62#14583826—41no—
63C2×Dic346D411yescharacteristic
64C2×D692—21no—
65C2×D692—21no—
66C2×D692D411yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173D411yescharacteristic
72C2×C22⋊D173D411yescharacteristic
73C22×Dic173—21no—
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76C2×C4×D173C2311yescharacteristic
77C2×C4×D173C2311yescharacteristic
78#14583813—41no—
79#14583811—41no—
80#14583809—41no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C8×D173—41no—
84C4⋊Dic173—21no—
85C4⋊Dic173C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87C4⋊Dic173C2311yescharacteristic
88#14583805—21no—
89#14583805—21no—
90#14583805—21no—
91#14583805—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#41×C173—41no—
100#41×C173—41no—
101C2×Q8×C173C2311yescharacteristic
102C2×Q8×C173C2311yescharacteristic
103C2×D4×C173—21no—
104C2×D4×C173C2311yescharacteristic
105C22×C4×C173—21no—
106C22×C4×C173C2311yescharacteristic
107Q16×C173—41no—
108SD16×C173—41no—
109C2×C8×C173C2311yescharacteristic
110C2×C8×C173C2311yescharacteristic
111C4⋊C4×C173—21no—
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173—21no—
115C22×D173C2×D411yescharacteristic
116C22×D173C2×D411yescharacteristic
117Dic346—41no—
118D692—21no—
119D692—21no—
120D692—41no—
121D692—42no—
122D692—81no—
123C22⋊D173—41no—
124C22⋊D173—41no—
125C22⋊D173—81no—
126C22⋊D173—81no—
127C22⋊D173—81no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—81no—
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C4×D173—21no—
135C4×D173—21no—
136C4×D173—21no—
137C4×D173—21no—
138C4×D173—42no—
139C4×D173—81no—
140#13063436—21no—
141#13063436—21no—
142C23×C173—21no—
143C23×C173C2×D411yescharacteristic
144Q8×C173—21no—
145Q8×C173—21no—
146Q8×C173—21no—
147Q8×C173—21no—
148D4×C173—21no—
149D4×C173—21no—
150D4×C173—42no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—42no—
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2411yescharacteristic
159C8×C173—21no—
160C8×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—21no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173C22×D411yescharacteristic
168C2×D173—21no—
169C2×D173—21no—
170C2×D173—21no—
171C2×D173—21no—
172C2×D173—21no—
173C2×D173—21no—
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C4×C173—21no—
178C4×C173—41no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173H(5,𝔽2)11yescharacteristic
182C4×C173H(5,𝔽2)11yescharacteristic
183Dic173—21no—
184Dic173—21no—
185Dic173—41no—
186Dic173—41no—
187Dic173—41no—
188C2×C173—41no—
189C2×C173—81no—
190C2×C173#53411yescharacteristic
191C2×C173#57611yescharacteristic
192C2×C173#57811yescharacteristic
193D173—41no—
194D173—41no—
195D173—81no—
196C173#320311yescharacteristic, complemented, Sylow
197#3203—1731nomaximal, Sylow
198#537—1731no—
199#537—1731no—
200#484—1731no—
201#484—1731no—
202#480—1731no—
203#480—1731no—
204#468—1731no—
205#468—1731no—
206#467—1731no—
207#466—1731no—
208#452—1731no—
209#452D17311yescharacteristic, complemented
210#449—1731no—
211#449—1731no—
212#408—1731no—
213C2×#41—1731no—
214C2×#41—1731no—
215C2×#41C2×D17311yescharacteristic
216C2×Q16C2×D17311yescharacteristic
217C2×SD16—1731no—
218C2×SD16C2×D17311yescharacteristic
219C2×D8—1731no—
220C2×M16—1731no—
221C22×C8—1731no—
222C22⋊Q8—3461no—
223C22⋊Q8—3461no—
224C22⋊Q8—1731no—
225C22⋊Q8—1731no—
226C22⋊Q8—1731no—
227C22⋊Q8C2×D17311yescharacteristic
228#121—3461no—
229#121—3461no—
230#121—3461no—
231#121—3461no—
232#121—3461no—
233#121—3461no—
234#121—1731no—
235#121—1731no—
236#121—1731no—
237#121—1731no—
238C2×C4⋊C4—1731no—
239C2×C4⋊C4—1731no—
240#107—1731no—
241#107—1731no—
242#107—1731no—
243#107—1731no—
244#103—1731no—
245#103—1731no—
246#103—1731no—
247#103C2×D17311yescharacteristic
248#102—1731no—
249#102—1731no—
250#102—1731no—
251#102C2×D17311yescharacteristic
252C22⋊C8—1731no—
253C22⋊C8—1731no—
254C22⋊C8—1731no—
255C22⋊C8C2×D17311yescharacteristic
256#41—41no—
257#41—41no—
258#41—6921no—
259#41—6921no—
260#41—6921no—
261#41—6921no—
262C2×Q8—1731no—
263C2×Q8C22×D17311yescharacteristic
264C2×Q8C22×D17311yescharacteristic
265C2×D4—21no—
266C2×D4—3461no—
267C2×D4—3461no—
268C2×D4—3461no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—1731no—
272C2×D4—1731no—
273C2×D4—1731no—
274C2×D4—1731no—
275C2×D4C22×D17311yescharacteristic
276C22×C4—21no—
277C22×C4—3461no—
278C22×C4—3461no—
279C22×C4—3461no—
280C22×C4—1731no—
281C22×C4—1731no—
282C22×C4C22×D17311yescharacteristic
283Q16—41no—
284SD16—41no—
285SD16—6921no—
286D8—6921no—
287M16—6921no—
288C2×C8—1731no—
289C2×C8—1731no—
290C2×C8—6921no—
291C2×C8C22×D17311yescharacteristic
292C2×C8C22×D17311yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—3461no—
295C4⋊C4—3461no—
296C4⋊C4—3461no—
297C4⋊C4—3461no—
298C4⋊C4—3461no—
299C4⋊C4—1731no—
300C4⋊C4—1731no—
301C4⋊C4—1731no—
302C4⋊C4C22×D17311yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C22⋊C4—3461no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C22⋊C4—3461no—
311C23—21no—
312C23—3461no—
313C23—1731no—
314C23—1731no—
315C23D4×D17311yescharacteristic
316Q8—21no—
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—6921no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—3461no—
325D4—3461no—
326D4—13841no—
327D4—13841no—
328D4—13841no—
329D4—13841no—
330D4—6921no—
331D4—6922no—
332D4—6921no—
333D4—6921no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—3461no—
347C2×C4—13841no—
348C2×C4—13841no—
349C2×C4—1731no—
350C2×C4—1731no—
351C2×C4—6922no—
352C2×C4D4×D17311yescharacteristic
353C2×C4C23×D17311yescharacteristic
354C8—21no—
355C8—21no—
356C8—3461no—
357C8—3461no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—3461no—
365C22—3461no—
366C22—3461no—
367C22—3461no—
368C22—3461no—
369C22—3461no—
370C22—6921no—
371C22—6921no—
372C22—6921no—
373C22C2×D4×D17311yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—3461no—
379C4—3461no—
380C4—6921no—
381C4—6921no—
382C4—6921no—
383C4#1914972411yescharacteristic
384C4#1914972411yescharacteristic
385C2—41no—
386C2—81no—
387C2—13841no—
388C2—6921no—
389C2—6921no—
390C2#2601205111yescharacteristic
391C2#2601229311yescharacteristic
392C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#452×C173C2
socleC22×C173C22×D4
hypercenter#452D173
hypocenterC173#3203
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #26012071C2semidirect
2. #26012069C2semidirect
3. #26011825C2semidirect
4. #26011824C2semidirect
5. #26011805C2semidirect
6. #26011803C2semidirect
7. #26011740C2semidirect
8. #26011734C2semidirect
9. #26011729C2semidirect
10. #26011726C2semidirect
11. #26011652C2semidirect
12. #26011636C2semidirect
13. #26011635C2semidirect
14. #26011456C2semidirect
15. #452×C173C2semidirect
196. C173#3203semidirect
209. #452D173semidirect

Maximal subgroups

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

subgroupindex
#260120712
#260120692
#260118252
#260118242
#260118052
#260118032
#260117402
#260117342
#260117292
#260117262
#260116522
#260116362
#260116352
#260114562
#452×C1732
#3203173