Finite Group Databasebeta

← #36364676

Subgroups — #36364676

Order 22144.

376 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012114C211yesmaximal, characteristic, complemented
2#26012062C211yesmaximal, characteristic, complemented
3#26011884C211yesmaximal, characteristic, complemented
4#26011872C211yesmaximal, characteristic, complemented
5#26011849C211yesmaximal, characteristic, complemented
6#26011840C211yesmaximal, characteristic, complemented
7#26011711C211yesmaximal, characteristic, complemented
8#26011705C211yesmaximal, characteristic, complemented
9#26011699C211yesmaximal, characteristic, complemented
10#26011695C211yesmaximal, characteristic, complemented
11#26011646C211yesmaximal, characteristic, complemented
12#26011636C211yesmaximal, characteristic
13#26011632C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#452×C173C211yesmaximal, characteristic, complemented
16C22×Dic346C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583815C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21C2×#14583813C2211yescharacteristic
22C2×#14583811C2211yescharacteristic
23#19149678C2211yescharacteristic
24#19149677C2211yescharacteristic
25#19149677C2211yescharacteristic
26#19149674—21no—
27#19149668C2211yescharacteristic
28#19149666C2211yescharacteristic
29#19149666C2211yescharacteristic
30#19149665—21no—
31#19149661—21no—
32#19149660—21no—
33#19149660—21no—
34#19149659—21no—
35#19149658—21no—
36#19149658—21no—
37#19149653C2211yescharacteristic
38Q8×Dic173C2211yescharacteristic
39#19149641C2211yescharacteristic
40#19149638C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149602C2211yescharacteristic
48#19149600C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346—21no—
61C2×Dic346—41no—
62C2×Dic346—41no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C22×Dic173—21no—
68C2×C4×D173—21no—
69#14583816—41no—
70#14583816—41no—
71#14583815—41no—
72#14583815—41no—
73#14583813—41no—
74#14583811—41no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C4⋊Dic173—21no—
78C4⋊Dic173D411yescharacteristic
79C4⋊Dic173D411yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86C22⋊Dic173—21no—
87C22⋊Dic173—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173D411yescharacteristic
91C22⋊Dic173D411yescharacteristic
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C4×Dic173C2311yescharacteristic
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—
115Dic346—21no—
116Dic346—21no—
117Dic346—21no—
118Dic346—21no—
119Dic346—42no—
120Dic346—81no—
121D692—21no—
122D692—21no—
123C22⋊D173—81no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—42no—
128C2×Dic173C2×D411yescharacteristic
129C2×Dic173C2×D411yescharacteristic
130C2×Dic173C2×D411yescharacteristic
131C2×Dic173C2×D411yescharacteristic
132C4×D173—81no—
133#13063436—21no—
134#13063436—21no—
135C23×C173—21no—
136C23×C173C2×D411yescharacteristic
137Q8×C173—21no—
138Q8×C173—21no—
139Q8×C173—21no—
140Q8×C173—21no—
141D4×C173—21no—
142D4×C173—21no—
143D4×C173—42no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—42no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2411yescharacteristic
152C8×C173—21no—
153C8×C173—21no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173C22×D411yescharacteristic
161C2×D173—41no—
162C2×D173—41no—
163C2×D173—41no—
164C4×C173—21no—
165C4×C173—41no—
166C4×C173—41no—
167C4×C173—41no—
168C4×C173H(5,𝔽2)11yescharacteristic
169C4×C173H(5,𝔽2)11yescharacteristic
170Dic173—21no—
171Dic173—21no—
172Dic173—21no—
173Dic173—21no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177C2×C173—41no—
178C2×C173—81no—
179C2×C173#53511yescharacteristic
180C2×C173#57711yescharacteristic
181C2×C173#57811yescharacteristic
182D173—81no—
183C173#322511yescharacteristic, complemented, Sylow
184#3225—1731nomaximal, Sylow
185#542—1731no—
186#536—1731no—
187#497—1731no—
188#495—1731no—
189#489—1731no—
190#487—1731no—
191#464—1731no—
192#463—1731no—
193#462—1731no—
194#461—1731no—
195#452D17311yescharacteristic, complemented
196#451—1731no—
197#449—1731no—
198#448—1731no—
199#433—1731no—
200C2×#41C2×D17311yescharacteristic
201C22×Q8—1731no—
202C2×Q16—1731no—
203C2×Q16—1731no—
204C2×Q16C2×D17311yescharacteristic
205C2×SD16—1731no—
206C2×SD16—1731no—
207C2×SD16—1731no—
208C2×SD16C2×D17311yescharacteristic
209C2×D8—1731no—
210C4⋊Q8—1731no—
211C4⋊Q8—1731no—
212C4⋊Q8—1731no—
213#126—3461no—
214#124—3461no—
215#124—1731no—
216#124—1731no—
217#124—1731no—
218#123—3461no—
219#123—3461no—
220#123—3461no—
221C22⋊Q8—3461no—
222C22⋊Q8—3461no—
223C22⋊Q8—3461no—
224C22⋊Q8C2×D17311yescharacteristic
225#121—1731no—
226C4×Q8—1731no—
227C4×D4—1731no—
228#117—1731no—
229C4⋊C8—1731no—
230C4⋊C8—1731no—
231#103—1731no—
232#103—1731no—
233#103—1731no—
234#103—1731no—
235#103—1731no—
236#103C2×D17311yescharacteristic
237#102—1731no—
238#102C2×D17311yescharacteristic
239C22⋊C8—1731no—
240C22⋊C8C2×D17311yescharacteristic
241#97—1731no—
242C4×C8—1731no—
243#41—41no—
244#41—41no—
245C2×Q8—3461no—
246C2×Q8—3461no—
247C2×Q8—1731no—
248C2×Q8—1731no—
249C2×Q8—6921no—
250C2×Q8—6921no—
251C2×Q8C22×D17311yescharacteristic
252C2×Q8C22×D17311yescharacteristic
253C2×D4—21no—
254C2×D4—3461no—
255C2×D4—1731no—
256C2×D4C22×D17311yescharacteristic
257C22×C4—21no—
258C22×C4—3461no—
259C22×C4—3461no—
260C22×C4C22×D17311yescharacteristic
261Q16—41no—
262Q16—6921no—
263Q16—6921no—
264SD16—41no—
265SD16—6921no—
266SD16—6921no—
267SD16—6921no—
268D8—6921no—
269C2×C8—1731no—
270C2×C8—1731no—
271C2×C8C22×D17311yescharacteristic
272C2×C8C22×D17311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—1731no—
281C4⋊C4—1731no—
282C4⋊C4—1731no—
283C4⋊C4C22×D17311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C22⋊C4—3461no—
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—1731no—
294C22⋊C4—1731no—
295C42—3461no—
296C42—1731no—
297C42—1731no—
298C23—21no—
299C23—3461no—
300C23D4×D17311yescharacteristic
301Q8—21no—
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—3461no—
306Q8—3461no—
307Q8—3461no—
308Q8—3461no—
309Q8—13841no—
310Q8—6922no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—3461no—
315D4—3461no—
316D4—13841no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—13841no—
327C2×C4—1731no—
328C2×C4—1731no—
329C2×C4—1731no—
330C2×C4—1731no—
331C2×C4—6922no—
332C2×C4D4×D17311yescharacteristic
333C2×C4C23×D17311yescharacteristic
334C8—21no—
335C8—21no—
336C8—3461no—
337C8—3461no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—6921no—
345C22—6921no—
346C22—6921no—
347C22C2×D4×D17311yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—3461no—
353C4—3461no—
354C4—3461no—
355C4—3461no—
356C4—6921no—
357C4—6921no—
358C4—6921no—
359C4#1914972411yescharacteristic
360C4#1914972411yescharacteristic
361C2—41no—
362C2—81no—
363C2—13841no—
364C2#2601205611yescharacteristic
365C2#2601229811yescharacteristic
366C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#452×C173C2
socleC22×C173C22×D4
prime radical#26011636C2
hypercenter#452D173
hypocenterC173#3225
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. #26012114C2semidirect
2. #26012062C2semidirect
3. #26011884C2semidirect
4. #26011872C2semidirect
5. #26011849C2semidirect
6. #26011840C2semidirect
7. #26011711C2semidirect
8. #26011705C2semidirect
9. #26011699C2semidirect
10. #26011695C2semidirect
11. #26011646C2semidirect
13. #26011632C2semidirect
14. #26011549C2semidirect
15. #452×C173C2semidirect
183. C173#3225semidirect
195. #452D173semidirect

Maximal subgroups

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

subgroupindex
#260121142
#260120622
#260118842
#260118722
#260118492
#260118402
#260117112
#260117052
#260116992
#260116952
#260116462
#260116362
#260116322
#260115492
#452×C1732
#3225173