Finite Group Databasebeta

← #36365646

Subgroups — #36365646

Order 22144.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012198C211yesmaximal, characteristic, complemented
2#26012137C211yesmaximal, characteristic
3#26011851C211yesmaximal, characteristic, complemented
4#26011827C211yesmaximal, characteristic, complemented
5#26011805C211yesmaximal, characteristic, complemented
6#26011772C211yesmaximal, characteristic, complemented
7#26011737C211yesmaximal, characteristic, complemented
8#26011732C211yesmaximal, characteristic, complemented
9#26011714C211yesmaximal, characteristic, complemented
10#26011655C211yesmaximal, characteristic, complemented
11#26011644C211yesmaximal, characteristic, complemented
12#26011593C211yesmaximal, characteristic, complemented
13#26011493C211yesmaximal, characteristic, complemented
14C2×#19149601C211yesmaximal, characteristic, complemented
15#435×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C2×Dic692C2211yescharacteristic, complemented
19C2×#14583815C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21#19149679C2211yescharacteristic, complemented
22#19149670—21no—
23#19149667C2211yescharacteristic
24#19149664—21no—
25#19149658C2211yescharacteristic
26#19149658C2211yescharacteristic
27#19149658C2211yescharacteristic, complemented
28#19149656—21no—
29#19149656—21no—
30#19149653C2211yescharacteristic
31#19149652—21no—
32#19149647—21no—
33C4×Dic346C2211yescharacteristic, complemented
34C4×D692C2211yescharacteristic
35D4×Dic173—21no—
36C2×C4⋊Dic173C2211yescharacteristic
37C2×C4⋊Dic173C2211yescharacteristic, complemented
38C22⋊C4×D173—21no—
39#19149613C2211yescharacteristic, complemented
40#19149613C2211yescharacteristic, complemented
41#19149613C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149604C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149601C2211yescharacteristic
50#19149601C2211yescharacteristic
51#19149601C2211yescharacteristic
52C2×M16×C173C2211yescharacteristic, complemented
53C22×C8×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55C4⋊C8×C173C2211yescharacteristic, complemented
56C22⋊C8×C173C2211yescharacteristic
57C22⋊C8×C173C2211yescharacteristic, complemented
58#97×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583826—22no—
61#14583826—41no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65C2×D692—22no—
66C2×D692—21no—
67C2×D692—41no—
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C22×Dic173—21no—
73C22×Dic173—21no—
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76Dic692—41no—
77#14583815—22no—
78#14583815—41no—
79C4⋊Dic173—21no—
80C4⋊Dic173—21no—
81C4⋊Dic173—21no—
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97C4×Dic173—21no—
98C2×D4×C173D411yescharacteristic
99C22×C4×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101M16×C173—41no—
102C2×C8×C173—41no—
103C2×C8×C173C2311yescharacteristic
104C2×C8×C173C2311yescharacteristic
105C2×C8×C173C2311yescharacteristic
106C2×C8×C173C2311yescharacteristic
107C4⋊C4×C173D411yescharacteristic
108C22⋊C4×C173D411yescharacteristic
109C22⋊C4×C173D411yescharacteristic
110C42×C173C2311yescharacteristic
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—42no—
114C22×D173—42no—
115C22×D173—81no—
116Dic346—21no—
117Dic346—21no—
118Dic346—21no—
119Dic346—21no—
120Dic346—81no—
121D692—21no—
122D692—21no—
123D692—21no—
124D692—42no—
125D692#4112yes—
126C22⋊D173—42no—
127C22⋊D173—81no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—81no—
135C2×Dic173—81no—
136C4×D173—42no—
137C4×D173—81no—
138C23×C173C2×D411yescharacteristic
139C23×C173C2×D411yescharacteristic
140D4×C173—22no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173C2411yescharacteristic
150C8×C173—21no—
151C8×C173—21no—
152C8×C173—21no—
153C8×C173—21no—
154C22×C173—21no—
155C22×C173—21no—
156C22×C173—21no—
157C22×C173—21no—
158C22×C173—21no—
159C22×C173#13712yes—
160C22×C173C22×D411yescharacteristic
161C2×D173—22no—
162C2×D173—41no—
163C2×D173—41no—
164C2×D173—41no—
165C2×D173—41no—
166C2×D173—41no—
167C2×D173—84no—
168C4×C173—21no—
169C4×C173—21no—
170C4×C173—21no—
171C4×C173—21no—
172C4×C173C2×#4111yescharacteristic
173C4×C173H(5,𝔽2)11yescharacteristic
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177Dic173—41no—
178Dic173—41no—
179Dic173—41no—
180C2×C173—22no—
181C2×C173—41no—
182C2×C173#54611yescharacteristic
183C2×C173C2×#13711yescharacteristic
184C2×C173#57611yescharacteristic
185D173—42no—
186D173—81no—
187C173#331411yescharacteristic, complemented, Sylow
188#3314—1731nomaximal, Sylow
189#554—1731no—
190#546—1731no—
191#489—1731no—
192#484—1731no—
193#480—1731no—
194#474—1731no—
195#468—1731no—
196#467—1731no—
197#464—1731no—
198#452—1731no—
199#450—1731no—
200#440—1731no—
201#435D17311yescharacteristic, complemented
202#417—1731no—
203C2×#102—1731no—
204C2×#41—1731no—
205C22×D4—1731no—
206C2×Q16—1731no—
207C2×SD16—1731no—
208C2×SD16—1731no—
209C2×M16C2×D17311yescharacteristic, complemented
210C22×C8C2×D17311yescharacteristic
211C4⋊Q8—1731no—
212#125—3461no—
213#124—1731no—
214#123—3461no—
215C22⋊Q8—3461no—
216C22⋊Q8—3461no—
217C22⋊Q8—1731no—
218C22⋊Q8—1731no—
219C22⋊Q8—1731no—
220#121—3461no—
221#121—1731no—
222#120—3461no—
223C4×Q8—1731no—
224C4×D4—3461no—
225C4×D4—1731no—
226C4×D4C2×D17311yescharacteristic
227C2×C4⋊C4—1731no—
228C2×C4⋊C4—1731no—
229C2×C22⋊C4—3461no—
230#107—1731no—
231#107—1731no—
232#107—1731no—
233C4⋊C8C2×D17311yescharacteristic, complemented
234#103—1731no—
235#103—1731no—
236#103—1731no—
237#103—1731no—
238#102—1731no—
239#102—1731no—
240#102—1731no—
241#102—1731no—
242#102—1731no—
243#102—1731no—
244C22⋊C8C2×D17311yescharacteristic
245C22⋊C8C2×D17311yescharacteristic, complemented
246#97C2×D17311yescharacteristic, complemented
247C24—3461no—
248#41—3462no—
249#41—6921no—
250C2×Q8—3461no—
251C2×Q8—1731no—
252C2×Q8—1731no—
253C2×D4—3462no—
254C2×D4—3461no—
255C2×D4—3461no—
256C2×D4—3461no—
257C2×D4—1731no—
258C2×D4—1731no—
259C2×D4—6921no—
260C2×D4D69211yescharacteristic
261C22×C4—3461no—
262C22×C4—3461no—
263C22×C4—3461no—
264C22×C4—3461no—
265C22×C4C22×D17311yescharacteristic
266C22×C4C22×D17311yescharacteristic
267Q16—6921no—
268SD16—3462no—
269SD16—6921no—
270M16—41no—
271C2×C8—41no—
272C2×C8C22×D17311yescharacteristic
273C2×C8C22×D17311yescharacteristic
274C2×C8C22×D17311yescharacteristic
275C2×C8C22×D17311yescharacteristic
276C4⋊C4—3461no—
277C4⋊C4—3461no—
278C4⋊C4—3461no—
279C4⋊C4—3461no—
280C4⋊C4—3461no—
281C4⋊C4—3461no—
282C4⋊C4—3461no—
283C4⋊C4—1731no—
284C4⋊C4—1731no—
285C4⋊C4—1731no—
286C4⋊C4—1731no—
287C4⋊C4D69211yescharacteristic
288C22⋊C4—3461no—
289C22⋊C4—3461no—
290C22⋊C4—3461no—
291C22⋊C4—3461no—
292C22⋊C4—3461no—
293C22⋊C4—3461no—
294C22⋊C4—3461no—
295C22⋊C4D69211yescharacteristic
296C22⋊C4D69211yescharacteristic
297C42—3461no—
298C42C22×D17311yescharacteristic
299C23—3461no—
300C23—3461no—
301C23—13841no—
302C23—6922no—
303C23—6922no—
304C23C2×D69211yescharacteristic
305C23C2×D69211yescharacteristic
306Q8—3461no—
307Q8—3461no—
308Q8—3461no—
309Q8—3461no—
310Q8—13841no—
311D4—22no—
312D4—3461no—
313D4—3461no—
314D4—3461no—
315D4—13841no—
316D4—1732no—
317D4—6922no—
318D4—6922no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—13841no—
330C2×C4—13841no—
331C2×C4—13841no—
332C2×C4—6922no—
333C2×C4C2×D69211yescharacteristic
334C2×C4C2×D69211yescharacteristic
335C2×C4C2×D69211yescharacteristic
336C2×C4C2×D69211yescharacteristic
337C2×C4C23×D17311yescharacteristic
338C8—21no—
339C8—21no—
340C8—21no—
341C8—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—3462no—
348C22—13844no—
349C22—6921no—
350C22—6921no—
351C22—6921no—
352C22—6921no—
353C22—6921no—
354C22#1914971212yes—
355C22C22×D69211yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—6921no—
361C4—6921no—
362C4—6921no—
363C4—6921no—
364C4—6921no—
365C4—6921no—
366C4#41×D17311yescharacteristic
367C4#1914972411yescharacteristic
368C2—22no—
369C2—41no—
370C2—13841no—
371C2—6922no—
372C2#2601213611yescharacteristic
373C2C2×#1914971211yescharacteristic
374C2#2601229611yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#435×C173C2
socleC22×C173C22×D4
prime radical#26012137C2
hypercenter#435D173
hypocenterC173#3314
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #26012198C2semidirect
3. #26011851C2semidirect
4. #26011827C2semidirect
5. #26011805C2semidirect
6. #26011772C2semidirect
7. #26011737C2semidirect
8. #26011732C2semidirect
9. #26011714C2semidirect
10. #26011655C2semidirect
11. #26011644C2semidirect
12. #26011593C2semidirect
13. #26011493C2semidirect
14. C2×#19149601C2semidirect
15. #435×C173C2semidirect
18. C2×Dic692C22semidirect
21. #19149679C22semidirect
27. #19149658C22semidirect
33. C4×Dic346C22semidirect
37. C2×C4⋊Dic173C22semidirect
39. #19149613C22semidirect
40. #19149613C22semidirect
41. #19149613C22semidirect
42. #19149604C22semidirect
43. #19149604C22semidirect
44. #19149604C22semidirect
45. #19149604C22semidirect
52. C2×M16×C173C22semidirect
55. C4⋊C8×C173C22semidirect
57. C22⋊C8×C173C22semidirect
58. #97×C173C22semidirect
187. C173#3314semidirect
201. #435D173semidirect
209. C2×M16C2×D173semidirect
233. C4⋊C8C2×D173semidirect
245. C22⋊C8C2×D173semidirect
246. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260121982
#260121372
#260118512
#260118272
#260118052
#260117722
#260117372
#260117322
#260117142
#260116552
#260116442
#260115932
#260114932
C2×#191496012
#435×C1732
#3314173