Finite Group Databasebeta

← #36363776

Subgroups — #36363776

Order 22144.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012102C211yesmaximal, characteristic, complemented
2#26011973C211yesmaximal, characteristic
3#26011825C211yesmaximal, characteristic, complemented
4#26011820C211yesmaximal, characteristic, complemented
5#26011798C211yesmaximal, characteristic, complemented
6#26011792C211yesmaximal, characteristic, complemented
7#26011693C211yesmaximal, characteristic, complemented
8#26011652C211yesmaximal, characteristic
9#26011652C211yesmaximal, characteristic
10#26011598C211yesmaximal, characteristic, complemented
11#26011594C211yesmaximal, characteristic, complemented
12Q16×Dic173C211yesmaximal, characteristic, complemented
13#26011576C211yesmaximal, characteristic, complemented
14#26011551C211yesmaximal, characteristic, complemented
15#462×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583813C2211yescharacteristic
18C2×#14583813C2211yescharacteristic
19#19149674—21no—
20#19149674—21no—
21#19149671C2211yescharacteristic
22#19149669—21no—
23#19149669—21no—
24#19149655C2211yescharacteristic
25#19149655C2211yescharacteristic
26Q8×Dic173C2211yescharacteristic
27Q8×Dic173C2211yescharacteristic
28Q8×Dic173C2211yescharacteristic
29C4×D692—21no—
30C4×D692C2211yescharacteristic
31#19149641—21no—
32#19149641C2211yescharacteristic
33#19149636—21no—
34#19149636C2211yescharacteristic
35C4⋊C4×D173—21no—
36C4⋊C4×D173C2211yescharacteristic
37#19149613C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149610C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic
42#19149603C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic
54C4⋊Q8×C173C2211yescharacteristic
55C4×Q8×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59#14583827—22no—
60#14583827—22no—
61C2×D692—21no—
62C2×D692C2311yescharacteristic
63C2×C4×D173—21no—
64C2×C4×D173—21no—
65C2×C4×D173—21no—
66C2×C4×D173—21no—
67C2×C4×D173C2311yescharacteristic
68#14583813—22no—
69#14583813—22no—
70C2×#13063436C2311yescharacteristic
71C2×#13063436C2311yescharacteristic
72C4⋊Dic173—21no—
73C4⋊Dic173—21no—
74C4⋊Dic173—21no—
75C4⋊Dic173C2311yescharacteristic
76C4⋊Dic173C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78#14583805—21no—
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805D411yescharacteristic
84#14583805D411yescharacteristic
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803D411yescharacteristic
90#14583803D411yescharacteristic
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173C2311yescharacteristic
95C2×Q8×C173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C2×Q8×C173C2311yescharacteristic
98Q16×C173—22no—
99Q16×C173—22no—
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173—21no—
104C4⋊C4×C173—21no—
105C4⋊C4×C173C2311yescharacteristic
106C42×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173C2×D411yescharacteristic
110D692—24no—
111D692—21no—
112D692#4112yes—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173C2×D411yescharacteristic
117C2×Dic173C2×D411yescharacteristic
118C2×Dic173C2×D411yescharacteristic
119C4×D173—24no—
120C4×D173—21no—
121C4×D173—21no—
122C4×D173—81no—
123C4×D173—81no—
124C4×D173—81no—
125#13063436—21no—
126#13063436—21no—
127Q8×C173—21no—
128Q8×C173—21no—
129Q8×C173—21no—
130Q8×C173—21no—
131Q8×C173—21no—
132Q8×C173#4112yes—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173C2×D411yescharacteristic
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2411yescharacteristic
140C8×C173—21no—
141C8×C173—21no—
142C22×C173C22×D411yescharacteristic
143C2×D173—21no—
144C2×D173—22no—
145C2×D173—21no—
146C2×D173—21no—
147C2×D173—41no—
148C2×D173—41no—
149C4×C173—22no—
150C4×C173—21no—
151C4×C173—21no—
152C4×C173—41no—
153C4×C173—41no—
154C4×C173—41no—
155C4×C173C2×#4111yescharacteristic
156C4×C173C2×#4111yescharacteristic
157Dic173—21no—
158Dic173—21no—
159Dic173—21no—
160Dic173—41no—
161Dic173—41no—
162Dic173—41no—
163C2×C173#52511yescharacteristic
164C2×C173#57711yescharacteristic
165C2×C173#57811yescharacteristic
166D173—41no—
167D173—42no—
168C173#312011yescharacteristic, complemented, Sylow
169#3120—1731nomaximal, Sylow
170#541—1731no—
171#520—1731no—
172#484—1731no—
173#483—1731no—
174#479—1731no—
175#478—1731no—
176#462D17311yescharacteristic, complemented
177#461—1731no—
178#452—1731no—
179#452—1731no—
180#441—1731no—
181#440—1731no—
182C4×Q16—1731no—
183C4×SD16—1731no—
184#433—1731no—
185C2×#41—1731no—
186C2×Q16C2×D17311yescharacteristic
187C2×Q16C2×D17311yescharacteristic
188C2×SD16—1731no—
189C2×SD16—1731no—
190C4⋊Q8C2×D17311yescharacteristic
191#126—3461no—
192#126—3461no—
193#125—3461no—
194#125—3461no—
195#125—1731no—
196C22⋊Q8—1731no—
197C22⋊Q8—1731no—
198C4×Q8—1731no—
199C4×Q8—1731no—
200C4×Q8—1731no—
201C4×Q8C2×D17311yescharacteristic
202C4×D4—3461no—
203C4×D4—3461no—
204C4×D4—1731no—
205C4×D4—1731no—
206#117—3461no—
207#117—1731no—
208C2×C4⋊C4—3461no—
209C2×C4⋊C4—1731no—
210#107—1731no—
211#107—1731no—
212#106—1731no—
213#106—1731no—
214C4⋊C8—1731no—
215C4⋊C8C2×D17311yescharacteristic
216#103—1731no—
217#103—1731no—
218#103—1731no—
219#103—1731no—
220#103C2×D17311yescharacteristic
221#103C2×D17311yescharacteristic
222#102—1731no—
223#102—1731no—
224C22⋊C8—1731no—
225C22⋊C8—1731no—
226#97—1731no—
227C4×C8—1731no—
228#41—3462no—
229#41—3462no—
230C2×Q8C22×D17311yescharacteristic
231C2×Q8C22×D17311yescharacteristic
232C2×Q8C22×D17311yescharacteristic
233C2×D4—3461no—
234C2×D4—1731no—
235C22×C4—3461no—
236C22×C4—3461no—
237C22×C4—3461no—
238C22×C4—3461no—
239C22×C4—1731no—
240Q16—22no—
241Q16—22no—
242SD16—3462no—
243SD16—3462no—
244C2×C8—1731no—
245C2×C8—1731no—
246C2×C8C22×D17311yescharacteristic
247C2×C8C22×D17311yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—3461no—
252C4⋊C4—3461no—
253C4⋊C4—3461no—
254C4⋊C4—3461no—
255C4⋊C4—3461no—
256C4⋊C4—3461no—
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—1731no—
260C4⋊C4—1731no—
261C4⋊C4—1731no—
262C4⋊C4—1731no—
263C4⋊C4—1731no—
264C4⋊C4C22×D17311yescharacteristic
265C22⋊C4—3461no—
266C22⋊C4—3461no—
267C22⋊C4—3461no—
268C22⋊C4—3461no—
269C22⋊C4—1731no—
270C22⋊C4—1731no—
271C42—21no—
272C42—3461no—
273C42—3461no—
274C42—3461no—
275C42—1731no—
276C42C22×D17311yescharacteristic
277C23—3461no—
278C23—1731no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8#1458382512yes—
285D4—3464no—
286D4—3461no—
287D4—1732no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—3464no—
293C2×C4—3461no—
294C2×C4—3461no—
295C2×C4—3461no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—13841no—
299C2×C4—13841no—
300C2×C4—13841no—
301C2×C4—1731no—
302C2×C4—1731no—
303C2×C4—1731no—
304C2×C4D4×D17311yescharacteristic
305C2×C4D4×D17311yescharacteristic
306C2×C4C23×D17311yescharacteristic
307C8—21no—
308C8—21no—
309C8—3461no—
310C8—3461no—
311C22—3461no—
312C22—3462no—
313C22—3461no—
314C22—3461no—
315C22—6921no—
316C22—6921no—
317C22C2×D4×D17311yescharacteristic
318C4—22no—
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—3461no—
325C4—3461no—
326C4—3461no—
327C4—6921no—
328C4—6921no—
329C4—6921no—
330C4#41×D17311yescharacteristic
331C4C2×#1458382511yescharacteristic
332C2—6921no—
333C2—6922no—
334C2#2601200111yescharacteristic
335C2#2601230011yescharacteristic
336C2#2601230411yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#462×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583827C22
hypercenter#462D173
hypocenterC173#3120
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. #26012102C2semidirect
3. #26011825C2semidirect
4. #26011820C2semidirect
5. #26011798C2semidirect
6. #26011792C2semidirect
7. #26011693C2semidirect
10. #26011598C2semidirect
11. #26011594C2semidirect
12. Q16×Dic173C2semidirect
13. #26011576C2semidirect
14. #26011551C2semidirect
15. #462×C173C2semidirect
168. C173#3120semidirect
176. #462D173semidirect

Maximal subgroups

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

subgroupindex
#260121022
#260119732
#260118252
#260118202
#260117982
#260117922
#260116932
#260116522
#260115982
#260115942
Q16×Dic1732
#260115762
#260115512
#462×C1732
#3120173