Finite Group Databasebeta

← #36363733

Subgroups — #36363733

Order 22144.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012020C211yesmaximal, characteristic
2C4×Q8×D173C211yesmaximal, characteristic, complemented
3#26011825C211yesmaximal, characteristic, complemented
4#26011808C211yesmaximal, characteristic
5#26011799C211yesmaximal, characteristic, complemented
6#26011798C211yesmaximal, characteristic, complemented
7#26011710C211yesmaximal, characteristic, complemented
8#26011645C211yesmaximal, characteristic, complemented
9#26011628C211yesmaximal, characteristic
10#26011598C211yesmaximal, characteristic, complemented
11#26011595C211yesmaximal, characteristic, complemented
12#26011592C211yesmaximal, characteristic, complemented
13#26011588C211yesmaximal, characteristic, complemented
14#26011547C211yesmaximal, characteristic, complemented
15#464×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×#14583816C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19#19149672—21no—
20#19149671C2211yescharacteristic
21#19149661—21no—
22#19149655C2211yescharacteristic
23#19149650C2211yescharacteristic
24Q8×Dic173C2211yescharacteristic
25Q8×Dic173C2211yescharacteristic
26C4×Dic346—21no—
27C4×Dic346C2211yescharacteristic
28#19149644—21no—
29#19149644C2211yescharacteristic
30D4×Dic173C2211yescharacteristic
31#19149639—21no—
32#19149635—21no—
33C4⋊C4×D173—21no—
34C4⋊C4×D173C2211yescharacteristic
35C42×D173—21no—
36C42×D173C2211yescharacteristic
37#19149613C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149611C2211yescharacteristic
40#19149610C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic
51#19149589C2211yescharacteristic
52C2×Q16×C173C2211yescharacteristic
53C2×SD16×C173C2211yescharacteristic
54#124×C173C2211yescharacteristic
55C4×Q8×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61Q8×D173—22no—
62Q8×D173—22no—
63C2×C22⋊D173—21no—
64C22×Dic173—21no—
65C2×C4×D173—21no—
66C2×C4×D173—21no—
67C2×C4×D173D411yescharacteristic
68C2×C4×D173D411yescharacteristic
69C2×C4×D173C2311yescharacteristic
70#14583816—22no—
71#14583814—22no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C4⋊Dic173—21no—
75C4⋊Dic173—21no—
76C4⋊Dic173C2311yescharacteristic
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85C22⋊Dic173—21no—
86C22⋊Dic173—21no—
87#14583803—21no—
88C4×Dic173—21no—
89C4×Dic173—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173D411yescharacteristic
93C4×Dic173D411yescharacteristic
94C4×Dic173C2311yescharacteristic
95C2×Q8×C173C2311yescharacteristic
96C2×Q8×C173C2311yescharacteristic
97C2×D4×C173C2311yescharacteristic
98Q16×C173—22no—
99SD16×C173—22no—
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C22⋊C4×C173—21no—
105C22⋊C4×C173—21no—
106C42×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173C2×D411yescharacteristic
109Dic346—24no—
110Dic346—21no—
111Dic346#4112yes—
112C22⋊D173—42no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—42no—
118C2×Dic173C2×D411yescharacteristic
119C2×Dic173C2×D411yescharacteristic
120C2×Dic173C2×D411yescharacteristic
121C4×D173—24no—
122C4×D173—22no—
123C4×D173—21no—
124C4×D173—21no—
125C4×D173—41no—
126C4×D173—42no—
127#13063436—21no—
128#13063436—21no—
129C23×C173—21no—
130Q8×C173—21no—
131Q8×C173—21no—
132Q8×C173—21no—
133Q8×C173#4112yes—
134D4×C173—21no—
135D4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2411yescharacteristic
142C8×C173—21no—
143C8×C173—21no—
144C22×C173—41no—
145C22×C173—41no—
146C22×C173—41no—
147C22×C173C22×D411yescharacteristic
148C2×D173—21no—
149C2×D173—21no—
150C2×D173#13712yes—
151C4×C173—22no—
152C4×C173—21no—
153C4×C173—21no—
154C4×C173—41no—
155C4×C173—41no—
156C4×C173C2×#4111yescharacteristic
157C4×C173C2×#4111yescharacteristic
158Dic173—22no—
159Dic173—21no—
160Dic173—21no—
161Dic173—21no—
162Dic173—41no—
163Dic173—41no—
164Dic173—41no—
165C2×C173—81no—
166C2×C173#52511yescharacteristic
167C2×C173C2×#13711yescharacteristic
168C2×C173#57511yescharacteristic
169D173—22no—
170C173#311711yescharacteristic, complemented, Sylow
171#3117—1731nomaximal, Sylow
172#529—1731no—
173C2×C4×Q8—1731no—
174#484—1731no—
175#481—1731no—
176#479—1731no—
177#479—1731no—
178#464—1731no—
179#464D17311yescharacteristic, complemented
180#451—1731no—
181#448—1731no—
182#441—1731no—
183#441—1731no—
184#440—1731no—
185#440—1731no—
186#432—1731no—
187C22×Q8—1731no—
188C2×Q16—1731no—
189C2×Q16C2×D17311yescharacteristic
190C2×SD16—1731no—
191C2×SD16C2×D17311yescharacteristic
192#126—3461no—
193#125—1731no—
194#124C2×D17311yescharacteristic
195#123—3461no—
196C22⋊Q8—1731no—
197#121—1731no—
198C4×Q8—3461no—
199C4×Q8—3461no—
200C4×Q8—1731no—
201C4×Q8—1731no—
202C4×Q8—1731no—
203C4×Q8—1731no—
204C4×Q8C2×D17311yescharacteristic
205C4×D4—3461no—
206C4×D4—1731no—
207#117—3461no—
208C2×C4⋊C4—3461no—
209C2×C4⋊C4—1731no—
210C2×C42—3461no—
211C2×C42—1731no—
212#107—1731no—
213#107—1731no—
214#106—1731no—
215#106—1731no—
216C4⋊C8—1731no—
217C4⋊C8C2×D17311yescharacteristic
218#103—1731no—
219#103—1731no—
220#103—1731no—
221#103—1731no—
222#103—1731no—
223#103C2×D17311yescharacteristic
224#102—1731no—
225#102C2×D17311yescharacteristic
226C22⋊C8—1731no—
227C22⋊C8—1731no—
228#97—1731no—
229#97—1731no—
230C2×Q8—3461no—
231C2×Q8—3462no—
232C2×Q8—3462no—
233C2×Q8—1731no—
234C2×Q8C22×D17311yescharacteristic
235C2×Q8C22×D17311yescharacteristic
236C2×D4—3461no—
237C2×D4C22×D17311yescharacteristic
238C22×C4—3461no—
239C22×C4—3461no—
240C22×C4—3461no—
241C22×C4—1731no—
242C22×C4—1731no—
243C22×C4—1731no—
244Q16—22no—
245Q16—3462no—
246SD16—22no—
247SD16—3462no—
248C2×C8—1731no—
249C2×C8—1731no—
250C2×C8C22×D17311yescharacteristic
251C2×C8C22×D17311yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—3461no—
254C4⋊C4—3461no—
255C4⋊C4—3461no—
256C4⋊C4—3461no—
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—3461no—
260C4⋊C4—3461no—
261C4⋊C4—1731no—
262C4⋊C4—1731no—
263C4⋊C4—1731no—
264C4⋊C4C22×D17311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—3461no—
268C22⋊C4—3461no—
269C22⋊C4—3461no—
270C42—21no—
271C42—3461no—
272C42—3461no—
273C42—3461no—
274C42—3461no—
275C42—1731no—
276C42—1731no—
277C42—1731no—
278C42C22×D17311yescharacteristic
279C23—21no—
280C23—1731no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—3464no—
285Q8—3461no—
286Q8—1732no—
287Q8#1458382512yes—
288D4—21no—
289D4—21no—
290D4—6922no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—3464no—
295C2×C4—3462no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—3461no—
300C2×C4—3461no—
301C2×C4—3461no—
302C2×C4—1731no—
303C2×C4—1731no—
304C2×C4—1731no—
305C2×C4—6921no—
306C2×C4—6922no—
307C2×C4—6922no—
308C2×C4D4×D17311yescharacteristic
309C2×C4D4×D17311yescharacteristic
310C2×C4C23×D17311yescharacteristic
311C8—21no—
312C8—21no—
313C8—3461no—
314C8—3461no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—3461no—
319C22—3461no—
320C22—1732no—
321C22C2×D4×D17311yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—3462no—
328C4—3461no—
329C4—3461no—
330C4—3461no—
331C4—6921no—
332C4—6921no—
333C4—6921no—
334C4#41×D17311yescharacteristic
335C4C2×#1458382511yescharacteristic
336C2—81no—
337C2—3462no—
338C2#2601200111yescharacteristic
339C2#137×D17311yescharacteristic
340C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#464×C173C2
socleC22×C173C22×D4
prime radical#19149650C22
hypercenter#464D173
hypocenterC173#3117
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D173C2semidirect
3. #26011825C2semidirect
5. #26011799C2semidirect
6. #26011798C2semidirect
7. #26011710C2semidirect
8. #26011645C2semidirect
10. #26011598C2semidirect
11. #26011595C2semidirect
12. #26011592C2semidirect
13. #26011588C2semidirect
14. #26011547C2semidirect
15. #464×C173C2semidirect
170. C173#3117semidirect
179. #464D173semidirect

Maximal subgroups

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

subgroupindex
#260120202
C4×Q8×D1732
#260118252
#260118082
#260117992
#260117982
#260117102
#260116452
#260116282
#260115982
#260115952
#260115922
#260115882
#260115472
#464×C1732
#3117173