Finite Group Databasebeta

← #36363542

Subgroups — #36363542

Order 22144.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012071C211yesmaximal, characteristic
2#26011972C211yesmaximal, characteristic, complemented
3#26011830C211yesmaximal, characteristic, complemented
4#26011829C211yesmaximal, characteristic, complemented
5#26011825C211yesmaximal, characteristic, complemented
6#26011818C211yesmaximal, characteristic, complemented
7#26011816C211yesmaximal, characteristic, complemented
8#26011806C211yesmaximal, characteristic, complemented
9#26011804C211yesmaximal, characteristic, complemented
10#26011532C211yesmaximal, characteristic, complemented
11#107×D173C211yesmaximal, characteristic, complemented
12#26011491C211yesmaximal, characteristic, complemented
13#26011486C211yesmaximal, characteristic, complemented
14#26011456C211yesmaximal, characteristic, complemented
15#484×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×#14583809C2211yescharacteristic
19C2×C8×D173C2211yescharacteristic
20#19149657—21no—
21#19149655C2211yescharacteristic
22#19149654—21no—
23#19149654C2211yescharacteristic
24#19149653C2211yescharacteristic
25#19149652—21no—
26#19149652—21no—
27#19149644—21no—
28#19149641—21no—
29C4×C22⋊D173—21no—
30C4×C22⋊D173—21no—
31#19149638C2211yescharacteristic
32#19149637C2211yescharacteristic
33#19149637C2211yescharacteristic
34C4⋊C4×D173C2211yescharacteristic
35C4⋊C4×D173C2211yescharacteristic
36C4⋊C4×D173C2211yescharacteristic
37#19149613C2211yescharacteristic
38#19149612C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149611C2211yescharacteristic
42#19149610C2211yescharacteristic
43#19149603C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C22⋊Q8×C173C2211yescharacteristic
53C2×C4⋊C4×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#107×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583827—22no—
60#14583827—22no—
61#14583826—22no—
62#14583826—22no—
63C2×Dic346D411yescharacteristic
64C2×D692—21no—
65C2×D692—21no—
66C2×D692D411yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173D411yescharacteristic
70C2×C22⋊D173D411yescharacteristic
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74C2×C4×D173—21no—
75C2×C4×D173C2311yescharacteristic
76C2×C4×D173C2311yescharacteristic
77#14583809—22no—
78C2×#13063436C2311yescharacteristic
79C2×#13063436C2311yescharacteristic
80C8×D173—22no—
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83C4⋊Dic173C2311yescharacteristic
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94C4×Dic173—21no—
95C4×Dic173—21no—
96C4×Dic173—21no—
97C2×Q8×C173C2311yescharacteristic
98C22×C4×C173—21no—
99C22×C4×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173—21no—
103C4⋊C4×C173—21no—
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C4⋊C4×C173C2311yescharacteristic
107C22⋊C4×C173—21no—
108C22×D173—21no—
109C22×D173C2×D411yescharacteristic
110C22×D173C2×D411yescharacteristic
111Dic346—22no—
112D692—22no—
113D692—21no—
114D692—21no—
115D692—42no—
116D692—81no—
117C22⋊D173—22no—
118C22⋊D173—22no—
119C22⋊D173—81no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C4×D173—21no—
126C4×D173—21no—
127C4×D173—42no—
128C4×D173—81no—
129C4×D173—81no—
130C4×D173—81no—
131C4×D173#4112yes—
132C4×D173#4112yes—
133#13063436—21no—
134#13063436—21no—
135C23×C173C2×D411yescharacteristic
136Q8×C173—21no—
137Q8×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—81no—
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2411yescharacteristic
147C8×C173—21no—
148C8×C173—21no—
149C22×C173—21no—
150C22×C173—21no—
151C22×C173—21no—
152C22×C173C22×D411yescharacteristic
153C2×D173—21no—
154C2×D173—21no—
155C2×D173—21no—
156C2×D173—21no—
157C2×D173—21no—
158C2×D173—41no—
159C2×D173—41no—
160C2×D173—41no—
161C2×D173#12312yes—
162C4×C173—21no—
163C4×C173—41no—
164C4×C173—41no—
165C4×C173—41no—
166C4×C173—41no—
167C4×C173C2×#4111yescharacteristic
168C4×C173C2×#4111yescharacteristic
169Dic173—21no—
170Dic173—41no—
171Dic173—41no—
172Dic173—41no—
173Dic173#12312yes—
174C2×C173—41no—
175C2×C173C2×#12311yescharacteristic
176C2×C173#57611yescharacteristic
177C2×C173#57811yescharacteristic
178D173—22no—
179D173—41no—
180D173—81no—
181C173#309811yescharacteristic, complemented, Sylow
182#3098—1731nomaximal, Sylow
183#537—1731no—
184#520—1731no—
185#485—1731no—
186#485—1731no—
187#484—1731no—
188#484D17311yescharacteristic, complemented
189#482—1731no—
190#482—1731no—
191#480—1731no—
192#480—1731no—
193#428—1731no—
194C2×#107—1731no—
195#417—1731no—
196#416—1731no—
197#408—1731no—
198C2×#41—1731no—
199C2×#41—1731no—
200C2×M16—1731no—
201C22×C8—1731no—
202C22⋊Q8—3461no—
203C22⋊Q8—1731no—
204C22⋊Q8C2×D17311yescharacteristic
205#121—3461no—
206#121—3461no—
207#121—3461no—
208#121—1731no—
209#121—1731no—
210C4×Q8—3461no—
211C4×D4—3461no—
212C4×D4—3461no—
213C4×D4—3461no—
214#117—1731no—
215#117—1731no—
216#117—1731no—
217C2×C4⋊C4—1731no—
218C2×C4⋊C4—1731no—
219C2×C4⋊C4—1731no—
220C2×C4⋊C4C2×D17311yescharacteristic
221#107—1731no—
222#107—1731no—
223#107—1731no—
224#107—1731no—
225#107C2×D17311yescharacteristic
226#107C2×D17311yescharacteristic
227#106—1731no—
228#106—1731no—
229#103—1731no—
230#103—1731no—
231#103C2×D17311yescharacteristic
232#103C2×D17311yescharacteristic
233#102—1731no—
234#102—1731no—
235#102—1731no—
236#102—1731no—
237C22⋊C8—1731no—
238C22⋊C8—1731no—
239C22⋊C8—1731no—
240C22⋊C8C2×D17311yescharacteristic
241#41—3462no—
242#41—3462no—
243#41—3462no—
244#41—3462no—
245C2×Q8—1731no—
246C2×Q8C22×D17311yescharacteristic
247C2×D4—3461no—
248C2×D4—3461no—
249C2×D4—3461no—
250C2×D4—1731no—
251C2×D4—1731no—
252C2×D4—1731no—
253C2×D4—1731no—
254C22×C4—21no—
255C22×C4—3461no—
256C22×C4—3461no—
257C22×C4—3461no—
258C22×C4—3461no—
259C22×C4—1731no—
260C22×C4—1731no—
261C22×C4C22×D17311yescharacteristic
262M16—3462no—
263C2×C8—3462no—
264C2×C8—1731no—
265C2×C8—1731no—
266C2×C8C22×D17311yescharacteristic
267C2×C8C22×D17311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—3461no—
271C4⋊C4—3461no—
272C4⋊C4—3461no—
273C4⋊C4—3461no—
274C4⋊C4—1731no—
275C4⋊C4—1731no—
276C4⋊C4—1731no—
277C4⋊C4C22×D17311yescharacteristic
278C4⋊C4C22×D17311yescharacteristic
279C4⋊C4C22×D17311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—3461no—
287C42—3461no—
288C42—3461no—
289C42—3461no—
290C23—3461no—
291C23—1731no—
292C23—1731no—
293C23D4×D17311yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—3462no—
297D4—3462no—
298D4—3462no—
299D4—3462no—
300D4—3461no—
301D4—3461no—
302D4—13841no—
303D4—13841no—
304D4—6922no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—13841no—
318C2×C4—13841no—
319C2×C4—13841no—
320C2×C4—1731no—
321C2×C4—1731no—
322C2×C4—1732no—
323C2×C4—1732no—
324C2×C4—6922no—
325C2×C4D4×D17311yescharacteristic
326C2×C4C23×D17311yescharacteristic
327C8—21no—
328C8—21no—
329C8—3461no—
330C8—3461no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—3461no—
335C22—3461no—
336C22—3461no—
337C22—3461no—
338C22—3461no—
339C22—1732no—
340C22—6921no—
341C22—6921no—
342C22—6921no—
343C22C2×D4×D17311yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—3461no—
350C4—1732no—
351C4—6921no—
352C4—6921no—
353C4—6921no—
354C4#41×D17311yescharacteristic
355C4#41×D17311yescharacteristic
356C2—41no—
357C2—3462no—
358C2—13841no—
359C2—6921no—
360C2#123×D17311yescharacteristic
361C2#2601229511yescharacteristic
362C2#2601230411yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#484×C173C2
socleC22×C173C22×D4
prime radical#26012071C2
hypercenter#484D173
hypocenterC173#3098
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. #26011972C2semidirect
3. #26011830C2semidirect
4. #26011829C2semidirect
5. #26011825C2semidirect
6. #26011818C2semidirect
7. #26011816C2semidirect
8. #26011806C2semidirect
9. #26011804C2semidirect
10. #26011532C2semidirect
11. #107×D173C2semidirect
12. #26011491C2semidirect
13. #26011486C2semidirect
14. #26011456C2semidirect
15. #484×C173C2semidirect
181. C173#3098semidirect
188. #484D173semidirect

Maximal subgroups

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

subgroupindex
#260120712
#260119722
#260118302
#260118292
#260118252
#260118182
#260118162
#260118062
#260118042
#260115322
#107×D1732
#260114912
#260114862
#260114562
#484×C1732
#3098173