Finite Group Databasebeta

← #36366271

Subgroups — #36366271

Order 22144.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012200C211yesmaximal, characteristic, complemented
2#26012162C211yesmaximal, characteristic, complemented
3#26011856C211yesmaximal, characteristic, complemented
4#26011854C211yesmaximal, characteristic, complemented
5#26011836C211yesmaximal, characteristic, complemented
6#26011791C211yesmaximal, characteristic, complemented
7#26011790C211yesmaximal, characteristic, complemented
8#26011780C211yesmaximal, characteristic, complemented
9#26011686C211yesmaximal, characteristic, complemented
10#26011684C211yesmaximal, characteristic, complemented
11#26011684C211yesmaximal, characteristic, complemented
12#26011619C211yesmaximal, characteristic, complemented
13#26011600C211yesmaximal, characteristic, complemented
14#26011571C211yesmaximal, characteristic, complemented
15#442×C173C211yesmaximal, characteristic, complemented
16C2×#14583826—21no—
17C2×#14583811C2211yescharacteristic
18C2×#14583811C2211yescharacteristic
19#19149678C2211yescharacteristic
20#19149676C2211yescharacteristic
21#19149675C2211yescharacteristic
22#19149675C2211yescharacteristic
23#19149671C2211yescharacteristic
24#19149670C2211yescharacteristic
25#19149658—21no—
26#19149656—21no—
27#19149656—21no—
28#19149653—21no—
29#19149652—21no—
30#19149652—21no—
31C4×Dic346C2211yescharacteristic
32D4×Dic173C2211yescharacteristic
33D4×Dic173C2211yescharacteristic
34#19149641C2211yescharacteristic
35#19149641C2211yescharacteristic
36C2×#14583805—21no—
37#19149612C2211yescharacteristic
38#19149610C2211yescharacteristic
39C4⋊#13063436C2211yescharacteristic
40#19149608C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149601C2211yescharacteristic
43#19149601C2211yescharacteristic
44#19149601C2211yescharacteristic
45#19149601C2211yescharacteristic
46#19149600C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149599C2211yescharacteristic
49#19149599C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×D8×C173C2211yescharacteristic
53C4×D4×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#102×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58#97×C173C2211yescharacteristic
59#14583826—44no—
60C2×Dic346—21no—
61C2×Dic346D411yescharacteristic
62C2×D692—21no—
63C2×D692—21no—
64C2×D692C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C22×Dic173—21no—
71C22×Dic173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—21no—
74#14583811—22no—
75#14583811—22no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C4⋊Dic173—21no—
79C4⋊Dic173D411yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805D411yescharacteristic
87#14583805D411yescharacteristic
88C22⋊Dic173—21no—
89C22⋊Dic173—21no—
90#14583803—21no—
91#14583803—21no—
92C4×Dic173C2311yescharacteristic
93C4×Dic173C2311yescharacteristic
94C2×D4×C173C2311yescharacteristic
95C2×D4×C173C2311yescharacteristic
96C22×C4×C173—21no—
97C22×C4×C173—21no—
98D8×C173—22no—
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C4⋊C4×C173C2311yescharacteristic
102C4⋊C4×C173C2311yescharacteristic
103C22⋊C4×C173—21no—
104C22⋊C4×C173—21no—
105C42×C173C2311yescharacteristic
106C22×D173—21no—
107C22×D173—21no—
108Dic346—22no—
109Dic346—81no—
110D692—21no—
111D692—21no—
112D692—21no—
113D692—21no—
114D692—42no—
115D692—81no—
116C22⋊D173—42no—
117C22⋊D173—42no—
118C22⋊D173—81no—
119C22⋊D173—81no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—42no—
123C2×Dic173—81no—
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C4×D173—42no—
129C4×D173—42no—
130#13063436—21no—
131#13063436—21no—
132C23×C173—21no—
133C23×C173—21no—
134D4×C173—21no—
135D4×C173—21no—
136D4×C173—21no—
137D4×C173#4112yescomplemented
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—42no—
141C2×C4×C173—81no—
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2411yescharacteristic
145C8×C173—21no—
146C8×C173—21no—
147C22×C173—22no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173—41no—
153C22×C173C22×D411yescharacteristic
154C2×D173—41no—
155C2×D173—41no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C2×D173—41no—
160C4×C173—21no—
161C4×C173—21no—
162C4×C173—41no—
163C4×C173—41no—
164C4×C173C2×#4111yescharacteristic
165C4×C1732−1+411yescharacteristic
166Dic173—21no—
167Dic173—21no—
168Dic173—21no—
169Dic173—41no—
170Dic173—41no—
171Dic173AGL(1,ℤ/8)12yes—
172C2×C173—42no—
173C2×C173—81no—
174C2×C173#54811yescharacteristic
175C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
176C2×C173#57611yescharacteristic
177D173—81no—
178D173—81no—
179C173#336611yescharacteristic, complemented, Sylow
180#3366—1731nomaximal, Sylow
181#554—1731no—
182#550—1731no—
183#490—1731no—
184#490—1731no—
185#486—1731no—
186#478—1731no—
187#478—1731no—
188#476—1731no—
189#459—1731no—
190#459—1731no—
191#459—1731no—
192#446—1731no—
193#442—1731no—
194#442D17311yescharacteristic, complemented
195C4×D8—1731no—
196C2×#41—3461no—
197C2×D8—1731no—
198C2×D8—1731no—
199C2×D8C2×D17311yescharacteristic
200C4⋊Q8—1731no—
201#127—1731no—
202#127—1731no—
203#127—1731no—
204#125—1731no—
205#125—1731no—
206C22⋊Q8—3461no—
207C22⋊Q8—3461no—
208C22⋊Q8—3461no—
209#121—3461no—
210#121—3461no—
211#121—3461no—
212C4×Q8—1731no—
213C4×D4—1731no—
214C4×D4—1731no—
215C4×D4—1731no—
216C4×D4—1731no—
217C4×D4C2×D17311yescharacteristic
218C4×D4C2×D17311yescharacteristic
219C2×C4⋊C4—3461no—
220#107—1731no—
221#106—1731no—
222#106C2×D17311yescharacteristic
223C4⋊C8—1731no—
224C4⋊C8—1731no—
225C4⋊C8—1731no—
226#102—1731no—
227#102—1731no—
228#102—1731no—
229#102—1731no—
230#102—1731no—
231#102—1731no—
232#102—1731no—
233#102—1731no—
234#102C2×D17311yescharacteristic
235#102C2×D17311yescharacteristic
236#97—1731no—
237#97C2×D17311yescharacteristic
238C4×C8—1731no—
239#41—6924no—
240C2×Q8—3461no—
241C2×Q8—1731no—
242C2×D4—3461no—
243C2×D4—3461no—
244C2×D4—3461no—
245C2×D4—3461no—
246C2×D4—3461no—
247C2×D4—3461no—
248C2×D4—1731no—
249C2×D4—1731no—
250C2×D4C22×D17311yescharacteristic
251C2×D4C22×D17311yescharacteristic
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—3461no—
255C22×C4—3461no—
256C22×C4—3461no—
257C22×C4—3461no—
258D8—22no—
259D8—3462no—
260D8—3462no—
261C2×C8—1731no—
262C2×C8—1731no—
263C2×C8C22×D17311yescharacteristic
264C2×C8C22×D17311yescharacteristic
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—1731no—
271C4⋊C4—1731no—
272C4⋊C4—1731no—
273C4⋊C4—1731no—
274C4⋊C4—1731no—
275C4⋊C4C22×D17311yescharacteristic
276C4⋊C4C22×D17311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C42—1731no—
284C42—1731no—
285C42C22×D17311yescharacteristic
286C23—21no—
287C23—21no—
288C23—3461no—
289C23—3461no—
290Q8—3462no—
291Q8—13841no—
292D4—21no—
293D4—21no—
294D4—21no—
295D4—3461no—
296D4—3461no—
297D4—3461no—
298D4—3461no—
299D4—13841no—
300D4—13841no—
301D4—13841no—
302D4—6922no—
303D4—6922no—
304D4—6922no—
305D4#1458382612yescomplemented
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—81no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—13841no—
313C2×C4—1731no—
314C2×C4—1731no—
315C2×C4—1731no—
316C2×C4—1731no—
317C2×C4—6922no—
318C2×C4—6922no—
319C2×C4—6922no—
320C2×C4D4×D17311yescharacteristic
321C2×C4D4×D17311yescharacteristic
322C2×C4C23×D17311yescharacteristic
323C8—21no—
324C8—21no—
325C8—3461no—
326C8—3461no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—6921no—
334C22—6921no—
335C22—6921no—
336C22—6921no—
337C22—6921no—
338C22—6921no—
339C22C2×D4×D17311yescharacteristic
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—3461no—
345C4—3461no—
346C4—3461no—
347C4—1732no—
348C4—6921no—
349C4—6921no—
350C4C2×#1458382611yescharacteristic
351C4#1914972611yescharacteristic
352C2—42no—
353C2—81no—
354C2—13841no—
355C2—13841no—
356C2#2601215011yescharacteristic
357C2AGL(1,ℤ/8)×D17311yescharacteristic
358C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#442×C173C2
socleC22×C173C22×D4
hypercenter#442D173
hypocenterC173#3366
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #26012200C2semidirect
2. #26012162C2semidirect
3. #26011856C2semidirect
4. #26011854C2semidirect
5. #26011836C2semidirect
6. #26011791C2semidirect
7. #26011790C2semidirect
8. #26011780C2semidirect
9. #26011686C2semidirect
10. #26011684C2semidirect
11. #26011684C2semidirect
12. #26011619C2semidirect
13. #26011600C2semidirect
14. #26011571C2semidirect
15. #442×C173C2semidirect
137. D4×C173#41semidirect
179. C173#3366semidirect
194. #442D173semidirect
305. D4#14583826semidirect

Maximal subgroups

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

subgroupindex
#260122002
#260121622
#260118562
#260118542
#260118362
#260117912
#260117902
#260117802
#260116862
#260116842
#260116192
#260116002
#260115712
#442×C1732
#3366173