Finite Group Databasebeta

← #36363739

Subgroups — #36363739

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#26012026C211yesmaximal, characteristic
2C4×Q8×D173C211yesmaximal, characteristic, complemented
3#26011828C211yesmaximal, characteristic, complemented
4#26011811C211yesmaximal, characteristic
5#26011795C211yesmaximal, characteristic, complemented
6#26011714C211yesmaximal, characteristic, complemented
7#26011713C211yesmaximal, characteristic, complemented
8#26011648C211yesmaximal, characteristic, complemented
9#26011631C211yesmaximal, characteristic
10#26011597C211yesmaximal, characteristic, complemented
11#26011595C211yesmaximal, characteristic, complemented
12#26011594C211yesmaximal, characteristic, complemented
13#26011590C211yesmaximal, characteristic, complemented
14#26011547C211yesmaximal, characteristic, complemented
15#479×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic
17C2×Dic692C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583815C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21#19149674—21no—
22#19149667C2211yescharacteristic
23#19149664—21no—
24#19149659C2211yescharacteristic
25#19149654C2211yescharacteristic
26Q8×Dic173C2211yescharacteristic
27C4×Dic346—21no—
28C4×Dic346C2211yescharacteristic
29#19149644—21no—
30#19149644C2211yescharacteristic
31#19149644C2211yescharacteristic
32#19149641—21no—
33#19149641C2211yescharacteristic
34#19149637—21no—
35C4⋊C4×D173—21no—
36C4⋊C4×D173C2211yescharacteristic
37C42×D173—21no—
38C42×D173C2211yescharacteristic
39#19149612C2211yescharacteristic
40#19149610C2211yescharacteristic
41C4⋊#13063436C2211yescharacteristic
42#19149604C2211yescharacteristic
43#19149602C2211yescharacteristic
44#19149602C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149599C2211yescharacteristic
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149589C2211yescharacteristic
51#19149589C2211yescharacteristic
52#125×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54#107×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61C2×Dic346C2311yescharacteristic
62Q8×D173—22no—
63Q8×D173—22no—
64C2×D692—21no—
65C2×D692C2311yescharacteristic
66C2×C4×D173—21no—
67C2×C4×D173—21no—
68C2×C4×D173—21no—
69C2×C4×D173D411yescharacteristic
70C2×C4×D173D411yescharacteristic
71C2×C4×D173C2311yescharacteristic
72Dic692—22no—
73#14583816—22no—
74#14583815—22no—
75#14583813—22no—
76C2×#13063436C2311yescharacteristic
77C2×#13063436C2311yescharacteristic
78C4⋊Dic173—21no—
79C4⋊Dic173—21no—
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583803—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90C4×Dic173—21no—
91C4×Dic173—21no—
92C4×Dic173—21no—
93C4×Dic173—21no—
94C4×Dic173D411yescharacteristic
95C4×Dic173D411yescharacteristic
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173C2311yescharacteristic
98C2×C8×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C4⋊C4×C173—21no—
101C4⋊C4×C173—21no—
102C4⋊C4×C173—21no—
103C4⋊C4×C173C2311yescharacteristic
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C42×C173—21no—
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173C2×D411yescharacteristic
110Dic346—24no—
111Dic346—21no—
112Dic346—21no—
113Dic346—21no—
114Dic346#4112yes—
115D692—21no—
116D692—21no—
117D692—42no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173C2×D411yescharacteristic
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C4×D173—24no—
125C4×D173—22no—
126C4×D173—21no—
127C4×D173—21no—
128C4×D173—41no—
129C4×D173—42no—
130C4×D173—42no—
131#13063436—21no—
132#13063436—21no—
133Q8×C173—21no—
134Q8×C173#4112yes—
135C2×C4×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×C173C22×D411yescharacteristic
145C2×D173—21no—
146C2×D173—21no—
147C2×D173—41no—
148C2×D173—41no—
149C2×D173—41no—
150C2×D173#13712yes—
151C4×C173—22no—
152C4×C173—21no—
153C4×C173—21no—
154C4×C173—41no—
155C4×C173—41no—
156C4×C173—41no—
157C4×C173C2×#4111yescharacteristic
158C4×C173C2×#4111yescharacteristic
159Dic173—22no—
160Dic173—21no—
161Dic173—21no—
162Dic173—21no—
163Dic173—41no—
164Dic173—41no—
165C2×C173#52511yescharacteristic
166C2×C173C2×#13711yescharacteristic
167C2×C173#57511yescharacteristic
168D173—22no—
169D173—81no—
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#479D17311yescharacteristic, complemented
178#464—1731no—
179#464—1731no—
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×Q16—1731no—
190C2×SD16—1731no—
191C2×SD16—1731no—
192#126—3461no—
193#125C2×D17311yescharacteristic
194#124—1731no—
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#107C2×D17311yescharacteristic
214#106—1731no—
215#106C2×D17311yescharacteristic
216C4⋊C8—1731no—
217C4⋊C8C2×D17311yescharacteristic
218#103—1731no—
219#103—1731no—
220#103—1731no—
221#103—1731no—
222#103C2×D17311yescharacteristic
223#103C2×D17311yescharacteristic
224#102—1731no—
225#102—1731no—
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×Q8—1731no—
235C2×Q8C22×D17311yescharacteristic
236C2×D4—3461no—
237C2×D4—1731no—
238C22×C4—3461no—
239C22×C4—3461no—
240C22×C4—3461no—
241C22×C4—1731no—
242C22×C4—1731no—
243C22×C4—1731no—
244Q16—3462no—
245Q16—3462no—
246SD16—3462no—
247SD16—3462no—
248C2×C8—1731no—
249C2×C8—1731no—
250C2×C8C22×D17311yescharacteristic
251C2×C8C22×D17311yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—3461no—
256C4⋊C4—3461no—
257C4⋊C4—3461no—
258C4⋊C4—3461no—
259C4⋊C4—3461no—
260C4⋊C4—3461no—
261C4⋊C4—1731no—
262C4⋊C4C22×D17311yescharacteristic
263C4⋊C4C22×D17311yescharacteristic
264C4⋊C4C22×D17311yescharacteristic
265C22⋊C4—3461no—
266C22⋊C4—3461no—
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—3461no—
280C23—1731no—
281Q8—21no—
282Q8—3464no—
283Q8—3461no—
284Q8—3461no—
285Q8—3461no—
286Q8—1732no—
287Q8#1458382712yes—
288D4—3461no—
289D4—3461no—
290D4—6922no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—3464no—
296C2×C4—3462no—
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—3461no—
316C22—3461no—
317C22—1732no—
318C22—6921no—
319C22—6921no—
320C22—6921no—
321C22C2×D4×D17311yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—3462no—
329C4—3461no—
330C4—3461no—
331C4—3461no—
332C4—6921no—
333C4—6921no—
334C4#41×D17311yescharacteristic
335C4C2×#1458382711yescharacteristic
336C2—3462no—
337C2—13841no—
338C2#2601200411yescharacteristic
339C2#137×D17311yescharacteristic
340C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#479×C173C2
socleC22×C173C22×D4
prime radical#19149654C22
hypercenter#479D173
hypocenterC173#3117
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D173C2semidirect
3. #26011828C2semidirect
5. #26011795C2semidirect
6. #26011714C2semidirect
7. #26011713C2semidirect
8. #26011648C2semidirect
10. #26011597C2semidirect
11. #26011595C2semidirect
12. #26011594C2semidirect
13. #26011590C2semidirect
14. #26011547C2semidirect
15. #479×C173C2semidirect
170. C173#3117semidirect
177. #479D173semidirect

Maximal subgroups

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

subgroupindex
#260120262
C4×Q8×D1732
#260118282
#260118112
#260117952
#260117142
#260117132
#260116482
#260116312
#260115972
#260115952
#260115942
#260115902
#260115472
#479×C1732
#3117173