Finite Group Databasebeta

← #36364119

Subgroups — #36364119

Order 22144.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012045C211yesmaximal, characteristic, complemented
2#26012038C211yesmaximal, characteristic
3#26011888C211yesmaximal, characteristic, complemented
4#26011862C211yesmaximal, characteristic, complemented
5#26011840C211yesmaximal, characteristic, complemented
6#26011836C211yesmaximal, characteristic, complemented
7#26011723C211yesmaximal, characteristic, complemented
8#26011721C211yesmaximal, characteristic, complemented
9#26011719C211yesmaximal, characteristic, complemented
10#26011717C211yesmaximal, characteristic, complemented
11C4×#14583815C211yesmaximal, characteristic, complemented
12C4×#14583813C211yesmaximal, characteristic, complemented
13C4×#14583814C211yesmaximal, characteristic, complemented
14C4×C8×D173C211yesmaximal, characteristic, complemented
15C4×SD16×C173C211yesmaximal, characteristic, complemented
16C2×#14583826—21no—
17C2×#14583815C2211yescharacteristic
18C2×#14583814C2211yescharacteristic
19C2×#14583813C2211yescharacteristic
20C2×C8×D173C2211yescharacteristic
21C2×C8×D173C2211yescharacteristic
22#19149678C2211yescharacteristic
23#19149678C2211yescharacteristic
24#19149677C2211yescharacteristic
25#19149675C2211yescharacteristic
26#19149669—21no—
27#19149665—21no—
28#19149659C2211yescharacteristic
29#19149657—21no—
30#19149655C2211yescharacteristic
31#19149654C2211yescharacteristic
32#19149652—21no—
33#19149650C2211yescharacteristic
34C4×Dic346—21no—
35C4×Dic346C2211yescharacteristic
36C4×D692C2211yescharacteristic
37C4×C22⋊D173—21no—
38#19149636—21no—
39C42×D173C2211yescharacteristic
40#19149611C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149610C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149602C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149600C2211yescharacteristic
48#19149599C2211yescharacteristic
49C4×#13063436C2211yescharacteristic
50C8×Dic173C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C4×Q8×C173C2211yescharacteristic
54C4×D4×C173C2211yescharacteristic
55#106×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#102×C173C2211yescharacteristic
58C4×C8×C173C2211yescharacteristic
59#14583826—44no—
60C2×Dic346—21no—
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63C2×D692—21no—
64C2×D692C2311yescharacteristic
65C2×C22⋊D173—21no—
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C4×D173—21no—
69C2×C4×D173D411yescharacteristic
70C2×C4×D173D411yescharacteristic
71C2×C4×D173C2311yescharacteristic
72#14583815—22no—
73#14583814—22no—
74#14583813—22no—
75C2×#13063436C2311yescharacteristic
76C2×#13063436C2311yescharacteristic
77C8×D173—22no—
78C8×D173—22no—
79C4⋊Dic173—21no—
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82#14583805—21no—
83#14583805—21no—
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—
93C4×Dic173—21no—
94C4×Dic173D411yescharacteristic
95C4×Dic173D411yescharacteristic
96C4×Dic173C2311yescharacteristic
97C2×Q8×C173C2311yescharacteristic
98C2×D4×C173C2311yescharacteristic
99C22×C4×C173—21no—
100SD16×C173—22no—
101C2×C8×C173C2311yescharacteristic
102C2×C8×C173C2311yescharacteristic
103C4⋊C4×C173—21no—
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C42×C173—21no—
108C42×C173C2311yescharacteristic
109C22×D173—21no—
110C22×D173C2×D411yescharacteristic
111Dic346—21no—
112Dic346—21no—
113Dic346—42no—
114Dic346—42no—
115D692—21no—
116D692—21no—
117D692—42no—
118C22⋊D173—42no—
119C22⋊D173—42no—
120C22⋊D173—42no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C4×D173—21no—
128C4×D173—21no—
129C4×D173—22no—
130C4×D173—42no—
131C4×D173SD1614yescomplemented
132#13063436—21no—
133#13063436#4112yescomplemented
134C23×C173—21no—
135Q8×C173—21no—
136Q8×C173—21no—
137D4×C173—21no—
138D4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—42no—
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2411yescharacteristic
146C8×C173—21no—
147C8×C173#4112yescomplemented
148C22×C173—41no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173C22×D411yescharacteristic
152C2×D173—21no—
153C2×D173—21no—
154C2×D173—41no—
155C2×D173—41no—
156C2×D173—41no—
157C2×D173C2×SD1612yes—
158C4×C173—21no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173C2×SD1612yes—
163C4×C173C2×#4111yescharacteristic
164C4×C173C2×#4111yescharacteristic
165Dic173—21no—
166Dic173—41no—
167Dic173—41no—
168Dic173—41no—
169Dic173C2×SD1612yes—
170Dic173#13512yes—
171C2×C173—81no—
172C2×C173#53211yescharacteristic
173C2×C173C22×SD1611yescharacteristic
174C2×C173C2×#13511yescharacteristic
175D173—22no—
176D173—81no—
177C173#316711yescharacteristic, complemented, Sylow
178#3167—1731nomaximal, Sylow
179#533—1731no—
180#532—1731no—
181#498—1731no—
182#492—1731no—
183#487—1731no—
184#486—1731no—
185#465—1731no—
186#465—1731no—
187#465—1731no—
188#465—1731no—
189C4×SD16—1731no—
190C4×SD16—1731no—
191C4×SD16—1731no—
192C4×SD16D17311yescharacteristic, complemented
193C2×C4×C8—1731no—
194C2×#41—3461no—
195C2×SD16—1731no—
196C2×SD16—1731no—
197C2×SD16—1731no—
198C2×SD16C2×D17311yescharacteristic
199C22×C8—1731no—
200C22×C8—1731no—
201C4⋊Q8—1731no—
202C4⋊Q8—1731no—
203C4⋊Q8—1731no—
204#127—1731no—
205#125—3461no—
206#124—3461no—
207C22⋊Q8—3461no—
208C22⋊Q8—1731no—
209C22⋊Q8—1731no—
210#121—3461no—
211#121—1731no—
212#121—1731no—
213C4×Q8—3461no—
214C4×Q8—1731no—
215C4×Q8C2×D17311yescharacteristic
216C4×D4—3461no—
217C4×D4—1731no—
218C4×D4C2×D17311yescharacteristic
219#117—3461no—
220C2×C42—1731no—
221#106—1731no—
222#106—1731no—
223#106—1731no—
224#106C2×D17311yescharacteristic
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103C2×D17311yescharacteristic
229#102—1731no—
230#102—1731no—
231#102—1731no—
232#102C2×D17311yescharacteristic
233C4×C8—1731no—
234C4×C8—1731no—
235C4×C8—1731no—
236C4×C8C2×D17311yescharacteristic
237#41—6924no—
238C2×Q8—3461no—
239C2×Q8—3461no—
240C2×Q8—1731no—
241C2×Q8C22×D17311yescharacteristic
242C2×D4—3461no—
243C2×D4—3461no—
244C2×D4—3461no—
245C2×D4—3461no—
246C2×D4—1731no—
247C2×D4C22×D17311yescharacteristic
248C22×C4—21no—
249C22×C4—3461no—
250C22×C4—1731no—
251C22×C4—1731no—
252C22×C4—1731no—
253SD16—22no—
254SD16—3462no—
255SD16—3462no—
256SD16—3462no—
257C2×C8—3462no—
258C2×C8—3462no—
259C2×C8—1731no—
260C2×C8—1731no—
261C2×C8C22×D17311yescharacteristic
262C2×C8C22×D17311yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—3461no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—3461no—
271C4⋊C4—1731no—
272C4⋊C4—1731no—
273C4⋊C4C22×D17311yescharacteristic
274C4⋊C4C22×D17311yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C42—21no—
282C42—3461no—
283C42—1731no—
284C42—1731no—
285C42—1731no—
286C42C22×D17311yescharacteristic
287C23—21no—
288C23—3461no—
289C23—1731no—
290Q8—21no—
291Q8—21no—
292Q8—3461no—
293Q8—3461no—
294Q8—6922no—
295Q8—6922no—
296D4—21no—
297D4—21no—
298D4—3461no—
299D4—3461no—
300D4—6922no—
301D4—6922no—
302D4—6922no—
303D4—6922no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—3462no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—1731no—
315C2×C4—1731no—
316C2×C4—1731no—
317C2×C4—1734no—
318C2×C4—6922no—
319C2×C4D4×D17311yescharacteristic
320C2×C4D4×D17311yescharacteristic
321C2×C4C23×D17311yescharacteristic
322C8—21no—
323C8—3461no—
324C8—1732no—
325C8#1458382612yescomplemented
326C22—41no—
327C22—41no—
328C22—41no—
329C22—3461no—
330C22—3461no—
331C22—1732no—
332C22—6921no—
333C22—6921no—
334C22—6921no—
335C22C2×D4×D17311yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—3461no—
341C4—1732no—
342C4—1732no—
343C4—6921no—
344C4—6921no—
345C4—6921no—
346C4SD16×D17312yes—
347C4#41×D17311yescharacteristic
348C4C2×#1458382611yescharacteristic
349C2—81no—
350C2—3462no—
351C2—13841no—
352C2#2601203811yescharacteristic
353C2C2×SD16×D17311yescharacteristic
354C2#135×D17311yescharacteristic
subgroupG/H
centerC2×C4D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC4×SD16×C173C2
socleC22×C173C22×D4
prime radical#26012038C2
hypercenterC4×SD16D173
hypocenterC173#3167
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #26012045C2semidirect
3. #26011888C2semidirect
4. #26011862C2semidirect
5. #26011840C2semidirect
6. #26011836C2semidirect
7. #26011723C2semidirect
8. #26011721C2semidirect
9. #26011719C2semidirect
10. #26011717C2semidirect
11. C4×#14583815C2semidirect
12. C4×#14583813C2semidirect
13. C4×#14583814C2semidirect
14. C4×C8×D173C2semidirect
15. C4×SD16×C173C2semidirect
131. C4×D173SD16semidirect
133. #13063436#41semidirect
147. C8×C173#41semidirect
177. C173#3167semidirect
192. C4×SD16D173semidirect
325. C8#14583826semidirect

Maximal subgroups

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

subgroupindex
#260120452
#260120382
#260118882
#260118622
#260118402
#260118362
#260117232
#260117212
#260117192
#260117172
C4×#145838152
C4×#145838132
C4×#145838142
C4×C8×D1732
C4×SD16×C1732
#3167173