Finite Group Databasebeta

← #36364642

Subgroups — #36364642

Order 22144.

364 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#26012101C211yesmaximal, characteristic, complemented
2#26012065C211yesmaximal, characteristic
3#26011884C211yesmaximal, characteristic, complemented
4#26011869C211yesmaximal, characteristic, complemented
5#26011858C211yesmaximal, characteristic, complemented
6#26011847C211yesmaximal, characteristic, complemented
7#26011716C211yesmaximal, characteristic, complemented
8#26011715C211yesmaximal, characteristic, complemented
9#26011700C211yesmaximal, characteristic, complemented
10#26011699C211yesmaximal, characteristic, complemented
11#26011655C211yesmaximal, characteristic
12#26011651C211yesmaximal, characteristic
13#26011647C211yesmaximal, characteristic, complemented
14#26011549C211yesmaximal, characteristic, complemented
15#451×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C2×Dic692C2211yescharacteristic
18C2×#14583816C2211yescharacteristic
19C2×#14583816C2211yescharacteristic
20C2×#14583816C2211yescharacteristic
21C2×#14583815C2211yescharacteristic
22C2×#14583813C2211yescharacteristic
23#19149677C2211yescharacteristic
24#19149677C2211yescharacteristic
25#19149677C2211yescharacteristic
26#19149672—21no—
27#19149672—21no—
28#19149670—21no—
29#19149670C2211yescharacteristic
30#19149669—21no—
31#19149668C2211yescharacteristic
32#19149668C2211yescharacteristic
33#19149662—21no—
34#19149662—21no—
35#19149658C2211yescharacteristic
36#19149655—21no—
37#19149655—21no—
38Q8×Dic173C2211yescharacteristic
39#19149644C2211yescharacteristic
40#19149638C2211yescharacteristic
41#19149608C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149604C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149602C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149599C2211yescharacteristic
49C22⋊#13063436C2211yescharacteristic
50#19149589C2211yescharacteristic
51C8×Dic173C2211yescharacteristic
52C22×Q8×C173C2211yescharacteristic
53C2×Q16×C173C2211yescharacteristic
54C2×Q16×C173C2211yescharacteristic
55C22⋊Q8×C173C2211yescharacteristic
56#103×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—41no—
60#14583826—41no—
61C2×Dic346C2311yescharacteristic
62C2×Dic346C2311yescharacteristic
63C2×D692C2311yescharacteristic
64C2×C22⋊D173—21no—
65C2×C4×D173—21no—
66Dic692—41no—
67#14583816—41no—
68#14583816—41no—
69#14583816—41no—
70#14583815—41no—
71#14583813—41no—
72C2×#13063436C2311yescharacteristic
73C2×#13063436C2311yescharacteristic
74C4⋊Dic173—21no—
75C4⋊Dic173—21no—
76C4⋊Dic173D411yescharacteristic
77C4⋊Dic173D411yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79#14583805—21no—
80#14583805—21no—
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88C22⋊Dic173—21no—
89C22⋊Dic173D411yescharacteristic
90C22⋊Dic173D411yescharacteristic
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95C4×Dic173—21no—
96C4×Dic173—21no—
97C4×Dic173C2311yescharacteristic
98C4×Dic173C2311yescharacteristic
99C2×Q8×C173—21no—
100C2×Q8×C173—41no—
101C2×Q8×C173—41no—
102C2×Q8×C173C2311yescharacteristic
103C2×Q8×C173C2311yescharacteristic
104C2×Q8×C173C2311yescharacteristic
105C22×C4×C173—21no—
106C22×C4×C173C2311yescharacteristic
107Q16×C173—41no—
108Q16×C173—41no—
109C2×C8×C173C2311yescharacteristic
110C2×C8×C173C2311yescharacteristic
111C4⋊C4×C173—21no—
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22×D173—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—21no—
118Dic346—21no—
119D692—21no—
120D692—21no—
121C22⋊D173—42no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173C2×D411yescharacteristic
126C2×Dic173C2×D411yescharacteristic
127C2×Dic173C2×D411yescharacteristic
128C2×Dic173C2×D411yescharacteristic
129C4×D173—42no—
130#13063436—21no—
131#13063436—21no—
132C23×C173C2×D411yescharacteristic
133Q8×C173—21no—
134Q8×C173—21no—
135Q8×C173—21no—
136Q8×C173—21no—
137Q8×C173—21no—
138Q8×C173—21no—
139Q8×C173—42no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—42no—
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2411yescharacteristic
149C8×C173—21no—
150C8×C173—21no—
151C22×C173—21no—
152C22×C173—21no—
153C22×C173—21no—
154C22×C173C22×D411yescharacteristic
155C2×D173—41no—
156C2×D173—41no—
157C2×D173—41no—
158C4×C173—21no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173—41no—
163C4×C173H(5,𝔽2)11yescharacteristic
164C4×C173H(5,𝔽2)11yescharacteristic
165Dic173—21no—
166Dic173—21no—
167Dic173—21no—
168Dic173—21no—
169Dic173—41no—
170Dic173—41no—
171Dic173—41no—
172C2×C173—41no—
173C2×C173#53511yescharacteristic
174C2×C173#57811yescharacteristic
175C2×C173#57811yescharacteristic
176D173—81no—
177C173#322311yescharacteristic, complemented, Sylow
178#3223—1731nomaximal, Sylow
179#541—1731no—
180#536—1731no—
181#497—1731no—
182#494—1731no—
183#491—1731no—
184#488—1731no—
185#464—1731no—
186#464—1731no—
187#462—1731no—
188#462—1731no—
189#452—1731no—
190#452—1731no—
191#451—1731no—
192#451D17311yescharacteristic, complemented
193#433—1731no—
194C2×#41—1731no—
195C22×Q8C2×D17311yescharacteristic
196C2×Q16—1731no—
197C2×Q16—1731no—
198C2×Q16—1731no—
199C2×Q16—1731no—
200C2×Q16C2×D17311yescharacteristic
201C2×Q16C2×D17311yescharacteristic
202C2×SD16—1731no—
203C2×SD16—1731no—
204C4⋊Q8—1731no—
205C4⋊Q8—1731no—
206C4⋊Q8—1731no—
207#126—3461no—
208#126—3461no—
209#125—3461no—
210#125—3461no—
211#125—1731no—
212#124—1731no—
213#124—1731no—
214#123—3461no—
215#123—3461no—
216C22⋊Q8—3461no—
217C22⋊Q8—3461no—
218C22⋊Q8—1731no—
219C22⋊Q8C2×D17311yescharacteristic
220C4×Q8—1731no—
221C4×Q8—1731no—
222#117—1731no—
223C4⋊C8—1731no—
224C4⋊C8—1731no—
225#103—1731no—
226#103—1731no—
227#103—1731no—
228#103—1731no—
229#103C2×D17311yescharacteristic
230#103C2×D17311yescharacteristic
231#102—1731no—
232#102—1731no—
233C22⋊C8—1731no—
234C22⋊C8C2×D17311yescharacteristic
235#97—1731no—
236C4×C8—1731no—
237#41—6921no—
238#41—6921no—
239C2×Q8—21no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—1731no—
243C2×Q8—1731no—
244C2×Q8C22×D17311yescharacteristic
245C2×Q8C22×D17311yescharacteristic
246C2×Q8C22×D17311yescharacteristic
247C2×D4—3461no—
248C2×D4—1731no—
249C22×C4—21no—
250C22×C4—3461no—
251C22×C4C22×D17311yescharacteristic
252Q16—41no—
253Q16—41no—
254Q16—6921no—
255Q16—6921no—
256Q16—6921no—
257Q16—6921no—
258SD16—6921no—
259SD16—6921no—
260C2×C8—1731no—
261C2×C8—1731no—
262C2×C8C22×D17311yescharacteristic
263C2×C8C22×D17311yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—3461no—
268C4⋊C4—3461no—
269C4⋊C4—3461no—
270C4⋊C4—3461no—
271C4⋊C4—3461no—
272C4⋊C4—3461no—
273C4⋊C4—3461no—
274C4⋊C4—3461no—
275C4⋊C4—3461no—
276C4⋊C4—1731no—
277C4⋊C4—1731no—
278C4⋊C4—1731no—
279C4⋊C4C22×D17311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—3461no—
282C22⋊C4—3461no—
283C22⋊C4—3461no—
284C22⋊C4—3461no—
285C22⋊C4—3461no—
286C22⋊C4—1731no—
287C22⋊C4—1731no—
288C42—3461no—
289C42—3461no—
290C42—1731no—
291C42—1731no—
292C23—3461no—
293C23D4×D17311yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300Q8—42no—
301Q8—3461no—
302Q8—3461no—
303Q8—3461no—
304Q8—3461no—
305D4—3461no—
306D4—3461no—
307D4—6922no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—1731no—
319C2×C4—1731no—
320C2×C4—1731no—
321C2×C4—1731no—
322C2×C4—6922no—
323C2×C4D4×D17311yescharacteristic
324C2×C4C23×D17311yescharacteristic
325C8—21no—
326C8—21no—
327C8—3461no—
328C8—3461no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—6921no—
333C22—6921no—
334C22—6921no—
335C22C2×D4×D17311yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—3461no—
342C4—3461no—
343C4—3461no—
344C4—3461no—
345C4—6921no—
346C4—6921no—
347C4—6921no—
348C4#1914972411yescharacteristic
349C4#1914972411yescharacteristic
350C2—41no—
351C2—13841no—
352C2#2601205611yescharacteristic
353C2#2601230411yescharacteristic
354C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#451×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583826C22
hypercenter#451D173
hypocenterC173#3223
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #26012101C2semidirect
3. #26011884C2semidirect
4. #26011869C2semidirect
5. #26011858C2semidirect
6. #26011847C2semidirect
7. #26011716C2semidirect
8. #26011715C2semidirect
9. #26011700C2semidirect
10. #26011699C2semidirect
13. #26011647C2semidirect
14. #26011549C2semidirect
15. #451×C173C2semidirect
177. C173#3223semidirect
192. #451D173semidirect

Maximal subgroups

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

subgroupindex
#260121012
#260120652
#260118842
#260118692
#260118582
#260118472
#260117162
#260117152
#260117002
#260116992
#260116552
#260116512
#260116472
#260115492
#451×C1732
#3223173