Finite Group Databasebeta

← #36364977

Subgroups — #36364977

Order 22144.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012114C211yesmaximal, characteristic, complemented
2#26012094C211yesmaximal, characteristic, complemented
3#26011800C211yesmaximal, characteristic, complemented
4#26011790C211yesmaximal, characteristic, complemented
5#26011780C211yesmaximal, characteristic, complemented
6#26011769C211yesmaximal, characteristic, complemented
7#26011742C211yesmaximal, characteristic, complemented
8#26011739C211yesmaximal, characteristic, complemented
9#26011727C211yesmaximal, characteristic, complemented
10#26011709C211yesmaximal, characteristic, complemented
11#26011708C211yesmaximal, characteristic, complemented
12#26011689C211yesmaximal, characteristic, complemented
13#26011685C211yesmaximal, characteristic, complemented
14#26011517C211yesmaximal, characteristic, complemented
15#465×C173C211yesmaximal, characteristic, complemented
16C2×#14583813C2211yescharacteristic
17C2×D1384C2211yescharacteristic
18C2×#14583811C2211yescharacteristic
19C2×#14583810C2211yescharacteristic
20#19149678C2211yescharacteristic
21#19149675C2211yescharacteristic
22#19149674—21no—
23#19149670C2211yescharacteristic
24#19149668C2211yescharacteristic
25#19149665—21no—
26#19149663—21no—
27#19149661—21no—
28#19149661—21no—
29#19149659—21no—
30#19149653C2211yescharacteristic
31#19149653C2211yescharacteristic
32#19149651—21no—
33#19149650—21no—
34Q8×Dic173C2211yescharacteristic
35D4×Dic173C2211yescharacteristic
36#19149641C2211yescharacteristic
37#19149641C2211yescharacteristic
38#19149638C2211yescharacteristic
39#19149613C2211yescharacteristic
40#19149612C2211yescharacteristic
41#19149610C2211yescharacteristic
42#19149608C2211yescharacteristic
43#19149608C2211yescharacteristic
44#19149608C2211yescharacteristic
45#19149608C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149600C2211yescharacteristic
50#19149599C2211yescharacteristic
51#19149599C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#106×C173C2211yescharacteristic
57#103×C173C2211yescharacteristic
58#102×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×D692C2311yescharacteristic
61C2×D692C2311yescharacteristic
62C2×C22⋊D173—21no—
63C2×C22⋊D173—21no—
64C2×C22⋊D173—21no—
65C2×C22⋊D173—21no—
66C22×Dic173—21no—
67C2×C4×D173—21no—
68C2×C4×D173—21no—
69#14583813—41no—
70D1384—41no—
71#14583811—41no—
72#14583810—41no—
73C2×#13063436C2311yescharacteristic
74C2×#13063436C2311yescharacteristic
75C4⋊Dic173—21no—
76C4⋊Dic173—21no—
77C4⋊Dic173D411yescharacteristic
78C4⋊Dic173D411yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83C22⋊Dic173—21no—
84C22⋊Dic173D411yescharacteristic
85C22⋊Dic173D411yescharacteristic
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92C4×Dic173—21no—
93C4×Dic173C2311yescharacteristic
94C4×Dic173C2311yescharacteristic
95C2×Q8×C173C2311yescharacteristic
96C2×D4×C173—21no—
97C2×D4×C173C2311yescharacteristic
98C22×C4×C173C2311yescharacteristic
99SD16×C173—41no—
100C2×C8×C173—41no—
101C2×C8×C173C2311yescharacteristic
102C2×C8×C173C2311yescharacteristic
103C4⋊C4×C173—21no—
104C4⋊C4×C173C2311yescharacteristic
105C4⋊C4×C173C2311yescharacteristic
106C22⋊C4×C173—21no—
107C22⋊C4×C173—21no—
108C22×D173—21no—
109C22×D173—21no—
110Dic346—81no—
111D692—21no—
112D692—21no—
113D692—21no—
114D692—21no—
115C22⋊D173—81no—
116C22⋊D173—81no—
117C22⋊D173—81no—
118C22⋊D173—81no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—81no—
122C2×Dic173C2×D411yescharacteristic
123C2×Dic173C2×D411yescharacteristic
124C2×Dic173C2×D411yescharacteristic
125C2×Dic173C2×D411yescharacteristic
126C4×D173—81no—
127C4×D173—81no—
128#13063436—21no—
129#13063436—21no—
130C23×C173—21no—
131C23×C173C2×D411yescharacteristic
132Q8×C173—21no—
133Q8×C173—21no—
134D4×C173—21no—
135D4×C173—21no—
136D4×C173—81no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2411yescharacteristic
144C8×C173—21no—
145C8×C173—21no—
146C22×C173—21no—
147C22×C173—21no—
148C22×C173—21no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173C22×D411yescharacteristic
153C2×D173—41no—
154C2×D173—41no—
155C2×D173—41no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C4×C173—21no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173—41no—
163C4×C173H(5,𝔽2)11yescharacteristic
164C4×C1732−1+411yescharacteristic
165Dic173—21no—
166Dic173—21no—
167Dic173—21no—
168Dic173—21no—
169Dic173—41no—
170Dic173—41no—
171C2×C173—41no—
172C2×C173—81no—
173C2×C173#53711yescharacteristic
174C2×C173#57611yescharacteristic
175C2×C173#57711yescharacteristic
176D173—81no—
177D173—81no—
178C173#325711yescharacteristic, complemented, Sylow
179#3257—1731nomaximal, Sylow
180#542—1731no—
181#540—1731no—
182#479—1731no—
183#478—1731no—
184#476—1731no—
185#474—1731no—
186#469—1731no—
187#468—1731no—
188#466—1731no—
189#465D17311yescharacteristic, complemented
190#463—1731no—
191#463—1731no—
192#460—1731no—
193#459—1731no—
194#424—1731no—
195C2×SD16—1731no—
196C2×SD16C2×D17311yescharacteristic
197C2×D8—1731no—
198C2×D8—1731no—
199C2×M16—1731no—
200C22×C8C2×D17311yescharacteristic
201C4⋊Q8—1731no—
202#127—1731no—
203#126—3461no—
204#125—1731no—
205#124—3461no—
206#124—1731no—
207#123—3461no—
208#123—3461no—
209#123—3461no—
210C22⋊Q8—3461no—
211C22⋊Q8C2×D17311yescharacteristic
212#121—3461no—
213#121—3461no—
214#121—1731no—
215#121—1731no—
216#121C2×D17311yescharacteristic
217C4×Q8—1731no—
218C4×D4—1731no—
219C4×D4—1731no—
220C4×D4—1731no—
221#117—1731no—
222#107—1731no—
223#107—1731no—
224#106—1731no—
225#106C2×D17311yescharacteristic
226C4⋊C8—1731no—
227C4⋊C8—1731no—
228C4⋊C8—1731no—
229C4⋊C8—1731no—
230#103—1731no—
231#103C2×D17311yescharacteristic
232#102—1731no—
233#102—1731no—
234#102—1731no—
235#102—1731no—
236#102—1731no—
237#102C2×D17311yescharacteristic
238C2×Q8—3461no—
239C2×Q8C22×D17311yescharacteristic
240C2×D4—21no—
241C2×D4—3461no—
242C2×D4—3461no—
243C2×D4—3461no—
244C2×D4—3461no—
245C2×D4—1731no—
246C2×D4—1731no—
247C2×D4C22×D17311yescharacteristic
248C22×C4—3461no—
249C22×C4—3461no—
250C22×C4—3461no—
251C22×C4C22×D17311yescharacteristic
252SD16—41no—
253SD16—6921no—
254D8—6921no—
255D8—6921no—
256M16—6921no—
257C2×C8—41no—
258C2×C8—1731no—
259C2×C8—1731no—
260C2×C8C22×D17311yescharacteristic
261C2×C8C22×D17311yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—3461no—
264C4⋊C4—3461no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—1731no—
268C4⋊C4—1731no—
269C4⋊C4—1731no—
270C4⋊C4—1731no—
271C4⋊C4C22×D17311yescharacteristic
272C4⋊C4C22×D17311yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—3461no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4—1731no—
283C22⋊C4—1731no—
284C42—3461no—
285C42—1731no—
286C42—1731no—
287C23—21no—
288C23—3461no—
289C23—3461no—
290C23D4×D17311yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—13841no—
294D4—21no—
295D4—21no—
296D4—81no—
297D4—3461no—
298D4—3461no—
299D4—3461no—
300D4—3461no—
301D4—13841no—
302D4—13841no—
303D4—13841no—
304D4—13841no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—13841no—
313C2×C4—13841no—
314C2×C4—13841no—
315C2×C4—1731no—
316C2×C4—1731no—
317C2×C4—1731no—
318C2×C4—1731no—
319C2×C4D4×D17311yescharacteristic
320C2×C4C23×D17311yescharacteristic
321C8—21no—
322C8—21no—
323C8—3461no—
324C8—3461no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—6921no—
332C22—6921no—
333C22—6921no—
334C22—6921no—
335C22—6921no—
336C22—6921no—
337C22C2×D4×D17311yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—3461no—
343C4—3461no—
344C4—3461no—
345C4—3461no—
346C4—6921no—
347C4—6921no—
348C4#1914972411yescharacteristic
349C4#1914972611yescharacteristic
350C2—41no—
351C2—81no—
352C2—13841no—
353C2—13841no—
354C2#2601206811yescharacteristic
355C2#2601229311yescharacteristic
356C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#465×C173C2
socleC22×C173C22×D4
hypercenter#465D173
hypocenterC173#3257
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #26012114C2semidirect
2. #26012094C2semidirect
3. #26011800C2semidirect
4. #26011790C2semidirect
5. #26011780C2semidirect
6. #26011769C2semidirect
7. #26011742C2semidirect
8. #26011739C2semidirect
9. #26011727C2semidirect
10. #26011709C2semidirect
11. #26011708C2semidirect
12. #26011689C2semidirect
13. #26011685C2semidirect
14. #26011517C2semidirect
15. #465×C173C2semidirect
178. C173#3257semidirect
189. #465D173semidirect

Maximal subgroups

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

subgroupindex
#260121142
#260120942
#260118002
#260117902
#260117802
#260117692
#260117422
#260117392
#260117272
#260117092
#260117082
#260116892
#260116852
#260115172
#465×C1732
#3257173