Finite Group Databasebeta

← #36361799

Subgroups — #36361799

Order 22144.

366 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C173⋊#401C211yesmaximal, characteristic, complemented
2#26011433C211yesmaximal, characteristic, complemented
3#26011426C211yesmaximal, characteristic
4#26011421C211yesmaximal, characteristic, complemented
5#26011420C211yesmaximal, characteristic
6#26011420C211yesmaximal, characteristic
7#26011418C211yesmaximal, characteristic
8#26011413C211yesmaximal, characteristic
9#26011412C211yesmaximal, characteristic, complemented
10#26011411C211yesmaximal, characteristic, complemented
11#26011404C211yesmaximal, characteristic
12#26011388C211yesmaximal, characteristic, complemented
13#26011383C211yesmaximal, characteristic
14#26011374C211yesmaximal, characteristic, complemented
15#400×C173C211yesmaximal, characteristic, complemented
16C22×Dic346C2211yescharacteristic
17C2×C4⋊Dic173C2211yescharacteristic
18C2×C4⋊Dic173C2211yescharacteristic
19C2×C4⋊Dic173C2211yescharacteristic
20C2×#14583805C2211yescharacteristic
21C2×#14583805C2211yescharacteristic
22C2×#14583805C2211yescharacteristic
23C2×#14583805C2211yescharacteristic
24C2×#14583805C2211yescharacteristic
25C2×#14583803C2211yescharacteristic
26C2×#14583803C2211yescharacteristic
27C2×#14583803C2211yescharacteristic
28C2×#14583803C2211yescharacteristic
29C2×#14583803C2211yescharacteristic
30C2×#14583803C2211yescharacteristic
31C2×#14583803C2211yescharacteristic
32C2×C4×Dic173C2211yescharacteristic
33#19149586C2211yescharacteristic
34#19149586C2211yescharacteristic
35#19149586C2211yescharacteristic
36#19149586C2211yescharacteristic
37#19149586C2211yescharacteristic
38#19149586C2211yescharacteristic
39#19149586C2211yescharacteristic
40#19149586C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44C2×C4⋊C4×C173C2211yescharacteristic
45C2×C4⋊C4×C173C2211yescharacteristic
46#95×C173C2211yescharacteristic
47#95×C173C2211yescharacteristic
48#95×C173C2211yescharacteristic
49#95×C173C2211yescharacteristic
50#95×C173C2211yescharacteristic
51C23×D173C2311yescharacteristic
52C2×Dic346—41no—
53C2×Dic346—41no—
54C2×Dic346—41no—
55C22×Dic173C2311yescharacteristic
56C22×Dic173C2311yescharacteristic
57C22×Dic173C2311yescharacteristic
58C22×Dic173C2311yescharacteristic
59C22×Dic173C2311yescharacteristic
60C22×Dic173C2311yescharacteristic
61C22×Dic173C2311yescharacteristic
62C4⋊Dic173—22no—
63C4⋊Dic173—41no—
64C4⋊Dic173—41no—
65#14583805—22no—
66#14583805—41no—
67#14583805—41no—
68#14583805—41no—
69#14583805—41no—
70#14583803—22no—
71#14583803—41no—
72#14583803—41no—
73#14583803—41no—
74#14583803—41no—
75#14583803—41no—
76#14583803—41no—
77C4×Dic173—41no—
78C22×C4×C173C2311yescharacteristic
79C22×C4×C173C2311yescharacteristic
80C22×C4×C173C2311yescharacteristic
81C22×C4×C173C2311yescharacteristic
82C22×C4×C173C2311yescharacteristic
83C22×C4×C173C2311yescharacteristic
84C22×C4×C173C2311yescharacteristic
85C4⋊C4×C173—41no—
86C4⋊C4×C173—41no—
87C22×D173—21no—
88C22×D173—21no—
89C22×D173—21no—
90C22×D173—21no—
91C22×D173—21no—
92C22×D173—21no—
93C22×D173—21no—
94Dic346—44no—
95C2×Dic173—21no—
96C2×Dic173—21no—
97C2×Dic173—21no—
98C2×Dic173—21no—
99C2×Dic173—21no—
100C2×Dic173—21no—
101C2×Dic173—21no—
102C2×Dic173—21no—
103C2×Dic173—21no—
104C2×Dic173—21no—
105C2×Dic173—21no—
106C2×Dic173—21no—
107C2×Dic173—21no—
108C2×Dic173—21no—
109C2×Dic173—21no—
110C2×Dic173—21no—
111C2×Dic173—21no—
112C2×Dic173—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173#4112yes—
116C23×C173C2411yescharacteristic
117C2×C4×C173—21no—
118C2×C4×C173—21no—
119C2×C4×C173—21no—
120C2×C4×C173—21no—
121C2×C4×C173—21no—
122C2×C4×C173—21no—
123C2×C4×C173—21no—
124C2×C4×C173—21no—
125C2×C4×C173—21no—
126C2×C4×C173—21no—
127C2×C4×C173—21no—
128C2×C4×C173—21no—
129C2×C4×C173—21no—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C22×C173C2×#4111yescharacteristic
139C22×C173H(5,𝔽2)11yescharacteristic
140C22×C173H(5,𝔽2)11yescharacteristic
141C22×C173H(5,𝔽2)11yescharacteristic
142C22×C1732−1+411yescharacteristic
143C22×C1732−1+411yescharacteristic
144C22×C1732−1+411yescharacteristic
145C2×D173—41no—
146C2×D173—41no—
147C2×D173—41no—
148C2×D173—41no—
149C2×D173—41no—
150C2×D173—41no—
151C2×D173—41no—
152C4×C173—41no—
153C4×C173—41no—
154C4×C173—41no—
155C4×C173—41no—
156C4×C173—41no—
157C4×C173—41no—
158C4×C173—41no—
159Dic173—22no—
160Dic173—41no—
161Dic173—41no—
162Dic173—41no—
163Dic173—41no—
164Dic173—41no—
165Dic173—41no—
166C2×C173#53911yescharacteristic
167C2×C173#54211yescharacteristic
168C2×C173#54211yescharacteristic
169C2×C173#56011yescharacteristic
170C2×C173#56311yescharacteristic
171C2×C173#56311yescharacteristic
172C2×C173#56311yescharacteristic
173D173—81no—
174C173#283811yescharacteristic, complemented, Sylow
175#2838—1731nomaximal, Sylow
176#401—1731no—
177#400—1731no—
178#400D17311yescharacteristic, complemented
179#399—1731no—
180#398—1731no—
181#397—1731no—
182#397—1731no—
183#397—1731no—
184#396—1731no—
185#395—1731no—
186#395—1731no—
187#393—1731no—
188#389—1731no—
189#388—1731no—
190#387—1731no—
191C22×Q8—1731no—
192C2×C4⋊C4—1731no—
193C2×C4⋊C4—1731no—
194C2×C4⋊C4—1731no—
195C2×C4⋊C4—1731no—
196C2×C4⋊C4—1731no—
197C2×C4⋊C4—1731no—
198C2×C4⋊C4—1731no—
199C2×C4⋊C4—1731no—
200C2×C4⋊C4C2×D17311yescharacteristic
201C2×C4⋊C4C2×D17311yescharacteristic
202C2×C22⋊C4—1731no—
203C2×C22⋊C4—1731no—
204C2×C22⋊C4—1731no—
205C2×C22⋊C4—1731no—
206C2×C22⋊C4—1731no—
207C2×C22⋊C4—1731no—
208C2×C22⋊C4—1731no—
209C2×C42—1731no—
210#95—1731no—
211#95—1731no—
212#95—1731no—
213#95—1731no—
214#95—1731no—
215#95—1731no—
216#95—1731no—
217#95—1731no—
218#95—1731no—
219#95—1731no—
220#95—1731no—
221#95C2×D17311yescharacteristic
222#95C2×D17311yescharacteristic
223#95C2×D17311yescharacteristic
224#95C2×D17311yescharacteristic
225#95C2×D17311yescharacteristic
226C24—1731no—
227C2×Q8—6921no—
228C2×Q8—6921no—
229C2×Q8—6921no—
230C22×C4—1731no—
231C22×C4—1731no—
232C22×C4—1731no—
233C22×C4—1731no—
234C22×C4—1731no—
235C22×C4—1731no—
236C22×C4—1731no—
237C22×C4C22×D17311yescharacteristic
238C22×C4C22×D17311yescharacteristic
239C22×C4C22×D17311yescharacteristic
240C22×C4C22×D17311yescharacteristic
241C22×C4C22×D17311yescharacteristic
242C22×C4C22×D17311yescharacteristic
243C22×C4C22×D17311yescharacteristic
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—3462no—
247C4⋊C4—3462no—
248C4⋊C4—6921no—
249C4⋊C4—6921no—
250C4⋊C4—6921no—
251C4⋊C4—6921no—
252C4⋊C4—6921no—
253C4⋊C4—6921no—
254C22⋊C4—3462no—
255C22⋊C4—6921no—
256C22⋊C4—6921no—
257C22⋊C4—6921no—
258C22⋊C4—6921no—
259C22⋊C4—6921no—
260C22⋊C4—6921no—
261C42—6921no—
262C23—3461no—
263C23—3461no—
264C23—3461no—
265C23—3461no—
266C23—3461no—
267C23—3461no—
268C23—3461no—
269C23C23×D17311yescharacteristic
270Q8—6924no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—3461no—
293C2×C4—3461no—
294C2×C4—3461no—
295C2×C4—3461no—
296C2×C4—3461no—
297C2×C4—3461no—
298C2×C4—3461no—
299C2×C4—3461no—
300C2×C4—3461no—
301C2×C4—3461no—
302C2×C4—3461no—
303C2×C4—3461no—
304C2×C4—3461no—
305C2×C4—3461no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—1732no—
313C22—6921no—
314C22—6921no—
315C22—6921no—
316C22—6921no—
317C22—6921no—
318C22—6921no—
319C22—6921no—
320C22#41×D17311yescharacteristic
321C22#1914972411yescharacteristic
322C22#1914972511yescharacteristic
323C22#1914972511yescharacteristic
324C22#1914972611yescharacteristic
325C22#1914972611yescharacteristic
326C22#1914972711yescharacteristic
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—3462no—
335C4—6921no—
336C4—6921no—
337C4—6921no—
338C4—6921no—
339C4—6921no—
340C4—6921no—
341C2—13841no—
342C2#2601208911yescharacteristic
343C2#2601210511yescharacteristic
344C2#2601210711yescharacteristic
345C2#2601223111yescharacteristic
346C2#2601224211yescharacteristic
347C2#2601224311yescharacteristic
348C2#2601224511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#400×C173C2
prime radicalC23×D173C23
hypercenter#400D173
hypocenterC173#2838
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C173⋊#401C2semidirect
2. #26011433C2semidirect
4. #26011421C2semidirect
9. #26011412C2semidirect
10. #26011411C2semidirect
12. #26011388C2semidirect
14. #26011374C2semidirect
15. #400×C173C2semidirect
174. C173#2838semidirect
178. #400D173semidirect

Maximal subgroups

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

subgroupindex
C173⋊#4012
#260114332
#260114262
#260114212
#260114202
#260114182
#260114132
#260114122
#260114112
#260114042
#260113882
#260113832
#260113742
#400×C1732
#2838173