Finite Group Databasebeta

← #36361780

Subgroups — #36361780

Order 22144.

400 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26011432C211yesmaximal, characteristic, complemented
2#26011427C211yesmaximal, characteristic, complemented
3#26011427C211yesmaximal, characteristic, complemented
4#26011424C211yesmaximal, characteristic, complemented
5#26011422C211yesmaximal, characteristic, complemented
6#26011419C211yesmaximal, characteristic, complemented
7#26011419C211yesmaximal, characteristic, complemented
8#26011414C211yesmaximal, characteristic, complemented
9#26011414C211yesmaximal, characteristic, complemented
10#26011405C211yesmaximal, characteristic
11#26011405C211yesmaximal, characteristic
12#26011388C211yesmaximal, characteristic, complemented
13#26011385C211yesmaximal, characteristic, complemented
14#26011379C211yesmaximal, characteristic, complemented
15#394×C173C211yesmaximal, characteristic
16C2×C4⋊Dic173C2211yescharacteristic
17C2×C4⋊Dic173C2211yescharacteristic
18C2×#14583805C2211yescharacteristic
19C2×#14583805C2211yescharacteristic
20C2×#14583805C2211yescharacteristic
21C2×#14583805C2211yescharacteristic
22C2×#14583805C2211yescharacteristic
23C2×#14583805C2211yescharacteristic
24C2×#14583805C2211yescharacteristic
25C2×#14583805C2211yescharacteristic
26C2×C22⋊Dic173C2211yescharacteristic
27C2×C22⋊Dic173C2211yescharacteristic
28C2×C22⋊Dic173C2211yescharacteristic
29C2×C22⋊Dic173C2211yescharacteristic
30C2×C22⋊Dic173C2211yescharacteristic
31C2×C22⋊Dic173C2211yescharacteristic
32C2×C22⋊Dic173C2211yescharacteristic
33C2×C22⋊Dic173C2211yescharacteristic
34C2×C4×Dic173C2211yescharacteristic
35#19149586C2211yescharacteristic
36#19149586C2211yescharacteristic
37#19149586C2211yescharacteristic
38#19149586C2211yescharacteristic
39#19149586C2211yescharacteristic
40#19149586C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44C22×D4×C173C2211yescharacteristic
45C2×C4⋊C4×C173C2211yescharacteristic
46C2×C22⋊C4×C173C2211yescharacteristic
47C2×C22⋊C4×C173C2211yescharacteristic
48C2×C22⋊C4×C173C2211yescharacteristic
49C2×C22⋊C4×C173C2211yescharacteristic
50#95×C173C2211yescharacteristic
51C22×Dic173C2311yescharacteristic
52C22×Dic173C2311yescharacteristic
53C22×Dic173C2311yescharacteristic
54C22×Dic173C2311yescharacteristic
55C22×Dic173C2311yescharacteristic
56C22×Dic173C2311yescharacteristic
57C22×Dic173C2311yescharacteristic
58C22×Dic173C2311yescharacteristic
59C4⋊Dic173—41no—
60C4⋊Dic173—41no—
61#14583805—22no—
62#14583805—41no—
63#14583805—41no—
64#14583805—41no—
65#14583805—41no—
66#14583805—41no—
67#14583805—41no—
68#14583805—41no—
69C22⋊Dic173—22no—
70C22⋊Dic173—22no—
71C22⋊Dic173—41no—
72C22⋊Dic173—41no—
73C22⋊Dic173—41no—
74C22⋊Dic173—41no—
75C22⋊Dic173—41no—
76C22⋊Dic173—41no—
77C4×Dic173—41no—
78C24×C173C2311yescharacteristic
79C24×C173C2311yescharacteristic
80C2×D4×C173—41no—
81C2×D4×C173—41no—
82C2×D4×C173—41no—
83C22×C4×C173C2311yescharacteristic
84C22×C4×C173C2311yescharacteristic
85C22×C4×C173C2311yescharacteristic
86C22×C4×C173C2311yescharacteristic
87C22×C4×C173C2311yescharacteristic
88C4⋊C4×C173—41no—
89C22⋊C4×C173—41no—
90C22⋊C4×C173—41no—
91C22⋊C4×C173—41no—
92C22⋊C4×C173—41no—
93C2×Dic173—21no—
94C2×Dic173—21no—
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—21no—
116C2×Dic173#4112yes—
117C23×C173—21no—
118C23×C173—21no—
119C23×C173—21no—
120C23×C173—21no—
121C23×C173—21no—
122C23×C173—21no—
123C23×C173—21no—
124C23×C173—21no—
125C23×C173—21no—
126C23×C173—21no—
127C23×C173—21no—
128C23×C173—21no—
129C23×C173—21no—
130C23×C173—21no—
131C23×C173C2411yescharacteristic
132D4×C173—44no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
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—21no—
147C2×C4×C173—21no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173—41no—
151C22×C173—41no—
152C22×C173—41no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173—41no—
156C22×C173—41no—
157C22×C173—41no—
158C22×C173—41no—
159C22×C173—41no—
160C22×C173—41no—
161C22×C173—41no—
162C22×C173C2×#4111yescharacteristic
163C22×C173H(5,𝔽2)11yescharacteristic
164C22×C173H(5,𝔽2)11yescharacteristic
165C22×C173H(5,𝔽2)11yescharacteristic
166C22×C173H(5,𝔽2)11yescharacteristic
167C22×C1732−1+411yescharacteristic
168C22×C1732−1+411yescharacteristic
169C4×C173—41no—
170C4×C173—41no—
171C4×C173—41no—
172C4×C173—41no—
173C4×C173—41no—
174Dic173—22no—
175Dic173—41no—
176Dic173—41no—
177Dic173—41no—
178Dic173—41no—
179Dic173—41no—
180Dic173—41no—
181Dic173—41no—
182C2×C173—81no—
183C2×C173—81no—
184C2×C173#53911yescharacteristic
185C2×C173#54011yescharacteristic
186C2×C173#54211yescharacteristic
187C2×C173#56011yescharacteristic
188C2×C173#56211yescharacteristic
189C2×C173#56211yescharacteristic
190C2×C173#56311yescharacteristic
191C173#283711yescharacteristic, complemented, Sylow
192#2837—1731nomaximal, Sylow
193#400—1731no—
194#399—1731no—
195#399—1731no—
196#398—1731no—
197#398—1731no—
198#397—1731no—
199#397—1731no—
200#396—1731no—
201#396—1731no—
202#394—1731no—
203#394—1731no—
204#394D17311yescharacteristic
205#389—1731no—
206#388—1731no—
207#388—1731no—
208C22×D4C2×D17311yescharacteristic
209C2×C4⋊C4—1731no—
210C2×C4⋊C4—1731no—
211C2×C4⋊C4—1731no—
212C2×C4⋊C4—1731no—
213C2×C4⋊C4—1731no—
214C2×C4⋊C4—1731no—
215C2×C4⋊C4—1731no—
216C2×C4⋊C4—1731no—
217C2×C4⋊C4—1731no—
218C2×C4⋊C4—1731no—
219C2×C4⋊C4C2×D17311yescharacteristic
220C2×C22⋊C4—1731no—
221C2×C22⋊C4—1731no—
222C2×C22⋊C4—1731no—
223C2×C22⋊C4—1731no—
224C2×C22⋊C4—1731no—
225C2×C22⋊C4—1731no—
226C2×C22⋊C4—1731no—
227C2×C22⋊C4—1731no—
228C2×C22⋊C4C2×D17311yescharacteristic
229C2×C22⋊C4C2×D17311yescharacteristic
230C2×C22⋊C4C2×D17311yescharacteristic
231C2×C22⋊C4C2×D17311yescharacteristic
232C2×C42—1731no—
233#95—1731no—
234#95—1731no—
235#95—1731no—
236#95—1731no—
237#95—1731no—
238#95—1731no—
239#95—1731no—
240#95—1731no—
241#95—1731no—
242#95C2×D17311yescharacteristic
243C24C22×D17311yescharacteristic
244C24C22×D17311yescharacteristic
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C22×C4—1731no—
249C22×C4—1731no—
250C22×C4—1731no—
251C22×C4—1731no—
252C22×C4—1731no—
253C22×C4—1731no—
254C22×C4—1731no—
255C22×C4—1731no—
256C22×C4C22×D17311yescharacteristic
257C22×C4C22×D17311yescharacteristic
258C22×C4C22×D17311yescharacteristic
259C22×C4C22×D17311yescharacteristic
260C22×C4C22×D17311yescharacteristic
261C4⋊C4—41no—
262C4⋊C4—3462no—
263C4⋊C4—6921no—
264C4⋊C4—6921no—
265C4⋊C4—6921no—
266C4⋊C4—6921no—
267C4⋊C4—6921no—
268C4⋊C4—6921no—
269C4⋊C4—6921no—
270C4⋊C4—6921no—
271C4⋊C4—6921no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—3462no—
277C22⋊C4—3462no—
278C22⋊C4—6921no—
279C22⋊C4—6921no—
280C22⋊C4—6921no—
281C22⋊C4—6921no—
282C22⋊C4—6921no—
283C22⋊C4—6921no—
284C42—6921no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23C23×D17311yescharacteristic
300D4—44no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—3461no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—3461no—
339C2×C4—1732no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22#41×D17311yescharacteristic
355C22#1914972411yescharacteristic
356C22#1914972411yescharacteristic
357C22#1914972411yescharacteristic
358C22#1914972411yescharacteristic
359C22#1914972611yescharacteristic
360C22#1914972611yescharacteristic
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—3462no—
367C4—6921no—
368C4—6921no—
369C4—6921no—
370C4—6921no—
371C4—6921no—
372C4—6921no—
373C4—6921no—
374C2—81no—
375C2—81no—
376C2#2601208411yescharacteristic
377C2#2601209111yescharacteristic
378C2#2601210511yescharacteristic
379C2#2601223011yescharacteristic
380C2#2601223611yescharacteristic
381C2#2601223611yescharacteristic
382C2#2601224411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#394×C173C2
prime radicalC22×D4×C173C22
hypercenter#394D173
hypocenterC173#2837
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #26011432C2semidirect
2. #26011427C2semidirect
3. #26011427C2semidirect
4. #26011424C2semidirect
5. #26011422C2semidirect
6. #26011419C2semidirect
7. #26011419C2semidirect
8. #26011414C2semidirect
9. #26011414C2semidirect
12. #26011388C2semidirect
13. #26011385C2semidirect
14. #26011379C2semidirect
191. C173#2837semidirect

Maximal subgroups

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

subgroupindex
#260114322
#260114272
#260114242
#260114222
#260114192
#260114142
#260114052
#260113882
#260113852
#260113792
#394×C1732
#2837173