Finite Group Databasebeta

← #36361824

Subgroups — #36361824

Order 22144.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26011432C211yesmaximal, characteristic, complemented
2#26011429C211yesmaximal, characteristic, complemented
3#26011427C211yesmaximal, characteristic
4#26011424C211yesmaximal, characteristic, complemented
5#26011422C211yesmaximal, characteristic, complemented
6#26011422C211yesmaximal, characteristic, complemented
7#26011421C211yesmaximal, characteristic, complemented
8#26011414C211yesmaximal, characteristic
9#26011412C211yesmaximal, characteristic, complemented
10#26011411C211yesmaximal, characteristic, complemented
11#26011403C211yesmaximal, characteristic
12#26011352C211yesmaximal, characteristic
13#26011349C211yesmaximal, characteristic
14#26011349C211yesmaximal, characteristic
15#396×C173C211yesmaximal, characteristic
16C22×Dic346C2211yescharacteristic
17C23×Dic173C2211yescharacteristic
18C2×C4⋊Dic173C2211yescharacteristic
19C2×C4⋊Dic173C2211yescharacteristic
20C2×#14583805C2211yescharacteristic
21C2×#14583805C2211yescharacteristic
22C2×#14583805C2211yescharacteristic
23C2×#14583805C2211yescharacteristic
24C2×#14583805C2211yescharacteristic
25C2×#14583805C2211yescharacteristic
26C2×#14583805C2211yescharacteristic
27C2×#14583805C2211yescharacteristic
28C2×C22⋊Dic173C2211yescharacteristic
29C2×C22⋊Dic173C2211yescharacteristic
30C2×C22⋊Dic173C2211yescharacteristic
31#19149586C2211yescharacteristic
32#19149586C2211yescharacteristic
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
46C2×C4⋊C4×C173C2211yescharacteristic
47C2×C22⋊C4×C173C2211yescharacteristic
48C2×C22⋊C4×C173C2211yescharacteristic
49C2×C22⋊C4×C173C2211yescharacteristic
50#95×C173C2211yescharacteristic
51C2×Dic346—41no—
52C2×Dic346—41no—
53C2×Dic346—41no—
54C22×Dic173—41no—
55C22×Dic173—41no—
56C22×Dic173Q814yescomplemented
57C22×Dic173C2311yescharacteristic
58C22×Dic173C2311yescharacteristic
59C22×Dic173C2311yescharacteristic
60C22×Dic173C2311yescharacteristic
61C22×Dic173C2311yescharacteristic
62C22×Dic173C2311yescharacteristic
63C22×Dic173C2311yescharacteristic
64C22×Dic173C2311yescharacteristic
65C4⋊Dic173—41no—
66C4⋊Dic173—41no—
67#14583805—41no—
68#14583805—41no—
69#14583805—41no—
70#14583805—41no—
71#14583805—41no—
72#14583805—41no—
73#14583805—41no—
74#14583805—41no—
75C22⋊Dic173—41no—
76C22⋊Dic173—41no—
77C22⋊Dic173—41no—
78C24×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—
87C4⋊C4×C173—41no—
88C22⋊C4×C173—41no—
89C22⋊C4×C173—41no—
90C22⋊C4×C173—41no—
91Dic346—44no—
92C2×Dic173—21no—
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—44no—
115C2×Dic173C2×Q812yes—
116C2×Dic173C2×Q812yes—
117C23×C173—21no—
118C23×C173—21no—
119C23×C173—21no—
120C23×C173—21no—
121C23×C173—21no—
122C23×C173—21no—
123C23×C173C2×Q812yes—
124C23×C173C2411yescharacteristic
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—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C22×C173—22no—
144C22×C173—22no—
145C22×C173—22no—
146C22×C173—41no—
147C22×C173—41no—
148C22×C173—41no—
149C22×C173—41no—
150C22×C173C22×Q811yescharacteristic
151C22×C173H(5,𝔽2)11yescharacteristic
152C22×C173H(5,𝔽2)11yescharacteristic
153C22×C173H(5,𝔽2)11yescharacteristic
154C22×C173H(5,𝔽2)11yescharacteristic
155C22×C1732−1+411yescharacteristic
156C22×C1732−1+411yescharacteristic
157C4×C173—41no—
158C4×C173—41no—
159C4×C173—41no—
160C4×C173—41no—
161C4×C173—41no—
162C4×C173—41no—
163Dic173—22no—
164Dic173—22no—
165Dic173—41no—
166Dic173—41no—
167Dic173—41no—
168Dic173—41no—
169Dic173—41no—
170Dic173—41no—
171C2×C173—42no—
172C2×C173#54311yescharacteristic
173C2×C173#54411yescharacteristic
174C2×C173#54411yescharacteristic
175C2×C173#56011yescharacteristic
176C2×C173#56211yescharacteristic
177C2×C173#56211yescharacteristic
178C2×C173#56311yescharacteristic
179C173#284111yescharacteristic, complemented, Sylow
180#2841—1731nomaximal, Sylow
181#400—1731no—
182#400—1731no—
183#399—1731no—
184#398—1731no—
185#398—1731no—
186#398—1731no—
187#398—1731no—
188#396—1731no—
189#396D17311yescharacteristic
190#395—1731no—
191#395—1731no—
192#393—1731no—
193#381—1731no—
194#380—1731no—
195#380—1731no—
196C22×Q8—1731no—
197C23×C4—1731no—
198C2×C4⋊C4—1731no—
199C2×C4⋊C4—1731no—
200C2×C4⋊C4—1731no—
201C2×C4⋊C4—1731no—
202C2×C4⋊C4—1731no—
203C2×C4⋊C4—1731no—
204C2×C4⋊C4—1731no—
205C2×C4⋊C4—1731no—
206C2×C4⋊C4—1731no—
207C2×C4⋊C4—1731no—
208C2×C4⋊C4C2×D17311yescharacteristic
209C2×C4⋊C4C2×D17311yescharacteristic
210C2×C4⋊C4C2×D17311yescharacteristic
211C2×C22⋊C4—1731no—
212C2×C22⋊C4—1731no—
213C2×C22⋊C4—1731no—
214C2×C22⋊C4C2×D17311yescharacteristic
215C2×C22⋊C4C2×D17311yescharacteristic
216C2×C22⋊C4C2×D17311yescharacteristic
217#95—1731no—
218#95—1731no—
219#95—1731no—
220#95—1731no—
221#95—1731no—
222#95—1731no—
223#95—1731no—
224#95—1731no—
225#95—1731no—
226#95—1731no—
227#95—1731no—
228#95—1731no—
229#95—1731no—
230#95C2×D17311yescharacteristic
231C24C22×D17311yescharacteristic
232C2×Q8—6921no—
233C2×Q8—6921no—
234C2×Q8—6921no—
235C22×C4—1731no—
236C22×C4—1731no—
237C22×C4—1731no—
238C22×C4—1731no—
239C22×C4—1731no—
240C22×C4—1731no—
241C22×C4—1731no—
242C22×C4—1731no—
243C22×C4—1734no—
244C22×C4—6921no—
245C22×C4—6921no—
246C22×C4C22×D17311yescharacteristic
247C22×C4C22×D17311yescharacteristic
248C22×C4C22×D17311yescharacteristic
249C22×C4C22×D17311yescharacteristic
250C22×C4C22×D17311yescharacteristic
251C22×C4C22×D17311yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—6921no—
256C4⋊C4—6921no—
257C4⋊C4—6921no—
258C4⋊C4—6921no—
259C4⋊C4—6921no—
260C4⋊C4—6921no—
261C4⋊C4—6921no—
262C4⋊C4—6921no—
263C4⋊C4—6921no—
264C4⋊C4—6921no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—6921no—
269C22⋊C4—6921no—
270C22⋊C4—6921no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23Q8×D17312yes—
278C23C23×D17311yescharacteristic
279Q8—6924no—
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—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
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—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—1732no—
321C2×C4—1732no—
322C2×C4—6924no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C2×Q8×D17311yescharacteristic
331C22#1914972411yescharacteristic
332C22#1914972411yescharacteristic
333C22#1914972411yescharacteristic
334C22#1914972511yescharacteristic
335C22#1914972611yescharacteristic
336C22#1914972611yescharacteristic
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3462no—
344C4—3462no—
345C4—6921no—
346C4—6921no—
347C4—6921no—
348C4—6921no—
349C4—6921no—
350C4—6921no—
351C2—42no—
352C2#2601211611yescharacteristic
353C2#2601212011yescharacteristic
354C2#2601212011yescharacteristic
355C2#2601222911yescharacteristic
356C2#2601223611yescharacteristic
357C2#2601223611yescharacteristic
358C2#2601224311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#396×C173C2
prime radicalC24×C173C23
hypercenter#396D173
hypocenterC173#2841
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. #26011432C2semidirect
2. #26011429C2semidirect
4. #26011424C2semidirect
5. #26011422C2semidirect
6. #26011422C2semidirect
7. #26011421C2semidirect
9. #26011412C2semidirect
10. #26011411C2semidirect
56. C22×Dic173Q8semidirect
179. C173#2841semidirect

Maximal subgroups

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

subgroupindex
#260114322
#260114292
#260114272
#260114242
#260114222
#260114212
#260114142
#260114122
#260114112
#260114032
#260113522
#260113492
#396×C1732
#2841173