Finite Group Databasebeta

← #36356517

Subgroups — #36356517

Order 22144.

480 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149644C211yesmaximal, characteristic
2#26011395C211yesmaximal, characteristic
3#26011395C211yesmaximal, characteristic
4#26011378C211yesmaximal, characteristic
5#26011377C211yesmaximal, characteristic
6#26011372C211yesmaximal, characteristic
7#26011360C211yesmaximal, characteristic
8#26011355C211yesmaximal, characteristic
9#26011355C211yesmaximal, characteristic
10C4×#14583805C211yesmaximal, characteristic
11C4×#14583805C211yesmaximal, characteristic
12#26011342C211yesmaximal, characteristic
13#26011342C211yesmaximal, characteristic
14#26011342C211yesmaximal, characteristic
15#387×C173C211yesmaximal, characteristic
16C22×Dic346C2211yescharacteristic
17#19149644C418yescomplemented
18C2×C4⋊Dic173C2211yescharacteristic
19C2×#14583805C2211yescharacteristic
20C2×#14583805C2211yescharacteristic
21C2×#14583805C2211yescharacteristic
22C2×#14583805C2211yescharacteristic
23C2×#14583805C2211yescharacteristic
24C2×#14583805C2211yescharacteristic
25C2×#14583805C2211yescharacteristic
26C2×#14583805C2211yescharacteristic
27C2×C4×Dic173C2211yescharacteristic
28C2×C4×Dic173C2211yescharacteristic
29C2×C4×Dic173C2211yescharacteristic
30C2×C4×Dic173C2211yescharacteristic
31C2×C4×Dic173C2211yescharacteristic
32C2×C4×Dic173C2211yescharacteristic
33C2×C4×Dic173C2211yescharacteristic
34C2×C4×Dic173C2211yescharacteristic
35#19149586C2211yescharacteristic
36#19149586C2211yescharacteristic
37#19149586C2211yescharacteristic
38#19149586C2211yescharacteristic
39#19149586C2211yescharacteristic
40#19149586C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44#19149586C2211yescharacteristic
45C2×C4⋊C4×C173C2211yescharacteristic
46C2×C4⋊C4×C173C2211yescharacteristic
47C2×C42×C173C2211yescharacteristic
48#95×C173C2211yescharacteristic
49#95×C173C2211yescharacteristic
50#95×C173C2211yescharacteristic
51#95×C173C2211yescharacteristic
52C2×Dic346—22no—
53C2×Dic346—22no—
54C2×Dic346C2×C414yes—
55C22×Dic173C2311yescharacteristic
56C22×Dic173C2311yescharacteristic
57C22×Dic173C2311yescharacteristic
58C22×Dic173C2311yescharacteristic
59C22×Dic173C2311yescharacteristic
60C22×Dic173C2311yescharacteristic
61C22×Dic173C2311yescharacteristic
62C22×Dic173C2311yescharacteristic
63C4⋊Dic173—41no—
64#14583805—22no—
65#14583805—22no—
66#14583805—22no—
67#14583805—22no—
68#14583805—22no—
69#14583805C2×C414yes—
70#14583805C2×C414yes—
71#14583805Q814yescomplemented
72C4×Dic173—22no—
73C4×Dic173—22no—
74C4×Dic173—41no—
75C4×Dic173—41no—
76C4×Dic173—41no—
77C4×Dic173C2×C414yes—
78C4×Dic173C2×C414yes—
79C4×Dic173C2×C414yes—
80C22×C4×C173C2311yescharacteristic
81C22×C4×C173C2311yescharacteristic
82C22×C4×C173C2311yescharacteristic
83C22×C4×C173C2311yescharacteristic
84C22×C4×C173C2311yescharacteristic
85C22×C4×C173C2311yescharacteristic
86C22×C4×C173C2311yescharacteristic
87C4⋊C4×C173—41no—
88C4⋊C4×C173C2×C414yes—
89C42×C173—41no—
90Dic346—28no—
91C2×Dic173—21no—
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×Dic173C22×C412yes—
107C2×Dic173C22×C412yes—
108C2×Dic173C22×C412yes—
109C2×Dic173C22×C412yes—
110C2×Dic173C2×Q812yes—
111C2×Dic173C2×Q812yes—
112C2×Dic173#4112yes—
113C2×Dic173#4112yes—
114C2×Dic173#4112yes—
115C23×C173C2411yescharacteristic
116C2×C4×C173—21no—
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×C173C22×C412yes—
134C2×C4×C173C22×C412yes—
135C2×C4×C173C22×C412yes—
136C2×C4×C173C2×Q812yes—
137C22×C173C23×C411yescharacteristic
138C22×C173C22×Q811yescharacteristic
139C22×C173C2×#4111yescharacteristic
140C22×C173C2×#4111yescharacteristic
141C22×C173C2×#4111yescharacteristic
142C22×C173H(5,𝔽2)11yescharacteristic
143C22×C1732−1+411yescharacteristic
144C4×C173—22no—
145C4×C173—22no—
146C4×C173—22no—
147C4×C173—22no—
148C4×C173—41no—
149C4×C173—41no—
150C4×C173—41no—
151Dic173—22no—
152Dic173—22no—
153Dic173—22no—
154Dic173—41no—
155Dic173—41no—
156Dic173—41no—
157Dic173#11714yes—
158Dic173C4×Q814yes—
159C2×C173C2×#11711yescharacteristic
160C2×C173C2×C4×Q811yescharacteristic
161C2×C173#52011yescharacteristic
162C2×C173#55411yescharacteristic
163C2×C173#55611yescharacteristic
164C2×C173#55711yescharacteristic
165C2×C173#55911yescharacteristic
166C173#235511yescharacteristic, complemented, Sylow
167#2355—1731nomaximal, Sylow
168C2×C4×Q8—1731no—
169#391—1731no—
170#391—1731no—
171#387—1731no—
172#387—1731no—
173#387—1731no—
174#387D17311yescharacteristic
175#384—1731no—
176#382—1731no—
177#382—1731no—
178C4×C4⋊C4—1731no—
179C4×C4⋊C4—1731no—
180C4×C4⋊C4—1731no—
181C4×C4⋊C4—1731no—
182C4×C4⋊C4—1731no—
183C22×Q8—1731no—
184C4×Q8—1738no—
185C2×C4⋊C4—1731no—
186C2×C4⋊C4—1731no—
187C2×C4⋊C4—1731no—
188C2×C4⋊C4—1731no—
189C2×C4⋊C4—1731no—
190C2×C4⋊C4—1731no—
191C2×C4⋊C4—1731no—
192C2×C4⋊C4—1731no—
193C2×C4⋊C4—1731no—
194C2×C4⋊C4C2×D17311yescharacteristic
195C2×C4⋊C4C2×D17311yescharacteristic
196C2×C42—1731no—
197C2×C42—1731no—
198C2×C42—1731no—
199C2×C42—1731no—
200C2×C42—1731no—
201C2×C42—1731no—
202C2×C42—1731no—
203C2×C42—1731no—
204C2×C42C2×D17311yescharacteristic
205#95—1731no—
206#95—1731no—
207#95—1731no—
208#95—1731no—
209#95—1731no—
210#95—1731no—
211#95—1731no—
212#95—1731no—
213#95—1731no—
214#95—1731no—
215#95C2×D17311yescharacteristic
216#95C2×D17311yescharacteristic
217#95C2×D17311yescharacteristic
218#95C2×D17311yescharacteristic
219C2×Q8—3462no—
220C2×Q8—3462no—
221C2×Q8—1734no—
222C22×C4—1731no—
223C22×C4—1731no—
224C22×C4—1731no—
225C22×C4—1731no—
226C22×C4—1731no—
227C22×C4—1731no—
228C22×C4—1731no—
229C22×C4—1731no—
230C22×C4C22×D17311yescharacteristic
231C22×C4C22×D17311yescharacteristic
232C22×C4C22×D17311yescharacteristic
233C22×C4C22×D17311yescharacteristic
234C22×C4C22×D17311yescharacteristic
235C22×C4C22×D17311yescharacteristic
236C22×C4C22×D17311yescharacteristic
237C4⋊C4—41no—
238C4⋊C4—3462no—
239C4⋊C4—3462no—
240C4⋊C4—3462no—
241C4⋊C4—3462no—
242C4⋊C4—3462no—
243C4⋊C4—1734no—
244C4⋊C4—1734no—
245C4⋊C4—1734no—
246C4⋊C4—6921no—
247C4⋊C4C4×D17314yes—
248C42—41no—
249C42—3462no—
250C42—3462no—
251C42—1734no—
252C42—1734no—
253C42—1734no—
254C42—6921no—
255C42—6921no—
256C42—6921no—
257C23C23×D17311yescharacteristic
258Q8—3468no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—3461no—
277C2×C4—3461no—
278C2×C4—3461no—
279C2×C4—3461no—
280C2×C4—3461no—
281C2×C4—3461no—
282C2×C4—3461no—
283C2×C4—3461no—
284C2×C4—3461no—
285C2×C4—3461no—
286C2×C4—3461no—
287C2×C4—3461no—
288C2×C4—3461no—
289C2×C4—3461no—
290C2×C4—3461no—
291C2×C4—1732no—
292C2×C4—1732no—
293C2×C4—1732no—
294C2×C4—1732no—
295C2×C4—1732no—
296C2×C4—1732no—
297C2×C4—1732no—
298C2×C4—1732no—
299C2×C4—1732no—
300C2×C4C2×C4×D17312yes—
301C2×C4C2×C4×D17312yes—
302C2×C4C2×C4×D17312yes—
303C2×C4Q8×D17312yes—
304C22C22×C4×D17311yescharacteristic
305C22C2×Q8×D17311yescharacteristic
306C22#41×D17311yescharacteristic
307C22#41×D17311yescharacteristic
308C22#41×D17311yescharacteristic
309C22#1914972411yescharacteristic
310C22#1914972611yescharacteristic
311C4—22no—
312C4—22no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—3462no—
319C4—3462no—
320C4—3462no—
321C4—1734no—
322C4—1734no—
323C4—6921no—
324C4—6921no—
325C4—6921no—
326C2#117×D17311yescharacteristic
327C2C4×Q8×D17311yescharacteristic
328C2#2601197011yescharacteristic
329C2#2601219511yescharacteristic
330C2#2601220811yescharacteristic
331C2#2601221811yescharacteristic
332C2#2601222511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#387×C173C2
socle, prime radical, cosocleC23×C173C24
hypercenter#387D173
hypocenterC173#2355
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
17. #19149644C4semidirect
71. #14583805Q8semidirect
166. C173#2355semidirect

Maximal subgroups

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

subgroupindex
C2×#191496442
#260113952
#260113782
#260113772
#260113722
#260113602
#260113552
C4×#145838052
#260113422
#387×C1732
#2355173