Finite Group Databasebeta

← #36354102

Subgroups — #36354102

Order 22144.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012050C211yesmaximal, characteristic
2#26011964C211yesmaximal, characteristic, complemented
3C2×#19149594C211yesmaximal, characteristic, complemented
4C2×#19149593C211yesmaximal, characteristic, complemented
5#26011258C211yesmaximal, characteristic, complemented
6#26011257C211yesmaximal, characteristic, complemented
7C2×#99×C173C211yesmaximal, characteristic, complemented
8C22×C22⋊D173C2211yescharacteristic
9#19149662C412yescomplemented
10#19149660—21no—
11#19149651—22no—
12#19149648C412yescomplemented
13#19149647—21no—
14C4×C22⋊D173—22no—
15#19149639—22no—
16#19149635—21no—
17C2×C22⋊Dic173C2211yescharacteristic
18C2×C22⋊Dic173C2211yescharacteristic
19C2×C22⋊Dic173C2211yescharacteristic
20C22⋊C4×D173—21no—
21#19149594—22no—
22#19149593—22no—
23C22×D4×C173C2211yescharacteristic
24C2×C22⋊C4×C173C2211yescharacteristic
25C2×C22⋊C4×C173C2211yescharacteristic
26#99×C173—22no—
27C23×D173D411yescharacteristic
28C2×C22⋊D173—22no—
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C2×C22⋊D173—21no—
33C2×C22⋊D173C2×C412yes—
34C2×C22⋊D173D412yes—
35C22×Dic173—21no—
36C22×Dic173—21no—
37C22×Dic173—21no—
38C22×Dic173D411yescharacteristic
39C2×C4×D173—22no—
40#14583805—22no—
41#14583805—22no—
42C22⋊Dic173—22no—
43C22⋊Dic173—21no—
44C22⋊Dic173—21no—
45C22⋊Dic173—21no—
46C22⋊Dic173—21no—
47C22⋊Dic173C2×C412yescomplemented
48C22⋊Dic173D412yescomplemented
49#14583803—22no—
50#14583803—22no—
51C4×Dic173—22no—
52C24×C173C2×C411yescharacteristic, complemented
53C24×C173C2311yescharacteristic
54C2×D4×C173—21no—
55C2×D4×C173—21no—
56C2×D4×C173—21no—
57C2×D4×C173—21no—
58C2×D4×C173—21no—
59C2×D4×C173—21no—
60C22×C4×C173—21no—
61C22×C4×C173—21no—
62C22×C4×C173C2×C411yescharacteristic, complemented
63C22⋊C4×C173—21no—
64C22⋊C4×C173—21no—
65C22⋊C4×C173—21no—
66C22⋊C4×C173D412yescomplemented
67C22×D173—21no—
68C22×D173—21no—
69C22×D173—81no—
70C22×D173C22⋊C412yes—
71C22⋊D173—22no—
72C22⋊D173—22no—
73C22⋊D173—22no—
74C22⋊D173—22no—
75C22⋊D173—82no—
76C2×Dic173—21no—
77C2×Dic173—21no—
78C2×Dic173—22no—
79C2×Dic173—22no—
80C2×Dic173—41no—
81C2×Dic173—81no—
82C2×Dic173—81no—
83C2×Dic173—81no—
84C2×Dic173C22⋊C412yes—
85C4×D173—82no—
86C23×C173—21no—
87C23×C173—21no—
88C23×C173—21no—
89C23×C173—81no—
90C23×C173—81no—
91C23×C173C22×C411yescharacteristic
92C23×C173C2×D411yescharacteristic
93C23×C173C2×D411yescharacteristic
94C23×C173C2×D411yescharacteristic
95C23×C173C2×D411yescharacteristic
96C23×C173#4111yescharacteristic
97C23×C173#4111yescharacteristic
98D4×C173—41no—
99D4×C173—41no—
100D4×C173—41no—
101D4×C173—41no—
102C2×C4×C173—21no—
103C2×C4×C173—21no—
104C2×C4×C173—21no—
105C2×C4×C173—22no—
106C2×C4×C173—41no—
107C2×C4×C173—81no—
108C2×C4×C173—81no—
109C22×C173—41no—
110C22×C173—41no—
111C22×C173—41no—
112C22×C173—41no—
113C22×C173—41no—
114C22×C173—41no—
115C22×C173—41no—
116C22×C173—42no—
117C22×C173—41no—
118C22×C173—81no—
119C22×C173—81no—
120C22×C173C2×C22⋊C411yescharacteristic
121C22×C173C4×D411yescharacteristic
122C22×C173C4×D411yescharacteristic
123C22×C173#12011yescharacteristic
124C22×C173#12111yescharacteristic
125C22×C173#12111yescharacteristic
126C22×C173#12311yescharacteristic
127C2×D173—22no—
128C2×D173—42no—
129C2×D173—42no—
130C2×D173—81no—
131C4×C173—42no—
132C4×C173—41no—
133C4×C173—81no—
134Dic173—22no—
135Dic173—42no—
136Dic173—42no—
137Dic173—81no—
138C2×C173—21no—
139C2×C173—21no—
140C2×C173—21no—
141C2×C173—21no—
142C2×C173—21no—
143C2×C173—21no—
144C2×C173—21no—
145C2×C173—81no—
146C2×C173#38611yescharacteristic
147D173—42no—
148C173#191311yescharacteristic, complemented, Sylow
149#1913—1731nomaximal, Sylow
150#534—1731no—
151#518—1731no—
152C2×#99—1731no—
153C2×#99—1731no—
154C2×#99D17311yescharacteristic, complemented
155#342—1731no—
156#342—1731no—
157C22×D4—1731no—
158C22×D4C2×D17311yescharacteristic
159#123—3461no—
160#123—1732no—
161#121—3462no—
162#120—3461no—
163#120—1732no—
164C4×D4—3462no—
165C4×D4—3462no—
166#117—3461no—
167C2×C22⋊C4—3461no—
168C2×C22⋊C4—1731no—
169C2×C22⋊C4—1731no—
170C2×C22⋊C4—1731no—
171C2×C22⋊C4C2×D17311yescharacteristic
172C2×C22⋊C4C2×D17311yescharacteristic
173#99—22no—
174#99—3462no—
175#99—3462no—
176C24—1731no—
177C24C4×D17311yescharacteristic, complemented
178C24C22×D17311yescharacteristic
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—3461no—
186C2×D4—3461no—
187C2×D4—3461no—
188C2×D4—3461no—
189C2×D4—3462no—
190C2×D4—1732no—
191C2×D4—1732no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—3462no—
195C22×C4—3461no—
196C22×C4—3461no—
197C22×C4—3461no—
198C22×C4—1731no—
199C22×C4C4×D17311yescharacteristic, complemented
200C4⋊C4—3462no—
201C4⋊C4—3462no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—3462no—
206C22⋊C4—3462no—
207C22⋊C4—3461no—
208C22⋊C4—3461no—
209C22⋊C4—3462no—
210C22⋊C4—3461no—
211C22⋊C4—3461no—
212C22⋊C4—1732no—
213C22⋊C4—1732no—
214C22⋊C4C22⋊D17312yescomplemented
215C42—3462no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—81no—
220C23—81no—
221C23—3461no—
222C23—3461no—
223C23—13841no—
224C23—1732no—
225C23C2×C4×D17311yescharacteristic
226C23D4×D17311yescharacteristic
227C23D4×D17311yescharacteristic
228C23D4×D17311yescharacteristic
229C23C2×C22⋊D17311yescharacteristic
230C23#1458382511yescharacteristic
231C23#1458382611yescharacteristic
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—3462no—
237D4—3462no—
238D4—3462no—
239D4—3462no—
240D4—13842no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—3461no—
249C2×C4—3461no—
250C2×C4—3462no—
251C2×C4—3462no—
252C2×C4—13842no—
253C2×C4—13841no—
254C2×C4—13841no—
255C2×C4—13841no—
256C2×C4—1732no—
257C2×C4—6921no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—42no—
266C22—41no—
267C22—81no—
268C22—81no—
269C22—3462no—
270C22—13841no—
271C22—6922no—
272C22—6922no—
273C22C22⋊C4×D17311yescharacteristic
274C22#1914963911yescharacteristic
275C22C4×C22⋊D17311yescharacteristic
276C22#1914964711yescharacteristic
277C22#1914965011yescharacteristic
278C22#1914965211yescharacteristic
279C22#1914966111yescharacteristic
280C4—42no—
281C4—41no—
282C4—81no—
283C4—3462no—
284C4—13841no—
285C4—6922no—
286C4—6922no—
287C2—21no—
288C2—21no—
289C2—21no—
290C2—21no—
291C2—21no—
292C2—21no—
293C2—21no—
294C2—81no—
295C2—6922no—
296C2#2601136711yescharacteristic
subgroupG/H
centerC2#26011367
commutatorC23×C173C22×C4
FrattiniC24C22×D173
FittingC2×#99×C173C2
socleC2×C173#386
prime radical#26012050C2
cosocleC24×C173C23
hypercenterC2×#99D173
hypocenterC173#1913
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #26011964C2semidirect
3. C2×#19149594C2semidirect
4. C2×#19149593C2semidirect
5. #26011258C2semidirect
6. #26011257C2semidirect
7. C2×#99×C173C2semidirect
9. #19149662C4semidirect
12. #19149648C4semidirect
47. C22⋊Dic173C2×C4semidirect
48. C22⋊Dic173D4semidirect
52. C24×C173C2×C4semidirect
62. C22×C4×C173C2×C4semidirect
66. C22⋊C4×C173D4semidirect
148. C173#1913semidirect
154. C2×#99D173semidirect
177. C24C4×D173semidirect
199. C22×C4C4×D173semidirect
214. C22⋊C4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260120502
#260119642
C2×#191495942
C2×#191495932
#260112582
#260112572
C2×#99×C1732
#1913173