Finite Group Databasebeta

← #36354691

Subgroups — #36354691

Order 22144.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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
2C2×#19149662C211yesmaximal, characteristic, complemented
3C2×#19149594C211yesmaximal, characteristic, complemented
4C2×#19149593C211yesmaximal, characteristic, complemented
5#26011354C211yesmaximal, characteristic, complemented
6#26011257C211yesmaximal, characteristic, complemented
7C2×#123×C173C211yesmaximal, characteristic, complemented
8C22×C22⋊D173C2211yescharacteristic
9#19149662—22no—
10#19149662—21no—
11#19149662—22no—
12#19149660—21no—
13#19149651—22no—
14#19149648—21no—
15#19149647—21no—
16C2×#14583805—21no—
17C2×C22⋊Dic173C2211yescharacteristic
18C2×C22⋊Dic173C2211yescharacteristic
19C2×C22⋊Dic173C2211yescharacteristic
20C2×#14583803—21no—
21#19149594—22no—
22#19149593—22no—
23#19149586—22no—
24C22×D4×C173C2211yescharacteristic
25C23×C4×C173C2211yescharacteristic
26#123×C173—22no—
27#123×C173—22no—
28C2×C4⋊C4×C173—21no—
29C2×C22⋊C4×C173—21no—
30C2×C22⋊C4×C173C2211yescharacteristic
31C23×D173D411yescharacteristic
32C2×C22⋊D173—22no—
33C2×C22⋊D173—21no—
34C2×C22⋊D173—21no—
35C2×C22⋊D173—21no—
36C2×C22⋊D173—21no—
37C2×C22⋊D173—21no—
38C2×C22⋊D173—21no—
39C22×Dic173—21no—
40C22×Dic173—21no—
41C22×Dic173D411yescharacteristic
42C22×Dic173D412yes—
43#14583805—24no—
44#14583805—22no—
45C22⋊Dic173—22no—
46C22⋊Dic173—21no—
47C22⋊Dic173—21no—
48C22⋊Dic173—21no—
49C22⋊Dic173—21no—
50C22⋊Dic173—21no—
51C22⋊Dic173D412yescomplemented
52#14583803—24no—
53#14583803—22no—
54C24×C173D411yescharacteristic
55C24×C173C2311yescharacteristic
56C2×D4×C173—21no—
57C2×D4×C173—21no—
58C2×D4×C173—21no—
59C2×D4×C173—21no—
60C2×D4×C173—21no—
61C2×D4×C173—21no—
62C22×C4×C173—21no—
63C22×C4×C173—21no—
64C22×C4×C173—21no—
65C22×C4×C173—21no—
66C22×C4×C173—21no—
67C22×C4×C173—21no—
68C22×C4×C173—21no—
69C22×C4×C173D411yescharacteristic
70C22×C4×C173D412yescomplemented
71C4⋊C4×C173—24no—
72C22⋊C4×C173—24no—
73C22⋊C4×C173—21no—
74C22⋊C4×C173D412yescomplemented
75C22×D173—21no—
76C22×D173—21no—
77C22×D173—21no—
78C22×D173—42no—
79C22⋊D173—41no—
80C22⋊D173—41no—
81C22⋊D173—41no—
82C22⋊D173—41no—
83C22⋊D173—82no—
84C2×Dic173—22no—
85C2×Dic173—21no—
86C2×Dic173—21no—
87C2×Dic173—22no—
88C2×Dic173—21no—
89C2×Dic173—42no—
90C2×Dic173—42no—
91C2×Dic173—42no—
92C2×Dic173—41no—
93C23×C173—21no—
94C23×C173—21no—
95C23×C173—21no—
96C23×C173—42no—
97C23×C173—42no—
98C23×C173C2×D411yescharacteristic
99C23×C173C2×D411yescharacteristic
100C23×C173C2×D411yescharacteristic
101C23×C173C2×D411yescharacteristic
102C23×C173C2×D411yescharacteristic
103C23×C173C2×D411yescharacteristic
104C23×C173#4111yescharacteristic
105D4×C173—41no—
106D4×C173—41no—
107D4×C173—41no—
108D4×C173—41no—
109C2×C4×C173—22no—
110C2×C4×C173—22no—
111C2×C4×C173—21no—
112C2×C4×C173—22no—
113C2×C4×C173—21no—
114C2×C4×C173—22no—
115C2×C4×C173—21no—
116C2×C4×C173—42no—
117C2×C4×C173—41no—
118C2×C4×C173—41no—
119C2×C4×C173—41no—
120C2×C4×C173—41no—
121C22×C173—22no—
122C22×C173—22no—
123C22×C173—22no—
124C22×C173—41no—
125C22×C173—41no—
126C22×C173—41no—
127C22×C173—41no—
128C22×C173—42no—
129C22×C173—41no—
130C22×C173—42no—
131C22×C173—81no—
132C22×C173#12011yescharacteristic
133C22×C173#12011yescharacteristic
134C22×C173#12011yescharacteristic
135C22×C173#12111yescharacteristic
136C22×C173#12111yescharacteristic
137C22×C173#12111yescharacteristic
138C22×C173#12711yescharacteristic
139C2×D173—42no—
140C2×D173—41no—
141C2×D173—42no—
142C2×D173—81no—
143C4×C173—42no—
144C4×C173—42no—
145C4×C173—41no—
146Dic173—42no—
147Dic173—41no—
148Dic173—42no—
149Dic173—81no—
150C2×C173—21no—
151C2×C173—21no—
152C2×C173—21no—
153C2×C173—21no—
154C2×C173—21no—
155C2×C173—21no—
156C2×C173—81no—
157C2×C173#39211yescharacteristic
158C2×C173#45812yes—
159D173—81no—
160C173#202211yescharacteristic, complemented, Sylow
161#2022—1731nomaximal, Sylow
162#534—1731no—
163C2×#123—1731no—
164C2×#123D17311yescharacteristic, complemented
165C2×#99—1731no—
166C2×#99—1731no—
167#381—1731no—
168#342—1731no—
169C22×D4—1731no—
170C22×D4C2×D17311yescharacteristic
171C23×C4C2×D17311yescharacteristic
172#123—22no—
173#123—22no—
174#123—3461no—
175#123—3461no—
176#123—3462no—
177#123—3462no—
178#121—3462no—
179#120—3461no—
180#120—3461no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—3461no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—3461no—
185C2×C22⋊C4—1731no—
186C2×C22⋊C4—1731no—
187C2×C22⋊C4—1731no—
188C2×C22⋊C4C2×D17311yescharacteristic
189#99—3462no—
190#99—3462no—
191#95—3462no—
192C24—1731no—
193C24C22⋊D17311yescharacteristic
194C24C22×D17311yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—3462no—
202C2×D4—3461no—
203C2×D4—3461no—
204C2×D4—3461no—
205C2×D4—3461no—
206C2×D4—3461no—
207C2×D4—3461no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—3461no—
216C22×C4—3461no—
217C22×C4—1731no—
218C22×C4—1732no—
219C22×C4C22⋊D17311yescharacteristic
220C22×C4C22⋊D17312yescomplemented
221C4⋊C4—24no—
222C4⋊C4—3464no—
223C4⋊C4—3462no—
224C22⋊C4—24no—
225C22⋊C4—21no—
226C22⋊C4—3464no—
227C22⋊C4—3462no—
228C22⋊C4—3462no—
229C22⋊C4—3461no—
230C22⋊C4—3461no—
231C22⋊C4—3461no—
232C22⋊C4—3461no—
233C22⋊C4—3461no—
234C22⋊C4—1732no—
235C22⋊C4C22⋊D17312yescomplemented
236C23—21no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—42no—
241C23—3461no—
242C23—3461no—
243C23—3461no—
244C23—6922no—
245C23D4×D17311yescharacteristic
246C23D4×D17311yescharacteristic
247C23D4×D17311yescharacteristic
248C23C2×C22⋊D17311yescharacteristic
249C23C2×C22⋊D17311yescharacteristic
250C23C2×C22⋊D17311yescharacteristic
251C23#1458382511yescharacteristic
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—13842no—
257D4—6921no—
258D4—6921no—
259D4—6921no—
260D4—6921no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—3462no—
274C2×C4—3461no—
275C2×C4—3462no—
276C2×C4—3461no—
277C2×C4—3461no—
278C2×C4—6922no—
279C2×C4—6921no—
280C2×C4—6922no—
281C2×C4—6922no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—41no—
291C22—42no—
292C22—81no—
293C22—13841no—
294C22—6922no—
295C22—6921no—
296C22—6922no—
297C22#1914964711yescharacteristic
298C22#1914964711yescharacteristic
299C22#1914964811yescharacteristic
300C22#1914965011yescharacteristic
301C22#1914965111yescharacteristic
302C22#1914965111yescharacteristic
303C22#1914967511yescharacteristic
304C4—42no—
305C4—42no—
306C4—41no—
307C4—13841no—
308C4—6922no—
309C4—6921no—
310C4—6922no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—81no—
318C2—13841no—
319C2#2601139911yescharacteristic
320C2#2601168012yes—
subgroupG/H
centerC22#19149647
commutator, cosocleC24×C173C23
FrattiniC24C22×D173
FittingC2×#123×C173C2
socleC22×C173#120
prime radical#26012050C2
hypercenterC2×#123D173
hypocenterC173#2022
LCSG ▷ C24×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#19149662C2semidirect
3. C2×#19149594C2semidirect
4. C2×#19149593C2semidirect
5. #26011354C2semidirect
6. #26011257C2semidirect
7. C2×#123×C173C2semidirect
51. C22⋊Dic173D4semidirect
70. C22×C4×C173D4semidirect
74. C22⋊C4×C173D4semidirect
160. C173#2022semidirect
164. C2×#123D173semidirect
220. C22×C4C22⋊D173semidirect
235. C22⋊C4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260120502
C2×#191496622
C2×#191495942
C2×#191495932
#260113542
#260112572
C2×#123×C1732
#2022173