Finite Group Databasebeta

← #13076618

Subgroups — #13076618

Order 1408.

578 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×C11×#123C211yesmaximal, characteristic, complemented
2C2×#66983C211yesmaximal, characteristic, complemented
3#10622836C212yesmaximal, complemented
4#10622825C211yesmaximal, characteristic, complemented
5#10622607C212yesmaximal, complemented
6#10622604C211yesmaximal, characteristic, complemented
7#10622599C211yesmaximal, characteristic, complemented
8#10622574C212yesmaximal, complemented
9#10622573C211yesmaximal, characteristic, complemented
10#10622572C212yesmaximal, complemented
11#10622562C211yesmaximal, characteristic, complemented
12C22×D4×C11C2211yescharacteristic
13C23×C4×C11C2211yescharacteristic
14C22×C22⋊D11C2212yes—
15C22×C22⋊D11C2212yes—
16C22×D44C2211yescharacteristic
17C11×#123—24no—
18C2×C11×C4⋊C4C2211yescharacteristic
19C2×C11×C4⋊C4C2211yescharacteristic
20C2×C11×C22⋊C4C2211yescharacteristic
21C2×C11×C22⋊C4C2211yescharacteristic
22C2×C11×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic11C2212yes—
24C2×C22⋊Dic11C2212yes—
25#66983—24no—
26C2×(C2×D11)⋊C4C2212yes—
27C2×(C2×D11)⋊C4C2212yes—
28C2×(C2×D11)⋊C4C2212yes—
29C2×(C2×D11)⋊C4C2212yes—
30C2×C4⋊Dic11C2211yescharacteristic
31C2×C4⋊Dic11C2212yes—
32C2×Dic11⋊C4C2212yes—
33C11⋊#95C2211yescharacteristic
34C11⋊#95C2211yescharacteristic
35C11⋊#95C2212yes—
36C11⋊#95C2212yes—
37C24×C11C2311yescharacteristic
38C24×C11C2311yescharacteristic
39C23×D11C2312yes—
40C2×D4×C11—22no—
41C2×D4×C11—42no—
42C22×C4×C11—42no—
43C22×C4×C11D414yescomplemented
44C22×C4×C11C2311yescharacteristic
45C22×C4×C11C2311yescharacteristic
46C22×C4×C11C2311yescharacteristic
47C22×C4×C11C2311yescharacteristic
48C22×C4×C11C2311yescharacteristic
49C2×C22⋊D11—24no—
50C2×C22⋊D11—42no—
51C2×C22⋊D11—42no—
52C2×C22⋊D11—42no—
53C2×C22⋊D11—42no—
54C2×C22⋊D11—42no—
55C22×Dic11C2312yes—
56C22×Dic11C2312yes—
57C22×Dic11C2312yes—
58C2×D44—22no—
59C2×D44—42no—
60C11×C4⋊C4—22no—
61C11×C4⋊C4—22no—
62C11×C22⋊C4—22no—
63C11×C22⋊C4—22no—
64C11×C22⋊C4—22no—
65C22⋊Dic11—42no—
66C22⋊Dic11—42no—
67(C2×D11)⋊C4—24no—
68(C2×D11)⋊C4—42no—
69(C2×D11)⋊C4—42no—
70(C2×D11)⋊C4—42no—
71C4⋊Dic11—22no—
72C4⋊Dic11—42no—
73Dic11⋊C4—42no—
74#2813—111nomaximal, Sylow
75C23×C11—21no—
76C23×C11—22no—
77C23×C11—22no—
78C23×C11—21no—
79C23×C11—21no—
80C23×C11—22no—
81C23×C11—21no—
82C23×C11—22no—
83C23×C11—21no—
84C23×C11C2×D412yes—
85C23×C11C2411yescharacteristic
86C22×D11—22no—
87C22×D11—22no—
88C22×D11—22no—
89C22×D11—22no—
90C22×D11—22no—
91C22×D11—22no—
92C22×D11—22no—
93D4×C11—44no—
94C2×C4×C11—22no—
95C2×C4×C11—22no—
96C2×C4×C11—21no—
97C2×C4×C11—21no—
98C2×C4×C11—21no—
99C2×C4×C11—22no—
100C2×C4×C11—21no—
101C2×C4×C11—22no—
102C2×C4×C11—21no—
103C2×C4×C11—44no—
104C2×C4×C11C2×D412yes—
105C2×C4×C11C2×D412yes—
106C22⋊D11—48no—
107C22⋊D11—48no—
108C2×Dic11—22no—
109C2×Dic11—22no—
110C2×Dic11—22no—
111C2×Dic11—22no—
112C2×Dic11—22no—
113C2×Dic11—22no—
114C2×Dic11—22no—
115C2×Dic11—22no—
116C2×Dic11—22no—
117D44—44no—
118C2×#123D1111yescharacteristic, complemented
119C2×#121—111no—
120#400—111no—
121#399—111no—
122#396—112no—
123#394—111no—
124#394—112no—
125#394—112no—
126#392—112no—
127#392—111no—
128#380—111no—
129C22×C11—22no—
130C22×C11—22no—
131C22×C11—22no—
132C22×C11—42no—
133C22×C11—42no—
134C22×C11—41no—
135C22×C11—41no—
136C22×C11—41no—
137C22×C11—42no—
138C22×C11—42no—
139C22×C11C22×D411yescharacteristic
140C22×C11H(5,𝔽2)12yes—
141C22×C11H(5,𝔽2)12yes—
142C22×C11H(5,𝔽2)11yescharacteristic
143C22×C112−1+411yescharacteristic
144C2×D11—42no—
145C2×D11—42no—
146C2×D11—42no—
147C2×D11—42no—
148C2×D11—42no—
149C2×D11—42no—
150C2×D11—42no—
151C4×C11—22no—
152C4×C11—22no—
153C4×C11—41no—
154C4×C11—41no—
155C4×C11—41no—
156Dic11—42no—
157Dic11—42no—
158Dic11—42no—
159C22×D4—112no—
160C22×D4—112no—
161C22×D4—111no—
162C22×D4C2×D1111yescharacteristic
163C23×C4C2×D1111yescharacteristic
164#123—24no—
165#121—224no—
166C2×C4⋊C4—111no—
167C2×C4⋊C4—112no—
168C2×C4⋊C4—112no—
169C2×C4⋊C4C2×D1111yescharacteristic
170C2×C4⋊C4C2×D1111yescharacteristic
171C2×C22⋊C4—112no—
172C2×C22⋊C4—112no—
173C2×C22⋊C4—112no—
174C2×C22⋊C4—112no—
175C2×C22⋊C4—112no—
176C2×C22⋊C4—112no—
177C2×C22⋊C4C2×D1111yescharacteristic
178C2×C22⋊C4C2×D1111yescharacteristic
179C2×C22⋊C4C2×D1111yescharacteristic
180#95—111no—
181#95—111no—
182#95—112no—
183#95—112no—
184C2×C11—42no—
185C2×C11—81no—
186C2×C11#53411yescharacteristic
187C2×C11#53511yescharacteristic
188C2×C11#53711yescharacteristic
189C2×C11#56012yes—
190C2×C11#56212yes—
191D11—82no—
192C24—112no—
193C24C22×D1111yescharacteristic
194C24C22×D1111yescharacteristic
195C2×D4—22no—
196C2×D4—42no—
197C2×D4—224no—
198C2×D4—222no—
199C2×D4—442no—
200C2×D4—442no—
201C2×D4—442no—
202C2×D4—442no—
203C2×D4—442no—
204C2×D4—442no—
205C22×C4—42no—
206C22×C4—112no—
207C22×C4—112no—
208C22×C4—112no—
209C22×C4C22⋊D1114yescomplemented
210C22×C4C22×D1111yescharacteristic
211C22×C4C22×D1111yescharacteristic
212C22×C4C22×D1111yescharacteristic
213C22×C4C22×D1111yescharacteristic
214C22×C4C22×D1111yescharacteristic
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—222no—
218C4⋊C4—442no—
219C4⋊C4—442no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—224no—
224C22⋊C4—442no—
225C22⋊C4—442no—
226C22⋊C4—442no—
227C22⋊C4—442no—
228C22⋊C4—442no—
229C11#281311yescharacteristic, complemented, Sylow
230C23—22no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—222no—
240C23—222no—
241C23—222no—
242C23—222no—
243C23—222no—
244C23—222no—
245C23—222no—
246C23C2×C22⋊D1112yes—
247C23C23×D1111yescharacteristic
248D4—44no—
249D4—444no—
250D4—448no—
251D4—448no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—44no—
262C2×C4—222no—
263C2×C4—222no—
264C2×C4—222no—
265C2×C4—222no—
266C2×C4—222no—
267C2×C4—222no—
268C2×C4—222no—
269C2×C4—222no—
270C2×C4—222no—
271C2×C4C2×C22⋊D1112yes—
272C2×C4C2×C22⋊D1112yes—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—442no—
284C22—442no—
285C22—442no—
286C22—442no—
287C22—442no—
288C22—442no—
289C22—442no—
290C22#6703712yes—
291C22#6704212yes—
292C22#6704211yescharacteristic
293C22#6704311yescharacteristic
294C22C22×C22⋊D1111yescharacteristic
295C4—22no—
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—442no—
301C4—442no—
302C4—442no—
303C2—42no—
304C2—81no—
305C2—882no—
306C2#1062319712yes—
307C2#1062319812yes—
308C2#1062333511yescharacteristic
309C2#1062336111yescharacteristic
310C2#1062336411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC2×C11×#123C2
hypercenterC2×#123D11
hypocenterC11#2813
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C11×#123C2semidirect
2. C2×#66983C2semidirect
3. #10622836C2semidirect
4. #10622825C2semidirect
5. #10622607C2semidirect
6. #10622604C2semidirect
7. #10622599C2semidirect
8. #10622574C2semidirect
9. #10622573C2semidirect
10. #10622572C2semidirect
11. #10622562C2semidirect
43. C22×C4×C11D4semidirect
118. C2×#123D11semidirect
209. C22×C4C22⋊D11semidirect
229. C11#2813semidirect

Maximal subgroups

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

subgroupindex
C2×C11×#1232
C2×#669832
#106228362
#106228252
#106226072
#106226042
#106225992
#106225742
#106225732
#106225722
#106225622
#281311