Finite Group Databasebeta

← #11740089

Subgroups — #11740089

Order 896.

652 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#121C211yesmaximal, characteristic
2C2×#6498C212yesmaximal, complemented
3C2×D4×Dic7C211yesmaximal, characteristic, complemented
4#89994C212yesmaximal, complemented
5#89993C211yesmaximal, characteristic, complemented
6#89984C211yesmaximal, characteristic, complemented
7C4×C22⋊Dic7C211yesmaximal, characteristic, complemented
8#89748C211yesmaximal, characteristic, complemented
9#89745C211yesmaximal, characteristic, complemented
10#89724C212yesmaximal, complemented
11#89718C212yesmaximal, complemented
12C22×C7×D4C2212yes—
13C22×C7×D4C2211yescharacteristic
14C23×C4×C7C2211yescharacteristic
15C23×Dic7C2212yes—
16C7×#121—24no—
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2212yes—
21C2×C22⋊Dic7C2212yes—
22C2×C22⋊Dic7C2212yes—
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25#6498—28no—
26D4×Dic7—24no—
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×Dic7⋊C4C2212yes—
30C2×Dic7⋊C4C2212yes—
31C2×Dic7⋊C4C2212yes—
32C2×C4×Dic7C2211yescharacteristic
33C2×C4×Dic7C2211yescharacteristic
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37C7⋊#95C2212yes—
38#2422—71nomaximal, Sylow
39C24×C7C2311yescharacteristic
40C24×C7C2312yes—
41C2×C7×D4—24no—
42C2×C7×D4—24no—
43C2×C7×D4—22no—
44C2×C7×D4—22no—
45C2×C7×D4—42no—
46C2×C7×D4D414yes—
47C22×C4×C7—22no—
48C22×C4×C7—22no—
49C22×C4×C7—22no—
50C22×C4×C7C2311yescharacteristic
51C22×C4×C7C2312yes—
52C22×C4×C7C2311yescharacteristic
53C22×Dic7—24no—
54C22×Dic7—24no—
55C22×Dic7—42no—
56C22×Dic7C2311yescharacteristic
57C22×Dic7C2312yes—
58C22×Dic7C2312yes—
59C22×Dic7C2311yescharacteristic
60C22×Dic7C2311yescharacteristic
61C22×Dic7C2311yescharacteristic
62C7×C4⋊C4—22no—
63C7×C22⋊C4—24no—
64C22⋊Dic7—24no—
65C22⋊Dic7—24no—
66C22⋊Dic7—24no—
67C22⋊Dic7—24no—
68C22⋊Dic7—22no—
69C22⋊Dic7D414yescomplemented
70C4⋊Dic7—22no—
71C4⋊Dic7—22no—
72Dic7⋊C4—24no—
73Dic7⋊C4—24no—
74Dic7⋊C4—42no—
75C4×Dic7—22no—
76C4×Dic7—22no—
77C2×#123—72no—
78C2×#121D711yescharacteristic
79C2×C4×D4—71no—
80#397—72no—
81#394—72no—
82#390—71no—
83#389—71no—
84#385—72no—
85#384—71no—
86#380—71no—
87C4×C22⋊C4—71no—
88C23×C7—22no—
89C23×C7—21no—
90C23×C7—21no—
91C23×C7—22no—
92C23×C7—21no—
93C23×C7—21no—
94C23×C7—22no—
95C23×C7—22no—
96C23×C7—22no—
97C23×C7—21no—
98C23×C7—21no—
99C23×C7—22no—
100C23×C7C2×D412yes—
101C23×C7C2×D414yes—
102C23×C7C2411yescharacteristic
103C7×D4—28no—
104C7×D4—48no—
105C2×C4×C7—22no—
106C2×C4×C7—21no—
107C2×C4×C7—22no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—22no—
111C2×C4×C7—44no—
112C2×C4×C7C2×D412yes—
113C2×C4×C7#4112yes—
114C2×C4×C7#4112yes—
115C2×Dic7—21no—
116C2×Dic7—21no—
117C2×Dic7—21no—
118C2×Dic7—21no—
119C2×Dic7—21no—
120C2×Dic7—21no—
121C2×Dic7—21no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—22no—
125C2×Dic7—22no—
126C2×Dic7—22no—
127C2×Dic7—21no—
128C2×Dic7—22no—
129C2×Dic7—21no—
130C2×Dic7—48no—
131C2×Dic7C2×D412yes—
132C2×Dic7C2×D412yes—
133C2×Dic7#4112yes—
134C22×D4C2×D712yes—
135C22×D4C2×D711yescharacteristic
136C23×C4—72no—
137C23×C4C2×D711yescharacteristic
138#123—148no—
139#121—24no—
140C4×D4—144no—
141C2×C4⋊C4—71no—
142C2×C4⋊C4—72no—
143C2×C4⋊C4—72no—
144C2×C4⋊C4—71no—
145C2×C4⋊C4—72no—
146C2×C4⋊C4C2×D711yescharacteristic
147C2×C22⋊C4—71no—
148C2×C22⋊C4—72no—
149C2×C22⋊C4—72no—
150C2×C22⋊C4—72no—
151C2×C22⋊C4—71no—
152C2×C22⋊C4—72no—
153C2×C22⋊C4C2×D712yes—
154C2×C42—71no—
155C2×C42—71no—
156#95—71no—
157#95—71no—
158#95—72no—
159#95—72no—
160C22×C7—24no—
161C22×C7—22no—
162C22×C7—24no—
163C22×C7—22no—
164C22×C7—24no—
165C22×C7—22no—
166C22×C7—41no—
167C22×C7—42no—
168C22×C7—41no—
169C22×C7—42no—
170C22×C7—41no—
171C22×C7—42no—
172C22×C7—41no—
173C22×C7—42no—
174C22×C7C22×D411yescharacteristic
175C22×C7C22×D411yescharacteristic
176C22×C7C2×#4111yescharacteristic
177C22×C7C2×#4111yescharacteristic
178C22×C7C2×#4111yescharacteristic
179C22×C7H(5,𝔽2)11yescharacteristic
180C22×C72−1+411yescharacteristic
181C4×C7—41no—
182C4×C7—42no—
183C4×C7#12314yescomplemented
184Dic7—22no—
185Dic7—22no—
186Dic7—22no—
187Dic7—42no—
188Dic7—42no—
189Dic7—41no—
190C24C22×D711yescharacteristic
191C24C22×D712yes—
192C2×D4—24no—
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—42no—
197C2×D4C22⋊D714yes—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—71no—
202C22×C4—71no—
203C22×C4—71no—
204C22×C4—71no—
205C22×C4—72no—
206C22×C4—72no—
207C22×C4—144no—
208C22×C4—144no—
209C22×C4—282no—
210C22×C4C22×D711yescharacteristic
211C22×C4C22×D712yes—
212C22×C4C22×D711yescharacteristic
213C4⋊C4—22no—
214C4⋊C4—144no—
215C4⋊C4—144no—
216C4⋊C4—142no—
217C4⋊C4—142no—
218C4⋊C4—282no—
219C22⋊C4—24no—
220C22⋊C4—74no—
221C22⋊C4—142no—
222C22⋊C4—144no—
223C22⋊C4—144no—
224C22⋊C4—144no—
225C22⋊C4—144no—
226C42—142no—
227C42—142no—
228C2×C7—44no—
229C2×C7—42no—
230C2×C7C2×#12311yescharacteristic
231C2×C7#53211yescharacteristic
232C2×C7#54211yescharacteristic
233C2×C7D4211yescharacteristic
234C2×C7#54711yescharacteristic
235C2×C7#54711yescharacteristic
236C2×C7#55311yescharacteristic
237C23—22no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—21no—
244C23—22no—
245C23—21no—
246C23—22no—
247C23—21no—
248C23—21no—
249C23D4×D712yes—
250C23C2×C22⋊D714yes—
251C23C23×D711yescharacteristic
252D4—28no—
253D4—48no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—44no—
261C2×C4—72no—
262C2×C4—72no—
263C2×C4—72no—
264C2×C4—141no—
265C2×C4—141no—
266C2×C4—141no—
267C2×C4—141no—
268C2×C4—141no—
269C2×C4—141no—
270C2×C4—142no—
271C2×C4—142no—
272C2×C4—142no—
273C2×C4—142no—
274C2×C4—141no—
275C2×C4—142no—
276C2×C4—141no—
277C2×C4—141no—
278C2×C4—142no—
279C2×C4—288no—
280C2×C4#115912yes—
281C2×C4#115912yes—
282C2×C4C2×C22⋊D712yes—
283C7#242211yescharacteristic, complemented, Sylow
284C22—24no—
285C22—24no—
286C22—22no—
287C22—24no—
288C22—22no—
289C22—22no—
290C22—41no—
291C22—41no—
292C22—42no—
293C22—41no—
294C22—42no—
295C22—42no—
296C22—41no—
297C22—42no—
298C22C2×D4×D711yescharacteristic
299C22C2×#115911yescharacteristic
300C22C2×#115911yescharacteristic
301C22#654811yescharacteristic
302C22D7×#4111yescharacteristic
303C22#655411yescharacteristic
304C22C22×C22⋊D711yescharacteristic
305C4—41no—
306C4—42no—
307C4—142no—
308C4—142no—
309C4—142no—
310C4—282no—
311C4—281no—
312C4—282no—
313C4#649814yescomplemented
314C2—44no—
315C2—42no—
316C2#9028711yescharacteristic
317C2#9029411yescharacteristic
318C2#9030211yescharacteristic
319C2#9030511yescharacteristic
320C2C2×#649811yescharacteristic
321C2D4×C22⋊D711yescharacteristic
322C2#9051511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
Fitting, prime radicalC2×C7×#121C2
hypercenterC2×#121D7
hypocenterC7#2422
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#6498C2semidirect
3. C2×D4×Dic7C2semidirect
4. #89994C2semidirect
5. #89993C2semidirect
6. #89984C2semidirect
7. C4×C22⋊Dic7C2semidirect
8. #89748C2semidirect
9. #89745C2semidirect
10. #89724C2semidirect
11. #89718C2semidirect
69. C22⋊Dic7D4semidirect
183. C4×C7#123semidirect
283. C7#2422semidirect
313. C4#6498semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1212
C2×#64982
C2×D4×Dic72
#899942
#899932
#899842
C4×C22⋊Dic72
#897482
#897452
#897242
#897182
#24227