Finite Group Databasebeta

← #11740091

Subgroups — #11740091

Order 896.

632 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#121C211yesmaximal, characteristic
2C2×#6499C211yesmaximal, characteristic, complemented
3C2×#6487C211yesmaximal, characteristic, complemented
4C2×#6450C211yesmaximal, characteristic, complemented
5#89994C212yesmaximal, complemented
6#89993C212yesmaximal, complemented
7#89993C211yesmaximal, characteristic, complemented
8C4×C22⋊Dic7C211yesmaximal, characteristic, complemented
9#89757C211yesmaximal, characteristic, complemented
10#89720C212yesmaximal, complemented
11#89716C212yesmaximal, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C7×D4C2212yes—
14C23×C4×C7C2211yescharacteristic
15C22×Dic14C2211yescharacteristic
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
25C2×C22⋊Dic7C2212yes—
26#6499—24no—
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2212yes—
29#6487—24no—
30C2×Dic7⋊C4C2212yes—
31C2×Dic7⋊C4C2212yes—
32C2×C4×Dic7C2212yes—
33C2×C4×Dic7C2211yescharacteristic
34C2×C4×Dic7C2211yescharacteristic
35#6450—24no—
36C7⋊#95C2211yescharacteristic
37C7⋊#95C2212yes—
38C7⋊#95C2211yescharacteristic
39#2426—71nomaximal, Sylow
40C24×C7C2312yes—
41C24×C7C2311yescharacteristic
42C2×C7×D4—24no—
43C2×C7×D4—22no—
44C2×C7×D4—24no—
45C2×C7×D4—22no—
46C2×C7×D4—22no—
47C2×C7×D4—42no—
48C22×C4×C7—22no—
49C22×C4×C7—22no—
50C22×C4×C7—22no—
51C22×C4×C7C2311yescharacteristic
52C22×C4×C7C2312yes—
53C22×C4×C7C2311yescharacteristic
54C22×Dic7C2311yescharacteristic
55C22×Dic7C2311yescharacteristic
56C22×Dic7C2311yescharacteristic
57C22×Dic7C2312yes—
58C22×Dic7C2312yes—
59C22×Dic7C2311yescharacteristic
60C2×Dic14—22no—
61C2×Dic14—22no—
62C2×Dic14D414yescomplemented
63C7×C4⋊C4D414yes—
64C7×C22⋊C4—24no—
65C22⋊Dic7—22no—
66C22⋊Dic7—24no—
67C22⋊Dic7—24no—
68C22⋊Dic7—24no—
69C22⋊Dic7—22no—
70C22⋊Dic7—24no—
71C22⋊Dic7—42no—
72C4⋊Dic7—22no—
73C4⋊Dic7—42no—
74Dic7⋊C4—24no—
75Dic7⋊C4—24no—
76C4×Dic7—24no—
77C4×Dic7—22no—
78C4×Dic7—22no—
79C2×#124—71no—
80C2×C22⋊Q8—71no—
81C2×#121D711yescharacteristic
82C2×C4×Q8—71no—
83#396—72no—
84#394—72no—
85#390—71no—
86#390—72no—
87#389—71no—
88#388—72no—
89C4×C22⋊C4—71no—
90C23×C7—21no—
91C23×C7—21no—
92C23×C7—21no—
93C23×C7—21no—
94C23×C7—21no—
95C23×C7—21no—
96C23×C7—22no—
97C23×C7—22no—
98C23×C7—22no—
99C23×C7—22no—
100C23×C7—22no—
101C23×C7—22no—
102C23×C7—22no—
103C23×C7#4112yes—
104C23×C7C2411yescharacteristic
105C7×D4—48no—
106C7×D4—44no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—21no—
112C2×C4×C7—44no—
113C2×C4×C7C2×D412yes—
114C2×C4×C7C2×D414yes—
115C2×C4×C7C2×D412yes—
116C2×C4×C7#4112yes—
117C2×Dic7—21no—
118C2×Dic7—22no—
119C2×Dic7—22no—
120C2×Dic7—21no—
121C2×Dic7—21no—
122C2×Dic7—21no—
123C2×Dic7—21no—
124C2×Dic7—22no—
125C2×Dic7—21no—
126C2×Dic7—21no—
127C2×Dic7—21no—
128C2×Dic7—22no—
129C2×Dic7—21no—
130C2×Dic7—22no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7C2×D414yes—
134C2×Dic7#4112yes—
135Dic14—28no—
136C22×Q8—71no—
137C22×D4C2×D711yescharacteristic
138C22×D4C2×D712yes—
139C23×C4C2×D711yescharacteristic
140#124—144no—
141C22⋊Q8—144no—
142#121—24no—
143C4×Q8—144no—
144C2×C4⋊C4—72no—
145C2×C4⋊C4—72no—
146C2×C4⋊C4—71no—
147C2×C4⋊C4—72no—
148C2×C4⋊C4C2×D711yescharacteristic
149C2×C22⋊C4—72no—
150C2×C22⋊C4—72no—
151C2×C22⋊C4—72no—
152C2×C22⋊C4—71no—
153C2×C22⋊C4—71no—
154C2×C22⋊C4—72no—
155C2×C22⋊C4—72no—
156C2×C22⋊C4C2×D712yes—
157C2×C42—71no—
158C2×C42—72no—
159C2×C42—71no—
160#95—71no—
161#95—71no—
162#95—72no—
163C22×C7—22no—
164C22×C7—22no—
165C22×C7—22no—
166C22×C7—41no—
167C22×C7—41no—
168C22×C7—41no—
169C22×C7—41no—
170C22×C7—42no—
171C22×C7—42no—
172C22×C7—42no—
173C22×C7—42no—
174C22×C7—42no—
175C22×C7—42no—
176C22×C7—42no—
177C22×C7C22×D411yescharacteristic
178C22×C7C22×D411yescharacteristic
179C22×C7C2×#4111yescharacteristic
180C22×C7C2×#4111yescharacteristic
181C22×C7C2×#4111yescharacteristic
182C22×C7H(5,𝔽2)11yescharacteristic
183C22×C72−1+411yescharacteristic
184C4×C7—24no—
185C4×C7—41no—
186C4×C7#12114yes—
187Dic7—24no—
188Dic7—22no—
189Dic7—41no—
190Dic7—41no—
191Dic7—41no—
192Dic7—42no—
193C24C22×D712yes—
194C24C22×D711yescharacteristic
195C2×Q8—74no—
196C2×Q8—142no—
197C2×Q8—142no—
198C2×D4—24no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—42no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—71no—
208C22×C4—72no—
209C22×C4—71no—
210C22×C4—71no—
211C22×C4—71no—
212C22×C4—72no—
213C22×C4C22×D711yescharacteristic
214C22×C4C22×D712yes—
215C22×C4C22×D711yescharacteristic
216C4⋊C4—144no—
217C4⋊C4—142no—
218C4⋊C4—144no—
219C4⋊C4—282no—
220C4⋊C4C22⋊D714yes—
221C22⋊C4—24no—
222C22⋊C4—142no—
223C22⋊C4—144no—
224C22⋊C4—144no—
225C22⋊C4—144no—
226C22⋊C4—144no—
227C22⋊C4—142no—
228C22⋊C4—282no—
229C42—142no—
230C42—144no—
231C42—142no—
232C2×C7—42no—
233C2×C7—82no—
234C2×C7C2×#12111yescharacteristic
235C2×C7#52911yescharacteristic
236C2×C7#54211yescharacteristic
237C2×C7#54611yescharacteristic
238C2×C7#54811yescharacteristic
239C2×C7#54811yescharacteristic
240C2×C7#55011yescharacteristic
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23#115912yes—
255C23C23×D711yescharacteristic
256Q8—148no—
257D4—48no—
258D4—44no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—44no—
265C2×C4—74no—
266C2×C4—72no—
267C2×C4—142no—
268C2×C4—141no—
269C2×C4—141no—
270C2×C4—142no—
271C2×C4—141no—
272C2×C4—141no—
273C2×C4—142no—
274C2×C4—141no—
275C2×C4—141no—
276C2×C4—141no—
277C2×C4—142no—
278C2×C4—141no—
279C2×C4—141no—
280C2×C4—142no—
281C2×C4—141no—
282C2×C4—141no—
283C2×C4D4×D712yes—
284C2×C4#115912yes—
285C2×C4C2×C22⋊D714yes—
286C2×C4C2×C22⋊D712yes—
287C7#242611yescharacteristic, complemented, Sylow
288C22—22no—
289C22—22no—
290C22—22no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22C2×D4×D711yescharacteristic
303C22C2×#115911yescharacteristic
304C22C2×#115911yescharacteristic
305C22#654811yescharacteristic
306C22D7×#4111yescharacteristic
307C22#655411yescharacteristic
308C22C22×C22⋊D711yescharacteristic
309C4—24no—
310C4—41no—
311C4—144no—
312C4—142no—
313C4—281no—
314C4—281no—
315C4—281no—
316C4—282no—
317C4#650114yes—
318C2—42no—
319C2—82no—
320C2#9028911yescharacteristic
321C2#9029011yescharacteristic
322C2#9029111yescharacteristic
323C2#9029411yescharacteristic
324C2C2×#650111yescharacteristic
325C2#9049711yescharacteristic
326C2#9052211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
Fitting, prime radicalC2×C7×#121C2
hypercenterC2×#121D7
hypocenterC7#2426
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#6499C2semidirect
3. C2×#6487C2semidirect
4. C2×#6450C2semidirect
5. #89994C2semidirect
6. #89993C2semidirect
7. #89993C2semidirect
8. C4×C22⋊Dic7C2semidirect
9. #89757C2semidirect
10. #89720C2semidirect
11. #89716C2semidirect
62. C2×Dic14D4semidirect
287. C7#2426semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1212
C2×#64992
C2×#64872
C2×#64502
#899942
#899932
C4×C22⋊Dic72
#897572
#897202
#897162
#24267