Finite Group Databasebeta

← #11740222

Subgroups — #11740222

Order 896.

604 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#6509C211yesmaximal, characteristic, complemented
3C2×Q8×Dic7C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊D7C211yesmaximal, characteristic
5#89984C211yesmaximal, characteristic, complemented
6#89762C212yesmaximal, complemented
7#89759C211yesmaximal, characteristic, complemented
8#89756C212yesmaximal, complemented
9#89749C211yesmaximal, characteristic, complemented
10#89733C212yesmaximal
11#89717C212yesmaximal, complemented
12C22×C7×Q8C2211yescharacteristic
13C23×C4×C7C2211yescharacteristic
14C22×C22⋊D7C2211yescharacteristic
15C22×C4×D7C2211yescharacteristic
16C7×C22⋊Q8—24no—
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C4⋊C4C2212yes—
19C2×C7×C22⋊C4C2212yes—
20C2×C22⋊Dic7C2211yescharacteristic
21C2×C22⋊Dic7C2212yes—
22#6509—24no—
23Q8×Dic7—24no—
24C4×C22⋊D7—24no—
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2212yes—
27C2×(C2×D7)⋊C4C2212yes—
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×C4⋊Dic7C2212yes—
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2212yes—
33C2×C4×Dic7C2212yes—
34C2×C4×Dic7C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37C7⋊#95C2212yes—
38C7⋊#95C2212yes—
39#2536—71nomaximal, Sylow
40C24×C7C2311yescharacteristic
41C23×D7C2311yescharacteristic
42C2×C7×Q8—22no—
43C2×C7×Q8—22no—
44C2×C7×Q8D414yescomplemented
45C22×C4×C7—22no—
46C22×C4×C7—22no—
47C22×C4×C7—22no—
48C22×C4×C7C2312yes—
49C22×C4×C7C2311yescharacteristic
50C22×C4×C7C2312yes—
51C22×C4×C7C2311yescharacteristic
52C2×C22⋊D7—22no—
53C2×C22⋊D7—22no—
54C2×C22⋊D7—22no—
55C22×Dic7C2312yes—
56C22×Dic7C2312yes—
57C22×Dic7C2311yescharacteristic
58C22×Dic7C2311yescharacteristic
59C22×Dic7C2311yescharacteristic
60C2×C4×D7—22no—
61C2×C4×D7—22no—
62C2×C4×D7—22no—
63C7×C4⋊C4—22no—
64C7×C4⋊C4—24no—
65C7×C22⋊C4—24no—
66C22⋊Dic7—24no—
67C22⋊Dic7—22no—
68(C2×D7)⋊C4—24no—
69(C2×D7)⋊C4—24no—
70(C2×D7)⋊C4—22no—
71C4⋊Dic7—22no—
72C4⋊Dic7—22no—
73C4⋊Dic7—24no—
74Dic7⋊C4—24no—
75Dic7⋊C4—22no—
76C4×Dic7—24no—
77C4×Dic7D414yes—
78C2×C22⋊Q8—71no—
79C2×C22⋊Q8D711yescharacteristic, complemented
80C2×C4×Q8—71no—
81C2×C4×D4—71no—
82#398—72no—
83#394—72no—
84#388—72no—
85#388—72no—
86#382—71no—
87#380—71no—
88#380—71no—
89C23×C7—21no—
90C23×C7—21no—
91C23×C7—21no—
92C23×C7—21no—
93C23×C7—21no—
94C23×C7—21no—
95C23×C7#4112yes—
96C23×C7C2411yescharacteristic
97C22×D7—21no—
98C22×D7—21no—
99C22×D7—21no—
100C22×D7—21no—
101C22×D7—21no—
102C22×D7—21no—
103C22×D7#4112yes—
104C7×Q8—28no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—21no—
108C2×C4×C7—22no—
109C2×C4×C7—21no—
110C2×C4×C7—22no—
111C2×C4×C7—21no—
112C2×C4×C7—22no—
113C2×C4×C7—44no—
114C2×C4×C7C2×D412yes—
115C2×C4×C7C2×D414yes—
116C2×C4×C7C2×D412yes—
117C2×C4×C7#4112yes—
118C22⋊D7—28no—
119C2×Dic7—21no—
120C2×Dic7—21no—
121C2×Dic7—21no—
122C2×Dic7—21no—
123C2×Dic7—22no—
124C2×Dic7—21no—
125C2×Dic7—22no—
126C2×Dic7—21no—
127C2×Dic7—22no—
128C2×Dic7—22no—
129C2×Dic7—22no—
130C2×Dic7—21no—
131C2×Dic7—22no—
132C2×Dic7C2×D412yes—
133C2×Dic7C2×D412yes—
134C4×D7—44no—
135C22×Q8C2×D711yescharacteristic
136C22×D4—71no—
137C23×C4—71no—
138C23×C4C2×D711yescharacteristic
139C22⋊Q8—24no—
140C22⋊Q8—144no—
141C4×Q8—144no—
142C4×D4—144no—
143C2×C4⋊C4—71no—
144C2×C4⋊C4—71no—
145C2×C4⋊C4—72no—
146C2×C4⋊C4—72no—
147C2×C4⋊C4—71no—
148C2×C4⋊C4C2×D711yescharacteristic
149C2×C4⋊C4C2×D712yes—
150C2×C22⋊C4—71no—
151C2×C22⋊C4—72no—
152C2×C22⋊C4—71no—
153C2×C22⋊C4—72no—
154C2×C22⋊C4—72no—
155C2×C22⋊C4C2×D712yes—
156C2×C42—71no—
157C2×C42—72no—
158#95—72no—
159#95—72no—
160#95—72no—
161#95—72no—
162C22×C7—22no—
163C22×C7—22no—
164C22×C7—22no—
165C22×C7—41no—
166C22×C7—41no—
167C22×C7—41no—
168C22×C7—41no—
169C22×C7C22×D411yescharacteristic
170C22×C7C22×D411yescharacteristic
171C22×C7C2×#4111yescharacteristic
172C22×C7C2×#4111yescharacteristic
173C22×C7C2×#4111yescharacteristic
174C22×C72−1+411yescharacteristic
175C22×C72−1+411yescharacteristic
176C2×D7—22no—
177C2×D7—22no—
178C2×D7—22no—
179C2×D7—41no—
180C2×D7—41no—
181C2×D7—41no—
182C2×D7—41no—
183C4×C7—24no—
184C4×C7—41no—
185C4×C7—42no—
186C4×C7#12114yes—
187Dic7—22no—
188Dic7—22no—
189Dic7—41no—
190Dic7—42no—
191Dic7—42no—
192C24—71no—
193C24C22×D711yescharacteristic
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8C22⋊D714yescomplemented
197C2×D4—142no—
198C2×D4—142no—
199C2×D4—142no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—71no—
204C22×C4—71no—
205C22×C4—72no—
206C22×C4—72no—
207C22×C4—71no—
208C22×C4—142no—
209C22×C4—142no—
210C22×C4—142no—
211C22×C4C22×D712yes—
212C22×C4C22×D711yescharacteristic
213C22×C4C22×D712yes—
214C22×C4C22×D711yescharacteristic
215C4⋊C4—22no—
216C4⋊C4—24no—
217C4⋊C4—142no—
218C4⋊C4—144no—
219C4⋊C4—142no—
220C4⋊C4—144no—
221C4⋊C4—142no—
222C22⋊C4—24no—
223C22⋊C4—142no—
224C22⋊C4—144no—
225C22⋊C4—142no—
226C22⋊C4—144no—
227C22⋊C4—144no—
228C42—74no—
229C42—144no—
230C2×C7—42no—
231C2×C7C2×#12111yescharacteristic
232C2×C7#52511yescharacteristic
233C2×C7#53611yescharacteristic
234C2×C7#54811yescharacteristic
235C2×C7#54811yescharacteristic
236C2×C7#55211yescharacteristic
237C2×C7#55211yescharacteristic
238D7—42no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—72no—
246C23—141no—
247C23—141no—
248C23—141no—
249C23—141no—
250C23—141no—
251C23—141no—
252C23#116112yes—
253C23C23×D711yescharacteristic
254Q8—28no—
255D4—148no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—44no—
265C2×C4—72no—
266C2×C4—72no—
267C2×C4—141no—
268C2×C4—141no—
269C2×C4—141no—
270C2×C4—141no—
271C2×C4—142no—
272C2×C4—141no—
273C2×C4—141no—
274C2×C4—142no—
275C2×C4—142no—
276C2×C4—142no—
277C2×C4—142no—
278C2×C4—142no—
279C2×C4—141no—
280C2×C4—284no—
281C2×C4D4×D712yes—
282C2×C4#115912yes—
283C2×C4C2×C22⋊D714yes—
284C2×C4C2×C22⋊D712yes—
285C7#253611yescharacteristic, complemented, Sylow
286C22—22no—
287C22—22no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—142no—
294C22—142no—
295C22—142no—
296C22—281no—
297C22—281no—
298C22—281no—
299C22—281no—
300C22C2×D4×D711yescharacteristic
301C22C2×#115911yescharacteristic
302C22C2×#116111yescharacteristic
303C22#655111yescharacteristic
304C22D7×#4111yescharacteristic
305C22#655411yescharacteristic
306C22C22×C22⋊D711yescharacteristic
307C4—24no—
308C4—41no—
309C4—42no—
310C4—142no—
311C4—142no—
312C4—281no—
313C4—282no—
314C4—282no—
315C4#650214yes—
316C2—42no—
317C2—282no—
318C2#9031811yescharacteristic
319C2#9031911yescharacteristic
320C2#9033011yescharacteristic
321C2#9033311yescharacteristic
322C2C2×#650211yescharacteristic
323C2#9050811yescharacteristic
324C2#9052211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×C22⋊Q8C2
prime radicalC22×C22⋊D7C22
hypercenterC2×C22⋊Q8D7
hypocenterC7#2536
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C7×C22⋊Q8C2semidirect
2. C2×#6509C2semidirect
3. C2×Q8×Dic7C2semidirect
5. #89984C2semidirect
6. #89762C2semidirect
7. #89759C2semidirect
8. #89756C2semidirect
9. #89749C2semidirect
11. #89717C2semidirect
44. C2×C7×Q8D4semidirect
79. C2×C22⋊Q8D7semidirect
196. C2×Q8C22⋊D7semidirect
285. C7#2536semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C22⋊Q82
C2×#65092
C2×Q8×Dic72
C2×C4×C22⋊D72
#899842
#897622
#897592
#897562
#897492
#897332
#897172
#25367