Finite Group Databasebeta

← #11738654

Subgroups — #11738654

Order 896.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90388C211yesmaximal, characteristic, complemented
2#90265C211yesmaximal, characteristic
3C7×#479C211yesmaximal, characteristic, complemented
4#89838C211yesmaximal, characteristic, complemented
5#89803C211yesmaximal, characteristic, complemented
6#89661C211yesmaximal, characteristic, complemented
7#89656C211yesmaximal, characteristic, complemented
8#89634C211yesmaximal, characteristic, complemented
9#89624C211yesmaximal, characteristic, complemented
10#89613C211yesmaximal, characteristic, complemented
11#89612C211yesmaximal, characteristic, complemented
12#89582C211yesmaximal, characteristic
13#89582C211yesmaximal, characteristic
14#89562C211yesmaximal, characteristic, complemented
15#89561C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C7×#125C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19Q8×Dic7C2211yescharacteristic
20C2×#1144C2211yescharacteristic
21C2×#1144C2211yescharacteristic
22C2×#1132C2211yescharacteristic
23C2×#1132C2211yescharacteristic
24#6461—21no—
25#6461—21no—
26#6460C2211yescharacteristic
27#6460C2211yescharacteristic
28#6456—21no—
29#6456C2211yescharacteristic
30D7×C4⋊C4—21no—
31D7×C4⋊C4C2211yescharacteristic
32#6453—21no—
33#6453—21no—
34#6450C2211yescharacteristic
35#6450C2211yescharacteristic
36C4×D28—21no—
37C4×D28C2211yescharacteristic
38#6435—21no—
39#6435C2211yescharacteristic
40#6432C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#106C2211yescharacteristic
43C7×C4⋊C8C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46(C2×D7)⋊C8C2211yescharacteristic
47(C2×D7)⋊C8C2211yescharacteristic
48#6390C2211yescharacteristic
49#6390C2211yescharacteristic
50#6389C2211yescharacteristic
51C8×Dic7C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6383C2211yescharacteristic
55#6383C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3120—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61#1161—22no—
62#1161—22no—
63C2×D28—21no—
64C2×D28C2311yescharacteristic
65C2×C4×D7—21no—
66C2×C4×D7—21no—
67C2×C4×D7—21no—
68C2×C4×D7—21no—
69C2×C4×D7C2311yescharacteristic
70C2×Dic14C2311yescharacteristic
71C2×Dic14C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C7×C4⋊C4—21no—
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C42×C7—21no—
81C42×C7C2311yescharacteristic
82#1144—22no—
83#1144—22no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4D411yescharacteristic
89(C2×D7)⋊C4D411yescharacteristic
90C4⋊Dic7—21no—
91C4⋊Dic7—21no—
92C4⋊Dic7—21no—
93C4⋊Dic7—21no—
94C4⋊Dic7—21no—
95C4⋊Dic7C2311yescharacteristic
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4D411yescharacteristic
100Dic7⋊C4D411yescharacteristic
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#1132—22no—
108#1132—22no—
109#541—71no—
110#520—71no—
111#484—71no—
112#483—71no—
113#479D711yescharacteristic, complemented
114#478—71no—
115#462—71no—
116#461—71no—
117#452—71no—
118#452—71no—
119#441—71no—
120#440—71no—
121C4×Q16—71no—
122C4×SD16—71no—
123#433—71no—
124C22×D7—21no—
125C22×D7C2×D411yescharacteristic
126C7×Q8—21no—
127C7×Q8#4112yes—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2411yescharacteristic
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141D28—24no—
142D28—21no—
143D28#4112yes—
144C4×D7—24no—
145C4×D7—21no—
146C4×D7—21no—
147C4×D7—81no—
148C4×D7—81no—
149C4×D7—81no—
150Dic14—21no—
151Dic14—21no—
152Dic14—21no—
153Dic14—21no—
154C7×C8—21no—
155C7×C8—21no—
156C7⋊C8—21no—
157C7⋊C8—21no—
158C2×#41—71no—
159C2×Q16—71no—
160C2×Q16—71no—
161C2×SD16—71no—
162C2×SD16—71no—
163C4⋊Q8—71no—
164#126—141no—
165#126—141no—
166#125—141no—
167#125—141no—
168#125C2×D711yescharacteristic
169C22⋊Q8—71no—
170C22⋊Q8—71no—
171C4×Q8—71no—
172C4×Q8—71no—
173C4×Q8—71no—
174C4×Q8C2×D711yescharacteristic
175C4×D4—71no—
176C4×D4—71no—
177C4×D4—141no—
178C4×D4—141no—
179#117—71no—
180#117—141no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—141no—
183#107—71no—
184#107C2×D711yescharacteristic
185#106—71no—
186#106C2×D711yescharacteristic
187C4⋊C8—71no—
188C4⋊C8C2×D711yescharacteristic
189#103—71no—
190#103—71no—
191#103—71no—
192#103—71no—
193#103C2×D711yescharacteristic
194#103C2×D711yescharacteristic
195#102—71no—
196#102—71no—
197C22⋊C8—71no—
198C22⋊C8—71no—
199#97—71no—
200C4×C8—71no—
201C22×C7C22×D411yescharacteristic
202C2×D7—21no—
203C2×D7—22no—
204C2×D7—21no—
205C2×D7—21no—
206C2×D7—41no—
207C2×D7—41no—
208C4×C7—22no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7C2×#4111yescharacteristic
215C4×C7C2×#4111yescharacteristic
216Dic7—21no—
217Dic7—21no—
218Dic7—21no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222#41—142no—
223#41—142no—
224C2×Q8—71no—
225C2×Q8—71no—
226C2×Q8C22×D711yescharacteristic
227C2×D4—71no—
228C2×D4—141no—
229C22×C4—71no—
230C22×C4—141no—
231C22×C4—141no—
232C22×C4—141no—
233C22×C4—141no—
234Q16—142no—
235Q16—142no—
236SD16—142no—
237SD16—142no—
238C2×C8—71no—
239C2×C8—71no—
240C2×C8C22×D711yescharacteristic
241C2×C8C22×D711yescharacteristic
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—71no—
246C4⋊C4—71no—
247C4⋊C4—71no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4C22×D711yescharacteristic
257C4⋊C4C22×D711yescharacteristic
258C4⋊C4C22×D711yescharacteristic
259C22⋊C4—71no—
260C22⋊C4—71no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C42—21no—
266C42—71no—
267C42—141no—
268C42—141no—
269C42—141no—
270C42C22×D711yescharacteristic
271C2×C7#52511yescharacteristic
272C2×C7#57711yescharacteristic
273C2×C7#57811yescharacteristic
274D7—41no—
275D7—42no—
276C23—71no—
277C23—141no—
278Q8—21no—
279Q8—141no—
280Q8—141no—
281Q8—141no—
282Q8—141no—
283Q8#116112yes—
284D4—72no—
285D4—144no—
286D4—141no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—71no—
292C2×C4—71no—
293C2×C4—71no—
294C2×C4—144no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—561no—
301C2×C4—561no—
302C2×C4—561no—
303C2×C4D4×D711yescharacteristic
304C2×C4D4×D711yescharacteristic
305C2×C4C23×D711yescharacteristic
306C8—21no—
307C8—21no—
308C8—141no—
309C8—141no—
310C7#312011yescharacteristic, complemented, Sylow
311C22—141no—
312C22—141no—
313C22—142no—
314C22—141no—
315C22—281no—
316C22—281no—
317C22C2×D4×D711yescharacteristic
318C4—21no—
319C4—21no—
320C4—22no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—141no—
325C4—141no—
326C4—141no—
327C4—281no—
328C4—281no—
329C4—281no—
330C4C2×#116111yescharacteristic
331C4D7×#4111yescharacteristic
332C2—281no—
333C2—282no—
334C2#9031811yescharacteristic
335C2#9046111yescharacteristic
336C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#479C2
socleC22×C7C22×D4
prime radicalC2×#1161C22
hypercenter#479D7
hypocenterC7#3120
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. #90388C2semidirect
3. C7×#479C2semidirect
4. #89838C2semidirect
5. #89803C2semidirect
6. #89661C2semidirect
7. #89656C2semidirect
8. #89634C2semidirect
9. #89624C2semidirect
10. #89613C2semidirect
11. #89612C2semidirect
14. #89562C2semidirect
15. #89561C2semidirect
113. #479D7semidirect
310. C7#3120semidirect

Maximal subgroups

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

subgroupindex
#903882
#902652
C7×#4792
#898382
#898032
#896612
#896562
#896342
#896242
#896132
#896122
#895822
#895622
#895612
#31207