Finite Group Databasebeta

← #11738612

Subgroups — #11738612

Order 896.

384 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×#125C211yesmaximal, characteristic, complemented
2C4×#1159C211yesmaximal, characteristic
3C7×#478C211yesmaximal, characteristic, complemented
4#89835C211yesmaximal, characteristic, complemented
5#89781C211yesmaximal, characteristic, complemented
6#89653C211yesmaximal, characteristic, complemented
7#89649C211yesmaximal, characteristic, complemented
8#89633C211yesmaximal, characteristic, complemented
9#89629C211yesmaximal, characteristic, complemented
10#89609C211yesmaximal, characteristic, complemented
11#89601C211yesmaximal, characteristic, complemented
12#89543C211yesmaximal, characteristic
13#89543C211yesmaximal, characteristic
14#89538C211yesmaximal, characteristic, complemented
15#89535C211yesmaximal, characteristic, complemented
16C2×#1159C2211yescharacteristic
17C7×#125C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19D4×Dic7C2211yescharacteristic
20C4×C22⋊D7—21no—
21C2×C4×Dic7—21no—
22C2×#1131C2211yescharacteristic
23C2×#1131C2211yescharacteristic
24#6455C2211yescharacteristic
25D7×C4⋊C4—21no—
26D7×C4⋊C4—21no—
27D7×C4⋊C4C2211yescharacteristic
28D7×C4⋊C4C2211yescharacteristic
29#6453C2211yescharacteristic
30#6453C2211yescharacteristic
31#6452—21no—
32#6452—21no—
33#6450C2211yescharacteristic
34(C22⋊D7)⋊C4—21no—
35#6440—21no—
36C42×D7C2211yescharacteristic
37#6433C2211yescharacteristic
38C4×Dic14C2211yescharacteristic
39C7×#107C2211yescharacteristic
40C7×#106C2211yescharacteristic
41C7×C4⋊C8C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×#102C2211yescharacteristic
44#6406C2211yescharacteristic
45#6406C2211yescharacteristic
46#6392C2211yescharacteristic
47#6391C2211yescharacteristic
48#6390C2211yescharacteristic
49#6390C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3080—71nomaximal, Sylow
60C2×C7×D4C2311yescharacteristic
61C22×C4×C7—21no—
62C2×C22⋊D7—21no—
63C22×Dic7—21no—
64C22×Dic7—21no—
65#1159—22no—
66#1159Q814yes—
67C2×C4×D7—21no—
68C2×C4×D7—21no—
69C2×C4×D7D411yescharacteristic
70C2×C4×D7D411yescharacteristic
71C2×C4×D7C2311yescharacteristic
72C2×Dic14C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C22⋊C4—21no—
81C42×C7C2311yescharacteristic
82C22⋊Dic7—21no—
83(C2×D7)⋊C4—21no—
84C4⋊Dic7—21no—
85C4⋊Dic7—21no—
86C4⋊Dic7C2311yescharacteristic
87C4⋊Dic7C2311yescharacteristic
88C4⋊Dic7C2311yescharacteristic
89Dic7⋊C4—21no—
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97C4×Dic7—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7D411yescharacteristic
101C4×Dic7D411yescharacteristic
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105#1131—22no—
106#1131—22no—
107C2×#125—71no—
108C4×#41—71no—
109#479—71no—
110#479—71no—
111#479—71no—
112#479—71no—
113#478—71no—
114#478—71no—
115#478—71no—
116#478D711yescharacteristic, complemented
117#428—71no—
118#428—71no—
119#423—71no—
120#417—71no—
121#417—71no—
122C23×C7—21no—
123C22×D7C2×D411yescharacteristic
124C7×D4—21no—
125C7×D4C2×Q812yes—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—42no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2411yescharacteristic
133C22⋊D7—24no—
134C2×Dic7—21no—
135C2×Dic7—24no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—42no—
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143C4×D7—41no—
144C4×D7—41no—
145C4×D7—42no—
146C4×D7—42no—
147C4×D7C2×Q812yes—
148C4×D7#4112yes—
149Dic14—21no—
150Dic14C2×Q812yes—
151C7×C8—21no—
152C7×C8—21no—
153C7⋊C8—21no—
154C7⋊C8—21no—
155C2×#41—71no—
156C2×M16—71no—
157C2×M16—71no—
158#125—71no—
159#125—71no—
160#125—71no—
161#125—141no—
162#125—141no—
163#125C2×D711yescharacteristic
164C4×Q8—71no—
165C4×Q8—71no—
166C4×D4—71no—
167C4×D4—141no—
168C4×D4—141no—
169C4×D4C2×D711yescharacteristic
170#117—71no—
171#117—141no—
172C2×C4⋊C4—71no—
173C2×C4⋊C4—71no—
174C2×C4⋊C4—141no—
175C2×C4⋊C4—141no—
176C2×C42—71no—
177C2×C42—141no—
178#107—71no—
179#107—71no—
180#107—71no—
181#107C2×D711yescharacteristic
182#106—71no—
183#106—71no—
184#106—71no—
185#106C2×D711yescharacteristic
186C4⋊C8—71no—
187C4⋊C8—71no—
188C4⋊C8—71no—
189C4⋊C8C2×D711yescharacteristic
190#103—71no—
191#103—71no—
192#103—71no—
193#103—71no—
194#102—71no—
195#102—71no—
196#102C2×D711yescharacteristic
197#102C2×D711yescharacteristic
198C22×C7—22no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7C22×D411yescharacteristic
202C2×D7—21no—
203C2×D7—21no—
204C2×D7C22⋊Q812yes—
205C4×C7—21no—
206C4×C7—21no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7C22×Q811yescharacteristic
211C4×C7C2×#4111yescharacteristic
212Dic7—22no—
213Dic7—21no—
214Dic7—21no—
215Dic7—41no—
216Dic7—41no—
217Dic7—41no—
218Dic7C22⋊Q812yes—
219#41—74no—
220#41—142no—
221C2×Q8—71no—
222C2×D4—141no—
223C2×D4C22×D711yescharacteristic
224C22×C4—21no—
225C22×C4—71no—
226C22×C4—71no—
227C22×C4—71no—
228C22×C4—141no—
229C22×C4—141no—
230C22×C4—141no—
231C22×C4—141no—
232M16—142no—
233M16—142no—
234C2×C8—71no—
235C2×C8—71no—
236C2×C8C22×D711yescharacteristic
237C2×C8C22×D711yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—71no—
241C4⋊C4—71no—
242C4⋊C4—71no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4C22×D711yescharacteristic
254C4⋊C4C22×D711yescharacteristic
255C4⋊C4C22×D711yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C42—71no—
260C42—71no—
261C42—71no—
262C42—141no—
263C42—141no—
264C42—141no—
265C42C22×D711yescharacteristic
266C2×C7—42no—
267C2×C7C2×C22⋊Q811yescharacteristic
268C2×C7#57511yescharacteristic
269C2×C7#57511yescharacteristic
270D7—22no—
271C23—21no—
272C23—71no—
273Q8—72no—
274Q8—141no—
275D4—21no—
276D4—144no—
277D4Q8×D712yes—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—71no—
283C2×C4—72no—
284C2×C4—72no—
285C2×C4—71no—
286C2×C4—71no—
287C2×C4—144no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—281no—
293C2×C4—281no—
294C2×C4—282no—
295C2×C4—282no—
296C2×C4—282no—
297C2×C4D4×D711yescharacteristic
298C2×C4D4×D711yescharacteristic
299C2×C4C23×D711yescharacteristic
300C8—21no—
301C8—21no—
302C8—141no—
303C8—141no—
304C7#308011yescharacteristic, complemented, Sylow
305C22—22no—
306C22—41no—
307C22—41no—
308C22—72no—
309C22—141no—
310C22—141no—
311C22C2×D4×D711yescharacteristic
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—72no—
318C4—141no—
319C4—141no—
320C4—142no—
321C4—281no—
322C4—281no—
323C4—281no—
324C4C2×Q8×D711yescharacteristic
325C4D7×#4111yescharacteristic
326C2—42no—
327C2—142no—
328C2D7×C22⋊Q811yescharacteristic
329C2#9045911yescharacteristic
330C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#478C2
socleC22×C7C22×D4
prime radicalC2×#1159C22
hypercenter#478D7
hypocenterC7#3080
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. D7×#125C2semidirect
3. C7×#478C2semidirect
4. #89835C2semidirect
5. #89781C2semidirect
6. #89653C2semidirect
7. #89649C2semidirect
8. #89633C2semidirect
9. #89629C2semidirect
10. #89609C2semidirect
11. #89601C2semidirect
14. #89538C2semidirect
15. #89535C2semidirect
116. #478D7semidirect
304. C7#3080semidirect

Maximal subgroups

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

subgroupindex
D7×#1252
C4×#11592
C7×#4782
#898352
#897812
#896532
#896492
#896332
#896292
#896092
#896012
#895432
#895382
#895352
#30807