Finite Group Databasebeta

← #11738803

Subgroups — #11738803

Order 896.

366 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#90287C211yesmaximal, characteristic
2#90208C211yesmaximal, characteristic, complemented
3C7×#482C211yesmaximal, characteristic, complemented
4#89807C211yesmaximal, characteristic, complemented
5#89775C211yesmaximal, characteristic, complemented
6#89662C211yesmaximal, characteristic, complemented
7#89648C211yesmaximal, characteristic, complemented
8#89642C211yesmaximal, characteristic, complemented
9#89628C211yesmaximal, characteristic, complemented
10#89536C211yesmaximal, characteristic, complemented
11#89536C211yesmaximal, characteristic, complemented
12#89533C211yesmaximal, characteristic, complemented
13#89531C211yesmaximal, characteristic, complemented
14#89509C211yesmaximal, characteristic, complemented
15#89491C211yesmaximal, characteristic, complemented
16C2×#1159—21no—
17C7×#121C2211yescharacteristic
18C7×#117C2211yescharacteristic
19#6503C2211yescharacteristic
20#6502—21no—
21D4×Dic7C2211yescharacteristic
22#6493C2211yescharacteristic
23#6489C2211yescharacteristic
24C2×C4×Dic7C2211yescharacteristic
25#6456C2211yescharacteristic
26#6453C2211yescharacteristic
27#6453C2211yescharacteristic
28#6451C2211yescharacteristic
29#6451C2211yescharacteristic
30#6451C2211yescharacteristic
31#6448—21no—
32(C22⋊D7)⋊C4—21no—
33#6441—21no—
34#6441—21no—
35C4×Dic14—21no—
36C4×Dic14—21no—
37C7×#107C2211yescharacteristic
38C7×#106C2211yescharacteristic
39C7×#102C2211yescharacteristic
40C7×#102C2211yescharacteristic
41C7×C22⋊C8C2211yescharacteristic
42#6406C2211yescharacteristic
43#6406C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6395C2211yescharacteristic
46#6395C2211yescharacteristic
47#6392C2211yescharacteristic
48#6391C2211yescharacteristic
49#6389C2211yescharacteristic
50#6389C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53#6383C2211yescharacteristic
54#6383C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3235—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C22×Dic7—21no—
67C22×Dic7D411yescharacteristic
68C22×Dic7D411yescharacteristic
69#1159—44no—
70C2×D28C2311yescharacteristic
71C2×C4×D7—21no—
72C2×Dic14—21no—
73C2×Dic14—21no—
74C2×Dic14—21no—
75C2×C7×C8C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C22⋊C4—21no—
81C7×C22⋊C4—21no—
82C42×C7—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7C2311yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97C4×Dic7—21no—
98C4×Dic7D411yescharacteristic
99C4×Dic7D411yescharacteristic
100C4×Dic7C2311yescharacteristic
101C4×Dic7C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104#533—71no—
105#532—71no—
106#501—71no—
107#501—71no—
108#490—71no—
109#490—71no—
110#482—71no—
111#482—71no—
112#482—71no—
113#482D711yescharacteristic, complemented
114#476—71no—
115#476—71no—
116#474—71no—
117#474—71no—
118#432—71no—
119C23×C7—21no—
120C23×C7C2×D411yescharacteristic
121C22×D7—21no—
122C7×D4—21no—
123C7×D4—21no—
124C7×D4—42no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2411yescharacteristic
132C22⋊D7—42no—
133C22⋊D7—42no—
134C22⋊D7—42no—
135C2×Dic7—22no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—41no—
140C2×Dic7—42no—
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145D28—21no—
146D28—21no—
147C4×D7—42no—
148Dic14—42no—
149Dic14—42no—
150Dic14—42no—
151C7×C8—21no—
152C7×C8—21no—
153C7⋊C8—21no—
154C7⋊C8—21no—
155C2×#41—141no—
156C4⋊Q8—71no—
157C4⋊Q8—71no—
158C4⋊Q8—71no—
159#127—71no—
160#125—71no—
161#125—71no—
162#124—141no—
163C22⋊Q8—141no—
164C22⋊Q8—141no—
165#121—71no—
166#121—141no—
167#121C2×D711yescharacteristic
168C4×Q8—141no—
169C4×Q8—141no—
170C4×D4—71no—
171C4×D4—71no—
172C4×D4—141no—
173#117—71no—
174#117C2×D711yescharacteristic
175C2×C42—71no—
176#107—71no—
177#107—71no—
178#107—71no—
179#107C2×D711yescharacteristic
180#106—71no—
181#106—71no—
182#106—71no—
183#106C2×D711yescharacteristic
184C4⋊C8—71no—
185C4⋊C8—71no—
186#102—71no—
187#102—71no—
188#102—71no—
189#102—71no—
190#102—71no—
191#102—71no—
192#102C2×D711yescharacteristic
193#102C2×D711yescharacteristic
194C22⋊C8—71no—
195C22⋊C8C2×D711yescharacteristic
196#97—71no—
197#97—71no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7C22×D411yescharacteristic
205C2×D7—41no—
206C2×D7—41no—
207C2×D7—41no—
208C4×C7—21no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C72−1+411yescharacteristic
213C4×C72−1+411yescharacteristic
214Dic7—21no—
215Dic7—21no—
216Dic7—21no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220Dic7AGL(1,ℤ/8)12yes—
221#41—284no—
222C2×Q8—141no—
223C2×Q8—141no—
224C2×Q8—141no—
225C2×D4—21no—
226C2×D4—71no—
227C2×D4—141no—
228C2×D4—141no—
229C2×D4—141no—
230C2×D4C22×D711yescharacteristic
231C22×C4—71no—
232C22×C4—71no—
233C22×C4—141no—
234C22×C4—141no—
235C22×C4C22×D711yescharacteristic
236C2×C8—71no—
237C2×C8—71no—
238C2×C8C22×D711yescharacteristic
239C2×C8C22×D711yescharacteristic
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⋊C4C22×D711yescharacteristic
251C4⋊C4C22×D711yescharacteristic
252C4⋊C4C22×D711yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C42—21no—
260C42—71no—
261C42—71no—
262C42—71no—
263C42—71no—
264C42—141no—
265C2×C7—41no—
266C2×C7—81no—
267C2×C7#53611yescharacteristic
268C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
269C2×C7#57511yescharacteristic
270D7—81no—
271C23—21no—
272C23—141no—
273C23D4×D711yescharacteristic
274Q8—282no—
275Q8—282no—
276Q8—282no—
277D4—21no—
278D4—21no—
279D4—42no—
280D4—141no—
281D4—141no—
282D4—282no—
283D4—282no—
284D4—282no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—71no—
291C2×C4—71no—
292C2×C4—71no—
293C2×C4—71no—
294C2×C4—142no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—282no—
299C2×C4—282no—
300C2×C4—281no—
301C2×C4D4×D711yescharacteristic
302C2×C4C23×D711yescharacteristic
303C8—21no—
304C8—21no—
305C8—141no—
306C8—141no—
307C7#323511yescharacteristic, complemented, Sylow
308C22—21no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—281no—
315C22—281no—
316C22—281no—
317C22C2×D4×D711yescharacteristic
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—72no—
323C4—141no—
324C4—141no—
325C4—141no—
326C4—281no—
327C4—281no—
328C4—281no—
329C4#655411yescharacteristic
330C4#655411yescharacteristic
331C2—41no—
332C2—81no—
333C2—561no—
334C2#9033911yescharacteristic
335C2#9045911yescharacteristic
336C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#482C2
socleC22×C7C22×D4
prime radical#90287C2
hypercenter#482D7
hypocenterC7#3235
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #90208C2semidirect
3. C7×#482C2semidirect
4. #89807C2semidirect
5. #89775C2semidirect
6. #89662C2semidirect
7. #89648C2semidirect
8. #89642C2semidirect
9. #89628C2semidirect
10. #89536C2semidirect
11. #89536C2semidirect
12. #89533C2semidirect
13. #89531C2semidirect
14. #89509C2semidirect
15. #89491C2semidirect
113. #482D7semidirect
307. C7#3235semidirect

Maximal subgroups

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

subgroupindex
#902872
#902082
C7×#4822
#898072
#897752
#896622
#896482
#896422
#896282
#895362
#895332
#895312
#895092
#894912
#32357