Finite Group Databasebeta

← #11738086

Subgroups — #11738086

Order 896.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90315C211yesmaximal, characteristic, complemented
2#90287C211yesmaximal, characteristic, complemented
3C7×#465C211yesmaximal, characteristic, complemented
4#89937C211yesmaximal, characteristic, complemented
5#89935C211yesmaximal, characteristic, complemented
6#89887C211yesmaximal, characteristic, complemented
7C2×Dic7⋊C8C211yesmaximal, characteristic, complemented
8#89820C211yesmaximal, characteristic, complemented
9#89810C211yesmaximal, characteristic, complemented
10#89643C211yesmaximal, characteristic, complemented
11#89642C211yesmaximal, characteristic, complemented
12#89591C211yesmaximal, characteristic, complemented
13#89566C211yesmaximal, characteristic, complemented
14#89558C211yesmaximal, characteristic, complemented
15#89538C211yesmaximal, characteristic, complemented
16C2×#1159—21no—
17C2×C7×SD16C2211yescharacteristic
18C22×C7×C8C2211yescharacteristic
19C7×C22⋊Q8C2211yescharacteristic
20C7×#121C2211yescharacteristic
21#6510C2211yescharacteristic
22Q8×Dic7C2211yescharacteristic
23C2×#1143C2211yescharacteristic
24#6503C2211yescharacteristic
25#6502—21no—
26D4×Dic7C2211yescharacteristic
27C2×#1141C2211yescharacteristic
28#6493C2211yescharacteristic
29#6493C2211yescharacteristic
30C2×C4×Dic7C2211yescharacteristic
31C22×C7⋊C8C2211yescharacteristic
32C2×D56C2211yescharacteristic
33#6461—21no—
34#6456C2211yescharacteristic
35#6456C2211yescharacteristic
36#6455—21no—
37#6453C2211yescharacteristic
38#6451C2211yescharacteristic
39#6449—21no—
40#6448—21no—
41(C22⋊D7)⋊C4—21no—
42(C22⋊D7)⋊C4—21no—
43C7×#106C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#102C2211yescharacteristic
46#6409C2211yescharacteristic
47#6406C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6392C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic
58#6381C2211yescharacteristic
59#3247—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4—21no—
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C22×Dic7—21no—
69C22×Dic7D411yescharacteristic
70C22×Dic7D411yescharacteristic
71#1159—44no—
72C2×D28C2311yescharacteristic
73C2×D28C2311yescharacteristic
74C2×C4×D7—21no—
75C2×C4×D7—21no—
76C2×Dic14—21no—
77C7×SD16—41no—
78C2×C7×C8—41no—
79C2×C7×C8C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C7×C4⋊C4—21no—
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C22⋊C4—21no—
85C7×C22⋊C4—21no—
86C22⋊Dic7—21no—
87#1143—41no—
88#1141—41no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7—21no—
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101C4×Dic7—21no—
102C4×Dic7D411yescharacteristic
103C4×Dic7D411yescharacteristic
104C4×Dic7C2311yescharacteristic
105C4×Dic7C2311yescharacteristic
106C2×C7⋊C8—41no—
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109D56—41no—
110#532—71no—
111#529—71no—
112#478—71no—
113#478—71no—
114#477—71no—
115#476—71no—
116#466—71no—
117#466—71no—
118#465—71no—
119#465D711yescharacteristic, complemented
120#463—71no—
121#463—71no—
122#460—71no—
123#459—71no—
124C2×C4⋊C8—71no—
125C23×C7—21no—
126C23×C7C2×D411yescharacteristic
127C22×D7—21no—
128C22×D7—21no—
129C7×Q8—21no—
130C7×Q8—21no—
131C7×D4—21no—
132C7×D4—21no—
133C7×D4—42no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2411yescharacteristic
141C22⋊D7—42no—
142C22⋊D7—42no—
143C22⋊D7—42no—
144C22⋊D7—42no—
145C2×Dic7—22no—
146C2×Dic7—22no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—42no—
150C2×Dic7C2×D411yescharacteristic
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153C2×Dic7C2×D411yescharacteristic
154D28—21no—
155D28—21no—
156D28—21no—
157D28—21no—
158C4×D7—42no—
159C4×D7—42no—
160Dic14—42no—
161C7×C8—21no—
162C7×C8—21no—
163C7⋊C8—21no—
164C7⋊C8—21no—
165C2×#41—141no—
166C2×SD16—71no—
167C2×SD16C2×D711yescharacteristic
168C2×D8—71no—
169C2×D8—71no—
170C22×C8—71no—
171C22×C8C2×D711yescharacteristic
172C4⋊Q8—71no—
173#127—71no—
174#126—141no—
175#125—71no—
176#124—71no—
177#124—141no—
178#123—141no—
179C22⋊Q8C2×D711yescharacteristic
180#121—71no—
181#121—71no—
182#121—141no—
183#121C2×D711yescharacteristic
184C4×Q8—71no—
185C4×D4—71no—
186C4×D4—71no—
187C4×D4—71no—
188C4×D4—141no—
189C4×D4—141no—
190#117—141no—
191C2×C42—71no—
192#107—71no—
193#107—71no—
194#106—71no—
195#106C2×D711yescharacteristic
196C4⋊C8—71no—
197C4⋊C8—71no—
198C4⋊C8—71no—
199C4⋊C8—71no—
200#103—71no—
201#103C2×D711yescharacteristic
202#102—71no—
203#102—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207#102C2×D711yescharacteristic
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7AGL(1,ℤ/8)12yes—
214C22×C7C22×D411yescharacteristic
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C2×D7—41no—
221C4×C7—21no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7H(5,𝔽2)11yescharacteristic
226C4×C72−1+411yescharacteristic
227Dic7—21no—
228Dic7—21no—
229Dic7—21no—
230Dic7—41no—
231Dic7—41no—
232Dic7#13512yes—
233#41—284no—
234C2×Q8—141no—
235C2×Q8C22×D711yescharacteristic
236C2×D4—21no—
237C2×D4—71no—
238C2×D4—71no—
239C2×D4—141no—
240C2×D4—141no—
241C2×D4—141no—
242C2×D4—141no—
243C2×D4C22×D711yescharacteristic
244C22×C4—71no—
245C22×C4—71no—
246C22×C4—141no—
247C22×C4—141no—
248C22×C4—141no—
249C22×C4C22×D711yescharacteristic
250SD16—41no—
251SD16—281no—
252D8—281no—
253D8—281no—
254C2×C8—41no—
255C2×C8—71no—
256C2×C8—71no—
257C2×C8—281no—
258C2×C8C22×D711yescharacteristic
259C2×C8C22×D711yescharacteristic
260C4⋊C4—21no—
261C4⋊C4—71no—
262C4⋊C4—71no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4C22×D711yescharacteristic
268C4⋊C4C22×D711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C42—71no—
279C42—71no—
280C42—71no—
281C42—71no—
282C42—141no—
283C2×C7—22no—
284C2×C7—81no—
285C2×C7#53711yescharacteristic
286C2×C7C2×#13511yescharacteristic
287C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
288D7—81no—
289D7—81no—
290C23—21no—
291C23—141no—
292C23—141no—
293C23D4×D711yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—282no—
297D4—21no—
298D4—21no—
299D4—42no—
300D4—141no—
301D4—141no—
302D4—141no—
303D4—141no—
304D4—282no—
305D4—282no—
306D4—282no—
307D4—282no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—71no—
314C2×C4—71no—
315C2×C4—71no—
316C2×C4—71no—
317C2×C4—142no—
318C2×C4—142no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—282no—
322C2×C4—282no—
323C2×C4—282no—
324C2×C4D4×D711yescharacteristic
325C2×C4C23×D711yescharacteristic
326C8—21no—
327C8—21no—
328C8—141no—
329C8—141no—
330C7#324711yescharacteristic, complemented, Sylow
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—281no—
337C22—281no—
338C22—281no—
339C22—281no—
340C22—281no—
341C22—281no—
342C22#647712yes—
343C22C2×D4×D711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—72no—
349C4—141no—
350C4—141no—
351C4—141no—
352C4—281no—
353C4—281no—
354C4#654811yescharacteristic
355C4#655411yescharacteristic
356C2—22no—
357C2—81no—
358C2—561no—
359C2—561no—
360C2#9028811yescharacteristic
361C2C2×#647711yescharacteristic
362C2D7×#13511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#465C2
socleC22×C7C22×D4
hypercenter#465D7
hypocenterC7#3247
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #90315C2semidirect
2. #90287C2semidirect
3. C7×#465C2semidirect
4. #89937C2semidirect
5. #89935C2semidirect
6. #89887C2semidirect
7. C2×Dic7⋊C8C2semidirect
8. #89820C2semidirect
9. #89810C2semidirect
10. #89643C2semidirect
11. #89642C2semidirect
12. #89591C2semidirect
13. #89566C2semidirect
14. #89558C2semidirect
15. #89538C2semidirect
119. #465D7semidirect
330. C7#3247semidirect

Maximal subgroups

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

subgroupindex
#903152
#902872
C7×#4652
#899372
#899352
#898872
C2×Dic7⋊C82
#898202
#898102
#896432
#896422
#895912
#895662
#895582
#895382
#32477