Finite Group Databasebeta

← #11738239

Subgroups — #11738239

Order 896.

386 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90313C211yesmaximal, characteristic, complemented
2#90287C211yesmaximal, characteristic
3C7×#468C211yesmaximal, characteristic, complemented
4#89935C211yesmaximal, characteristic, complemented
5#89934C211yesmaximal, characteristic, complemented
6#89906C211yesmaximal, characteristic, complemented
7#89890C211yesmaximal, characteristic, complemented
8#89821C211yesmaximal, characteristic, complemented
9#89814C211yesmaximal, characteristic, complemented
10#89662C211yesmaximal, characteristic, complemented
11#89647C211yesmaximal, characteristic, complemented
12#89590C211yesmaximal, characteristic, complemented
13#89566C211yesmaximal, characteristic, complemented
14#89534C211yesmaximal, characteristic, complemented
15#89531C211yesmaximal, characteristic, complemented
16C2×#1159—21no—
17C2×C7×SD16C2211yescharacteristic
18C2×C7×M16C2211yescharacteristic
19C7×C22⋊Q8C2211yescharacteristic
20C7×#121C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22#6507C2211yescharacteristic
23C2×#1143C2211yescharacteristic
24#6503C2211yescharacteristic
25#6502—21no—
26D4×Dic7C2211yescharacteristic
27C2×#1142C2211yescharacteristic
28#6493C2211yescharacteristic
29#6487C2211yescharacteristic
30C2×C4×Dic7C2211yescharacteristic
31C2×#1136C2211yescharacteristic
32C2×#1132C2211yescharacteristic
33#6456C2211yescharacteristic
34#6452—21no—
35#6451C2211yescharacteristic
36#6451C2211yescharacteristic
37#6450—21no—
38#6450C2211yescharacteristic
39#6448—21no—
40(C22⋊D7)⋊C4—21no—
41#6441—21no—
42#6440—21no—
43C7×#107C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#102C2211yescharacteristic
46#6409C2211yescharacteristic
47#6406C2211yescharacteristic
48#6395C2211yescharacteristic
49#6392C2211yescharacteristic
50#6390C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55#6384C2211yescharacteristic
56#6383C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3254—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—
67C22×Dic7—21no—
68C22×Dic7D411yescharacteristic
69C22×Dic7D411yescharacteristic
70#1159—44no—
71C2×D28C2311yescharacteristic
72C2×C4×D7—21no—
73C2×Dic14—21no—
74C2×Dic14—21no—
75C2×Dic14C2311yescharacteristic
76C7×SD16—41no—
77C7×M16—41no—
78C2×C7×C8C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C7×C4⋊C4—21no—
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C22⋊C4—21no—
84C7×C22⋊C4—21no—
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87#1143—41no—
88#1142—41no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91C4⋊Dic7—21no—
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7D411yescharacteristic
103C4×Dic7D411yescharacteristic
104C4×Dic7C2311yescharacteristic
105C4×Dic7C2311yescharacteristic
106#1136—41no—
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#1132—41no—
110#533—71no—
111#532—71no—
112#477—71no—
113#477—71no—
114#474—71no—
115#474—71no—
116#468—71no—
117#468—71no—
118#468—71no—
119#468D711yescharacteristic, complemented
120#461—71no—
121#461—71no—
122#460—71no—
123#460—71no—
124#423—71no—
125C23×C7—21no—
126C23×C7C2×D411yescharacteristic
127C22×D7—21no—
128C7×Q8—21no—
129C7×Q8—21no—
130C7×D4—21no—
131C7×D4—21no—
132C7×D4—42no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2411yescharacteristic
140C22⋊D7—42no—
141C22⋊D7—42no—
142C22⋊D7—42no—
143C2×Dic7—22no—
144C2×Dic7—22no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—42no—
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153D28—21no—
154D28—21no—
155C4×D7—42no—
156Dic14—21no—
157Dic14—21no—
158Dic14—42no—
159Dic14—42no—
160C7×C8—21no—
161C7×C8—21no—
162C7⋊C8—21no—
163C7⋊C8—21no—
164C2×#41—141no—
165C2×SD16—71no—
166C2×SD16—71no—
167C2×SD16—71no—
168C2×SD16C2×D711yescharacteristic
169C2×M16—71no—
170C2×M16C2×D711yescharacteristic
171C4⋊Q8—71no—
172C4⋊Q8—71no—
173C4⋊Q8—71no—
174#127—71no—
175#125—141no—
176#124—141no—
177C22⋊Q8—71no—
178C22⋊Q8—141no—
179C22⋊Q8C2×D711yescharacteristic
180#121—71no—
181#121—141no—
182#121C2×D711yescharacteristic
183C4×Q8—71no—
184C4×Q8—71no—
185C4×Q8—141no—
186C4×D4—71no—
187C4×D4—71no—
188C4×D4—141no—
189#117—141no—
190C2×C42—71no—
191#107—71no—
192#107—71no—
193#107—71no—
194#107C2×D711yescharacteristic
195C4⋊C8—71no—
196C4⋊C8—71no—
197C4⋊C8—71no—
198C4⋊C8—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#103C2×D711yescharacteristic
203#102—71no—
204#102—71no—
205#102—71no—
206#102C2×D711yescharacteristic
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7C22×D411yescharacteristic
214C2×D7—41no—
215C2×D7—41no—
216C2×D7—41no—
217C4×C7—21no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7H(5,𝔽2)11yescharacteristic
222C4×C72−1+411yescharacteristic
223Dic7—21no—
224Dic7—21no—
225Dic7—41no—
226Dic7—41no—
227Dic7—41no—
228Dic7AGL(1,ℤ/8)12yes—
229Dic7#13712yes—
230#41—284no—
231C2×Q8—71no—
232C2×Q8—141no—
233C2×Q8—141no—
234C2×Q8C22×D711yescharacteristic
235C2×D4—21no—
236C2×D4—71no—
237C2×D4—141no—
238C2×D4—141no—
239C2×D4—141no—
240C2×D4C22×D711yescharacteristic
241C22×C4—71no—
242C22×C4—71no—
243C22×C4—141no—
244C22×C4—141no—
245C22×C4C22×D711yescharacteristic
246SD16—41no—
247SD16—281no—
248SD16—281no—
249SD16—281no—
250M16—41no—
251M16—281no—
252C2×C8—71no—
253C2×C8—71no—
254C2×C8C22×D711yescharacteristic
255C2×C8C22×D711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—71no—
258C4⋊C4—71no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4C22×D711yescharacteristic
267C4⋊C4C22×D711yescharacteristic
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C42—71no—
275C42—71no—
276C42—71no—
277C42—71no—
278C42—141no—
279C42—141no—
280C2×C7—41no—
281C2×C7—81no—
282C2×C7#53711yescharacteristic
283C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
284C2×C7C2×#13711yescharacteristic
285D7—81no—
286C23—21no—
287C23—141no—
288C23D4×D711yescharacteristic
289Q8—21no—
290Q8—21no—
291Q8—141no—
292Q8—141no—
293Q8—282no—
294Q8—282no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—141no—
299D4—141no—
300D4—282no—
301D4—282no—
302D4—282no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—71no—
309C2×C4—71no—
310C2×C4—71no—
311C2×C4—71no—
312C2×C4—142no—
313C2×C4—142no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—282no—
318C2×C4—282no—
319C2×C4D4×D711yescharacteristic
320C2×C4C23×D711yescharacteristic
321C8—21no—
322C8—21no—
323C8—141no—
324C8—141no—
325C7#325411yescharacteristic, complemented, Sylow
326C22—21no—
327C22—21no—
328C22—21no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—281no—
333C22—281no—
334C22—281no—
335C22C2×D4×D711yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—72no—
341C4—72no—
342C4—141no—
343C4—141no—
344C4—281no—
345C4—281no—
346C4—281no—
347C4#654811yescharacteristic
348C4#655411yescharacteristic
349C2—41no—
350C2—81no—
351C2—561no—
352C2#9028811yescharacteristic
353C2D7×AGL(1,ℤ/8)11yescharacteristic
354C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#468C2
socleC22×C7C22×D4
prime radical#90287C2
hypercenter#468D7
hypocenterC7#3254
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #90313C2semidirect
3. C7×#468C2semidirect
4. #89935C2semidirect
5. #89934C2semidirect
6. #89906C2semidirect
7. #89890C2semidirect
8. #89821C2semidirect
9. #89814C2semidirect
10. #89662C2semidirect
11. #89647C2semidirect
12. #89590C2semidirect
13. #89566C2semidirect
14. #89534C2semidirect
15. #89531C2semidirect
119. #468D7semidirect
325. C7#3254semidirect

Maximal subgroups

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

subgroupindex
#903132
#902872
C7×#4682
#899352
#899342
#899062
#898902
#898212
#898142
#896622
#896472
#895902
#895662
#895342
#895312
#32547