Finite Group Databasebeta

← #11738089

Subgroups — #11738089

Order 896.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90334C211yesmaximal, characteristic, complemented
2#90310C211yesmaximal, characteristic, complemented
3C7×#465C211yesmaximal, characteristic, complemented
4#89937C211yesmaximal, characteristic, complemented
5#89935C211yesmaximal, characteristic, complemented
6#89887C211yesmaximal, characteristic, complemented
7#89874C211yesmaximal, characteristic, complemented
8#89821C211yesmaximal, characteristic, complemented
9#89811C211yesmaximal, characteristic, complemented
10#89643C211yesmaximal, characteristic, complemented
11#89642C211yesmaximal, characteristic, complemented
12#89591C211yesmaximal, characteristic, complemented
13#89568C211yesmaximal, characteristic, complemented
14#89558C211yesmaximal, characteristic, complemented
15#89531C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C22×C7×C8C2211yescharacteristic
18C7×C22⋊Q8C2211yescharacteristic
19C7×#121C2211yescharacteristic
20#6510C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22C2×#1143C2211yescharacteristic
23#6503C2211yescharacteristic
24#6502—21no—
25#6501—21no—
26D4×Dic7C2211yescharacteristic
27C2×#1141C2211yescharacteristic
28#6493C2211yescharacteristic
29#6493C2211yescharacteristic
30#6489C2211yescharacteristic
31C2×#1136C2211yescharacteristic
32C2×D56C2211yescharacteristic
33#6461—21no—
34#6460—21no—
35#6456C2211yescharacteristic
36#6456C2211yescharacteristic
37#6453C2211yescharacteristic
38#6451C2211yescharacteristic
39#6449—21no—
40#6448—21no—
41#6446—21no—
42#6446—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#3257—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—
69C2×D28C2311yescharacteristic
70C2×D28C2311yescharacteristic
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×Dic14—21no—
74C7×SD16—41no—
75C2×C7×C8—41no—
76C2×C7×C8C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C7×C4⋊C4—21no—
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C22⋊C4—21no—
82C7×C22⋊C4—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7D411yescharacteristic
85C22⋊Dic7D411yescharacteristic
86#1143—41no—
87#1141—41no—
88(C2×D7)⋊C4—21no—
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—
94C4⋊Dic7—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7D411yescharacteristic
97C4⋊Dic7D411yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102C4×Dic7—21no—
103C4×Dic7C2311yescharacteristic
104C4×Dic7C2311yescharacteristic
105#1136—41no—
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108D56—41no—
109#542—71no—
110#540—71no—
111#479—71no—
112#478—71no—
113#476—71no—
114#474—71no—
115#469—71no—
116#468—71no—
117#466—71no—
118#465D711yescharacteristic, complemented
119#463—71no—
120#463—71no—
121#460—71no—
122#459—71no—
123#424—71no—
124C23×C7—21no—
125C23×C7C2×D411yescharacteristic
126C22×D7—21no—
127C22×D7—21no—
128C7×Q8—21no—
129C7×Q8—21no—
130C7×D4—21no—
131C7×D4—21no—
132C7×D4—81no—
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—81no—
141C22⋊D7—81no—
142C22⋊D7—81no—
143C22⋊D7—81no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—81no—
147C2×Dic7C2×D411yescharacteristic
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151D28—21no—
152D28—21no—
153D28—21no—
154D28—21no—
155C4×D7—81no—
156C4×D7—81no—
157Dic14—81no—
158C7×C8—21no—
159C7×C8—21no—
160C7⋊C8—21no—
161C7⋊C8—21no—
162C2×SD16—71no—
163C2×SD16C2×D711yescharacteristic
164C2×D8—71no—
165C2×D8—71no—
166C2×M16—71no—
167C22×C8C2×D711yescharacteristic
168C4⋊Q8—71no—
169#127—71no—
170#126—141no—
171#125—71no—
172#124—71no—
173#124—141no—
174#123—141no—
175#123—141no—
176#123—141no—
177C22⋊Q8—141no—
178C22⋊Q8C2×D711yescharacteristic
179#121—71no—
180#121—71no—
181#121—141no—
182#121—141no—
183#121C2×D711yescharacteristic
184C4×Q8—71no—
185C4×D4—71no—
186C4×D4—71no—
187C4×D4—71no—
188#117—71no—
189#107—71no—
190#107—71no—
191#106—71no—
192#106C2×D711yescharacteristic
193C4⋊C8—71no—
194C4⋊C8—71no—
195C4⋊C8—71no—
196C4⋊C8—71no—
197#103—71no—
198#103C2×D711yescharacteristic
199#102—71no—
200#102—71no—
201#102—71no—
202#102—71no—
203#102—71no—
204#102C2×D711yescharacteristic
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7C22×D411yescharacteristic
212C2×D7—41no—
213C2×D7—41no—
214C2×D7—41no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C4×C7—21no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7H(5,𝔽2)11yescharacteristic
223C4×C72−1+411yescharacteristic
224Dic7—21no—
225Dic7—21no—
226Dic7—21no—
227Dic7—21no—
228Dic7—41no—
229Dic7—41no—
230C2×Q8—141no—
231C2×Q8C22×D711yescharacteristic
232C2×D4—21no—
233C2×D4—71no—
234C2×D4—71no—
235C2×D4—141no—
236C2×D4—141no—
237C2×D4—141no—
238C2×D4—141no—
239C2×D4C22×D711yescharacteristic
240C22×C4—141no—
241C22×C4—141no—
242C22×C4—141no—
243C22×C4C22×D711yescharacteristic
244SD16—41no—
245SD16—281no—
246D8—281no—
247D8—281no—
248M16—281no—
249C2×C8—41no—
250C2×C8—71no—
251C2×C8—71no—
252C2×C8C22×D711yescharacteristic
253C2×C8C22×D711yescharacteristic
254C4⋊C4—21no—
255C4⋊C4—71no—
256C4⋊C4—71no—
257C4⋊C4—71no—
258C4⋊C4—71no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4C22×D711yescharacteristic
264C4⋊C4C22×D711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—71no—
268C22⋊C4—71no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C42—71no—
277C42—71no—
278C42—141no—
279C2×C7—41no—
280C2×C7—81no—
281C2×C7#53711yescharacteristic
282C2×C7#57611yescharacteristic
283C2×C7#57711yescharacteristic
284D7—81no—
285D7—81no—
286C23—21no—
287C23—141no—
288C23—141no—
289C23D4×D711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—561no—
293D4—21no—
294D4—21no—
295D4—81no—
296D4—141no—
297D4—141no—
298D4—141no—
299D4—141no—
300D4—561no—
301D4—561no—
302D4—561no—
303D4—561no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—71no—
310C2×C4—71no—
311C2×C4—71no—
312C2×C4—71no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—561no—
316C2×C4—561no—
317C2×C4—561no—
318C2×C4D4×D711yescharacteristic
319C2×C4C23×D711yescharacteristic
320C8—21no—
321C8—21no—
322C8—141no—
323C8—141no—
324C7#325711yescharacteristic, complemented, Sylow
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—281no—
332C22—281no—
333C22—281no—
334C22—281no—
335C22—281no—
336C22—281no—
337C22C2×D4×D711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—141no—
343C4—141no—
344C4—141no—
345C4—141no—
346C4—281no—
347C4—281no—
348C4#654811yescharacteristic
349C4#655411yescharacteristic
350C2—41no—
351C2—81no—
352C2—561no—
353C2—561no—
354C2#9028811yescharacteristic
355C2#9045311yescharacteristic
356C2#9046011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#465C2
socleC22×C7C22×D4
hypercenter#465D7
hypocenterC7#3257
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. #90334C2semidirect
2. #90310C2semidirect
3. C7×#465C2semidirect
4. #89937C2semidirect
5. #89935C2semidirect
6. #89887C2semidirect
7. #89874C2semidirect
8. #89821C2semidirect
9. #89811C2semidirect
10. #89643C2semidirect
11. #89642C2semidirect
12. #89591C2semidirect
13. #89568C2semidirect
14. #89558C2semidirect
15. #89531C2semidirect
118. #465D7semidirect
324. C7#3257semidirect

Maximal subgroups

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

subgroupindex
#903342
#903102
C7×#4652
#899372
#899352
#898872
#898742
#898212
#898112
#896432
#896422
#895912
#895682
#895582
#895312
#32577