Finite Group Databasebeta

← #11737113

Subgroups — #11737113

Order 896.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90274C211yesmaximal, characteristic, complemented
2#90252C211yesmaximal, characteristic
3C4×C7×SD16C211yesmaximal, characteristic, complemented
4#89946C211yesmaximal, characteristic
5#89939C211yesmaximal, characteristic, complemented
6#89802C211yesmaximal, characteristic, complemented
7#89790C211yesmaximal, characteristic, complemented
8#89640C211yesmaximal, characteristic, complemented
9#89628C211yesmaximal, characteristic, complemented
10#89585C211yesmaximal, characteristic, complemented
11#89564C211yesmaximal, characteristic, complemented
12#89550C211yesmaximal, characteristic
13#89540C211yesmaximal, characteristic, complemented
14C4×Dic28C211yesmaximal, characteristic, complemented
15#89461C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19#6509C2211yescharacteristic
20#6507C2211yescharacteristic
21C2×#1144C2211yescharacteristic
22#6501C2211yescharacteristic
23#6499C2211yescharacteristic
24C2×#1142C2211yescharacteristic
25#6492—21no—
26#6487—21no—
27C2×Dic28C2211yescharacteristic
28C2×#1131C2211yescharacteristic
29C2×C8×D7C2211yescharacteristic
30#6461—21no—
31#6460C2211yescharacteristic
32#6460C2211yescharacteristic
33#6453C2211yescharacteristic
34#6452—21no—
35#6451C2211yescharacteristic
36#6448—21no—
37#6442—21no—
38#6439—21no—
39#6435C2211yescharacteristic
40#6433—21no—
41C4×Dic14C2211yescharacteristic
42C4×Dic14C2211yescharacteristic
43C7×#106C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#102C2211yescharacteristic
46C4×C7×C8C2211yescharacteristic
47#6409C2211yescharacteristic
48#6406C2211yescharacteristic
49#6392C2211yescharacteristic
50#6390C2211yescharacteristic
51#6390C2211yescharacteristic
52#6389C2211yescharacteristic
53C8×Dic7C2211yescharacteristic
54#6384C2211yescharacteristic
55#6384C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3178—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C2×C22⋊D7—21no—
64C2×C4×D7C2311yescharacteristic
65C2×Dic14—21no—
66C2×Dic14C2311yescharacteristic
67C2×Dic14C2311yescharacteristic
68C7×SD16—22no—
69C2×C7×C8C2311yescharacteristic
70C2×C7×C8C2311yescharacteristic
71C7×C4⋊C4—21no—
72C7×C4⋊C4C2311yescharacteristic
73C7×C4⋊C4C2311yescharacteristic
74C7×C22⋊C4—21no—
75C42×C7—21no—
76C42×C7C2311yescharacteristic
77C22⋊Dic7—21no—
78C22⋊Dic7—21no—
79C22⋊Dic7—21no—
80#1144—22no—
81#1142—22no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4D411yescharacteristic
86(C2×D7)⋊C4D411yescharacteristic
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4D411yescharacteristic
100Dic7⋊C4D411yescharacteristic
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106Dic28—22no—
107#1131—22no—
108C8×D7—22no—
109#542—71no—
110#541—71no—
111#501—71no—
112#497—71no—
113#488—71no—
114#487—71no—
115#470—71no—
116#468—71no—
117#467—71no—
118#465—71no—
119#441—71no—
120#440—71no—
121C4×Q16—71no—
122C4×SD16D711yescharacteristic, complemented
123#405—71no—
124C23×C7—21no—
125C22×D7C2×D411yescharacteristic
126C7×Q8—21no—
127C7×Q8—21no—
128C7×D4—21no—
129C7×D4—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—81no—
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2411yescharacteristic
137C22⋊D7—81no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145C4×D7—21no—
146C4×D7—21no—
147Dic14—21no—
148Dic14—21no—
149Dic14—21no—
150Dic14—21no—
151Dic14—81no—
152C7×C8—21no—
153C7×C8#4112yes—
154C7⋊C8—21no—
155C7⋊C8#4112yes—
156C2×Q16—71no—
157C2×Q16—71no—
158C2×SD16—71no—
159C2×SD16C2×D711yescharacteristic
160C2×M16—71no—
161C22×C8—71no—
162C4⋊Q8—71no—
163C4⋊Q8—71no—
164#126—141no—
165#126—141no—
166#126—141no—
167#125—71no—
168#125—141no—
169#125—141no—
170#124—71no—
171#124—141no—
172#123—141no—
173C22⋊Q8—71no—
174C22⋊Q8—71no—
175C22⋊Q8—71no—
176C22⋊Q8—141no—
177#121—71no—
178C4×Q8—71no—
179C4×Q8—71no—
180C4×Q8C2×D711yescharacteristic
181C4×D4C2×D711yescharacteristic
182#117—71no—
183#107—71no—
184#107—71no—
185#106—71no—
186#106C2×D711yescharacteristic
187#103—71no—
188#103—71no—
189#103—71no—
190#103—71no—
191#103—71no—
192#103C2×D711yescharacteristic
193#102—71no—
194#102C2×D711yescharacteristic
195#97—71no—
196#97—71no—
197C4×C8—71no—
198C4×C8C2×D711yescharacteristic
199C22×C7—41no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7C22×D411yescharacteristic
203C2×D7—21no—
204C2×D7—21no—
205C2×D7—21no—
206C4×C7—21no—
207C4×C7—21no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7C2×#4111yescharacteristic
212C4×C7C2×#4111yescharacteristic
213Dic7—21no—
214Dic7—21no—
215Dic7—21no—
216Dic7—41no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220C2×Q8—71no—
221C2×Q8—71no—
222C2×Q8—141no—
223C2×Q8C22×D711yescharacteristic
224C2×D4—141no—
225C2×D4C22×D711yescharacteristic
226C22×C4—21no—
227C22×C4—71no—
228Q16—142no—
229Q16—142no—
230SD16—22no—
231SD16—142no—
232M16—142no—
233C2×C8—71no—
234C2×C8—71no—
235C2×C8—142no—
236C2×C8C22×D711yescharacteristic
237C2×C8C22×D711yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—71no—
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
255C22⋊C4—21no—
256C22⋊C4—71no—
257C22⋊C4—71no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C42—21no—
265C42—71no—
266C42—141no—
267C42—141no—
268C42C22×D711yescharacteristic
269C2×C7—81no—
270C2×C7#53211yescharacteristic
271C2×C7#57711yescharacteristic
272C2×C7#57811yescharacteristic
273D7—41no—
274C23—21no—
275C23—71no—
276Q8—21no—
277Q8—21no—
278Q8—141no—
279Q8—141no—
280Q8—141no—
281Q8—141no—
282Q8—561no—
283D4—21no—
284D4—21no—
285D4—561no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—71no—
290C2×C4—71no—
291C2×C4—71no—
292C2×C4—81no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4D4×D711yescharacteristic
300C2×C4D4×D711yescharacteristic
301C2×C4C23×D711yescharacteristic
302C8—21no—
303C8—72no—
304C8—141no—
305C8#115712yes—
306C7#317811yescharacteristic, complemented, Sylow
307C22—41no—
308C22—41no—
309C22—41no—
310C22—141no—
311C22—141no—
312C22—141no—
313C22C2×D4×D711yescharacteristic
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—141no—
320C4—141no—
321C4—141no—
322C4—281no—
323C4—281no—
324C4—281no—
325C4—281no—
326C4C2×#115711yescharacteristic
327C4D7×#4111yescharacteristic
328C2—81no—
329C2—281no—
330C2#9023911yescharacteristic
331C2#9045311yescharacteristic
332C2#9046211yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC4×C7×SD16C2
socleC22×C7C22×D4
prime radical#6501C22
hypercenterC4×SD16D7
hypocenterC7#3178
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #90274C2semidirect
3. C4×C7×SD16C2semidirect
5. #89939C2semidirect
6. #89802C2semidirect
7. #89790C2semidirect
8. #89640C2semidirect
9. #89628C2semidirect
10. #89585C2semidirect
11. #89564C2semidirect
13. #89540C2semidirect
14. C4×Dic28C2semidirect
15. #89461C2semidirect
122. C4×SD16D7semidirect
306. C7#3178semidirect

Maximal subgroups

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

subgroupindex
#902742
#902522
C4×C7×SD162
#899462
#899392
#898022
#897902
#896402
#896282
#895852
#895642
#895502
#895402
C4×Dic282
#894612
#31787