Finite Group Databasebeta

← #11737914

Subgroups — #11737914

Order 896.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90412C211yesmaximal, characteristic, complemented
2#90259C211yesmaximal, characteristic
3C7×#462C211yesmaximal, characteristic, complemented
4#89962C211yesmaximal, characteristic
5#89957C211yesmaximal, characteristic
6#89954C211yesmaximal, characteristic, complemented
7#89953C211yesmaximal, characteristic, complemented
8#89858C211yesmaximal, characteristic
9#89802C211yesmaximal, characteristic, complemented
10#89617C211yesmaximal, characteristic
11#89600C211yesmaximal, characteristic, complemented
12#89590C211yesmaximal, characteristic
13#89565C211yesmaximal, characteristic, complemented
14#89561C211yesmaximal, characteristic, complemented
15#89560C211yesmaximal, characteristic
16C2×C7×Q16C2211yescharacteristic
17C2×C7×Q16C2211yescharacteristic
18C7×C4⋊Q8C2211yescharacteristic
19C4×C7×Q8C2211yescharacteristic
20#6510C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22Q8×Dic7C2211yescharacteristic
23#6507C2211yescharacteristic
24#6507C2211yescharacteristic
25#6507C2211yescharacteristic
26C2×#1144C2211yescharacteristic
27C2×#1144C2211yescharacteristic
28C2×#1144C2211yescharacteristic
29C2×#1143C2211yescharacteristic
30C2×#1143C2211yescharacteristic
31C2×Dic28C2211yescharacteristic
32C2×#1132C2211yescharacteristic
33#6461—21no—
34#6459—21no—
35#6456C2211yescharacteristic
36#6453—21no—
37#6452—21no—
38#6452—21no—
39#6450—21no—
40#6450C2211yescharacteristic
41#6439—21no—
42#6438C2211yescharacteristic
43#6435—21no—
44C4×Dic14C2211yescharacteristic
45C4×Dic14C2211yescharacteristic
46C7×C4⋊C8C2211yescharacteristic
47C7×#103C2211yescharacteristic
48C7×#103C2211yescharacteristic
49#6409C2211yescharacteristic
50#6395C2211yescharacteristic
51#6390C2211yescharacteristic
52#6389C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55C8×Dic7C2211yescharacteristic
56#6384C2211yescharacteristic
57#6382C2211yescharacteristic
58#6377C2211yescharacteristic
59#3398—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×C7×Q8C2311yescharacteristic
63C2×D28C2311yescharacteristic
64C2×C4×D7—21no—
65C2×Dic14D411yescharacteristic
66C2×Dic14C2311yescharacteristic
67C2×Dic14C2311yescharacteristic
68C7×Q16—41no—
69C7×Q16—41no—
70C2×C7×C8C2311yescharacteristic
71C2×C7×C8C2311yescharacteristic
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4C2311yescharacteristic
76C42×C7—21no—
77C42×C7C2311yescharacteristic
78#1144—22no—
79#1144—22no—
80#1144—41no—
81#1143—22no—
82#1143—41no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7D411yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4D411yescharacteristic
101Dic7⋊C4D411yescharacteristic
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7—21no—
106C4×Dic7C2311yescharacteristic
107C4×Dic7C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110Dic28—41no—
111#1132—41no—
112#557—71no—
113#556—71no—
114#497—71no—
115#497—71no—
116#494—71no—
117#464—71no—
118#464—71no—
119#462—71no—
120#462—71no—
121#462D711yescharacteristic, complemented
122#461—71no—
123#446—71no—
124#441—71no—
125#441—71no—
126C4×SD16—71no—
127C22×D7—21no—
128C7×Q8—21no—
129C7×Q8—21no—
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2411yescharacteristic
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146C2×Dic7C2×D411yescharacteristic
147C2×Dic7C2×D411yescharacteristic
148D28—21no—
149D28—21no—
150C4×D7—42no—
151Dic14—22no—
152Dic14—21no—
153Dic14—21no—
154Dic14—21no—
155Dic14—21no—
156C7×C8—21no—
157C7×C8—21no—
158C7⋊C8—21no—
159C7⋊C8#4112yes—
160C2×Q16—71no—
161C2×Q16—71no—
162C2×Q16—71no—
163C2×Q16—71no—
164C2×Q16C2×D711yescharacteristic
165C2×Q16C2×D711yescharacteristic
166C2×SD16—71no—
167C2×SD16—71no—
168C2×SD16—71no—
169C4⋊Q8—71no—
170C4⋊Q8—71no—
171C4⋊Q8—71no—
172C4⋊Q8C2×D711yescharacteristic
173#126—141no—
174#126—141no—
175#125—141no—
176#125—141no—
177#125—141no—
178#124—71no—
179#124—71no—
180C22⋊Q8—141no—
181C4×Q8—71no—
182C4×Q8—71no—
183C4×Q8—71no—
184C4×Q8—71no—
185C4×Q8—71no—
186C4×Q8—141no—
187C4×Q8C2×D711yescharacteristic
188C4×D4—71no—
189#117—141no—
190#106—71no—
191C4⋊C8—71no—
192C4⋊C8—71no—
193C4⋊C8C2×D711yescharacteristic
194#103—71no—
195#103—71no—
196#103—71no—
197#103C2×D711yescharacteristic
198#103C2×D711yescharacteristic
199#102—71no—
200#97—71no—
201#97—71no—
202C4×C8—71no—
203C22×C7C22×D411yescharacteristic
204C2×D7—41no—
205C2×D7—41no—
206C2×D7—41no—
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7C2×#4111yescharacteristic
214C4×C7H(5,𝔽2)11yescharacteristic
215Dic7—21no—
216Dic7—21no—
217Dic7—21no—
218Dic7—41no—
219Dic7—41no—
220Dic7—41no—
221Dic7#13712yes—
222C2×Q8—71no—
223C2×Q8—71no—
224C2×Q8—71no—
225C2×Q8C22×D711yescharacteristic
226C2×Q8C22×D711yescharacteristic
227C2×Q8C22×D711yescharacteristic
228C2×D4—71no—
229C22×C4—141no—
230Q16—41no—
231Q16—41no—
232Q16—142no—
233Q16—142no—
234Q16—281no—
235Q16—281no—
236SD16—142no—
237SD16—281no—
238SD16—281no—
239C2×C8—71no—
240C2×C8—71no—
241C2×C8C22×D711yescharacteristic
242C2×C8C22×D711yescharacteristic
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—71no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—71no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4C22×D711yescharacteristic
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C42—21no—
267C42—71no—
268C42—71no—
269C42—141no—
270C42—141no—
271C42—141no—
272C42—141no—
273C42C22×D711yescharacteristic
274C2×C7#55011yescharacteristic
275C2×C7C2×#13711yescharacteristic
276C2×C7#57811yescharacteristic
277D7—81no—
278C23—141no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—21no—
285Q8—142no—
286Q8—141no—
287Q8—141no—
288Q8—141no—
289Q8—141no—
290D4—141no—
291D4—141no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—71no—
297C2×C4—71no—
298C2×C4—71no—
299C2×C4—71no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—282no—
304C2×C4D4×D711yescharacteristic
305C2×C4D4×D711yescharacteristic
306C2×C4C23×D711yescharacteristic
307C8—21no—
308C8—21no—
309C8—72no—
310C8—141no—
311C7#339811yescharacteristic, complemented, Sylow
312C22—281no—
313C22—281no—
314C22—281no—
315C22C2×D4×D711yescharacteristic
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—72no—
323C4—141no—
324C4—141no—
325C4—141no—
326C4—281no—
327C4—281no—
328C4—281no—
329C4#654811yescharacteristic
330C4D7×#4111yescharacteristic
331C2—561no—
332C2#9029011yescharacteristic
333C2#9045711yescharacteristic
334C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#462C2
socleC22×C7C22×D4
prime radicalC2×D28C23
hypercenter#462D7
hypocenterC7#3398
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. #90412C2semidirect
3. C7×#462C2semidirect
6. #89954C2semidirect
7. #89953C2semidirect
9. #89802C2semidirect
11. #89600C2semidirect
13. #89565C2semidirect
14. #89561C2semidirect
121. #462D7semidirect
311. C7#3398semidirect

Maximal subgroups

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

subgroupindex
#904122
#902592
C7×#4622
#899622
#899572
#899542
#899532
#898582
#898022
#896172
#896002
#895902
#895652
#895612
#895602
#33987