Finite Group Databasebeta

← #11738021

Subgroups — #11738021

Order 896.

354 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90371C211yesmaximal, characteristic
2#90269C211yesmaximal, characteristic, complemented
3C7×#464C211yesmaximal, characteristic, complemented
4#89960C211yesmaximal, characteristic, complemented
5#89954C211yesmaximal, characteristic, complemented
6#89943C211yesmaximal, characteristic
7#89934C211yesmaximal, characteristic, complemented
8#89826C211yesmaximal, characteristic, complemented
9#89794C211yesmaximal, characteristic, complemented
10D7×C4⋊C8C211yesmaximal, characteristic, complemented
11#89601C211yesmaximal, characteristic, complemented
12#89585C211yesmaximal, characteristic, complemented
13#89563C211yesmaximal, characteristic, complemented
14#89550C211yesmaximal, characteristic
15#89531C211yesmaximal, characteristic, complemented
16C2×C7×Q16C2211yescharacteristic
17C2×C7×SD16C2211yescharacteristic
18C7×#124C2211yescharacteristic
19C4×C7×Q8C2211yescharacteristic
20#6509C2211yescharacteristic
21#6509C2211yescharacteristic
22Q8×Dic7C2211yescharacteristic
23#6507C2211yescharacteristic
24C2×#1144C2211yescharacteristic
25#6501C2211yescharacteristic
26D4×Dic7C2211yescharacteristic
27C2×#1142C2211yescharacteristic
28C2×C8×D7C2211yescharacteristic
29C2×C8×D7C2211yescharacteristic
30#6460C2211yescharacteristic
31#6459—21no—
32#6452—21no—
33#6451C2211yescharacteristic
34#6446—21no—
35(C22⋊D7)⋊C4—21no—
36#6442—21no—
37#6440—21no—
38#6435—21no—
39C42×D7C2211yescharacteristic
40#6433C2211yescharacteristic
41C4×Dic14—21no—
42C4×Dic14C2211yescharacteristic
43C7×C4⋊C8C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#102C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6409C2211yescharacteristic
49#6406C2211yescharacteristic
50#6392C2211yescharacteristic
51#6391C2211yescharacteristic
52#6390C2211yescharacteristic
53#6390C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55Dic7⋊C8C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3248—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C2×C22⋊D7—21no—
64C22×Dic7—21no—
65C2×C4×D7D411yescharacteristic
66C2×C4×D7D411yescharacteristic
67C2×C4×D7C2311yescharacteristic
68C2×Dic14—21no—
69C2×Dic14C2311yescharacteristic
70C7×Q16—41no—
71C7×SD16—41no—
72C2×C7×C8C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C7×C4⋊C4—21no—
75C7×C4⋊C4C2311yescharacteristic
76C7×C22⋊C4—21no—
77C7×C22⋊C4—21no—
78C42×C7—21no—
79C42×C7C2311yescharacteristic
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82#1144—41no—
83#1142—41no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7C2311yescharacteristic
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⋊C4—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7D411yescharacteristic
103C4×Dic7D411yescharacteristic
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107C8×D7—41no—
108C8×D7—41no—
109#533—71no—
110#529—71no—
111#479—71no—
112#479—71no—
113#475—71no—
114#474—71no—
115#467—71no—
116#467—71no—
117#465—71no—
118#465—71no—
119#464—71no—
120#464D711yescharacteristic, complemented
121#462—71no—
122#461—71no—
123C2×C4⋊C8—71no—
124C23×C7—21no—
125C22×D7C2×D411yescharacteristic
126C7×Q8—21no—
127C7×Q8—21no—
128C7×Q8—21no—
129C7×Q8—21no—
130C7×D4—21no—
131C7×D4—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2411yescharacteristic
138C22⋊D7—42no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—42no—
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146C2×Dic7C2×D411yescharacteristic
147C4×D7—22no—
148C4×D7—21no—
149C4×D7—21no—
150C4×D7—22no—
151Dic14—21no—
152Dic14—21no—
153Dic14—42no—
154C7×C8—21no—
155C7×C8—21no—
156C7⋊C8—21no—
157C7⋊C8—21no—
158C2×Q16—71no—
159C2×Q16C2×D711yescharacteristic
160C2×SD16—71no—
161C2×SD16C2×D711yescharacteristic
162C22×C8—71no—
163C22×C8—71no—
164C4⋊Q8—71no—
165C4⋊Q8—71no—
166#126—141no—
167#125—71no—
168#125—141no—
169#124C2×D711yescharacteristic
170#123—141no—
171C22⋊Q8—71no—
172C22⋊Q8—71no—
173C22⋊Q8—71no—
174C22⋊Q8—141no—
175#121—71no—
176C4×Q8—71no—
177C4×Q8—71no—
178C4×Q8—141no—
179C4×Q8C2×D711yescharacteristic
180C4×D4—71no—
181C4×D4—141no—
182#117—141no—
183#117—141no—
184C2×C42—71no—
185#107—71no—
186#107—71no—
187#106—71no—
188#106—71no—
189C4⋊C8—71no—
190C4⋊C8—71no—
191C4⋊C8—71no—
192C4⋊C8C2×D711yescharacteristic
193#103—71no—
194#103—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103C2×D711yescharacteristic
199#102—71no—
200#102C2×D711yescharacteristic
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7C22×D411yescharacteristic
205C2×D7—21no—
206C2×D7—21no—
207C2×D7#13712yes—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7H(5,𝔽2)11yescharacteristic
214C4×C72−1+411yescharacteristic
215Dic7—21no—
216Dic7—21no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220Dic7—41no—
221Dic7#13512yes—
222C2×Q8—71no—
223C2×Q8—141no—
224C2×Q8C22×D711yescharacteristic
225C2×Q8C22×D711yescharacteristic
226C2×D4—141no—
227C2×D4C22×D711yescharacteristic
228C22×C4—71no—
229C22×C4—71no—
230C22×C4—71no—
231C22×C4—141no—
232Q16—41no—
233Q16—281no—
234SD16—41no—
235SD16—281no—
236C2×C8—71no—
237C2×C8—71no—
238C2×C8—281no—
239C2×C8—281no—
240C2×C8C22×D711yescharacteristic
241C2×C8C22×D711yescharacteristic
242C4⋊C4—21no—
243C4⋊C4—71no—
244C4⋊C4—71no—
245C4⋊C4—71no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4C22×D711yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C42—21no—
265C42—71no—
266C42—71no—
267C42—71no—
268C42—141no—
269C42—141no—
270C42C22×D711yescharacteristic
271C2×C7—81no—
272C2×C7#53711yescharacteristic
273C2×C7C2×#13511yescharacteristic
274C2×C7C2×#13711yescharacteristic
275D7—22no—
276C23—21no—
277C23—71no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—141no—
283Q8—141no—
284Q8—282no—
285D4—21no—
286D4—21no—
287D4—282no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—71no—
292C2×C4—71no—
293C2×C4—71no—
294C2×C4—142no—
295C2×C4—142no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—282no—
303C2×C4D4×D711yescharacteristic
304C2×C4D4×D711yescharacteristic
305C2×C4C23×D711yescharacteristic
306C8—21no—
307C8—21no—
308C8—141no—
309C8—141no—
310C7#324811yescharacteristic, complemented, Sylow
311C22—41no—
312C22—41no—
313C22—41no—
314C22—72no—
315C22—141no—
316C22—141no—
317C22C2×D4×D711yescharacteristic
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—72no—
324C4—141no—
325C4—141no—
326C4—281no—
327C4—281no—
328C4—281no—
329C4—281no—
330C4#654811yescharacteristic
331C4#655411yescharacteristic
332C2—81no—
333C2—142no—
334C2#9029911yescharacteristic
335C2D7×#13511yescharacteristic
336C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#464C2
socleC22×C7C22×D4
prime radical#6501C22
hypercenter#464D7
hypocenterC7#3248
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #90269C2semidirect
3. C7×#464C2semidirect
4. #89960C2semidirect
5. #89954C2semidirect
7. #89934C2semidirect
8. #89826C2semidirect
9. #89794C2semidirect
10. D7×C4⋊C8C2semidirect
11. #89601C2semidirect
12. #89585C2semidirect
13. #89563C2semidirect
15. #89531C2semidirect
120. #464D7semidirect
310. C7#3248semidirect

Maximal subgroups

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

subgroupindex
#903712
#902692
C7×#4642
#899602
#899542
#899432
#899342
#898262
#897942
D7×C4⋊C82
#896012
#895852
#895632
#895502
#895312
#32487