Finite Group Databasebeta

← #11739196

Subgroups — #11739196

Order 896.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90418C211yesmaximal, characteristic, complemented
2#90386C211yesmaximal, characteristic
3C7×#491C211yesmaximal, characteristic, complemented
4#89855C211yesmaximal, characteristic, complemented
5#89835C211yesmaximal, characteristic, complemented
6#89586C211yesmaximal, characteristic, complemented
7#89584C211yesmaximal, characteristic, complemented
8#89581C211yesmaximal, characteristic, complemented
9#89579C211yesmaximal, characteristic, complemented
10#89578C211yesmaximal, characteristic, complemented
11#89578C211yesmaximal, characteristic, complemented
12#89574C211yesmaximal, characteristic, complemented
13D7×#103C211yesmaximal, characteristic, complemented
14#89482C211yesmaximal, characteristic, complemented
15#89478C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C7×C4⋊Q8C2211yescharacteristic
19C7×#125C2211yescharacteristic
20#6509C2211yescharacteristic
21#6509C2211yescharacteristic
22C2×#1143C2211yescharacteristic
23C2×#1143C2211yescharacteristic
24C2×#1132C2211yescharacteristic
25C2×#1131C2211yescharacteristic
26C2×C8×D7C2211yescharacteristic
27#6461—21no—
28#6459—21no—
29#6459—21no—
30#6458—21no—
31#6458C2211yescharacteristic
32#6458C2211yescharacteristic
33#6457—21no—
34#6456—21no—
35#6456—21no—
36#6455C2211yescharacteristic
37D7×C4⋊C4C2211yescharacteristic
38#6451—21no—
39C4×D28C2211yescharacteristic
40C4×D28C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46C7×#97C2211yescharacteristic
47#6409C2211yescharacteristic
48#6409C2211yescharacteristic
49#6395C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51(C2×D7)⋊C8C2211yescharacteristic
52#6392C2211yescharacteristic
53#6390C2211yescharacteristic
54#6384C2211yescharacteristic
55#6384C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3320—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62Q8×D7—22no—
63Q8×D7—22no—
64Q8×D7—41no—
65Q8×D7—41no—
66C2×D28—21no—
67C2×D28—21no—
68C2×D28D411yescharacteristic
69C2×D28C2311yescharacteristic
70C2×C4×D7—21no—
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×C4×D7—21no—
74C2×C4×D7C2311yescharacteristic
75C2×C4×D7C2311yescharacteristic
76C2×Dic14—21no—
77C2×Dic14—21no—
78C2×Dic14C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C7×C4⋊C4—21no—
82C7×C4⋊C4—21no—
83C7×C4⋊C4—21no—
84C7×C4⋊C4—21no—
85C7×C4⋊C4C2311yescharacteristic
86C7×C4⋊C4C2311yescharacteristic
87C42×C7C2311yescharacteristic
88#1143—41no—
89#1143—41no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4D411yescharacteristic
97(C2×D7)⋊C4D411yescharacteristic
98C4⋊Dic7D411yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100C4⋊Dic7C2311yescharacteristic
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105C4×Dic7—21no—
106C4×Dic7—21no—
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#1132—22no—
110#1131—41no—
111C8×D7—41no—
112#553—71no—
113D4×Q8—71no—
114#491D711yescharacteristic, complemented
115#485—71no—
116#484—71no—
117#479—71no—
118#468—71no—
119#465—71no—
120#461—71no—
121#448—71no—
122#448—71no—
123#440—71no—
124#435—71no—
125#419—71no—
126C2×#103—71no—
127C22×D7—21no—
128C22×D7C2×D411yescharacteristic
129C22×D7C2×D411yescharacteristic
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
146D28—22no—
147D28—21no—
148D28—21no—
149D28—42no—
150D28—81no—
151C4×D7—21no—
152C4×D7—21no—
153C4×D7—21no—
154C4×D7—21no—
155C4×D7—42no—
156C4×D7—42no—
157C4×D7—81no—
158C4×D7—81no—
159Dic14—21no—
160Dic14—42no—
161Dic14—42no—
162Dic14#4112yes—
163C7×C8—21no—
164C7×C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C22×Q8—71no—
168C22×Q8—71no—
169C2×SD16—71no—
170C2×SD16—71no—
171C2×SD16—71no—
172C2×M16—71no—
173C22×C8—71no—
174C4⋊Q8—141no—
175C4⋊Q8C2×D711yescharacteristic
176#126—141no—
177#125C2×D711yescharacteristic
178#123—141no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182C22⋊Q8—141no—
183#121—71no—
184#121—71no—
185#121—141no—
186C4×Q8—71no—
187C4×D4—71no—
188C4×D4—71no—
189C4×D4—141no—
190C4×D4—141no—
191#117—71no—
192C2×C4⋊C4—71no—
193#107—71no—
194#107—71no—
195#106—71no—
196C4⋊C8—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#103C2×D711yescharacteristic
203#103C2×D711yescharacteristic
204#103C2×D711yescharacteristic
205#103C2×D711yescharacteristic
206#102—71no—
207C22⋊C8—71no—
208C22⋊C8—71no—
209#97C2×D711yescharacteristic
210C22×C7C22×D411yescharacteristic
211C2×D7—21no—
212C2×D7—21no—
213C2×D7—21no—
214C2×D7—21no—
215C2×D7—21no—
216C2×D7—41no—
217C2×D7—41no—
218C2×D7—41no—
219C2×D7#13712yes—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7C2×#4111yescharacteristic
227C4×C7H(5,𝔽2)11yescharacteristic
228Dic7—22no—
229Dic7—21no—
230Dic7—21no—
231Dic7—41no—
232Dic7—41no—
233C2×Q8—71no—
234C2×Q8—141no—
235C2×Q8—141no—
236C2×Q8—142no—
237C2×Q8—142no—
238C2×Q8—281no—
239C2×Q8—281no—
240C2×Q8C22×D711yescharacteristic
241C2×Q8C22×D711yescharacteristic
242C2×D4—71no—
243C2×D4—71no—
244C2×D4—141no—
245C2×D4—141no—
246C22×C4—71no—
247C22×C4—71no—
248C22×C4—141no—
249C22×C4—141no—
250C22×C4—141no—
251C22×C4—141no—
252SD16—142no—
253SD16—281no—
254SD16—281no—
255M16—281no—
256C2×C8—71no—
257C2×C8—71no—
258C2×C8—281no—
259C2×C8C22×D711yescharacteristic
260C2×C8C22×D711yescharacteristic
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—71no—
266C4⋊C4—71no—
267C4⋊C4—71no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4C22×D711yescharacteristic
273C4⋊C4C22×D711yescharacteristic
274C22⋊C4—71no—
275C22⋊C4—71no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C42—141no—
283C42—141no—
284C42C22×D711yescharacteristic
285C2×C7#54611yescharacteristic
286C2×C7C2×#13711yescharacteristic
287C2×C7#57711yescharacteristic
288D7—22no—
289D7—41no—
290D7—81no—
291C23—71no—
292C23—71no—
293C23—141no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—72no—
299Q8—141no—
300Q8—282no—
301Q8—282no—
302D4—142no—
303D4—141no—
304D4—141no—
305D4—282no—
306D4—561no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—71no—
312C2×C4—71no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—282no—
321C2×C4—282no—
322C2×C4—561no—
323C2×C4—561no—
324C2×C4D4×D711yescharacteristic
325C2×C4D4×D711yescharacteristic
326C2×C4C23×D711yescharacteristic
327C8—21no—
328C8—21no—
329C8—141no—
330C8—141no—
331C7#332011yescharacteristic, complemented, Sylow
332C22—72no—
333C22—141no—
334C22—141no—
335C22—141no—
336C22—141no—
337C22—141no—
338C22—281no—
339C22—281no—
340C22—281no—
341C22C2×D4×D711yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—141no—
349C4—141no—
350C4—142no—
351C4—281no—
352C4—281no—
353C4D7×#4111yescharacteristic
354C4#655311yescharacteristic
355C2—142no—
356C2—281no—
357C2—561no—
358C2#9036711yescharacteristic
359C2D7×#13711yescharacteristic
360C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#491C2
socleC22×C7C22×D4
prime radical#90386C2
hypercenter#491D7
hypocenterC7#3320
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #90418C2semidirect
3. C7×#491C2semidirect
4. #89855C2semidirect
5. #89835C2semidirect
6. #89586C2semidirect
7. #89584C2semidirect
8. #89581C2semidirect
9. #89579C2semidirect
10. #89578C2semidirect
11. #89578C2semidirect
12. #89574C2semidirect
13. D7×#103C2semidirect
14. #89482C2semidirect
15. #89478C2semidirect
114. #491D7semidirect
331. C7#3320semidirect

Maximal subgroups

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

subgroupindex
#904182
#903862
C7×#4912
#898552
#898352
#895862
#895842
#895812
#895792
#895782
#895742
D7×#1032
#894822
#894782
#33207