Finite Group Databasebeta

← #11739194

Subgroups — #11739194

Order 896.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90418C211yesmaximal, characteristic
2#90385C211yesmaximal, characteristic
3C7×#491C211yesmaximal, characteristic, complemented
4#89862C211yesmaximal, characteristic, complemented
5#89835C211yesmaximal, characteristic, complemented
6#89588C211yesmaximal, characteristic, complemented
7#89587C211yesmaximal, characteristic, complemented
8#89585C211yesmaximal, characteristic, complemented
9#89580C211yesmaximal, characteristic, complemented
10#89580C211yesmaximal, characteristic, complemented
11#89577C211yesmaximal, characteristic, complemented
12#89574C211yesmaximal, characteristic, complemented
13D7×#103C211yesmaximal, characteristic, complemented
14#89488C211yesmaximal, characteristic, complemented
15#89478C211yesmaximal, characteristic
16C2×Q8×D7C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C7×C4⋊Q8C2211yescharacteristic
19C7×#125C2211yescharacteristic
20#6509C2211yescharacteristic
21#6509C2211yescharacteristic
22C2×#1144C2211yescharacteristic
23C2×#1144C2211yescharacteristic
24C2×Dic28C2211yescharacteristic
25C2×#1131C2211yescharacteristic
26C2×C8×D7C2211yescharacteristic
27#6461—21no—
28#6460C2211yescharacteristic
29#6460C2211yescharacteristic
30#6459—21no—
31#6459—21no—
32#6457—21no—
33#6456—21no—
34#6455—21no—
35#6455C2211yescharacteristic
36D7×C4⋊C4C2211yescharacteristic
37#6453—21no—
38#6451—21no—
39C4×D28C2211yescharacteristic
40C4×Dic14C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46C7×#97C2211yescharacteristic
47#6409C2211yescharacteristic
48#6409C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6391C2211yescharacteristic
52#6390C2211yescharacteristic
53#6390C2211yescharacteristic
54#6384C2211yescharacteristic
55#6384C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3312—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62Q8×D7—22no—
63Q8×D7—22no—
64Q8×D7—41no—
65Q8×D7—41no—
66C2×D28D411yescharacteristic
67C2×C4×D7—21no—
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×C4×D7C2311yescharacteristic
71C2×C4×D7C2311yescharacteristic
72C2×Dic14—21no—
73C2×Dic14—21no—
74C2×Dic14C2311yescharacteristic
75C2×Dic14C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C7×C4⋊C4—21no—
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4—21no—
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C42×C7C2311yescharacteristic
85#1144—41no—
86#1144—41no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4D411yescharacteristic
93(C2×D7)⋊C4D411yescharacteristic
94C4⋊Dic7—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7D411yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104C4×Dic7—21no—
105C4×Dic7—21no—
106C4×Dic7—21no—
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109Dic28—22no—
110#1131—41no—
111C8×D7—41no—
112#552—71no—
113D4×Q8—71no—
114#491D711yescharacteristic, complemented
115#485—71no—
116#483—71no—
117#479—71no—
118#470—71no—
119#467—71no—
120#462—71no—
121#451—71no—
122#451—71no—
123#441—71no—
124#435—71no—
125#419—71no—
126C2×#103—71no—
127C22×D7C2×D411yescharacteristic
128C22×D7C2×D411yescharacteristic
129C7×Q8—21no—
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2411yescharacteristic
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146D28—22no—
147C4×D7—21no—
148C4×D7—21no—
149C4×D7—21no—
150C4×D7—21no—
151C4×D7—42no—
152C4×D7—42no—
153C4×D7—42no—
154Dic14—21no—
155Dic14—21no—
156Dic14—21no—
157Dic14—42no—
158Dic14—42no—
159Dic14#4112yes—
160C7×C8—21no—
161C7×C8—21no—
162C7⋊C8—21no—
163C7⋊C8—21no—
164C22×Q8—71no—
165C22×Q8—71no—
166C2×Q16—71no—
167C2×Q16—71no—
168C2×Q16—71no—
169C2×M16—71no—
170C22×C8—71no—
171C4⋊Q8—141no—
172C4⋊Q8C2×D711yescharacteristic
173#126—141no—
174#125—141no—
175#125C2×D711yescharacteristic
176#123—141no—
177C22⋊Q8—71no—
178C22⋊Q8—71no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182C22⋊Q8—141no—
183C4×Q8—71no—
184C4×Q8—71no—
185C4×D4—71no—
186C4×D4—141no—
187#117—71no—
188#117—141no—
189C2×C4⋊C4—71no—
190#107—71no—
191#106—71no—
192#106—71no—
193C4⋊C8—71no—
194#103—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103C2×D711yescharacteristic
201#103C2×D711yescharacteristic
202#103C2×D711yescharacteristic
203#103C2×D711yescharacteristic
204C22⋊C8—71no—
205C22⋊C8—71no—
206#97C2×D711yescharacteristic
207C22×C7C22×D411yescharacteristic
208C2×D7—21no—
209C2×D7—21no—
210C2×D7—21no—
211C2×D7—21no—
212C2×D7—21no—
213C2×D7#13712yes—
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7C2×#4111yescharacteristic
221C4×C7H(5,𝔽2)11yescharacteristic
222Dic7—22no—
223Dic7—21no—
224Dic7—21no—
225Dic7—41no—
226Dic7—41no—
227Dic7—41no—
228C2×Q8—71no—
229C2×Q8—71no—
230C2×Q8—141no—
231C2×Q8—141no—
232C2×Q8—142no—
233C2×Q8—142no—
234C2×Q8—281no—
235C2×Q8—281no—
236C2×Q8C22×D711yescharacteristic
237C2×Q8C22×D711yescharacteristic
238C2×D4—71no—
239C22×C4—71no—
240C22×C4—71no—
241C22×C4—141no—
242C22×C4—141no—
243C22×C4—141no—
244Q16—142no—
245Q16—281no—
246Q16—281no—
247M16—281no—
248C2×C8—71no—
249C2×C8—71no—
250C2×C8—281no—
251C2×C8C22×D711yescharacteristic
252C2×C8C22×D711yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—71no—
258C4⋊C4—71no—
259C4⋊C4—71no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4C22×D711yescharacteristic
268C4⋊C4C22×D711yescharacteristic
269C22⋊C4—71no—
270C22⋊C4—71no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C42—141no—
277C42—141no—
278C42—141no—
279C42C22×D711yescharacteristic
280C2×C7#54611yescharacteristic
281C2×C7C2×#13711yescharacteristic
282C2×C7#57811yescharacteristic
283D7—22no—
284D7—41no—
285C23—71no—
286C23—71no—
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—72no—
292Q8—141no—
293Q8—141no—
294Q8—141no—
295Q8—282no—
296Q8—282no—
297D4—142no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—71no—
303C2×C4—71no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—282no—
313C2×C4—282no—
314C2×C4—282no—
315C2×C4D4×D711yescharacteristic
316C2×C4D4×D711yescharacteristic
317C2×C4C23×D711yescharacteristic
318C8—21no—
319C8—21no—
320C8—141no—
321C8—141no—
322C7#331211yescharacteristic, complemented, Sylow
323C22—72no—
324C22—141no—
325C22—141no—
326C22—141no—
327C22—141no—
328C22—141no—
329C22C2×D4×D711yescharacteristic
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—141no—
337C4—141no—
338C4—142no—
339C4—281no—
340C4—281no—
341C4—281no—
342C4D7×#4111yescharacteristic
343C4#655311yescharacteristic
344C2—142no—
345C2—281no—
346C2#9036711yescharacteristic
347C2D7×#13711yescharacteristic
348C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#491C2
socleC22×C7C22×D4
prime radicalC2×D28D4
hypercenter#491D7
hypocenterC7#3312
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
3. C7×#491C2semidirect
4. #89862C2semidirect
5. #89835C2semidirect
6. #89588C2semidirect
7. #89587C2semidirect
8. #89585C2semidirect
9. #89580C2semidirect
10. #89580C2semidirect
11. #89577C2semidirect
12. #89574C2semidirect
13. D7×#103C2semidirect
14. #89488C2semidirect
114. #491D7semidirect
322. C7#3312semidirect

Maximal subgroups

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

subgroupindex
#904182
#903852
C7×#4912
#898622
#898352
#895882
#895872
#895852
#895802
#895772
#895742
D7×#1032
#894882
#894782
#33127