Finite Group Databasebeta

← #11738372

Subgroups — #11738372

Order 896.

374 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#90332C211yesmaximal, characteristic
2#90323C211yesmaximal, characteristic
3C7×#470C211yesmaximal, characteristic, complemented
4#89958C211yesmaximal, characteristic, complemented
5#89954C211yesmaximal, characteristic, complemented
6#89900C211yesmaximal, characteristic
7#89892C211yesmaximal, characteristic, complemented
8#89822C211yesmaximal, characteristic, complemented
9#89817C211yesmaximal, characteristic, complemented
10#89639C211yesmaximal, characteristic, complemented
11#89626C211yesmaximal, characteristic, complemented
12#89588C211yesmaximal, characteristic, complemented
13#89574C211yesmaximal, characteristic, complemented
14#89564C211yesmaximal, characteristic, complemented
15#89562C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic
19C2×C7×M16C2211yescharacteristic
20C7×C22⋊Q8C2211yescharacteristic
21C7×C22⋊Q8C2211yescharacteristic
22#6509C2211yescharacteristic
23Q8×Dic7C2211yescharacteristic
24#6507C2211yescharacteristic
25C2×#1144C2211yescharacteristic
26C2×#1144C2211yescharacteristic
27C4×C22⋊D7C2211yescharacteristic
28#6489C2211yescharacteristic
29C2×#1131C2211yescharacteristic
30#6461—21no—
31#6460C2211yescharacteristic
32#6459—21no—
33#6455C2211yescharacteristic
34#6451—21no—
35#6451C2211yescharacteristic
36#6450—21no—
37#6450C2211yescharacteristic
38#6448—21no—
39(C22⋊D7)⋊C4—21no—
40#6442—21no—
41#6441—21no—
42C7×#106C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#103C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47C22⋊(C7⋊C8)C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6390C2211yescharacteristic
50#6390C2211yescharacteristic
51#6390C2211yescharacteristic
52#6390C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54C8×Dic7C2211yescharacteristic
55#6384C2211yescharacteristic
56#6384C2211yescharacteristic
57#6382C2211yescharacteristic
58#6381C2211yescharacteristic
59#3328—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7D411yescharacteristic
64C22×Dic7—21no—
65Q8×D7—22no—
66Q8×D7—41no—
67C2×C4×D7—21no—
68C2×C4×D7C2311yescharacteristic
69C2×Dic14—22no—
70C2×Dic14—21no—
71C2×Dic14—21no—
72C2×Dic14—21no—
73C2×Dic14—41no—
74C2×Dic14C2311yescharacteristic
75C2×Dic14C2311yescharacteristic
76C7×Q16—41no—
77C7×M16—41no—
78C2×C7×C8C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C7×C4⋊C4—21no—
81C7×C4⋊C4—21no—
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C22⋊C4—21no—
85C7×C22⋊C4—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7D411yescharacteristic
88#1144—22no—
89#1144—41no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4D411yescharacteristic
94C4⋊Dic7—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4D411yescharacteristic
104C4×Dic7—21no—
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7—21no—
108C4×Dic7C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111#1131—41no—
112#556—71no—
113D4×Q8—71no—
114#487—71no—
115#485—71no—
116#485—71no—
117#475—71no—
118#470—71no—
119#470D711yescharacteristic, complemented
120#462—71no—
121#451—71no—
122#451—71no—
123C4×Q16—71no—
124#436—71no—
125#419—71no—
126#419—71no—
127C23×C7C2×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×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2411yescharacteristic
141C22⋊D7—22no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—42no—
147C2×Dic7C2×D411yescharacteristic
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150C4×D7—21no—
151C4×D7—21no—
152C4×D7—42no—
153Dic14—21no—
154Dic14—21no—
155Dic14—21no—
156Dic14—42no—
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×Q16C2×D711yescharacteristic
169C2×M16—71no—
170C2×M16C2×D711yescharacteristic
171C4⋊Q8—71no—
172C4⋊Q8—71no—
173C4⋊Q8—141no—
174#126—141no—
175#126—141no—
176#124—141no—
177C22⋊Q8—71no—
178C22⋊Q8—71no—
179C22⋊Q8—141no—
180C22⋊Q8—141no—
181C22⋊Q8C2×D711yescharacteristic
182C22⋊Q8C2×D711yescharacteristic
183C4×Q8—71no—
184C4×Q8—71no—
185C4×Q8—141no—
186C4×D4—71no—
187C4×D4—141no—
188#117—71no—
189#117—71no—
190#107—71no—
191#106—71no—
192#106C2×D711yescharacteristic
193C4⋊C8—71no—
194#103—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#103C2×D711yescharacteristic
203#103C2×D711yescharacteristic
204C22⋊C8—71no—
205C22⋊C8—71no—
206C4×C8—71no—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7C22×D411yescharacteristic
211C2×D7—21no—
212C2×D7—21no—
213C2×D7—21no—
214C4×C7—21no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7C2×#4111yescharacteristic
220C4×C7H(5,𝔽2)11yescharacteristic
221Dic7—22no—
222Dic7—21no—
223Dic7—21no—
224Dic7—41no—
225Dic7—41no—
226Dic7—41no—
227Dic7#13712yes—
228C2×Q8—71no—
229C2×Q8—71no—
230C2×Q8—141no—
231C2×Q8—141no—
232C2×Q8—142no—
233C2×Q8—141no—
234C2×Q8—142no—
235C2×Q8—281no—
236C2×Q8—281no—
237C2×Q8C22×D711yescharacteristic
238C2×Q8C22×D711yescharacteristic
239C2×D4—71no—
240C22×C4—71no—
241C22×C4—141no—
242C22×C4—141no—
243C22×C4C22×D711yescharacteristic
244Q16—41no—
245Q16—142no—
246Q16—281no—
247M16—41no—
248M16—281no—
249C2×C8—71no—
250C2×C8—71no—
251C2×C8C22×D711yescharacteristic
252C2×C8C22×D711yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—71no—
256C4⋊C4—71no—
257C4⋊C4—71no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4C22×D711yescharacteristic
266C4⋊C4C22×D711yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—71no—
270C22⋊C4—71no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C42—71no—
276C42—141no—
277C42—141no—
278C42—141no—
279C42—141no—
280C2×C7—41no—
281C2×C7#54611yescharacteristic
282C2×C7C2×#13711yescharacteristic
283C2×C7#57811yescharacteristic
284D7—41no—
285C23—71no—
286C23D4×D711yescharacteristic
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—72no—
292Q8—141no—
293Q8—141no—
294Q8—141no—
295Q8—282no—
296Q8—282no—
297Q8—282no—
298D4—142no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—71no—
306C2×C4—71no—
307C2×C4—71no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—282no—
315C2×C4—282no—
316C2×C4D4×D711yescharacteristic
317C2×C4C23×D711yescharacteristic
318C8—21no—
319C8—21no—
320C8—141no—
321C8—141no—
322C7#332811yescharacteristic, complemented, Sylow
323C22—21no—
324C22—21no—
325C22—21no—
326C22—141no—
327C22—141no—
328C22—141no—
329C22C2×D4×D711yescharacteristic
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—72no—
336C4—141no—
337C4—141no—
338C4—142no—
339C4—281no—
340C4—281no—
341C4—281no—
342C4#654811yescharacteristic
343C4D7×#4111yescharacteristic
344C2—41no—
345C2—281no—
346C2#9030111yescharacteristic
347C2D7×#13711yescharacteristic
348C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#470C2
socleC22×C7C22×D4
prime radicalC2×C22⋊D7D4
hypercenter#470D7
hypocenterC7#3328
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
3. C7×#470C2semidirect
4. #89958C2semidirect
5. #89954C2semidirect
7. #89892C2semidirect
8. #89822C2semidirect
9. #89817C2semidirect
10. #89639C2semidirect
11. #89626C2semidirect
12. #89588C2semidirect
13. #89574C2semidirect
14. #89564C2semidirect
15. #89562C2semidirect
119. #470D7semidirect
322. C7#3328semidirect

Maximal subgroups

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

subgroupindex
#903322
#903232
C7×#4702
#899582
#899542
#899002
#898922
#898222
#898172
#896392
#896262
#895882
#895742
#895642
#895622
#33287