Finite Group Databasebeta

← #11739312

Subgroups — #11739312

Order 896.

356 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90423C211yesmaximal, characteristic, complemented
2#90362C211yesmaximal, characteristic, complemented
3C7×#497C211yesmaximal, characteristic, complemented
4#89962C211yesmaximal, characteristic, complemented
5#89957C211yesmaximal, characteristic, complemented
6Q16×Dic7C211yesmaximal, characteristic, complemented
7#89954C211yesmaximal, characteristic, complemented
8#89940C211yesmaximal, characteristic
9#89938C211yesmaximal, characteristic, complemented
10#89937C211yesmaximal, characteristic
11#89936C211yesmaximal, characteristic, complemented
12#89862C211yesmaximal, characteristic, complemented
13#89829C211yesmaximal, characteristic
14#89482C211yesmaximal, characteristic, complemented
15#89473C211yesmaximal, characteristic, complemented
16C2×C7×Q16C2211yescharacteristic
17C2×C7×Q16C2211yescharacteristic
18C2×C7×SD16C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C7×C4⋊Q8C2211yescharacteristic
21C7×#124C2211yescharacteristic
22#6510C2211yescharacteristic
23#6510C2211yescharacteristic
24Q8×Dic7C2211yescharacteristic
25Q8×Dic7C2211yescharacteristic
26Q8×Dic7C2211yescharacteristic
27#6507C2211yescharacteristic
28C2×#1144C2211yescharacteristic
29C2×#1144C2211yescharacteristic
30C2×#1143C2211yescharacteristic
31#6499C2211yescharacteristic
32D4×Dic7C2211yescharacteristic
33C2×#1141C2211yescharacteristic
34C2×#1132C2211yescharacteristic
35#6461—21no—
36#6461—21no—
37#6460—21no—
38#6455—21no—
39#6442—21no—
40#6442—21no—
41#6441—21no—
42#6440—21no—
43C4×D28C2211yescharacteristic
44C4×Dic14C2211yescharacteristic
45C4×Dic14C2211yescharacteristic
46C7×#97C2211yescharacteristic
47#6409C2211yescharacteristic
48#6409C2211yescharacteristic
49#6409C2211yescharacteristic
50#6406C2211yescharacteristic
51#6395C2211yescharacteristic
52#6392C2211yescharacteristic
53#6390C2211yescharacteristic
54#6389C2211yescharacteristic
55Dic7⋊C8C2211yescharacteristic
56Dic7⋊C8C2211yescharacteristic
57C8×Dic7C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3402—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4C2311yescharacteristic
64C22×Dic7—21no—
65C2×D28C2311yescharacteristic
66C2×C4×D7—21no—
67C2×Dic14D411yescharacteristic
68C2×Dic14C2311yescharacteristic
69C7×Q16—22no—
70C7×Q16—41no—
71C7×SD16—22no—
72C7×SD16—41no—
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C22⋊C4—21no—
78C7×C22⋊C4—21no—
79C42×C7C2311yescharacteristic
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7—21no—
83#1144—41no—
84#1144—41no—
85#1143—41no—
86#1141—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—
92C4⋊Dic7—21no—
93C4⋊Dic7—21no—
94C4⋊Dic7—21no—
95C4⋊Dic7D411yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4D411yescharacteristic
102Dic7⋊C4D411yescharacteristic
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7C2311yescharacteristic
108C4×Dic7C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111#1132—22no—
112#556—71no—
113#556—71no—
114#497—71no—
115#497D711yescharacteristic, complemented
116#495—71no—
117#464—71no—
118#464—71no—
119#463—71no—
120#463—71no—
121#462—71no—
122#462—71no—
123#446—71no—
124#440—71no—
125#440—71no—
126C4×Q16—71no—
127C23×C7—21no—
128C22×D7—21no—
129C7×Q8—21no—
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—21no—
134C7×Q8—21no—
135C7×D4—21no—
136C7×D4—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2411yescharacteristic
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—42no—
147C2×Dic7C2×D411yescharacteristic
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151D28—21no—
152D28—21no—
153C4×D7—42no—
154Dic14—21no—
155Dic14—21no—
156Dic14—41no—
157C7×C8—21no—
158C7×C8#4112yes—
159C7⋊C8—21no—
160C7⋊C8—21no—
161C2×Q16—71no—
162C2×Q16—71no—
163C2×Q16C2×D711yescharacteristic
164C2×Q16C2×D711yescharacteristic
165C2×SD16—71no—
166C2×SD16—71no—
167C2×SD16C2×D711yescharacteristic
168C2×SD16C2×D711yescharacteristic
169C2×D8—71no—
170C4⋊Q8—71no—
171C4⋊Q8C2×D711yescharacteristic
172#126—141no—
173#126—141no—
174#126—141no—
175#126—141no—
176#124—71no—
177#124—71no—
178#124—71no—
179#124C2×D711yescharacteristic
180C22⋊Q8—141no—
181C22⋊Q8—141no—
182C4×Q8—71no—
183C4×Q8—71no—
184C4×Q8—71no—
185C4×Q8—71no—
186C4×Q8—71no—
187C4×D4—71no—
188C4×D4—71no—
189#117—141no—
190#117—141no—
191#107—71no—
192C4⋊C8—71no—
193C4⋊C8—71no—
194C4⋊C8—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#102—71no—
200#102—71no—
201#97—71no—
202#97C2×D711yescharacteristic
203C4×C8—71no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7C22×D411yescharacteristic
208C2×D7—41no—
209C2×D7—41no—
210C2×D7—41no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7C2×#4111yescharacteristic
217C4×C7H(5,𝔽2)11yescharacteristic
218Dic7—21no—
219Dic7—21no—
220Dic7—21no—
221Dic7—21no—
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225C2×Q8—71no—
226C2×Q8—71no—
227C2×Q8C22×D711yescharacteristic
228C2×Q8C22×D711yescharacteristic
229C2×Q8C22×D711yescharacteristic
230C2×D4—71no—
231C2×D4C22×D711yescharacteristic
232C22×C4—141no—
233C22×C4—141no—
234Q16—22no—
235Q16—41no—
236Q16—281no—
237Q16—281no—
238SD16—22no—
239SD16—41no—
240SD16—142no—
241SD16—281no—
242D8—281no—
243C2×C8—71no—
244C2×C8—71no—
245C2×C8C22×D711yescharacteristic
246C2×C8C22×D711yescharacteristic
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—71no—
250C4⋊C4—71no—
251C4⋊C4—71no—
252C4⋊C4—71no—
253C4⋊C4—71no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C42—71no—
271C42—71no—
272C42—141no—
273C42—141no—
274C42—141no—
275C42—141no—
276C42C22×D711yescharacteristic
277C2×C7—81no—
278C2×C7#55011yescharacteristic
279C2×C7#57511yescharacteristic
280C2×C7#57811yescharacteristic
281D7—81no—
282C23—21no—
283C23—141no—
284Q8—21no—
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—141no—
291Q8—141no—
292Q8—281no—
293D4—21no—
294D4—21no—
295D4—141no—
296D4—141no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—71no—
301C2×C4—71no—
302C2×C4—71no—
303C2×C4—71no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—282no—
308C2×C4—282no—
309C2×C4D4×D711yescharacteristic
310C2×C4D4×D711yescharacteristic
311C2×C4C23×D711yescharacteristic
312C8—21no—
313C8—141no—
314C8—141no—
315C8#115912yes—
316C7#340211yescharacteristic, complemented, Sylow
317C22—41no—
318C22—41no—
319C22—41no—
320C22—281no—
321C22—281no—
322C22—281no—
323C22C2×D4×D711yescharacteristic
324C4—21no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—141no—
330C4—141no—
331C4—141no—
332C4—141no—
333C4—281no—
334C4—281no—
335C4—281no—
336C4C2×#115911yescharacteristic
337C4#654811yescharacteristic
338C2—81no—
339C2—561no—
340C2#9040911yescharacteristic
341C2#9046811yescharacteristic
342C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#497C2
socleC22×C7C22×D4
prime radicalC2×#1141C22
hypercenter#497D7
hypocenterC7#3402
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #90423C2semidirect
2. #90362C2semidirect
3. C7×#497C2semidirect
4. #89962C2semidirect
5. #89957C2semidirect
6. Q16×Dic7C2semidirect
7. #89954C2semidirect
9. #89938C2semidirect
11. #89936C2semidirect
12. #89862C2semidirect
14. #89482C2semidirect
15. #89473C2semidirect
115. #497D7semidirect
316. C7#3402semidirect

Maximal subgroups

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

subgroupindex
#904232
#903622
C7×#4972
#899622
#899572
Q16×Dic72
#899542
#899402
#899382
#899372
#899362
#898622
#898292
#894822
#894732
#34027