Finite Group Databasebeta

← #11737518

Subgroups — #11737518

Order 896.

396 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#90518C211yesmaximal, characteristic, complemented
2#90310C211yesmaximal, characteristic, complemented
3C7×#449C211yesmaximal, characteristic, complemented
4#90010C211yesmaximal, characteristic, complemented
5#89935C211yesmaximal, characteristic, complemented
6SD16×Dic7C211yesmaximal, characteristic, complemented
7#89924C211yesmaximal, characteristic, complemented
8#89922C211yesmaximal, characteristic, complemented
9#89805C211yesmaximal, characteristic, complemented
10#89589C211yesmaximal, characteristic, complemented
11#89571C211yesmaximal, characteristic, complemented
12#89535C211yesmaximal, characteristic, complemented
13#89528C211yesmaximal, characteristic, complemented
14#89507C211yesmaximal, characteristic, complemented
15#89492C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×C7×SD16C2211yescharacteristic
18C2×C7×D8C2211yescharacteristic
19C7×#121C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22C2×#1143C2211yescharacteristic
23#6503C2211yescharacteristic
24#6502—21no—
25#6501—21no—
26D4×Dic7—21no—
27D4×Dic7—21no—
28D4×Dic7C2211yescharacteristic
29D4×Dic7C2211yescharacteristic
30D4×Dic7C2211yescharacteristic
31C2×#1141C2211yescharacteristic
32#6493C2211yescharacteristic
33#6489—21no—
34#6489C2211yescharacteristic
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7C2211yescharacteristic
37#6456C2211yescharacteristic
38#6453C2211yescharacteristic
39#6449—21no—
40#6446—21no—
41C7×#103C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×C22⋊C8C2211yescharacteristic
44#6409C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47#6406C2211yescharacteristic
48C22⋊(C7⋊C8)C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic
51#6391C2211yescharacteristic
52#6391C2211yescharacteristic
53#6389C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55Dic7⋊C8C2211yescharacteristic
56C8×Dic7C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3119—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—22no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4—21no—
64C2×C7×D4—21no—
65C2×C7×D4C2311yescharacteristic
66C2×C7×D4C2311yescharacteristic
67C22×C4×C7—21no—
68C22×C4×C7C2311yescharacteristic
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71C2×C22⋊D7—21no—
72C22×Dic7—21no—
73C22×Dic7—21no—
74C22×Dic7—21no—
75C22×Dic7—21no—
76C2×D28C2311yescharacteristic
77C2×C4×D7—21no—
78C7×SD16—22no—
79C7×D8—22no—
80C2×C7×C8C2311yescharacteristic
81C2×C7×C8C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C22⋊C4—21no—
84C22⋊Dic7—21no—
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7D411yescharacteristic
88C22⋊Dic7D411yescharacteristic
89#1143—22no—
90#1141—22no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93C4⋊Dic7—21no—
94C4⋊Dic7—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7—21no—
97C4⋊Dic7D411yescharacteristic
98C4⋊Dic7D411yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100C4⋊Dic7C2311yescharacteristic
101C4⋊Dic7C2311yescharacteristic
102Dic7⋊C4—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7C2311yescharacteristic
106C4×Dic7C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#540—71no—
110#520—71no—
111#481—71no—
112#480—71no—
113#479—71no—
114#478—71no—
115#460—71no—
116#459—71no—
117#449—71no—
118#449D711yescharacteristic, complemented
119#442—71no—
120#440—71no—
121C4×SD16—71no—
122C4×D8—71no—
123#433—71no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7C2×D411yescharacteristic
127C22×D7—21no—
128C7×Q8—21no—
129C7×Q8#4112yes—
130C7×D4—24no—
131C7×D4—21no—
132C7×D4—21no—
133C7×D4—21no—
134C7×D4—81no—
135C7×D4#4112yes—
136C2×C4×C7—24no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2411yescharacteristic
143C22⋊D7—81no—
144C22⋊D7—81no—
145C22⋊D7—81no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—81no—
150C2×Dic7—81no—
151C2×Dic7—81no—
152C2×Dic7—81no—
153C2×Dic7C2×D411yescharacteristic
154C2×Dic7C2×D411yescharacteristic
155C2×Dic7C2×D411yescharacteristic
156C2×Dic7C2×D411yescharacteristic
157D28—21no—
158D28—21no—
159C4×D7—81no—
160C7×C8—21no—
161C7×C8—21no—
162C7⋊C8—21no—
163C7⋊C8—21no—
164C2×#41C2×D711yescharacteristic
165C2×SD16—71no—
166C2×SD16C2×D711yescharacteristic
167C2×D8—71no—
168C2×D8C2×D711yescharacteristic
169#127—71no—
170#125—71no—
171#123—141no—
172#123—141no—
173#121—71no—
174#121—141no—
175#121—141no—
176#121C2×D711yescharacteristic
177C4×Q8—71no—
178C4×Q8—71no—
179C4×D4—71no—
180C4×D4—71no—
181C4×D4—71no—
182C4×D4—71no—
183C4×D4—141no—
184C4×D4—141no—
185#117—71no—
186#117—141no—
187C2×C4⋊C4—71no—
188C2×C4⋊C4—141no—
189#107—71no—
190#107—71no—
191#106—71no—
192#106—71no—
193C4⋊C8—71no—
194C4⋊C8—71no—
195#103—71no—
196#103C2×D711yescharacteristic
197#102—71no—
198#102—71no—
199#102—71no—
200#102—71no—
201#102—71no—
202#102C2×D711yescharacteristic
203C22⋊C8—71no—
204C22⋊C8C2×D711yescharacteristic
205#97—71no—
206C4×C8—71no—
207C22×C7—21no—
208C22×C7—22no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7C22×D411yescharacteristic
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C4×C7—21no—
221C4×C7—22no—
222C4×C7—41no—
223C4×C7C2×#4111yescharacteristic
224C4×C7C2×#4111yescharacteristic
225Dic7—21no—
226Dic7—21no—
227Dic7—21no—
228Dic7—21no—
229Dic7—41no—
230Dic7—41no—
231Dic7—41no—
232#41—22no—
233#41—22no—
234C2×Q8C22×D711yescharacteristic
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—71no—
238C2×D4—141no—
239C2×D4—141no—
240C2×D4—141no—
241C2×D4C22×D711yescharacteristic
242C2×D4C22×D711yescharacteristic
243C22×C4—21no—
244C22×C4—141no—
245C22×C4—141no—
246C22×C4—141no—
247C22×C4—141no—
248C22×C4—141no—
249C22×C4C22×D711yescharacteristic
250SD16—22no—
251SD16—142no—
252D8—22no—
253D8—142no—
254C2×C8—71no—
255C2×C8—71no—
256C2×C8C22×D711yescharacteristic
257C2×C8C22×D711yescharacteristic
258C4⋊C4—71no—
259C4⋊C4—71no—
260C4⋊C4—71no—
261C4⋊C4—71no—
262C4⋊C4—71no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4C22×D711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—71no—
271C22⋊C4—71no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C42—71no—
278C42—71no—
279C42—141no—
280C42—141no—
281C2×C7—41no—
282C2×C7—42no—
283C2×C7—81no—
284C2×C7#52511yescharacteristic
285C2×C7#57611yescharacteristic
286C2×C7#57711yescharacteristic
287D7—81no—
288C23—21no—
289C23—21no—
290C23—141no—
291C23D4×D711yescharacteristic
292Q8—21no—
293Q8#115912yes—
294D4—24no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—81no—
299D4—141no—
300D4—141no—
301D4—561no—
302D4—561no—
303D4—561no—
304D4#115912yes—
305C2×C4—24no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—71no—
311C2×C4—71no—
312C2×C4—71no—
313C2×C4—71no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—561no—
318C2×C4—561no—
319C2×C4—561no—
320C2×C4—561no—
321C2×C4—561no—
322C2×C4D4×D711yescharacteristic
323C2×C4C23×D711yescharacteristic
324C8—21no—
325C8—21no—
326C8—141no—
327C8—141no—
328C7#311911yescharacteristic, complemented, Sylow
329C22—21no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—281no—
339C22—281no—
340C22—281no—
341C22C2×D4×D711yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—141no—
346C4—141no—
347C4—141no—
348C4—141no—
349C4—281no—
350C4—281no—
351C4—281no—
352C4C2×#115911yescharacteristic
353C4C2×#115911yescharacteristic
354C2—41no—
355C2—42no—
356C2—81no—
357C2—561no—
358C2#9027711yescharacteristic
359C2#9046111yescharacteristic
360C2#9046511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#449C2
socleC22×C7C22×D4
hypercenter#449D7
hypocenterC7#3119
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #90518C2semidirect
2. #90310C2semidirect
3. C7×#449C2semidirect
4. #90010C2semidirect
5. #89935C2semidirect
6. SD16×Dic7C2semidirect
7. #89924C2semidirect
8. #89922C2semidirect
9. #89805C2semidirect
10. #89589C2semidirect
11. #89571C2semidirect
12. #89535C2semidirect
13. #89528C2semidirect
14. #89507C2semidirect
15. #89492C2semidirect
118. #449D7semidirect
328. C7#3119semidirect

Maximal subgroups

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

subgroupindex
#905182
#903102
C7×#4492
#900102
#899352
SD16×Dic72
#899242
#899222
#898052
#895892
#895712
#895352
#895282
#895072
#894922
#31197