Finite Group Databasebeta

← #11740013

Subgroups — #11740013

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
1C7×#520C211yesmaximal, characteristic, complemented
2#90226C211yesmaximal, characteristic, complemented
3#90010C211yesmaximal, characteristic, complemented
4#90010C211yesmaximal, characteristic, complemented
5#89799C211yesmaximal, characteristic, complemented
6#89798C211yesmaximal, characteristic, complemented
7C4×#1143C211yesmaximal, characteristic, complemented
8#89794C211yesmaximal, characteristic, complemented
9#89787C211yesmaximal, characteristic, complemented
10#89786C211yesmaximal, characteristic, complemented
11C4×#1141C211yesmaximal, characteristic, complemented
12#89781C211yesmaximal, characteristic, complemented
13#89772C211yesmaximal, characteristic, complemented
14#89740C211yesmaximal, characteristic, complemented
15#89739C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19C4×C7×D4—21no—
20C4×C7×D4—21no—
21C4×C7×D4C2211yescharacteristic
22C4×C7×D4C2211yescharacteristic
23C7×#117—21no—
24C7×#117C2211yescharacteristic
25C2×C7×C4⋊C4—21no—
26C2×C7×C4⋊C4C2211yescharacteristic
27C2×#1143C2211yescharacteristic
28C2×#1143C2211yescharacteristic
29C2×#1141C2211yescharacteristic
30C2×#1141C2211yescharacteristic
31#6493C2211yescharacteristic
32#6493C2211yescharacteristic
33#6457—21no—
34#6457—21no—
35#6447—21no—
36#6447—21no—
37#6437C2211yescharacteristic
38C4×D28C2211yescharacteristic
39C4×D28C2211yescharacteristic
40#6433C2211yescharacteristic
41#6409C2211yescharacteristic
42#6409C2211yescharacteristic
43#6406C2211yescharacteristic
44#6406C2211yescharacteristic
45C22⋊(C7⋊C8)C2211yescharacteristic
46C22⋊(C7⋊C8)C2211yescharacteristic
47#6383C2211yescharacteristic
48#6383C2211yescharacteristic
49#6383C2211yescharacteristic
50#6383C2211yescharacteristic
51#6382C2211yescharacteristic
52#6382C2211yescharacteristic
53#6381C2211yescharacteristic
54#6381C2211yescharacteristic
55C4⋊(C7⋊C8)C2211yescharacteristic
56C4⋊(C7⋊C8)C2211yescharacteristic
57#6377C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3119—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—22no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4—21no—
64C2×C7×D4C2311yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7C2311yescharacteristic
70C2×C22⋊D7—21no—
71C2×C22⋊D7—21no—
72C2×D28—21no—
73C2×D28—21no—
74C2×D28C2311yescharacteristic
75C2×D28C2311yescharacteristic
76C2×C4×D7—21no—
77C2×C4×D7—21no—
78C7×C4⋊C4—21no—
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4D411yescharacteristic
82C7×C4⋊C4D411yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C22⋊C4—21no—
86C7×C22⋊C4—21no—
87C7×C22⋊C4D411yescharacteristic
88C7×C22⋊C4D411yescharacteristic
89C42×C7—21no—
90C42×C7—21no—
91C42×C7C2311yescharacteristic
92C42×C7C2311yescharacteristic
93#1143—22no—
94#1143—22no—
95#1141—22no—
96#1141—22no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99(C2×D7)⋊C4—21no—
100(C2×D7)⋊C4—21no—
101C4⋊Dic7C2311yescharacteristic
102C4⋊Dic7C2311yescharacteristic
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#540—71no—
110#520D711yescharacteristic, complemented
111#481—71no—
112#480—71no—
113#479—71no—
114#478—71no—
115#460—71no—
116#459—71no—
117#449—71no—
118#449—71no—
119#442—71no—
120#440—71no—
121C4×SD16—71no—
122C4×D8—71no—
123#433—71no—
124C23×C7—21no—
125C23×C7C2×D411yescharacteristic
126C22×D7—21no—
127C22×D7—21no—
128C7×Q8—21no—
129C7×Q8#4112yes—
130C7×D4—24no—
131C7×D4—21no—
132C7×D4#4112yes—
133C2×C4×C7—24no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—81no—
140C2×C4×C7—81no—
141C2×C4×C7—81no—
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2411yescharacteristic
148C22⋊D7—81no—
149C22⋊D7—81no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152D28—21no—
153D28—21no—
154D28—21no—
155D28—21no—
156D28—81no—
157D28—81no—
158C4×D7—81no—
159C4×D7—81no—
160C7⋊C8—21no—
161C7⋊C8—21no—
162C7⋊C8—21no—
163C7⋊C8—21no—
164C2×#41C2×D711yescharacteristic
165C2×SD16—71no—
166C2×SD16—71no—
167C2×D8—71no—
168C2×D8—71no—
169#127—71no—
170#125—71no—
171#123—141no—
172#123—141no—
173#121—71no—
174#121—71no—
175#121—141no—
176#121—141no—
177C4×Q8C2×D711yescharacteristic
178C4×Q8C2×D711yescharacteristic
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—71no—
182C4×D4—71no—
183C4×D4C2×D711yescharacteristic
184C4×D4C2×D711yescharacteristic
185#117—21no—
186#117C2×D711yescharacteristic
187C2×C4⋊C4—21no—
188C2×C4⋊C4C2×D711yescharacteristic
189#107—71no—
190#107—71no—
191#106—71no—
192#106—71no—
193C4⋊C8—71no—
194C4⋊C8—71no—
195#103—71no—
196#103—71no—
197#102—71no—
198#102—71no—
199#102—71no—
200#102—71no—
201#102—71no—
202#102—71no—
203C22⋊C8—71no—
204C22⋊C8—71no—
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×C7C22×D411yescharacteristic
214C2×D7—41no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C4×C7—21no—
221C4×C7—22no—
222C4×C7—21no—
223C4×C7—21no—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7C2×#4111yescharacteristic
229C4×C7C2×#4111yescharacteristic
230Dic7—41no—
231Dic7—41no—
232#41—22no—
233#41—22no—
234C2×Q8C22×D711yescharacteristic
235C2×D4—21no—
236C2×D4—71no—
237C2×D4—71no—
238C2×D4—141no—
239C2×D4—141no—
240C2×D4—141no—
241C2×D4—141no—
242C2×D4C22×D711yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—141no—
248C22×C4—141no—
249C22×C4C22×D711yescharacteristic
250SD16—142no—
251SD16—142no—
252D8—142no—
253D8—142no—
254C2×C8—71no—
255C2×C8—71no—
256C2×C8—71no—
257C2×C8—71no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—71no—
262C4⋊C4—71no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4C22⋊D711yescharacteristic
266C4⋊C4C22⋊D711yescharacteristic
267C4⋊C4C22×D711yescharacteristic
268C4⋊C4C22×D711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4C22⋊D711yescharacteristic
276C22⋊C4C22⋊D711yescharacteristic
277C42—21no—
278C42—21no—
279C42C22×D711yescharacteristic
280C42C22×D711yescharacteristic
281C2×C7—41no—
282C2×C7—42no—
283C2×C7#52511yescharacteristic
284C2×C7#57611yescharacteristic
285C2×C7#57711yescharacteristic
286D7—81no—
287D7—81no—
288C23—21no—
289C23—141no—
290C23—141no—
291C23C2×C22⋊D711yescharacteristic
292Q8—21no—
293Q8#115712yes—
294D4—24no—
295D4—21no—
296D4—141no—
297D4—141no—
298D4—141no—
299D4—141no—
300D4—561no—
301D4—561no—
302D4—561no—
303D4—561no—
304D4#115712yes—
305C2×C4—24no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—561no—
317C2×C4—561no—
318C2×C4C2×C22⋊D711yescharacteristic
319C2×C4C2×C22⋊D711yescharacteristic
320C2×C4C2×C22⋊D711yescharacteristic
321C2×C4C2×C22⋊D711yescharacteristic
322C2×C4C2×C22⋊D711yescharacteristic
323C2×C4C23×D711yescharacteristic
324C8—141no—
325C8—141no—
326C8—141no—
327C8—141no—
328C7#311911yescharacteristic, complemented, Sylow
329C22—21no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—281no—
336C22—281no—
337C22—281no—
338C22—281no—
339C22—281no—
340C22—281no—
341C22C22×C22⋊D711yescharacteristic
342C4—21no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—41no—
350C4—281no—
351C4—281no—
352C4C2×#115711yescharacteristic
353C4C2×#115711yescharacteristic
354C2—41no—
355C2—42no—
356C2—561no—
357C2—561no—
358C2#9048211yescharacteristic
359C2#9052611yescharacteristic
360C2#9052811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#520C2
socleC22×C7C22×D4
hypercenter#520D7
hypocenterC7#3119
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. C7×#520C2semidirect
2. #90226C2semidirect
3. #90010C2semidirect
4. #90010C2semidirect
5. #89799C2semidirect
6. #89798C2semidirect
7. C4×#1143C2semidirect
8. #89794C2semidirect
9. #89787C2semidirect
10. #89786C2semidirect
11. C4×#1141C2semidirect
12. #89781C2semidirect
13. #89772C2semidirect
14. #89740C2semidirect
15. #89739C2semidirect
110. #520D7semidirect
328. C7#3119semidirect

Maximal subgroups

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

subgroupindex
C7×#5202
#902262
#900102
#897992
#897982
C4×#11432
#897942
#897872
#897862
C4×#11412
#897812
#897722
#897402
#897392
#31197