Finite Group Databasebeta

← #11741017

Subgroups — #11741017

Order 896.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#550C211yesmaximal, characteristic, complemented
2#90256C211yesmaximal, characteristic, complemented
3#90010C211yesmaximal, characteristic, complemented
4#90010C211yesmaximal, characteristic, complemented
5#90005C211yesmaximal, characteristic, complemented
6#90004C211yesmaximal, characteristic, complemented
7#89844C211yesmaximal, characteristic, complemented
8#89843C211yesmaximal, characteristic, complemented
9#89811C211yesmaximal, characteristic, complemented
10#89810C211yesmaximal, characteristic, complemented
11#89807C211yesmaximal, characteristic, complemented
12#89805C211yesmaximal, characteristic, complemented
13#89798C211yesmaximal, characteristic, complemented
14#89794C211yesmaximal, characteristic, complemented
15#89783C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×C7×#41C2211yescharacteristic
18C7×#127—21no—
19C7×#127C2211yescharacteristic
20C7×#121—21no—
21C7×#121—21no—
22C7×#121C2211yescharacteristic
23C7×#121C2211yescharacteristic
24C4×C7×Q8C2211yescharacteristic
25C4×C7×D4—21no—
26C4×C7×D4C2211yescharacteristic
27C2×#1143C2211yescharacteristic
28#6502—21no—
29C2×#1141C2211yescharacteristic
30C2×#1141C2211yescharacteristic
31#6493C2211yescharacteristic
32#6493C2211yescharacteristic
33#6489C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×#1136C2211yescharacteristic
36C22×C7⋊C8C2211yescharacteristic
37#6461—21no—
38#6446—21no—
39(C22⋊D7)⋊C4—21no—
40C4×D28C2211yescharacteristic
41#6433C2211yescharacteristic
42#6409C2211yescharacteristic
43#6409C2211yescharacteristic
44#6409C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47#6406C2211yescharacteristic
48#6406C2211yescharacteristic
49#6406C2211yescharacteristic
50#6406C2211yescharacteristic
51C22⋊(C7⋊C8)C2211yescharacteristic
52C22⋊(C7⋊C8)C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3309—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—22no—
62C7×#41—41no—
63C7×#41—41no—
64C2×C7×Q8C2311yescharacteristic
65C2×C7×D4—21no—
66C2×C7×D4—21no—
67C2×C7×D4—21no—
68C2×C7×D4—21no—
69C2×C7×D4—21no—
70C2×C7×D4—21no—
71C2×C7×D4D411yescharacteristic
72C2×C7×D4C2311yescharacteristic
73C2×C7×D4C2311yescharacteristic
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7C2311yescharacteristic
77C22×C4×C7C2311yescharacteristic
78C2×C22⋊D7—21no—
79C2×C22⋊D7—21no—
80C22×Dic7—21no—
81C2×D28C2311yescharacteristic
82C2×C4×D7—21no—
83C7×C4⋊C4—21no—
84C7×C4⋊C4D411yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C7×C22⋊C4—21no—
87C7×C22⋊C4—21no—
88C7×C22⋊C4D411yescharacteristic
89C7×C22⋊C4D411yescharacteristic
90C42×C7—21no—
91C42×C7C2311yescharacteristic
92C22⋊Dic7—21no—
93#1143—22no—
94#1141—41no—
95#1141—41no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99C4⋊Dic7—21no—
100C4⋊Dic7C2311yescharacteristic
101C4⋊Dic7C2311yescharacteristic
102C4⋊Dic7C2311yescharacteristic
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105C4×Dic7—21no—
106#1136—41no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112#553—71no—
113#550D711yescharacteristic, complemented
114#490—71no—
115#482—71no—
116#480—71no—
117#479—71no—
118#469—71no—
119#466—71no—
120#459—71no—
121#449—71no—
122#449—71no—
123#440—71no—
124#435—71no—
125#417—71no—
126#416—71no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7C2×D411yescharacteristic
130C23×C7C2×D411yescharacteristic
131C22×D7—21no—
132C7×Q8—21no—
133C7×Q8#4112yes—
134C7×D4—21no—
135C7×D4—21no—
136C7×D4—21no—
137C7×D4—21no—
138C7×D4—41no—
139C7×D4—42no—
140C7×D4—42no—
141C7×D4—42no—
142C7×D4—81no—
143C7×D4—81no—
144C7×D4—81no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
153C2×C4×C7C2×D411yescharacteristic
154C2×C4×C7C2×D411yescharacteristic
155C2×C4×C7C2×D411yescharacteristic
156C2×C4×C7C2×D411yescharacteristic
157C2×C4×C7C2411yescharacteristic
158C22⋊D7—42no—
159C22⋊D7—81no—
160C2×Dic7—21no—
161C2×Dic7—21no—
162C2×Dic7—21no—
163C2×Dic7—81no—
164D28—21no—
165D28—21no—
166C4×D7—81no—
167C7⋊C8—21no—
168C7⋊C8—21no—
169C7⋊C8—21no—
170C7⋊C8—21no—
171C2×#41C2×D711yescharacteristic
172C2×#41C2×D711yescharacteristic
173C2×SD16—71no—
174C2×D8—71no—
175C2×D8—71no—
176C2×M16—71no—
177C22×C8—71no—
178#127—21no—
179#127C2×D711yescharacteristic
180#126—141no—
181#125—71no—
182#123—141no—
183#121—21no—
184#121—21no—
185#121—71no—
186#121—71no—
187#121—141no—
188#121C2×D711yescharacteristic
189#121C2×D711yescharacteristic
190C4×Q8C2×D711yescharacteristic
191C4×D4—21no—
192C4×D4—71no—
193C4×D4—141no—
194C4×D4C2×D711yescharacteristic
195#117—71no—
196C2×C4⋊C4—71no—
197#107—71no—
198#107—71no—
199#106—71no—
200C4⋊C8—71no—
201#103—71no—
202#103—71no—
203#103—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207#102—71no—
208#102—71no—
209#102—71no—
210#102—71no—
211C22⋊C8—71no—
212C22⋊C8—71no—
213#97—71no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—21no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7C22×D411yescharacteristic
227C2×D7—41no—
228C2×D7—41no—
229C2×D7—41no—
230C4×C7—21no—
231C4×C7—21no—
232C4×C7—22no—
233C4×C7—21no—
234C4×C7—21no—
235C4×C7—41no—
236C4×C7C2×#4111yescharacteristic
237C4×C7H(5,𝔽2)11yescharacteristic
238Dic7—41no—
239Dic7—41no—
240Dic7—41no—
241#41—22no—
242#41—22no—
243#41—41no—
244#41—41no—
245C2×Q8C22×D711yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—71no—
253C2×D4—141no—
254C2×D4—141no—
255C2×D4C22⋊D711yescharacteristic
256C2×D4C22×D711yescharacteristic
257C2×D4C22×D711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—141no—
261C22×C4—141no—
262C22×C4C22×D711yescharacteristic
263C22×C4C22×D711yescharacteristic
264SD16—142no—
265D8—281no—
266D8—281no—
267M16—281no—
268C2×C8—71no—
269C2×C8—71no—
270C2×C8—71no—
271C2×C8—71no—
272C2×C8—281no—
273C4⋊C4—21no—
274C4⋊C4—71no—
275C4⋊C4—71no—
276C4⋊C4—71no—
277C4⋊C4—141no—
278C4⋊C4—141no—
279C4⋊C4—141no—
280C4⋊C4C22⋊D711yescharacteristic
281C4⋊C4C22×D711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4C22⋊D711yescharacteristic
289C22⋊C4C22⋊D711yescharacteristic
290C42—21no—
291C42—141no—
292C42C22×D711yescharacteristic
293C2×C7—41no—
294C2×C7—41no—
295C2×C7—81no—
296C2×C7—81no—
297C2×C7#54611yescharacteristic
298C2×C7#57511yescharacteristic
299C2×C7#57611yescharacteristic
300D7—81no—
301C23—21no—
302C23—21no—
303C23—141no—
304C23C2×C22⋊D711yescharacteristic
305C23C2×C22⋊D711yescharacteristic
306Q8—21no—
307Q8#115912yes—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—41no—
313D4—42no—
314D4—42no—
315D4—42no—
316D4—81no—
317D4—81no—
318D4—81no—
319D4—141no—
320D4—141no—
321D4—282no—
322D4—561no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—561no—
335C2×C4—561no—
336C2×C4C2×C22⋊D711yescharacteristic
337C2×C4C2×C22⋊D711yescharacteristic
338C2×C4C2×C22⋊D711yescharacteristic
339C2×C4C2×C22⋊D711yescharacteristic
340C2×C4C23×D711yescharacteristic
341C8—141no—
342C8—141no—
343C8—141no—
344C8—141no—
345C7#330911yescharacteristic, complemented, Sylow
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—281no—
359C22—281no—
360C22—281no—
361C22C22×C22⋊D711yescharacteristic
362C4—21no—
363C4—21no—
364C4—22no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—281no—
369C4—281no—
370C4—281no—
371C4C2×#115911yescharacteristic
372C4#654811yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—81no—
377C2—561no—
378C2#9049711yescharacteristic
379C2#9051311yescharacteristic
380C2#9052611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#550C2
socleC22×C7C22×D4
hypercenter#550D7
hypocenterC7#3309
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C7×#550C2semidirect
2. #90256C2semidirect
3. #90010C2semidirect
4. #90010C2semidirect
5. #90005C2semidirect
6. #90004C2semidirect
7. #89844C2semidirect
8. #89843C2semidirect
9. #89811C2semidirect
10. #89810C2semidirect
11. #89807C2semidirect
12. #89805C2semidirect
13. #89798C2semidirect
14. #89794C2semidirect
15. #89783C2semidirect
113. #550D7semidirect
345. C7#3309semidirect

Maximal subgroups

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

subgroupindex
C7×#5502
#902562
#900102
#900052
#900042
#898442
#898432
#898112
#898102
#898072
#898052
#897982
#897942
#897832
#33097