Finite Group Databasebeta

← #11741064

Subgroups — #11741064

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×#553C211yesmaximal, characteristic, complemented
2#90244C211yesmaximal, characteristic, complemented
3#89839C211yesmaximal, characteristic, complemented
4#89835C211yesmaximal, characteristic, complemented
5#89811C211yesmaximal, characteristic, complemented
6#89810C211yesmaximal, characteristic, complemented
7#89809C211yesmaximal, characteristic, complemented
8#89809C211yesmaximal, characteristic, complemented
9#89790C211yesmaximal, characteristic, complemented
10#89787C211yesmaximal, characteristic, complemented
11#89783C211yesmaximal, characteristic, complemented
12#89775C211yesmaximal, characteristic, complemented
13#89774C211yesmaximal, characteristic, complemented
14#89739C211yesmaximal, characteristic, complemented
15#89738C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C7×#126—21no—
19C7×#125C2211yescharacteristic
20C7×#123—21no—
21C7×#121—21no—
22C7×#121C2211yescharacteristic
23C7×#121C2211yescharacteristic
24C4×C7×D4—21no—
25C4×C7×D4C2211yescharacteristic
26C4×C7×D4C2211yescharacteristic
27C7×#117C2211yescharacteristic
28C2×C7×C4⋊C4C2211yescharacteristic
29#6503—21no—
30C2×#1142C2211yescharacteristic
31C2×#1141C2211yescharacteristic
32C2×#1141C2211yescharacteristic
33#6493C2211yescharacteristic
34#6493C2211yescharacteristic
35C2×#1136C2211yescharacteristic
36C22×C7⋊C8C2211yescharacteristic
37#6456—21no—
38#6447—21no—
39#6447—21no—
40#6437C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42#6406C2211yescharacteristic
43C22⋊(C7⋊C8)C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6384C2211yescharacteristic
46#6384C2211yescharacteristic
47#6384C2211yescharacteristic
48#6383C2211yescharacteristic
49#6383C2211yescharacteristic
50#6383C2211yescharacteristic
51#6383C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3309—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4—21no—
62C2×C7×D4D411yescharacteristic
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7—21no—
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71C2×C22⋊D7—21no—
72#1157—22no—
73#1157—22no—
74#1157—41no—
75#1157—41no—
76C2×D28—21no—
77C2×D28—21no—
78C2×D28C2311yescharacteristic
79C2×D28C2311yescharacteristic
80C2×C4×D7—21no—
81C2×C4×D7—21no—
82C2×Dic14C2311yescharacteristic
83C7×C4⋊C4—21no—
84C7×C4⋊C4—21no—
85C7×C4⋊C4—21no—
86C7×C4⋊C4D411yescharacteristic
87C7×C4⋊C4C2311yescharacteristic
88C7×C4⋊C4C2311yescharacteristic
89C7×C4⋊C4C2311yescharacteristic
90C7×C22⋊C4—21no—
91C7×C22⋊C4—21no—
92C7×C22⋊C4—21no—
93C7×C22⋊C4—21no—
94C7×C22⋊C4D411yescharacteristic
95C7×C22⋊C4D411yescharacteristic
96C42×C7—21no—
97C42×C7C2311yescharacteristic
98#1142—22no—
99#1141—41no—
100#1141—41no—
101(C2×D7)⋊C4—21no—
102(C2×D7)⋊C4—21no—
103C4⋊Dic7C2311yescharacteristic
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#553D711yescharacteristic, complemented
113#550—71no—
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×C7C2×D411yescharacteristic
129C23×C7C2×D411yescharacteristic
130C22×D7—21no—
131C22×D7—21no—
132C7×D4—21no—
133C7×D4—21no—
134C7×D4—41no—
135C7×D4—42no—
136C7×D4—81no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—81no—
145C2×C4×C7—81no—
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2×D411yescharacteristic
148C2×C4×C7C2×D411yescharacteristic
149C2×C4×C7C2×D411yescharacteristic
150C2×C4×C7C2411yescharacteristic
151C22⋊D7—42no—
152C22⋊D7—42no—
153C22⋊D7—81no—
154C22⋊D7—81no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157D28—21no—
158D28—21no—
159D28—21no—
160D28—21no—
161D28—42no—
162D28—81no—
163C4×D7—42no—
164C4×D7—42no—
165Dic14—21no—
166Dic14#4112yes—
167C7⋊C8—21no—
168C7⋊C8—21no—
169C7⋊C8—21no—
170C7⋊C8—21no—
171C2×#41—71no—
172C2×#41—71no—
173C2×SD16—71no—
174C2×D8—71no—
175C2×D8—71no—
176C2×M16—71no—
177C22×C8—71no—
178#127—71no—
179#127—141no—
180#126—21no—
181#125C2×D711yescharacteristic
182#123—21no—
183#121—21no—
184#121—71no—
185#121—71no—
186#121—141no—
187#121—141no—
188#121C2×D711yescharacteristic
189#121C2×D711yescharacteristic
190C4×Q8—71no—
191C4×D4—21no—
192C4×D4—141no—
193C4×D4C2×D711yescharacteristic
194C4×D4C2×D711yescharacteristic
195#117C2×D711yescharacteristic
196C2×C4⋊C4C2×D711yescharacteristic
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×C7C22×D411yescharacteristic
224C2×D7—41no—
225C2×D7—41no—
226C2×D7—41no—
227C2×D7—41no—
228C2×D7—41no—
229C2×D7—41no—
230C4×C7—21no—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—21no—
234C4×C7—41no—
235C4×C7—41no—
236C4×C7—41no—
237C4×C7C2×#4111yescharacteristic
238C4×C7H(5,𝔽2)11yescharacteristic
239Dic7—22no—
240Dic7—41no—
241#41—142no—
242#41—142no—
243#41—281no—
244#41—281no—
245C2×Q8—71no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—71no—
249C2×D4—71no—
250C2×D4—141no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4—141no—
255C2×D4—141no—
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—21no—
275C4⋊C4—21no—
276C4⋊C4—71no—
277C4⋊C4—141no—
278C4⋊C4C22⋊D711yescharacteristic
279C4⋊C4C22×D711yescharacteristic
280C4⋊C4C22×D711yescharacteristic
281C4⋊C4C22×D711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
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#54611yescharacteristic
297C2×C7#57511yescharacteristic
298C2×C7#57611yescharacteristic
299D7—81no—
300D7—81no—
301C23—21no—
302C23—141no—
303C23—141no—
304C23C2×C22⋊D711yescharacteristic
305C23C2×C22⋊D711yescharacteristic
306Q8—72no—
307Q8—141no—
308D4—21no—
309D4—21no—
310D4—41no—
311D4—42no—
312D4—81no—
313D4—141no—
314D4—141no—
315D4—141no—
316D4—141no—
317D4—282no—
318D4—282no—
319D4—282no—
320D4—561no—
321D4—561no—
322D4—561no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—282no—
335C2×C4—282no—
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—281no—
356C22—281no—
357C22—281no—
358C22—281no—
359C22—281no—
360C22—281no—
361C22C22×C22⋊D711yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—142no—
370C4—281no—
371C4D7×#4111yescharacteristic
372C4#655311yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—561no—
377C2—561no—
378C2#9051311yescharacteristic
379C2#9052011yescharacteristic
380C2#9052611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#553C2
socleC22×C7C22×D4
hypercenter#553D7
hypocenterC7#3309
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. C7×#553C2semidirect
2. #90244C2semidirect
3. #89839C2semidirect
4. #89835C2semidirect
5. #89811C2semidirect
6. #89810C2semidirect
7. #89809C2semidirect
8. #89809C2semidirect
9. #89790C2semidirect
10. #89787C2semidirect
11. #89783C2semidirect
12. #89775C2semidirect
13. #89774C2semidirect
14. #89739C2semidirect
15. #89738C2semidirect
112. #553D7semidirect
345. C7#3309semidirect

Maximal subgroups

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

subgroupindex
C7×#5532
#902442
#898392
#898352
#898112
#898102
#898092
#897902
#897872
#897832
#897752
#897742
#897392
#897382
#33097