Finite Group Databasebeta

← #11740939

Subgroups — #11740939

Order 896.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#546C211yesmaximal, characteristic
2#90232C211yesmaximal, characteristic, complemented
3#90011C211yesmaximal, characteristic, complemented
4#90005C211yesmaximal, characteristic, complemented
5#89990C211yesmaximal, characteristic, complemented
6C2×#6406C211yesmaximal, characteristic, complemented
7#89828C211yesmaximal, characteristic, complemented
8#89826C211yesmaximal, characteristic, complemented
9#89823C211yesmaximal, characteristic, complemented
10#89815C211yesmaximal, characteristic, complemented
11#89813C211yesmaximal, characteristic, complemented
12#89807C211yesmaximal, characteristic, complemented
13C4×#1142C211yesmaximal, characteristic, complemented
14D4×C7⋊C8C211yesmaximal, characteristic, complemented
15#89781C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C7×#124C2211yescharacteristic
19C7×#123—21no—
20C7×C22⋊Q8C2211yescharacteristic
21C7×#121—21no—
22C7×#121C2211yescharacteristic
23C7×#120—21no—
24C4×C7×D4C2211yescharacteristic
25C4×C7×D4C2211yescharacteristic
26C2×C7×C22⋊C4—21no—
27C2×#1144C2211yescharacteristic, complemented
28#6498—21no—
29C2×#1142C2211yescharacteristic
30C2×#1142C2211yescharacteristic
31#6489C2211yescharacteristic, complemented
32C2×C4⋊Dic7C2211yescharacteristic
33#6487C2211yescharacteristic
34#6487C2211yescharacteristic, complemented
35C22×C7⋊C8C2211yescharacteristic
36C22×C7⋊C8C2211yescharacteristic, complemented
37#6452—21no—
38#6442—21no—
39#6440—21no—
40#6433C2211yescharacteristic, complemented
41C4×Dic14C2211yescharacteristic, complemented
42#6409C2211yescharacteristic, complemented
43#6409C2211yescharacteristic, complemented
44#6409C2211yescharacteristic, complemented
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47#6406C2211yescharacteristic
48#6406C2211yescharacteristic
49#6406C2211yescharacteristic
50#6406C2211yescharacteristic
51C22⋊(C7⋊C8)C2211yescharacteristic
52C22⋊(C7⋊C8)C2211yescharacteristic, complemented
53#6384C2211yescharacteristic, complemented
54#6382C2211yescharacteristic, complemented
55#6382C2211yescharacteristic, complemented
56#6381C2211yescharacteristic, complemented
57C4⋊(C7⋊C8)C2211yescharacteristic, complemented
58C4×C7⋊C8C2211yescharacteristic, complemented
59#3296—71nomaximal, Sylow
60C24×C7—21no—
61C7×#41—22no—
62C7×#41—41no—
63C2×C7×Q8C2311yescharacteristic
64C2×C7×D4—22no—
65C2×C7×D4—21no—
66C2×C7×D4—21no—
67C2×C7×D4—21no—
68C2×C7×D4—41no—
69C2×C7×D4D411yescharacteristic
70C2×C7×D4C2311yescharacteristic
71C2×C7×D4C2311yescharacteristic
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7C2311yescharacteristic
75C22×C4×C7C2311yescharacteristic
76C22×Dic7—21no—
77C2×Dic14C2311yescharacteristic
78C7×C4⋊C4—21no—
79C7×C4⋊C4D411yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C22⋊C4—21no—
82C7×C22⋊C4—21no—
83C7×C22⋊C4—21no—
84C7×C22⋊C4—21no—
85C7×C22⋊C4—21no—
86C7×C22⋊C4D411yescharacteristic
87C7×C22⋊C4D411yescharacteristic
88C42×C7C2311yescharacteristic
89C22⋊Dic7—21no—
90C22⋊Dic7—21no—
91C22⋊Dic7—21no—
92#1144—41no—
93#1142—22no—
94#1142—41no—
95C4⋊Dic7—21no—
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104C4×Dic7—21no—
105C4×Dic7—21no—
106C2×C7⋊C8—41no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112#552—71no—
113#546D711yescharacteristic
114#488—71no—
115#483—71no—
116#482—71no—
117#478—71no—
118#467—71no—
119#465—71no—
120#464—71no—
121#452—71no—
122#450—71no—
123C4×SD16—71no—
124C8×D4—71no—
125#416—71no—
126C2×#102—71no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—42no—
130C23×C7—42no—
131C23×C7—81no—
132C23×C7C2×D411yescharacteristic
133C23×C7C2×D411yescharacteristic
134C7×Q8—21no—
135C7×Q8—21no—
136C7×D4—22no—
137C7×D4—21no—
138C7×D4—21no—
139C7×D4—21no—
140C7×D4—42no—
141C7×D4—42no—
142C7×D4—81no—
143C7×D4#4112yes—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—42no—
151C2×C4×C7—81no—
152C2×C4×C7C2×D411yescharacteristic
153C2×C4×C7C2×D411yescharacteristic
154C2×C4×C7C2×D411yescharacteristic
155C2×C4×C7C2×D411yescharacteristic
156C2×C4×C7C2411yescharacteristic
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7—21no—
160C2×Dic7—21no—
161C2×Dic7—42no—
162Dic14—21no—
163Dic14—21no—
164C7⋊C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C2×#41C2×D711yescharacteristic
169C22×D4C2×D711yescharacteristic
170C2×Q16—71no—
171C2×SD16—71no—
172C2×SD16—71no—
173C22×C8—71no—
174C22×C8—71no—
175#126—141no—
176#125—71no—
177#125—141no—
178#124C2×D711yescharacteristic
179#123—21no—
180#123—141no—
181C22⋊Q8—71no—
182C22⋊Q8—71no—
183C22⋊Q8C2×D711yescharacteristic
184#121—21no—
185#121C2×D711yescharacteristic
186#120—21no—
187C4×Q8—71no—
188C4×D4C2×D711yescharacteristic
189C4×D4C2×D711yescharacteristic
190#117—71no—
191#117—141no—
192C2×C4⋊C4—71no—
193C2×C22⋊C4—21no—
194#107—71no—
195#106—71no—
196#106—71no—
197C4⋊C8—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#102—71no—
203#102—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207#102—71no—
208C22⋊C8—71no—
209C22⋊C8—71no—
210C4×C8—71no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—22no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—84no—
223C22×C7#13512yes—
224C22×C7C22×D411yescharacteristic
225C4×C7—21no—
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—21no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7C2×#4111yescharacteristic
232C4×C7H(5,𝔽2)11yescharacteristic
233Dic7—41no—
234Dic7—41no—
235Dic7—41no—
236Dic7—41no—
237C24—21no—
238#41—22no—
239#41—41no—
240C2×Q8—71no—
241C2×Q8C22×D711yescharacteristic
242C2×D4—22no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—41no—
247C2×D4C22⋊D711yescharacteristic
248C2×D4C22×D711yescharacteristic
249C2×D4C22×D711yescharacteristic
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—141no—
253C22×C4C22×D711yescharacteristic
254C22×C4C22×D711yescharacteristic
255Q16—281no—
256SD16—142no—
257SD16—281no—
258C2×C8—71no—
259C2×C8—71no—
260C2×C8—71no—
261C2×C8—71no—
262C2×C8—281no—
263C2×C8—281no—
264C4⋊C4—21no—
265C4⋊C4—71no—
266C4⋊C4—71no—
267C4⋊C4—71no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4—141no—
274C4⋊C4C22⋊D711yescharacteristic
275C4⋊C4C22×D711yescharacteristic
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4C22⋊D711yescharacteristic
285C22⋊C4C22⋊D711yescharacteristic
286C42—141no—
287C42—141no—
288C42C22×D711yescharacteristic
289C2×C7—22no—
290C2×C7—41no—
291C2×C7—42no—
292C2×C7—81no—
293C2×C7#54611yescharacteristic
294C2×C7C2×#13511yescharacteristic
295C2×C7#57711yescharacteristic
296C23—21no—
297C23—21no—
298C23—42no—
299C23—42no—
300C23—81no—
301C23C2×C22⋊D711yescharacteristic
302C23C2×C22⋊D711yescharacteristic
303Q8—21no—
304Q8—21no—
305Q8—141no—
306Q8—141no—
307D4—22no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—42no—
313D4—81no—
314D4#115912yes—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—282no—
328C2×C4C2×C22⋊D711yescharacteristic
329C2×C4C2×C22⋊D711yescharacteristic
330C2×C4C2×C22⋊D711yescharacteristic
331C2×C4C2×C22⋊D711yescharacteristic
332C2×C4C23×D711yescharacteristic
333C8—141no—
334C8—141no—
335C8—141no—
336C8—141no—
337C7#329611yescharacteristic, complemented, Sylow
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—84no—
350C22#651312yes—
351C22C22×C22⋊D711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—281no—
359C4—281no—
360C4—281no—
361C4—281no—
362C4C2×#115911yescharacteristic
363C4#654811yescharacteristic
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#9049711yescharacteristic
369C2C2×#651311yescharacteristic
370C2#9052711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
Fitting, prime radicalC7×#546C2
socleC22×C7C22×D4
hypercenter#546D7
hypocenterC7#3296
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
2. #90232C2semidirect
3. #90011C2semidirect
4. #90005C2semidirect
5. #89990C2semidirect
6. C2×#6406C2semidirect
7. #89828C2semidirect
8. #89826C2semidirect
9. #89823C2semidirect
10. #89815C2semidirect
11. #89813C2semidirect
12. #89807C2semidirect
13. C4×#1142C2semidirect
14. D4×C7⋊C8C2semidirect
15. #89781C2semidirect
27. C2×#1144C22semidirect
31. #6489C22semidirect
34. #6487C22semidirect
36. C22×C7⋊C8C22semidirect
40. #6433C22semidirect
41. C4×Dic14C22semidirect
42. #6409C22semidirect
43. #6409C22semidirect
44. #6409C22semidirect
52. C22⋊(C7⋊C8)C22semidirect
53. #6384C22semidirect
54. #6382C22semidirect
55. #6382C22semidirect
56. #6381C22semidirect
57. C4⋊(C7⋊C8)C22semidirect
58. C4×C7⋊C8C22semidirect
337. C7#3296semidirect

Maximal subgroups

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

subgroupindex
C7×#5462
#902322
#900112
#900052
#899902
C2×#64062
#898282
#898262
#898232
#898152
#898132
#898072
C4×#11422
D4×C7⋊C82
#897812
#32967