Finite Group Databasebeta

← #11736341

Subgroups — #11736341

Order 896.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90522C211yesmaximal, characteristic, complemented
2#90198C211yesmaximal, characteristic, complemented
3C7×#417C211yesmaximal, characteristic, complemented
4#90010C211yesmaximal, characteristic, complemented
5#89906C211yesmaximal, characteristic, complemented
6#89899C211yesmaximal, characteristic, complemented
7#89742C211yesmaximal, characteristic, complemented
8#89589C211yesmaximal, characteristic, complemented
9#89578C211yesmaximal, characteristic, complemented
10D7×#103C211yesmaximal, characteristic, complemented
11#89566C211yesmaximal, characteristic, complemented
12#89559C211yesmaximal, characteristic, complemented
13#89547C211yesmaximal, characteristic, complemented
14#89546C211yesmaximal, characteristic, complemented
15#89537C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C2×#1157—21no—
19C2×C7×M16C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22C2×#1143C2211yescharacteristic
23#6501—21no—
24#6501C2211yescharacteristic
25#6499—21no—
26#6499C2211yescharacteristic
27C2×#1141C2211yescharacteristic
28#6493C2211yescharacteristic
29#6492—21no—
30C4×C22⋊D7—21no—
31C4×C22⋊D7C2211yescharacteristic
32#6487—21no—
33#6487C2211yescharacteristic
34C2×#1132C2211yescharacteristic
35C2×C8×D7C2211yescharacteristic
36#6459—21no—
37#6458C2211yescharacteristic
38#6457—21no—
39#6456C2211yescharacteristic
40D7×C4⋊C4C2211yescharacteristic
41#6451C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#102C2211yescharacteristic
45C7×#102C2211yescharacteristic
46#6409C2211yescharacteristic
47#6406C2211yescharacteristic
48C22⋊(C7⋊C8)C2211yescharacteristic
49#6395C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6392C2211yescharacteristic
52#6390C2211yescharacteristic
53#6389C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55#6384C2211yescharacteristic
56#6384C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3308—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—41no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4—21no—
64C2×C7×D4C2311yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7—21no—
69C2×C22⋊D7—21no—
70C2×C22⋊D7D411yescharacteristic
71Q8×D7—22no—
72Q8×D7—41no—
73#1157—44no—
74C2×D28—21no—
75C2×D28C2311yescharacteristic
76C2×C4×D7—21no—
77C2×C4×D7—21no—
78C2×C4×D7—21no—
79C2×C4×D7C2311yescharacteristic
80C2×Dic14—21no—
81C2×Dic14—21no—
82C2×Dic14C2311yescharacteristic
83C7×M16—41no—
84C2×C7×C8C2311yescharacteristic
85C2×C7×C8C2311yescharacteristic
86C7×C4⋊C4—21no—
87C7×C4⋊C4C2311yescharacteristic
88C7×C4⋊C4C2311yescharacteristic
89C22⋊Dic7—21no—
90C22⋊Dic7—21no—
91C22⋊Dic7—21no—
92C22⋊Dic7—21no—
93C22⋊Dic7D411yescharacteristic
94#1143—22no—
95#1141—41no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99(C2×D7)⋊C4D411yescharacteristic
100C4⋊Dic7—21no—
101C4⋊Dic7C2311yescharacteristic
102C4⋊Dic7C2311yescharacteristic
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106Dic7⋊C4D411yescharacteristic
107C4×Dic7—21no—
108C4×Dic7C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111#1132—41no—
112C8×D7—41no—
113#548—71no—
114#547—71no—
115#489—71no—
116#484—71no—
117#480—71no—
118#477—71no—
119#468—71no—
120#466—71no—
121#463—71no—
122#449—71no—
123#448—71no—
124#440—71no—
125#435—71no—
126#417D711yescharacteristic, complemented
127C2×#103—71no—
128C23×C7—21no—
129C23×C7C2×D411yescharacteristic
130C22×D7—21no—
131C22×D7C2×D411yescharacteristic
132C7×Q8—21no—
133C7×Q8#4112yescomplemented
134C7×D4—21no—
135C7×D4—21no—
136C7×D4—42no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C22⋊D7—22no—
147C22⋊D7—42no—
148C22⋊D7—42no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7C2×D411yescharacteristic
153C2×Dic7C2×D411yescharacteristic
154C2×Dic7C2×D411yescharacteristic
155D28—21no—
156D28—21no—
157D28—42no—
158C4×D7—21no—
159C4×D7—21no—
160C4×D7—42no—
161C4×D7—42no—
162C4×D7—42no—
163Dic14—21no—
164Dic14—21no—
165Dic14—42no—
166Dic14—42no—
167C7×C8—21no—
168C7×C8—21no—
169C7⋊C8—21no—
170C7⋊C8—21no—
171C2×#41—141no—
172C2×#41C2×D711yescharacteristic
173C22×Q8—71no—
174C2×SD16—71no—
175C2×SD16—71no—
176C2×D8—71no—
177C2×M16C2×D711yescharacteristic
178C22×C8—71no—
179C4⋊Q8—71no—
180#124—71no—
181#124—141no—
182#123—141no—
183#123—141no—
184C22⋊Q8—71no—
185C22⋊Q8—141no—
186C22⋊Q8—141no—
187#121—71no—
188#121—71no—
189#121—71no—
190#121—141no—
191C4×Q8—71no—
192C4×D4—71no—
193C4×D4—71no—
194C4×D4—141no—
195C2×C4⋊C4—71no—
196C2×C4⋊C4C2×D711yescharacteristic
197#107—71no—
198#107—71no—
199#107—71no—
200C4⋊C8—71no—
201#103—71no—
202#103—71no—
203#103—71no—
204#103—71no—
205#103C2×D711yescharacteristic
206#103C2×D711yescharacteristic
207#102—71no—
208#102—71no—
209#102C2×D711yescharacteristic
210#102C2×D711yescharacteristic
211C22⋊C8—71no—
212C22⋊C8—71no—
213#97—71no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7C22×D411yescharacteristic
221C2×D7—21no—
222C2×D7—21no—
223C2×D7—41no—
224C2×D7—41no—
225C2×D7—41no—
226C2×D7AGL(1,ℤ/8)12yes—
227C4×C7—22no—
228C4×C7—21no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7C2×#4111yescharacteristic
232C4×C7H(5,𝔽2)11yescharacteristic
233Dic7—21no—
234Dic7—21no—
235Dic7—21no—
236Dic7—41no—
237Dic7—41no—
238Dic7—41no—
239#41—22no—
240#41—41no—
241#41—284no—
242C2×Q8—71no—
243C2×Q8—141no—
244C2×Q8—141no—
245C2×Q8—142no—
246C2×Q8—281no—
247C2×Q8C22×D711yescharacteristic
248C2×D4—21no—
249C2×D4—71no—
250C2×D4—71no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4C22×D711yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—71no—
258C22×C4—141no—
259C22×C4—141no—
260C22×C4—141no—
261C22×C4C22×D711yescharacteristic
262SD16—142no—
263SD16—281no—
264D8—281no—
265M16—41no—
266C2×C8—71no—
267C2×C8—71no—
268C2×C8—281no—
269C2×C8C22×D711yescharacteristic
270C2×C8C22×D711yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—71no—
273C4⋊C4—71no—
274C4⋊C4—71no—
275C4⋊C4—141no—
276C4⋊C4—141no—
277C4⋊C4—141no—
278C4⋊C4—141no—
279C4⋊C4C22×D711yescharacteristic
280C4⋊C4C22×D711yescharacteristic
281C22⋊C4—71no—
282C22⋊C4—71no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C42—71no—
291C42—141no—
292C2×C7—41no—
293C2×C7—81no—
294C2×C7#54611yescharacteristic
295C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
296C2×C7#57811yescharacteristic
297D7—22no—
298D7—81no—
299C23—21no—
300C23—71no—
301C23—141no—
302C23D4×D711yescharacteristic
303Q8—21no—
304Q8—141no—
305Q8—141no—
306Q8—282no—
307Q8—282no—
308Q8#115712yescomplemented
309D4—21no—
310D4—21no—
311D4—42no—
312D4—142no—
313D4—141no—
314D4—141no—
315D4—282no—
316D4—282no—
317D4—282no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—71no—
326C2×C4—71no—
327C2×C4—71no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—282no—
334C2×C4—282no—
335C2×C4—282no—
336C2×C4D4×D711yescharacteristic
337C2×C4C23×D711yescharacteristic
338C8—21no—
339C8—21no—
340C8—141no—
341C8—141no—
342C7#330811yescharacteristic, complemented, Sylow
343C22—21no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—72no—
350C22—141no—
351C22—141no—
352C22—281no—
353C22—281no—
354C22—281no—
355C22C2×D4×D711yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—141no—
361C4—141no—
362C4—141no—
363C4—281no—
364C4—281no—
365C4—281no—
366C4C2×#115711yescharacteristic
367C4#654811yescharacteristic
368C2—41no—
369C2—81no—
370C2—142no—
371C2—561no—
372C2#9018111yescharacteristic
373C2D7×AGL(1,ℤ/8)11yescharacteristic
374C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#417C2
socleC22×C7C22×D4
hypercenter#417D7
hypocenterC7#3308
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #90522C2semidirect
2. #90198C2semidirect
3. C7×#417C2semidirect
4. #90010C2semidirect
5. #89906C2semidirect
6. #89899C2semidirect
7. #89742C2semidirect
8. #89589C2semidirect
9. #89578C2semidirect
10. D7×#103C2semidirect
11. #89566C2semidirect
12. #89559C2semidirect
13. #89547C2semidirect
14. #89546C2semidirect
15. #89537C2semidirect
126. #417D7semidirect
133. C7×Q8#41semidirect
308. Q8#1157semidirect
342. C7#3308semidirect

Maximal subgroups

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

subgroupindex
#905222
#901982
C7×#4172
#900102
#899062
#898992
#897422
#895892
#895782
D7×#1032
#895662
#895592
#895472
#895462
#895372
#33087