Finite Group Databasebeta

← #11738932

Subgroups — #11738932

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#90322C211yesmaximal, characteristic, complemented
2#90198C211yesmaximal, characteristic, complemented
3C7×#484C211yesmaximal, characteristic, complemented
4#89821C211yesmaximal, characteristic, complemented
5#89738C211yesmaximal, characteristic, complemented
6#89662C211yesmaximal, characteristic, complemented
7#89658C211yesmaximal, characteristic, complemented
8#89656C211yesmaximal, characteristic, complemented
9#89644C211yesmaximal, characteristic, complemented
10#89590C211yesmaximal, characteristic, complemented
11#89583C211yesmaximal, characteristic, complemented
12#89575C211yesmaximal, characteristic, complemented
13#89564C211yesmaximal, characteristic, complemented
14#89506C211yesmaximal, characteristic, complemented
15#89501C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×#1157—21no—
18C2×#1157C2211yescharacteristic
19C7×C22⋊Q8C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21#6507C2211yescharacteristic
22C2×#1143C2211yescharacteristic
23#6493C2211yescharacteristic
24#6492—21no—
25C4×C22⋊D7C2211yescharacteristic
26C2×#1136C2211yescharacteristic
27C2×D56C2211yescharacteristic
28C2×#1132C2211yescharacteristic
29C2×#1131C2211yescharacteristic
30#6460C2211yescharacteristic
31#6459—21no—
32#6458C2211yescharacteristic
33#6457—21no—
34#6457—21no—
35#6456C2211yescharacteristic
36#6456C2211yescharacteristic
37D7×C4⋊C4—21no—
38D7×C4⋊C4C2211yescharacteristic
39#6451C2211yescharacteristic
40#6447—21no—
41#6446—21no—
42C7×#107C2211yescharacteristic
43C7×#107C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46C7×C22⋊C8C2211yescharacteristic
47#6409C2211yescharacteristic
48#6395C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50#6390C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6383C2211yescharacteristic
58#6381C2211yescharacteristic
59#3330—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C22×C4×C7—21no—
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C2×C22⋊D7D411yescharacteristic
66#1161—22no—
67#1161—41no—
68#1157—22no—
69#1157—41no—
70#1157—44no—
71C2×D28—21no—
72C2×D28—21no—
73C2×D28C2311yescharacteristic
74C2×D28C2311yescharacteristic
75C2×C4×D7—21no—
76C2×C4×D7—21no—
77C2×C4×D7—21no—
78C2×C4×D7—21no—
79C2×C4×D7—21no—
80C2×C4×D7C2311yescharacteristic
81C2×Dic14—21no—
82C2×Dic14C2311yescharacteristic
83C2×C7×C8C2311yescharacteristic
84C2×C7×C8C2311yescharacteristic
85C7×C4⋊C4—21no—
86C7×C4⋊C4—21no—
87C7×C4⋊C4C2311yescharacteristic
88C7×C4⋊C4C2311yescharacteristic
89C7×C4⋊C4C2311yescharacteristic
90C7×C22⋊C4—21no—
91C22⋊Dic7D411yescharacteristic
92#1143—41no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4D411yescharacteristic
99C4⋊Dic7—21no—
100C4⋊Dic7C2311yescharacteristic
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105Dic7⋊C4D411yescharacteristic
106C4×Dic7C2311yescharacteristic
107#1136—41no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110D56—22no—
111#1132—41no—
112#1131—41no—
113#547—71no—
114#547—71no—
115#487—71no—
116#484—71no—
117#484D711yescharacteristic, complemented
118#474—71no—
119#468—71no—
120#468—71no—
121#461—71no—
122#449—71no—
123#449—71no—
124C4×D8—71no—
125#436—71no—
126#417—71no—
127#417—71no—
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C22×D7—21no—
131C22×D7C2×D411yescharacteristic
132C7×Q8—21no—
133C7×Q8—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—42no—
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2411yescharacteristic
143C22⋊D7—22no—
144C22⋊D7—42no—
145C22⋊D7—42no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151D28—21no—
152D28—21no—
153D28—21no—
154D28—42no—
155D28—42no—
156D28#4112yescomplemented
157C4×D7—21no—
158C4×D7—21no—
159C4×D7—42no—
160C4×D7—42no—
161C4×D7—42no—
162C4×D7—42no—
163C4×D7—42no—
164Dic14—21no—
165Dic14—21no—
166Dic14—42no—
167C7×C8—21no—
168C7×C8—21no—
169C7⋊C8—21no—
170C7⋊C8—21no—
171C2×#41—71no—
172C2×#41—71no—
173C2×#41—141no—
174C2×SD16—71no—
175C2×SD16—71no—
176C2×D8—71no—
177C2×M16—71no—
178C2×M16—71no—
179C4⋊Q8—71no—
180C4⋊Q8—71no—
181#123—141no—
182#123—141no—
183#123—141no—
184#123—141no—
185C22⋊Q8—71no—
186C22⋊Q8—141no—
187C22⋊Q8C2×D711yescharacteristic
188#121—71no—
189#121—71no—
190#121—141no—
191C4×D4—71no—
192C4×D4—71no—
193C4×D4—71no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4—141no—
196C2×C4⋊C4C2×D711yescharacteristic
197#107—71no—
198#107C2×D711yescharacteristic
199#107C2×D711yescharacteristic
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#102—71no—
210#102—71no—
211C22⋊C8—71no—
212C22⋊C8C2×D711yescharacteristic
213C4×C8—71no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7C22×D411yescharacteristic
218C2×D7—21no—
219C2×D7—22no—
220C2×D7—21no—
221C2×D7—21no—
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—41no—
225C2×D7—41no—
226C2×D7—41no—
227C4×C7—21no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—41no—
232C4×C7C2×#4111yescharacteristic
233C4×C7H(5,𝔽2)11yescharacteristic
234Dic7—21no—
235Dic7—21no—
236Dic7—41no—
237Dic7—41no—
238Dic7AGL(1,ℤ/8)12yes—
239#41—142no—
240#41—142no—
241#41—281no—
242#41—284no—
243#41—281no—
244C2×Q8—71no—
245C2×Q8—141no—
246C2×Q8C22×D711yescharacteristic
247C2×D4—71no—
248C2×D4—71no—
249C2×D4—71no—
250C2×D4—141no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—141no—
254C22×C4—21no—
255C22×C4—71no—
256C22×C4—141no—
257C22×C4—141no—
258C22×C4—141no—
259C22×C4—141no—
260C22×C4—141no—
261C22×C4C22×D711yescharacteristic
262SD16—281no—
263SD16—281no—
264D8—142no—
265M16—281no—
266M16—281no—
267C2×C8—71no—
268C2×C8—71no—
269C2×C8C22×D711yescharacteristic
270C2×C8C22×D711yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—71no—
274C4⋊C4—71no—
275C4⋊C4—141no—
276C4⋊C4—141no—
277C4⋊C4—141no—
278C4⋊C4—141no—
279C4⋊C4—141no—
280C4⋊C4C22×D711yescharacteristic
281C4⋊C4C22×D711yescharacteristic
282C4⋊C4C22×D711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—71no—
285C22⋊C4—71no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C22⋊C4—141no—
291C42—71no—
292C2×C7—41no—
293C2×C7#54611yescharacteristic
294C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
295C2×C7#57811yescharacteristic
296D7—41no—
297D7—42no—
298D7—81no—
299C23—71no—
300C23—141no—
301C23—141no—
302C23D4×D711yescharacteristic
303Q8—21no—
304Q8—21no—
305Q8—141no—
306Q8—141no—
307Q8—282no—
308D4—72no—
309D4—142no—
310D4—141no—
311D4—141no—
312D4—141no—
313D4—282no—
314D4—282no—
315D4—282no—
316D4—282no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—71no—
325C2×C4—71no—
326C2×C4—71no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—282no—
332C2×C4—282no—
333C2×C4—282no—
334C2×C4—282no—
335C2×C4—282no—
336C2×C4D4×D711yescharacteristic
337C2×C4C23×D711yescharacteristic
338C8—21no—
339C8—21no—
340C8—141no—
341C8—141no—
342C7#333011yescharacteristic, complemented, Sylow
343C22—21no—
344C22—21no—
345C22—21no—
346C22—141no—
347C22—141no—
348C22—141no—
349C22—142no—
350C22—281no—
351C22—281no—
352C22—281no—
353C22—281no—
354C22—281no—
355C22C2×D4×D711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—72no—
362C4—141no—
363C4—141no—
364C4—281no—
365C4—281no—
366C4D7×#4111yescharacteristic
367C4#655311yescharacteristic
368C2—41no—
369C2—281no—
370C2—282no—
371C2—561no—
372C2#9034511yescharacteristic
373C2#9045711yescharacteristic
374C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#484C2
socleC22×C7C22×D4
hypercenter#484D7
hypocenterC7#3330
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #90322C2semidirect
2. #90198C2semidirect
3. C7×#484C2semidirect
4. #89821C2semidirect
5. #89738C2semidirect
6. #89662C2semidirect
7. #89658C2semidirect
8. #89656C2semidirect
9. #89644C2semidirect
10. #89590C2semidirect
11. #89583C2semidirect
12. #89575C2semidirect
13. #89564C2semidirect
14. #89506C2semidirect
15. #89501C2semidirect
117. #484D7semidirect
156. D28#41semidirect
342. C7#3330semidirect

Maximal subgroups

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

subgroupindex
#903222
#901982
C7×#4842
#898212
#897382
#896622
#896582
#896562
#896442
#895902
#895832
#895752
#895642
#895062
#895012
#33307