Finite Group Databasebeta

← #11738306

Subgroups — #11738306

Order 896.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90309C211yesmaximal, characteristic, complemented
2#90305C211yesmaximal, characteristic, complemented
3C7×#469C211yesmaximal, characteristic, complemented
4#89931C211yesmaximal, characteristic, complemented
5D8×Dic7C211yesmaximal, characteristic, complemented
6#89900C211yesmaximal, characteristic, complemented
7#89893C211yesmaximal, characteristic, complemented
8#89807C211yesmaximal, characteristic, complemented
9#89805C211yesmaximal, characteristic, complemented
10#89633C211yesmaximal, characteristic, complemented
11#89630C211yesmaximal, characteristic, complemented
12#89551C211yesmaximal, characteristic, complemented
13#89546C211yesmaximal, characteristic, complemented
14#89538C211yesmaximal, characteristic, complemented
15#89535C211yesmaximal, characteristic, complemented
16C2×C7×D8C2211yescharacteristic
17C2×C7×M16C2211yescharacteristic
18C7×#121C2211yescharacteristic
19C7×#121C2211yescharacteristic
20#6502—21no—
21#6501—21no—
22#6501C2211yescharacteristic
23#6501C2211yescharacteristic
24#6498—21no—
25D4×Dic7—21no—
26D4×Dic7C2211yescharacteristic
27D4×Dic7C2211yescharacteristic
28C4×C22⋊D7C2211yescharacteristic
29#6489C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×#1131C2211yescharacteristic
32#6455C2211yescharacteristic
33D7×C4⋊C4C2211yescharacteristic
34#6453C2211yescharacteristic
35#6453C2211yescharacteristic
36#6449—21no—
37(C22⋊D7)⋊C4—21no—
38#6442—21no—
39#6442—21no—
40C7×#106C2211yescharacteristic
41C7×#102C2211yescharacteristic
42C7×#102C2211yescharacteristic
43#6406C2211yescharacteristic
44#6406C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47C22⋊(C7⋊C8)C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6392C2211yescharacteristic
50#6392C2211yescharacteristic
51#6391C2211yescharacteristic
52#6391C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54C8×Dic7C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3355—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4—21no—
62C2×C7×D4C2311yescharacteristic
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7D411yescharacteristic
68C22×Dic7—21no—
69C22×Dic7—21no—
70C22×Dic7—21no—
71C2×C4×D7—21no—
72C2×C4×D7C2311yescharacteristic
73C7×D8—22no—
74C7×M16—22no—
75C2×C7×C8C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C22⋊C4—21no—
80C7×C22⋊C4—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85C22⋊Dic7—21no—
86C22⋊Dic7D411yescharacteristic
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4D411yescharacteristic
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7—21no—
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4D411yescharacteristic
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105#1131—22no—
106#553—71no—
107#553—71no—
108#499—71no—
109#482—71no—
110#482—71no—
111#480—71no—
112#480—71no—
113#478—71no—
114#478—71no—
115#469—71no—
116#469D711yescharacteristic, complemented
117C4×D8—71no—
118#436—71no—
119#428—71no—
120#428—71no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7C2×D411yescharacteristic
124C22×D7C2×D411yescharacteristic
125C7×D4—21no—
126C7×D4—21no—
127C7×D4—21no—
128C7×D4—21no—
129C7×D4—42no—
130C7×D4—81no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2411yescharacteristic
137C22⋊D7—41no—
138C22⋊D7—42no—
139C22⋊D7—81no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—81no—
145C2×Dic7—81no—
146C2×Dic7—81no—
147C2×Dic7C2×D411yescharacteristic
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150C4×D7—21no—
151C4×D7—21no—
152C4×D7—81no—
153C7×C8—21no—
154C7×C8#4112yes—
155C7⋊C8—21no—
156C7⋊C8—21no—
157C2×D8C2×D711yescharacteristic
158C2×M16—71no—
159C2×M16C2×D711yescharacteristic
160#126—141no—
161#126—141no—
162#125—71no—
163#125—71no—
164#123—141no—
165#123—141no—
166#121—71no—
167#121—71no—
168#121—141no—
169#121—141no—
170#121C2×D711yescharacteristic
171#121C2×D711yescharacteristic
172C4×D4—71no—
173C4×D4—71no—
174C4×D4—71no—
175C4×D4—141no—
176C4×D4—141no—
177#117—71no—
178#117—71no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4—71no—
181#107—71no—
182#107—71no—
183#107—71no—
184#107—71no—
185#107—71no—
186#106—71no—
187#106—71no—
188#106—71no—
189#106C2×D711yescharacteristic
190C4⋊C8—71no—
191#102—71no—
192#102—71no—
193#102—71no—
194#102—71no—
195#102C2×D711yescharacteristic
196#102C2×D711yescharacteristic
197C22⋊C8—71no—
198C22⋊C8—71no—
199C4×C8—71no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7C22×D411yescharacteristic
210C2×D7—21no—
211C2×D7—21no—
212C2×D7—21no—
213C4×C7—21no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7C2×#4111yescharacteristic
217C4×C72−1+411yescharacteristic
218Dic7—21no—
219Dic7—21no—
220Dic7—21no—
221Dic7—41no—
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—71no—
228C2×D4—141no—
229C2×D4—141no—
230C2×D4C22×D711yescharacteristic
231C2×D4C22×D711yescharacteristic
232C22×C4—71no—
233C22×C4—141no—
234C22×C4—141no—
235C22×C4—141no—
236C22×C4—141no—
237C22×C4C22×D711yescharacteristic
238D8—22no—
239M16—22no—
240M16—142no—
241C2×C8—71no—
242C2×C8—71no—
243C2×C8C22×D711yescharacteristic
244C2×C8C22×D711yescharacteristic
245C4⋊C4—71no—
246C4⋊C4—71no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—71no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4C22×D711yescharacteristic
257C4⋊C4C22×D711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—71no—
261C22⋊C4—71no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C42—71no—
269C42—141no—
270C42—141no—
271C2×C7—41no—
272C2×C7—81no—
273C2×C7—81no—
274C2×C7#54711yescharacteristic
275C2×C7#57511yescharacteristic
276C2×C7#57611yescharacteristic
277D7—41no—
278C23—21no—
279C23—21no—
280C23—71no—
281C23D4×D711yescharacteristic
282D4—21no—
283D4—21no—
284D4—21no—
285D4—21no—
286D4—42no—
287D4—81no—
288D4—281no—
289D4—282no—
290D4—561no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—71no—
296C2×C4—71no—
297C2×C4—71no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—561no—
305C2×C4—561no—
306C2×C4—561no—
307C2×C4—561no—
308C2×C4D4×D711yescharacteristic
309C2×C4C23×D711yescharacteristic
310C8—21no—
311C8—141no—
312C8—141no—
313C8#115912yes—
314C7#335511yescharacteristic, complemented, Sylow
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—141no—
325C22—141no—
326C22—141no—
327C22C2×D4×D711yescharacteristic
328C4—21no—
329C4—41no—
330C4—41no—
331C4—141no—
332C4—141no—
333C4—141no—
334C4—281no—
335C4—281no—
336C4—281no—
337C4—281no—
338C4C2×#115911yescharacteristic
339C4#655411yescharacteristic
340C2—41no—
341C2—81no—
342C2—81no—
343C2—281no—
344C2#9030211yescharacteristic
345C2#9046411yescharacteristic
346C2#9046511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#469C2
socleC22×C7C22×D4
hypercenter#469D7
hypocenterC7#3355
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #90309C2semidirect
2. #90305C2semidirect
3. C7×#469C2semidirect
4. #89931C2semidirect
5. D8×Dic7C2semidirect
6. #89900C2semidirect
7. #89893C2semidirect
8. #89807C2semidirect
9. #89805C2semidirect
10. #89633C2semidirect
11. #89630C2semidirect
12. #89551C2semidirect
13. #89546C2semidirect
14. #89538C2semidirect
15. #89535C2semidirect
116. #469D7semidirect
314. C7#3355semidirect

Maximal subgroups

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

subgroupindex
#903092
#903052
C7×#4692
#899312
D8×Dic72
#899002
#898932
#898072
#898052
#896332
#896302
#895512
#895462
#895382
#895352
#33557