Finite Group Databasebeta

← #11740964

Subgroups — #11740964

Order 896.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#547C211yesmaximal, characteristic, complemented
2#90256C211yesmaximal, characteristic, complemented
3#89856C211yesmaximal, characteristic, complemented
4#89852C211yesmaximal, characteristic, complemented
5#89811C211yesmaximal, characteristic, complemented
6#89810C211yesmaximal, characteristic, complemented
7#89807C211yesmaximal, characteristic, complemented
8#89806C211yesmaximal, characteristic, complemented
9#89786C211yesmaximal, characteristic, complemented
10#89783C211yesmaximal, characteristic, complemented
11#89781C211yesmaximal, characteristic, complemented
12#89740C211yesmaximal, characteristic, complemented
13#89739C211yesmaximal, characteristic, complemented
14#89736C211yesmaximal, characteristic, complemented
15C2×#6382C211yesmaximal, characteristic, complemented
16C2×C7×#41—21no—
17C7×C4⋊Q8C2211yescharacteristic
18C7×#123—21no—
19C7×#123—21no—
20C7×C22⋊Q8—21no—
21C7×#121C2211yescharacteristic
22C7×#121C2211yescharacteristic
23C4×C7×D4C2211yescharacteristic
24C4×C7×D4C2211yescharacteristic
25C2×C7×C4⋊C4C2211yescharacteristic
26C2×C7×C4⋊C4C2211yescharacteristic
27#6502—21no—
28C2×#1141C2211yescharacteristic
29#6493C2211yescharacteristic
30#6493C2211yescharacteristic
31#6489C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×#1136C2211yescharacteristic
34C22×C7⋊C8C2211yescharacteristic
35#6461—21no—
36#6446—21no—
37(C22⋊D7)⋊C4—21no—
38C4×D28C2211yescharacteristic
39#6433C2211yescharacteristic
40#6406C2211yescharacteristic
41#6406C2211yescharacteristic
42#6406C2211yescharacteristic
43C22⋊(C7⋊C8)C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6383C2211yescharacteristic
46#6383C2211yescharacteristic
47#6383C2211yescharacteristic
48#6382C2211yescharacteristic
49#6382C2211yescharacteristic
50#6382C2211yescharacteristic
51#6382C2211yescharacteristic
52#6382C2211yescharacteristic
53#6382C2211yescharacteristic
54#6381C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3345—71nomaximal, Sylow
60C7×#41—44no—
61C2×C7×Q8—21no—
62C2×C7×D4—21no—
63C2×C7×D4—21no—
64C2×C7×D4D411yescharacteristic
65C2×C7×D4C2311yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7C2311yescharacteristic
70C22×C4×C7C2311yescharacteristic
71C2×C22⋊D7—21no—
72C2×C22⋊D7—21no—
73C22×Dic7—21no—
74C2×D28C2311yescharacteristic
75C2×C4×D7—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4—21no—
78C7×C4⋊C4—21no—
79C7×C4⋊C4D411yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C22⋊C4—21no—
84C7×C22⋊C4—21no—
85C7×C22⋊C4—21no—
86C7×C22⋊C4D411yescharacteristic
87C7×C22⋊C4D411yescharacteristic
88C42×C7C2311yescharacteristic
89C22⋊Dic7—21no—
90#1141—22no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94C4⋊Dic7—21no—
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100C4×Dic7—21no—
101#1136—22no—
102C2×C7⋊C8—22no—
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#553—71no—
108#547D711yescharacteristic, complemented
109#501—71no—
110#482—71no—
111#481—71no—
112#481—71no—
113#480—71no—
114#478—71no—
115#476—71no—
116#469—71no—
117#466—71no—
118#442—71no—
119#435—71no—
120#428—71no—
121C2×#106—71no—
122C23×C7—21no—
123C23×C7C2×D411yescharacteristic
124C23×C7C2×D411yescharacteristic
125C22×D7—21no—
126C7×Q8—42no—
127C7×D4—22no—
128C7×D4—21no—
129C7×D4—21no—
130C7×D4—42no—
131C7×D4—42no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2411yescharacteristic
147C22⋊D7—42no—
148C22⋊D7—81no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—81no—
153D28—21no—
154D28—21no—
155C4×D7—81no—
156C7⋊C8—21no—
157C7⋊C8—21no—
158C7⋊C8—21no—
159C7⋊C8#4112yescomplemented
160C2×#41—21no—
161C2×D8—71no—
162C2×M16—71no—
163C22×C8—71no—
164C4⋊Q8C2×D711yescharacteristic
165#126—141no—
166#125—71no—
167#123—21no—
168#123—21no—
169#123—141no—
170C22⋊Q8—21no—
171#121—71no—
172#121—71no—
173#121—141no—
174#121C2×D711yescharacteristic
175#121C2×D711yescharacteristic
176C4×D4—71no—
177C4×D4—141no—
178C4×D4C2×D711yescharacteristic
179C4×D4C2×D711yescharacteristic
180#117—71no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4C2×D711yescharacteristic
183C2×C4⋊C4C2×D711yescharacteristic
184#107—71no—
185#107—71no—
186#107—71no—
187#106—71no—
188#106—71no—
189#106—71no—
190#106—71no—
191#106—71no—
192#106—71no—
193C4⋊C8—71no—
194#102—71no—
195#102—71no—
196#102—71no—
197#102—71no—
198#102—71no—
199#102—71no—
200C22⋊C8—71no—
201C22⋊C8—71no—
202#97—71no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7AGL(1,ℤ/8)12yes—
212C22×C7C22×D411yescharacteristic
213C2×D7—41no—
214C2×D7—41no—
215C2×D7—41no—
216C4×C7—21no—
217C4×C7—21no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7C2×#4111yescharacteristic
224C4×C72−1+411yescharacteristic
225Dic7—41no—
226Dic7—41no—
227Dic7—41no—
228#41—44no—
229C2×Q8—21no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—71no—
233C2×D4—141no—
234C2×D4—141no—
235C2×D4C22⋊D711yescharacteristic
236C2×D4C22×D711yescharacteristic
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—141no—
241C22×C4—141no—
242C22×C4C22×D711yescharacteristic
243C22×C4C22×D711yescharacteristic
244D8—142no—
245M16—142no—
246C2×C8—71no—
247C2×C8—71no—
248C2×C8—71no—
249C2×C8—71no—
250C2×C8—142no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—71no—
255C4⋊C4—71no—
256C4⋊C4—71no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4C22⋊D711yescharacteristic
261C4⋊C4C22×D711yescharacteristic
262C4⋊C4C22×D711yescharacteristic
263C4⋊C4C22×D711yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C22⋊C4C22⋊D711yescharacteristic
272C22⋊C4C22⋊D711yescharacteristic
273C42—141no—
274C42C22×D711yescharacteristic
275C2×C7—22no—
276C2×C7—41no—
277C2×C7—81no—
278C2×C7#54711yescharacteristic
279C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
280C2×C7#57711yescharacteristic
281D7—81no—
282C23—21no—
283C23—141no—
284C23C2×C22⋊D711yescharacteristic
285C23C2×C22⋊D711yescharacteristic
286Q8—42no—
287D4—22no—
288D4—21no—
289D4—21no—
290D4—42no—
291D4—42no—
292D4—141no—
293D4—141no—
294D4—282no—
295D4—561no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—561no—
310C2×C4—561no—
311C2×C4C2×C22⋊D711yescharacteristic
312C2×C4C2×C22⋊D711yescharacteristic
313C2×C4C2×C22⋊D711yescharacteristic
314C2×C4C2×C22⋊D711yescharacteristic
315C2×C4C23×D711yescharacteristic
316C8—72no—
317C8—141no—
318C8—141no—
319C8—141no—
320C7#334511yescharacteristic, complemented, Sylow
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—281no—
330C22—281no—
331C22—281no—
332C22#649512yes—
333C22C22×C22⋊D711yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—281no—
342C4—281no—
343C4—281no—
344C4D7×#4111yescharacteristic
345C4#655411yescharacteristic
346C2—22no—
347C2—41no—
348C2—81no—
349C2—561no—
350C2C2×#649511yescharacteristic
351C2#9051511yescharacteristic
352C2#9052811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#547C2
socleC22×C7C22×D4
hypercenter#547D7
hypocenterC7#3345
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C7×#547C2semidirect
2. #90256C2semidirect
3. #89856C2semidirect
4. #89852C2semidirect
5. #89811C2semidirect
6. #89810C2semidirect
7. #89807C2semidirect
8. #89806C2semidirect
9. #89786C2semidirect
10. #89783C2semidirect
11. #89781C2semidirect
12. #89740C2semidirect
13. #89739C2semidirect
14. #89736C2semidirect
15. C2×#6382C2semidirect
108. #547D7semidirect
159. C7⋊C8#41semidirect
320. C7#3345semidirect

Maximal subgroups

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

subgroupindex
C7×#5472
#902562
#898562
#898522
#898112
#898102
#898072
#898062
#897862
#897832
#897812
#897402
#897392
#897362
C2×#63822
#33457