Finite Group Databasebeta

← #11737549

Subgroups — #11737549

Order 896.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90515C211yesmaximal, characteristic, complemented
2#90307C211yesmaximal, characteristic, complemented
3C7×#449C211yesmaximal, characteristic, complemented
4#90005C211yesmaximal, characteristic, complemented
5#89943C211yesmaximal, characteristic, complemented
6#89934C211yesmaximal, characteristic, complemented
7#89930C211yesmaximal, characteristic, complemented
8D8×Dic7C211yesmaximal, characteristic, complemented
9#89813C211yesmaximal, characteristic, complemented
10#89586C211yesmaximal, characteristic, complemented
11#89564C211yesmaximal, characteristic, complemented
12#89544C211yesmaximal, characteristic, complemented
13#89533C211yesmaximal, characteristic, complemented
14(C22⋊D7)⋊C8C211yesmaximal, characteristic, complemented
15#89495C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×#1159C2211yescharacteristic
18C2×C7×SD16C2211yescharacteristic
19C2×C7×D8C2211yescharacteristic
20C7×#121C2211yescharacteristic
21#6509C2211yescharacteristic
22#6507C2211yescharacteristic
23#6502—21no—
24#6502—21no—
25#6501C2211yescharacteristic
26#6499—21no—
27#6498—21no—
28D4×Dic7C2211yescharacteristic
29D4×Dic7C2211yescharacteristic
30C2×#1142C2211yescharacteristic
31#6492—21no—
32C4×C22⋊D7C2211yescharacteristic
33#6489C2211yescharacteristic
34#6487C2211yescharacteristic
35C2×Dic7⋊C4—21no—
36C22×C7⋊C8C2211yescharacteristic
37C2×C8×D7C2211yescharacteristic
38#6455C2211yescharacteristic
39#6451C2211yescharacteristic
40#6448—21no—
41#6440—21no—
42C7×#103C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6406C2211yescharacteristic
48#6406C2211yescharacteristic
49#6406C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6392C2211yescharacteristic
52#6391C2211yescharacteristic
53#6390C2211yescharacteristic
54#6390C2211yescharacteristic
55Dic7⋊C8C2211yescharacteristic
56C8×Dic7C2211yescharacteristic
57#6384C2211yescharacteristic
58#6382C2211yescharacteristic
59#3303—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—41no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4—21no—
64C2×C7×D4—21no—
65C2×C7×D4C2311yescharacteristic
66C2×C7×D4C2311yescharacteristic
67C22×C4×C7—21no—
68C22×C4×C7C2311yescharacteristic
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71C2×C22⋊D7D411yescharacteristic
72C22×Dic7—21no—
73C22×Dic7—21no—
74C22×Dic7—21no—
75#1159—22no—
76#1159—41no—
77C2×C4×D7C2311yescharacteristic
78C2×Dic14—21no—
79C2×Dic14C2311yescharacteristic
80C7×SD16—41no—
81C7×D8—22no—
82C2×C7×C8C2311yescharacteristic
83C2×C7×C8C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C22⋊C4—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7—21no—
89C22⋊Dic7—21no—
90C22⋊Dic7—21no—
91C22⋊Dic7D411yescharacteristic
92#1142—41no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4D411yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4D411yescharacteristic
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7C2311yescharacteristic
108C2×C7⋊C8—41no—
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C8×D7—41no—
112#555—71no—
113#547—71no—
114#487—71no—
115#485—71no—
116#485—71no—
117#476—71no—
118#465—71no—
119#465—71no—
120#461—71no—
121#449—71no—
122#449D711yescharacteristic, complemented
123C4×D8—71no—
124C8×D4—71no—
125#416—71no—
126#416—71no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7C2×D411yescharacteristic
130C22×D7C2×D411yescharacteristic
131C7×Q8—21no—
132C7×Q8—21no—
133C7×D4—21no—
134C7×D4—21no—
135C7×D4—21no—
136C7×D4—42no—
137C7×D4—81no—
138C7×D4#4112yes—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—42no—
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C22⋊D7—22no—
147C22⋊D7—42no—
148C22⋊D7—81no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—42no—
154C2×Dic7—42no—
155C2×Dic7—42no—
156C2×Dic7C2×D411yescharacteristic
157C2×Dic7C2×D411yescharacteristic
158C2×Dic7C2×D411yescharacteristic
159C4×D7—21no—
160C4×D7—21no—
161Dic14—21no—
162Dic14—21no—
163Dic14—81no—
164C7×C8—21no—
165C7×C8—21no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C2×#41—71no—
169C2×#41C2×D711yescharacteristic
170C2×SD16—71no—
171C2×SD16C2×D711yescharacteristic
172C2×D8C2×D711yescharacteristic
173C22×C8—71no—
174C22×C8—71no—
175C4⋊Q8—71no—
176C4⋊Q8—71no—
177#124—141no—
178#124—141no—
179#123—141no—
180#123—141no—
181C22⋊Q8—71no—
182C22⋊Q8—71no—
183#121—71no—
184#121—141no—
185#121—141no—
186#121C2×D711yescharacteristic
187C4×D4—71no—
188C4×D4—71no—
189C4×D4—71no—
190#117—71no—
191#117—71no—
192#117—141no—
193C2×C4⋊C4—141no—
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#103—71no—
203#103C2×D711yescharacteristic
204#102—71no—
205#102—71no—
206#102—71no—
207#102C2×D711yescharacteristic
208C22⋊C8—71no—
209C22⋊C8C2×D711yescharacteristic
210C4×C8—71no—
211C22×C7—22no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7C22×D411yescharacteristic
221C2×D7—21no—
222C2×D7—21no—
223C2×D7—21no—
224C4×C7—21no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7C2×#4111yescharacteristic
228C4×C7H(5,𝔽2)11yescharacteristic
229Dic7—21no—
230Dic7—21no—
231Dic7—41no—
232Dic7—41no—
233Dic7—41no—
234Dic7—41no—
235Dic7#13512yes—
236#41—22no—
237#41—41no—
238#41—142no—
239#41—281no—
240C2×Q8—71no—
241C2×Q8—141no—
242C2×Q8C22×D711yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—71no—
246C2×D4—141no—
247C2×D4—141no—
248C2×D4C22×D711yescharacteristic
249C2×D4C22×D711yescharacteristic
250C22×C4—21no—
251C22×C4—71no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4—141no—
255C22×C4C22×D711yescharacteristic
256SD16—41no—
257SD16—281no—
258D8—22no—
259C2×C8—71no—
260C2×C8—71no—
261C2×C8—281no—
262C2×C8—281no—
263C2×C8C22×D711yescharacteristic
264C2×C8C22×D711yescharacteristic
265C4⋊C4—71no—
266C4⋊C4—71no—
267C4⋊C4—71no—
268C4⋊C4—71no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4—141no—
274C4⋊C4C22×D711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—71no—
277C22⋊C4—71no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C42—71no—
286C42—141no—
287C42—141no—
288C2×C7—41no—
289C2×C7—42no—
290C2×C7—81no—
291C2×C7#54611yescharacteristic
292C2×C7C2×#13511yescharacteristic
293C2×C7#57711yescharacteristic
294D7—41no—
295C23—21no—
296C23—21no—
297C23—71no—
298C23D4×D711yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—141no—
302Q8—141no—
303Q8—561no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—81no—
309D4—142no—
310D4—282no—
311D4—561no—
312D4#115912yes—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—71no—
319C2×C4—71no—
320C2×C4—71no—
321C2×C4—141no—
322C2×C4—141no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—282no—
328C2×C4—282no—
329C2×C4—282no—
330C2×C4D4×D711yescharacteristic
331C2×C4C23×D711yescharacteristic
332C8—21no—
333C8—21no—
334C8—141no—
335C8—141no—
336C7#330311yescharacteristic, complemented, Sylow
337C22—21no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—141no—
347C22—141no—
348C22—141no—
349C22C2×D4×D711yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—72no—
354C4—141no—
355C4—141no—
356C4—281no—
357C4—281no—
358C4—281no—
359C4—281no—
360C4C2×#115911yescharacteristic
361C4#654811yescharacteristic
362C2—41no—
363C2—42no—
364C2—81no—
365C2—281no—
366C2#9028211yescharacteristic
367C2D7×#13511yescharacteristic
368C2#9046611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#449C2
socleC22×C7C22×D4
hypercenter#449D7
hypocenterC7#3303
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #90515C2semidirect
2. #90307C2semidirect
3. C7×#449C2semidirect
4. #90005C2semidirect
5. #89943C2semidirect
6. #89934C2semidirect
7. #89930C2semidirect
8. D8×Dic7C2semidirect
9. #89813C2semidirect
10. #89586C2semidirect
11. #89564C2semidirect
12. #89544C2semidirect
13. #89533C2semidirect
14. (C22⋊D7)⋊C8C2semidirect
15. #89495C2semidirect
122. #449D7semidirect
336. C7#3303semidirect

Maximal subgroups

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

subgroupindex
#905152
#903072
C7×#4492
#900052
#899432
#899342
#899302
D8×Dic72
#898132
#895862
#895642
#895442
#895332
(C22⋊D7)⋊C82
#894952
#33037