Finite Group Databasebeta

← #11740982

Subgroups — #11740982

Order 896.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#548C211yesmaximal, characteristic
2#90233C211yesmaximal, characteristic, complemented
3#90011C211yesmaximal, characteristic, complemented
4#90005C211yesmaximal, characteristic, complemented
5#90000C211yesmaximal, characteristic, complemented
6#89997C211yesmaximal, characteristic, complemented
7#89827C211yesmaximal, characteristic, complemented
8#89826C211yesmaximal, characteristic, complemented
9#89823C211yesmaximal, characteristic, complemented
10#89817C211yesmaximal, characteristic, complemented
11#89814C211yesmaximal, characteristic, complemented
12#89807C211yesmaximal, characteristic, complemented
13#89802C211yesmaximal, characteristic, complemented
14#89793C211yesmaximal, characteristic, complemented
15#89783C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic
18C7×#124—21no—
19C7×#124C2211yescharacteristic
20C7×C22⋊Q8—21no—
21C7×C22⋊Q8C2211yescharacteristic
22C7×#121—21no—
23C7×#121C2211yescharacteristic
24C4×C7×Q8C2211yescharacteristic
25C4×C7×D4—21no—
26C4×C7×D4C2211yescharacteristic
27C2×#1144C2211yescharacteristic
28C2×#1144C2211yescharacteristic
29#6499—21no—
30C2×#1142C2211yescharacteristic
31#6489C2211yescharacteristic
32#6489C2211yescharacteristic
33#6487C2211yescharacteristic
34#6487C2211yescharacteristic
35C2×#1136C2211yescharacteristic
36C22×C7⋊C8C2211yescharacteristic
37#6450—21no—
38#6442—21no—
39#6442—21no—
40#6432C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42#6409C2211yescharacteristic
43#6409C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6406C2211yescharacteristic
49#6406C2211yescharacteristic
50#6406C2211yescharacteristic
51C22⋊(C7⋊C8)C2211yescharacteristic
52C22⋊(C7⋊C8)C2211yescharacteristic
53#6384C2211yescharacteristic
54#6382C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3313—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—41no—
62C2×C7×Q8—21no—
63C2×C7×Q8—22no—
64C2×C7×Q8—41no—
65C2×C7×Q8C2311yescharacteristic
66C2×C7×Q8C2311yescharacteristic
67C2×C7×D4—21no—
68C2×C7×D4—21no—
69C2×C7×D4D411yescharacteristic
70C2×C7×D4C2311yescharacteristic
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7C2311yescharacteristic
74C22×C4×C7C2311yescharacteristic
75C2×Dic14—21no—
76C2×Dic14C2311yescharacteristic
77C7×C4⋊C4—21no—
78C7×C4⋊C4—21no—
79C7×C4⋊C4D411yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C22⋊C4—21no—
82C7×C22⋊C4—21no—
83C7×C22⋊C4—21no—
84C7×C22⋊C4—21no—
85C7×C22⋊C4D411yescharacteristic
86C7×C22⋊C4D411yescharacteristic
87C42×C7—21no—
88C42×C7C2311yescharacteristic
89C22⋊Dic7—21no—
90C22⋊Dic7—21no—
91C22⋊Dic7—21no—
92C22⋊Dic7—21no—
93#1144—22no—
94#1144—41no—
95#1142—41no—
96C4⋊Dic7—21no—
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7—21no—
106#1136—41no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112#556—71no—
113#548D711yescharacteristic
114#489—71no—
115#485—71no—
116#482—71no—
117#477—71no—
118#468—71no—
119#467—71no—
120#464—71no—
121#452—71no—
122#451—71no—
123#441—71no—
124#435—71no—
125#419—71no—
126#416—71no—
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C23×C7C2×D411yescharacteristic
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—42no—
134C7×Q8#4112yes—
135C7×D4—21no—
136C7×D4—21no—
137C7×D4—41no—
138C7×D4—42no—
139C7×D4—42no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7C2×D411yescharacteristic
150C2×C4×C7C2×D411yescharacteristic
151C2×C4×C7C2×D411yescharacteristic
152C2×C4×C7C2×D411yescharacteristic
153C2×C4×C7C2411yescharacteristic
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158Dic14—21no—
159Dic14—21no—
160Dic14—42no—
161C7⋊C8—21no—
162C7⋊C8—21no—
163C7⋊C8—21no—
164C7⋊C8—21no—
165C2×#41C2×D711yescharacteristic
166C22×Q8C2×D711yescharacteristic
167C2×Q16—71no—
168C2×Q16—71no—
169C2×SD16—71no—
170C2×M16—71no—
171C22×C8—71no—
172C4⋊Q8—71no—
173#126—141no—
174#126—141no—
175#124—21no—
176#124—141no—
177#124C2×D711yescharacteristic
178C22⋊Q8—21no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8C2×D711yescharacteristic
182#121—21no—
183#121C2×D711yescharacteristic
184C4×Q8—71no—
185C4×Q8—141no—
186C4×Q8C2×D711yescharacteristic
187C4×D4—21no—
188C4×D4C2×D711yescharacteristic
189#117—71no—
190#117—71no—
191#107—71no—
192#107—71no—
193#106—71no—
194C4⋊C8—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#102—71no—
203#102—71no—
204#102—71no—
205C22⋊C8—71no—
206C22⋊C8—71no—
207#97—71no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7C22×D411yescharacteristic
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—22no—
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7C2×#4111yescharacteristic
226C4×C7H(5,𝔽2)11yescharacteristic
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231#41—22no—
232#41—41no—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—41no—
236C2×Q8—71no—
237C2×Q8—141no—
238C2×Q8C22×D711yescharacteristic
239C2×Q8C22×D711yescharacteristic
240C2×D4—21no—
241C2×D4—21no—
242C2×D4C22⋊D711yescharacteristic
243C2×D4C22×D711yescharacteristic
244C22×C4—21no—
245C22×C4—21no—
246C22×C4C22×D711yescharacteristic
247C22×C4C22×D711yescharacteristic
248Q16—142no—
249Q16—281no—
250SD16—281no—
251M16—281no—
252C2×C8—71no—
253C2×C8—71no—
254C2×C8—71no—
255C2×C8—71no—
256C2×C8—281no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—71no—
260C4⋊C4—71no—
261C4⋊C4—71no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4C22⋊D711yescharacteristic
267C4⋊C4C22×D711yescharacteristic
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4C22⋊D711yescharacteristic
277C22⋊C4C22⋊D711yescharacteristic
278C42—21no—
279C42—141no—
280C42—141no—
281C42—141no—
282C42C22×D711yescharacteristic
283C2×C7—41no—
284C2×C7—41no—
285C2×C7—81no—
286C2×C7#54611yescharacteristic
287C2×C7#57511yescharacteristic
288C2×C7#57811yescharacteristic
289C23—21no—
290C23C2×C22⋊D711yescharacteristic
291C23C2×C22⋊D711yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—141no—
297Q8—141no—
298Q8—282no—
299Q8#115912yes—
300D4—21no—
301D4—21no—
302D4—41no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4C2×C22⋊D711yescharacteristic
319C2×C4C2×C22⋊D711yescharacteristic
320C2×C4C2×C22⋊D711yescharacteristic
321C2×C4C2×C22⋊D711yescharacteristic
322C2×C4C23×D711yescharacteristic
323C8—141no—
324C8—141no—
325C8—141no—
326C8—141no—
327C7#331311yescharacteristic, complemented, Sylow
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22C22×C22⋊D711yescharacteristic
338C4—21no—
339C4—21no—
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—281no—
346C4—281no—
347C4—281no—
348C4—281no—
349C4C2×#115911yescharacteristic
350C4#654811yescharacteristic
351C2—41no—
352C2—41no—
353C2—81no—
354C2#9049711yescharacteristic
355C2#9051311yescharacteristic
356C2#9052911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
Fitting, prime radicalC7×#548C2
socleC22×C7C22×D4
hypercenter#548D7
hypocenterC7#3313
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
2. #90233C2semidirect
3. #90011C2semidirect
4. #90005C2semidirect
5. #90000C2semidirect
6. #89997C2semidirect
7. #89827C2semidirect
8. #89826C2semidirect
9. #89823C2semidirect
10. #89817C2semidirect
11. #89814C2semidirect
12. #89807C2semidirect
13. #89802C2semidirect
14. #89793C2semidirect
15. #89783C2semidirect
327. C7#3313semidirect

Maximal subgroups

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

subgroupindex
C7×#5482
#902332
#900112
#900052
#900002
#899972
#898272
#898262
#898232
#898172
#898142
#898072
#898022
#897932
#897832
#33137