Finite Group Databasebeta

← #11736744

Subgroups — #11736744

Order 896.

396 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90170C211yesmaximal, characteristic, complemented
2C4×#1157C211yesmaximal, characteristic, complemented
3C7×#431C211yesmaximal, characteristic, complemented
4#89619C211yesmaximal, characteristic, complemented
5#89617C211yesmaximal, characteristic, complemented
6#89616C211yesmaximal, characteristic, complemented
7#89601C211yesmaximal, characteristic, complemented
8#89509C211yesmaximal, characteristic, complemented
9#89506C211yesmaximal, characteristic
10#89505C211yesmaximal, characteristic, complemented
11#89495C211yesmaximal, characteristic, complemented
12C4×Dic28C211yesmaximal, characteristic, complemented
13C4×D56C211yesmaximal, characteristic, complemented
14C4×#1132C211yesmaximal, characteristic, complemented
15C4×#1132C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C2×C42×C7C2211yescharacteristic
18#6493C2211yescharacteristic
19#6492—21no—
20C4×C22⋊D7—21no—
21C4×C22⋊D7—21no—
22C4×C22⋊D7—21no—
23#6489C2211yescharacteristic
24#6487C2211yescharacteristic
25C2×Dic28C2211yescharacteristic
26C2×D56C2211yescharacteristic
27C2×#1132C2211yescharacteristic
28C2×#1132C2211yescharacteristic
29#6439—21no—
30#6438C2211yescharacteristic
31C4×D28C2211yescharacteristic
32C4×D28C2211yescharacteristic
33C4×D28C2211yescharacteristic
34#6435—21no—
35#6435—21no—
36C42×D7—21no—
37#6433C2211yescharacteristic
38C4×Dic14C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C4×Dic14C2211yescharacteristic
41C7×C4⋊C8C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×C22⋊C8C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45C4×C7×C8C2211yescharacteristic
46C4×C7×C8C2211yescharacteristic
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic
51#6392C2211yescharacteristic
52#6392C2211yescharacteristic
53#6391C2211yescharacteristic
54#6391C2211yescharacteristic
55#6390C2211yescharacteristic
56#6390C2211yescharacteristic
57#6390C2211yescharacteristic
58#6390C2211yescharacteristic
59#3102—71nomaximal, Sylow
60C22×C4×C7D411yescharacteristic
61C22×C4×C7D411yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65#1157—22no—
66#1157—22no—
67C2×D28C2311yescharacteristic
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×Dic14C2311yescharacteristic
74C2×Dic14C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C42×C7D411yescharacteristic
80C42×C7D411yescharacteristic
81C42×C7C2311yescharacteristic
82C42×C7C2311yescharacteristic
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105Dic28—22no—
106D56—22no—
107#1132—22no—
108#1132—22no—
109#529—71no—
110C4×#41—71no—
111#485—71no—
112#482—71no—
113#479—71no—
114#478—71no—
115#464—71no—
116#463—71no—
117#452—71no—
118#449—71no—
119C4×Q16—71no—
120C4×SD16—71no—
121C4×SD16—71no—
122C4×D8—71no—
123#431D711yescharacteristic, complemented
124C23×C7C2×D411yescharacteristic
125C22×D7—21no—
126C22×D7—21no—
127C2×C4×C7—22no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—41no—
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2411yescharacteristic
137C22⋊D7—24no—
138C22⋊D7—42no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145D28—21no—
146D28—21no—
147D28—21no—
148D28#4112yes—
149C4×D7—24no—
150C4×D7—42no—
151C4×D7—42no—
152C4×D7—42no—
153Dic14—21no—
154Dic14—21no—
155Dic14—21no—
156Dic14#4112yes—
157C7×C8—21no—
158C7×C8—21no—
159C7×C8—21no—
160C7×C8—21no—
161C2×#41—71no—
162C2×Q16—71no—
163C2×SD16—71no—
164C2×SD16—71no—
165C2×D8—71no—
166#126—141no—
167#125—71no—
168#124—71no—
169#123—141no—
170C22⋊Q8—71no—
171#121—71no—
172C4×Q8—71no—
173C4×Q8—71no—
174C4×Q8—71no—
175C4×D4—71no—
176C4×D4—71no—
177C4×D4—71no—
178C4×D4—141no—
179C4×D4—141no—
180C4×D4—141no—
181#117—71no—
182#117—141no—
183#117—141no—
184C2×C42—141no—
185C2×C42C2×D711yescharacteristic
186#107—71no—
187#107—71no—
188#106—71no—
189#106—71no—
190C4⋊C8C2×D711yescharacteristic
191C4⋊C8C2×D711yescharacteristic
192#103—71no—
193#103—71no—
194#103—71no—
195#103—71no—
196#102—71no—
197#102—71no—
198#102—71no—
199#102—71no—
200C22⋊C8C2×D711yescharacteristic
201C22⋊C8C2×D711yescharacteristic
202C4×C8C2×D711yescharacteristic
203C4×C8C2×D711yescharacteristic
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7C22×D411yescharacteristic
208C2×D7—22no—
209C2×D7—41no—
210C2×D7—41no—
211C2×D7—41no—
212C2×D7—41no—
213C2×D7—41no—
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—21no—
217C4×C7—21no—
218C4×C7#13512yes—
219C4×C7C2×#4111yescharacteristic
220C4×C7C2×#4111yescharacteristic
221Dic7—22no—
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225Dic7—41no—
226Dic7—41no—
227#41—142no—
228#41—142no—
229C2×Q8—71no—
230C2×Q8—71no—
231C2×D4—71no—
232C2×D4—71no—
233C2×D4—141no—
234C2×D4—141no—
235C22×C4—141no—
236C22×C4—141no—
237C22×C4—141no—
238C22×C4—141no—
239C22×C4D2811yescharacteristic
240C22×C4D2811yescharacteristic
241C22×C4C22×D711yescharacteristic
242Q16—142no—
243SD16—142no—
244SD16—142no—
245D8—142no—
246C2×C8C22×D711yescharacteristic
247C2×C8C22×D711yescharacteristic
248C2×C8C22×D711yescharacteristic
249C2×C8C22×D711yescharacteristic
250C4⋊C4—71no—
251C4⋊C4—71no—
252C4⋊C4—71no—
253C4⋊C4—71no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C42—141no—
269C42—141no—
270C42—141no—
271C42—141no—
272C42D2811yescharacteristic
273C42D2811yescharacteristic
274C42C22×D711yescharacteristic
275C42C22×D711yescharacteristic
276C2×C7—41no—
277C2×C7#52511yescharacteristic
278C2×C7C2×#13511yescharacteristic
279C2×C7#57511yescharacteristic
280D7—42no—
281D7—81no—
282C23—141no—
283C23—141no—
284C23C2×D2811yescharacteristic
285Q8—72no—
286Q8—141no—
287Q8—141no—
288Q8—141no—
289D4—72no—
290D4—144no—
291D4—141no—
292D4—141no—
293D4—141no—
294D4—282no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—144no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—282no—
307C2×C4—282no—
308C2×C4—282no—
309C2×C4C2×D2811yescharacteristic
310C2×C4C2×D2811yescharacteristic
311C2×C4C2×D2811yescharacteristic
312C2×C4C2×D2811yescharacteristic
313C2×C4C2×D2811yescharacteristic
314C2×C4C23×D711yescharacteristic
315C8—21no—
316C8—21no—
317C8—21no—
318C8—21no—
319C7#310211yescharacteristic, complemented, Sylow
320C22—21no—
321C22—21no—
322C22—21no—
323C22—142no—
324C22—281no—
325C22—281no—
326C22—281no—
327C22—281no—
328C22—281no—
329C22C22×D2811yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—142no—
335C4—281no—
336C4—281no—
337C4—281no—
338C4—281no—
339C4—281no—
340C4#646712yes—
341C4D7×#4111yescharacteristic
342C4D7×#4111yescharacteristic
343C2—41no—
344C2—282no—
345C2—561no—
346C2#9021511yescharacteristic
347C2C2×#646711yescharacteristic
348C2#9043911yescharacteristic
subgroupG/H
centerC2×C4C2×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#431C2
socleC22×C7C22×D4
prime radical#89506C2
hypercenter#431D7
hypocenterC7#3102
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #90170C2semidirect
2. C4×#1157C2semidirect
3. C7×#431C2semidirect
4. #89619C2semidirect
5. #89617C2semidirect
6. #89616C2semidirect
7. #89601C2semidirect
8. #89509C2semidirect
10. #89505C2semidirect
11. #89495C2semidirect
12. C4×Dic28C2semidirect
13. C4×D56C2semidirect
14. C4×#1132C2semidirect
15. C4×#1132C2semidirect
123. #431D7semidirect
319. C7#3102semidirect

Maximal subgroups

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

subgroupindex
#901702
C4×#11572
C7×#4312
#896192
#896172
#896162
#896012
#895092
#895062
#895052
#894952
C4×Dic282
C4×D562
C4×#11322
#31027