Finite Group Databasebeta

← #11737293

Subgroups — #11737293

Order 896.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90269C211yesmaximal, characteristic, complemented
2C4×#1161C211yesmaximal, characteristic
3C7×#441C211yesmaximal, characteristic, complemented
4#89959C211yesmaximal, characteristic
5#89954C211yesmaximal, characteristic, complemented
6#89802C211yesmaximal, characteristic, complemented
7#89798C211yesmaximal, characteristic, complemented
8#89642C211yesmaximal, characteristic, complemented
9#89639C211yesmaximal, characteristic, complemented
10#89590C211yesmaximal, characteristic, complemented
11#89586C211yesmaximal, characteristic, complemented
12#89582C211yesmaximal, characteristic
13#89566C211yesmaximal, characteristic, complemented
14#89482C211yesmaximal, characteristic, complemented
15#89477C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×C7×Q16C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19C4×C7×Q8C2211yescharacteristic
20#6509C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22#6507C2211yescharacteristic
23C2×#1144C2211yescharacteristic
24C2×#1143C2211yescharacteristic
25C2×#1132C2211yescharacteristic
26#6460C2211yescharacteristic
27#6459—21no—
28#6456—21no—
29#6456C2211yescharacteristic
30#6455—21no—
31#6455C2211yescharacteristic
32#6452—21no—
33#6451C2211yescharacteristic
34C4×D28—21no—
35C4×D28C2211yescharacteristic
36#6435—21no—
37C42×D7—21no—
38C42×D7C2211yescharacteristic
39C4×Dic14—21no—
40C4×Dic14C2211yescharacteristic
41C7×#106C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#97C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6395C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50#6392C2211yescharacteristic
51#6390C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54#6384C2211yescharacteristic
55#6383C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3116—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62#1161—22no—
63#1161—22no—
64C2×D28—21no—
65C2×D28C2311yescharacteristic
66C2×C4×D7—21no—
67C2×C4×D7—21no—
68C2×C4×D7—21no—
69C2×C4×D7D411yescharacteristic
70C2×C4×D7D411yescharacteristic
71C2×C4×D7C2311yescharacteristic
72C2×Dic14—21no—
73C2×Dic14C2311yescharacteristic
74C7×Q16—22no—
75C2×C7×C8C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C7×C4⋊C4—21no—
78C7×C4⋊C4—21no—
79C7×C4⋊C4C2311yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C42×C7—21no—
82C42×C7—21no—
83C42×C7C2311yescharacteristic
84#1144—22no—
85#1143—22no—
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⋊Dic7—21no—
92C4⋊Dic7—21no—
93C4⋊Dic7C2311yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7D411yescharacteristic
104C4×Dic7D411yescharacteristic
105C4×Dic7C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108#1132—22no—
109#533—71no—
110C4×#41—71no—
111#485—71no—
112#485—71no—
113#477—71no—
114#476—71no—
115#462—71no—
116#461—71no—
117#452—71no—
118#452—71no—
119#441—71no—
120#441D711yescharacteristic, complemented
121#440—71no—
122#440—71no—
123#432—71no—
124C22×D7—21no—
125C22×D7C2×D411yescharacteristic
126C7×Q8—21no—
127C7×Q8—21no—
128C7×Q8—21no—
129C7×Q8#4112yes—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2411yescharacteristic
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143D28—24no—
144D28—21no—
145D28#4112yes—
146C4×D7—24no—
147C4×D7—21no—
148C4×D7—21no—
149C4×D7—22no—
150C4×D7—41no—
151C4×D7—42no—
152C4×D7—42no—
153Dic14—21no—
154Dic14—21no—
155Dic14—42no—
156C7×C8—21no—
157C7×C8—21no—
158C7⋊C8—21no—
159C7⋊C8—21no—
160C2×#41—71no—
161C2×Q16—71no—
162C2×Q16C2×D711yescharacteristic
163C2×SD16—71no—
164C2×SD16—71no—
165C4⋊Q8—71no—
166C4⋊Q8—71no—
167#125—141no—
168C22⋊Q8—71no—
169C22⋊Q8—71no—
170C22⋊Q8—141no—
171C4×Q8—71no—
172C4×Q8—71no—
173C4×Q8—141no—
174C4×Q8C2×D711yescharacteristic
175C4×Q8C2×D711yescharacteristic
176C4×D4—71no—
177C4×D4—71no—
178C4×D4—141no—
179C4×D4—141no—
180#117—71no—
181#117—141no—
182#117—141no—
183C2×C42—71no—
184C2×C42—141no—
185#107—71no—
186#107—71no—
187#106—71no—
188#106C2×D711yescharacteristic
189C4⋊C8—71no—
190C4⋊C8—71no—
191#103—71no—
192#103—71no—
193#103—71no—
194#103—71no—
195#103C2×D711yescharacteristic
196#103C2×D711yescharacteristic
197#102—71no—
198#102—71no—
199C22⋊C8—71no—
200C22⋊C8—71no—
201#97—71no—
202#97C2×D711yescharacteristic
203C22×C7C22×D411yescharacteristic
204C2×D7—21no—
205C2×D7—22no—
206C2×D7—21no—
207C2×D7—21no—
208C2×D7—41no—
209C2×D7—41no—
210C4×C7—22no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7C2×#4111yescharacteristic
217C4×C7C2×#4111yescharacteristic
218Dic7—21no—
219Dic7—21no—
220Dic7—41no—
221Dic7—41no—
222Dic7—41no—
223Dic7#13712yes—
224#41—142no—
225#41—142no—
226C2×Q8—71no—
227C2×Q8—141no—
228C2×Q8C22×D711yescharacteristic
229C2×Q8C22×D711yescharacteristic
230C2×D4—71no—
231C2×D4—141no—
232C22×C4—71no—
233C22×C4—71no—
234C22×C4—71no—
235C22×C4—141no—
236C22×C4—141no—
237C22×C4—141no—
238Q16—22no—
239Q16—142no—
240SD16—142no—
241SD16—142no—
242C2×C8—71no—
243C2×C8—71no—
244C2×C8C22×D711yescharacteristic
245C2×C8C22×D711yescharacteristic
246C4⋊C4—21no—
247C4⋊C4—21no—
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⋊C4—141no—
257C4⋊C4C22×D711yescharacteristic
258C4⋊C4C22×D711yescharacteristic
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C42—21no—
265C42—21no—
266C42—71no—
267C42—71no—
268C42—71no—
269C42—141no—
270C42—141no—
271C42—141no—
272C42C22×D711yescharacteristic
273C2×C7#52511yescharacteristic
274C2×C7C2×#13711yescharacteristic
275C2×C7#57511yescharacteristic
276D7—41no—
277D7—42no—
278C23—71no—
279C23—141no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—141no—
284Q8—141no—
285Q8—282no—
286Q8#115712yes—
287D4—72no—
288D4—144no—
289D4—141no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—71no—
295C2×C4—71no—
296C2×C4—71no—
297C2×C4—142no—
298C2×C4—144no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—281no—
305C2×C4—282no—
306C2×C4—282no—
307C2×C4D4×D711yescharacteristic
308C2×C4D4×D711yescharacteristic
309C2×C4C23×D711yescharacteristic
310C8—21no—
311C8—21no—
312C8—141no—
313C8—141no—
314C7#311611yescharacteristic, complemented, Sylow
315C22—141no—
316C22—141no—
317C22—142no—
318C22—141no—
319C22—281no—
320C22—281no—
321C22C2×D4×D711yescharacteristic
322C4—21no—
323C4—21no—
324C4—22no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—72no—
329C4—141no—
330C4—141no—
331C4—281no—
332C4—281no—
333C4—281no—
334C4C2×#115711yescharacteristic
335C4D7×#4111yescharacteristic
336C2—281no—
337C2—282no—
338C2#9023811yescharacteristic
339C2D7×#13711yescharacteristic
340C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#441C2
socleC22×C7C22×D4
prime radicalC2×#1161C22
hypercenter#441D7
hypocenterC7#3116
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. #90269C2semidirect
3. C7×#441C2semidirect
5. #89954C2semidirect
6. #89802C2semidirect
7. #89798C2semidirect
8. #89642C2semidirect
9. #89639C2semidirect
10. #89590C2semidirect
11. #89586C2semidirect
13. #89566C2semidirect
14. #89482C2semidirect
15. #89477C2semidirect
120. #441D7semidirect
314. C7#3116semidirect

Maximal subgroups

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

subgroupindex
#902692
C4×#11612
C7×#4412
#899592
#899542
#898022
#897982
#896422
#896392
#895902
#895862
#895822
#895662
#894822
#894772
#31167