Finite Group Databasebeta

← #11738556

Subgroups — #11738556

Order 896.

376 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90420C211yesmaximal, characteristic, complemented
2#90265C211yesmaximal, characteristic
3C7×#477C211yesmaximal, characteristic, complemented
4#89859C211yesmaximal, characteristic, complemented
5#89794C211yesmaximal, characteristic, complemented
6#89663C211yesmaximal, characteristic, complemented
7#89662C211yesmaximal, characteristic, complemented
8#89653C211yesmaximal, characteristic, complemented
9D7×#107C211yesmaximal, characteristic, complemented
10#89619C211yesmaximal, characteristic, complemented
11#89611C211yesmaximal, characteristic, complemented
12#89576C211yesmaximal, characteristic
13#89575C211yesmaximal, characteristic
14#89571C211yesmaximal, characteristic, complemented
15#89566C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C7×C4⋊Q8C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19#6509—21no—
20Q8×Dic7C2211yescharacteristic
21#6507—21no—
22C2×#1131C2211yescharacteristic
23C2×C8×D7C2211yescharacteristic
24#6459—21no—
25#6456—21no—
26#6456C2211yescharacteristic
27#6455C2211yescharacteristic
28D7×C4⋊C4—21no—
29D7×C4⋊C4C2211yescharacteristic
30D7×C4⋊C4C2211yescharacteristic
31#6453C2211yescharacteristic
32#6452—21no—
33#6451C2211yescharacteristic
34C4×D28—21no—
35C4×D28C2211yescharacteristic
36#6435—21no—
37#6435C2211yescharacteristic
38#6433C2211yescharacteristic
39C7×#107C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×C4⋊C8C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48#6392C2211yescharacteristic
49#6391C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3083—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62#1161—22no—
63#1161Q814yescomplemented
64C2×D28—21no—
65C2×D28C2311yescharacteristic
66C2×C4×D7—21no—
67C2×C4×D7—21no—
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7C2311yescharacteristic
72C2×Dic14—21no—
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C42×C7—21no—
81C42×C7C2311yescharacteristic
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4D411yescharacteristic
86(C2×D7)⋊C4D411yescharacteristic
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7C2311yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4D411yescharacteristic
99Dic7⋊C4D411yescharacteristic
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105#1131—22no—
106C8×D7—22no—
107#544—71no—
108#520—71no—
109#479—71no—
110#479—71no—
111#478—71no—
112#478—71no—
113#477—71no—
114#477D711yescharacteristic, complemented
115#474—71no—
116#474—71no—
117#428—71no—
118C2×#107—71no—
119#424—71no—
120#417—71no—
121#416—71no—
122C22×D7—21no—
123C22×D7C2×D411yescharacteristic
124C7×Q8—21no—
125C7×Q8—81no—
126C7×Q8C2×Q812yes—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2411yescharacteristic
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7C2×D411yescharacteristic
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140D28—24no—
141D28—21no—
142D28C2×Q812yes—
143C4×D7—24no—
144C4×D7—81no—
145C4×D7—81no—
146C4×D7—81no—
147C4×D7—81no—
148C4×D7C2×Q812yes—
149C4×D7#4112yes—
150Dic14—81no—
151C7×C8—21no—
152C7×C8—21no—
153C7⋊C8—21no—
154C7⋊C8—21no—
155C2×#41—71no—
156C2×M16—71no—
157C22×C8—71no—
158C4⋊Q8—71no—
159C4⋊Q8—141no—
160C4⋊Q8C2×D711yescharacteristic
161#125—71no—
162#125—71no—
163#125—141no—
164C22⋊Q8—141no—
165C22⋊Q8—141no—
166C4×Q8—71no—
167C4×Q8C2×D711yescharacteristic
168C4×D4—71no—
169C4×D4—71no—
170C4×D4—141no—
171C4×D4—141no—
172#117—71no—
173#117—71no—
174#117—141no—
175C2×C4⋊C4—71no—
176C2×C4⋊C4—71no—
177C2×C4⋊C4—141no—
178#107—71no—
179#107—71no—
180#107—71no—
181#107—71no—
182#107C2×D711yescharacteristic
183#107C2×D711yescharacteristic
184#106—71no—
185#106—71no—
186C4⋊C8—71no—
187C4⋊C8—71no—
188C4⋊C8—71no—
189C4⋊C8C2×D711yescharacteristic
190#103—71no—
191#103—71no—
192#103C2×D711yescharacteristic
193#103C2×D711yescharacteristic
194#102—71no—
195#102—71no—
196#102—71no—
197#102—71no—
198C22×C7C22×D411yescharacteristic
199C2×D7—21no—
200C2×D7—21no—
201C2×D7—22no—
202C2×D7—41no—
203C2×D7—41no—
204C2×D7C22⋊Q812yes—
205C4×C7—22no—
206C4×C7—21no—
207C4×C7—21no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7C22×Q811yescharacteristic
212C4×C7C2×#4111yescharacteristic
213Dic7—21no—
214Dic7—21no—
215Dic7—41no—
216Dic7—41no—
217Dic7—41no—
218Dic7C22⋊Q812yes—
219#41—74no—
220#41—142no—
221C2×Q8—21no—
222C2×Q8—141no—
223C2×Q8C22×D711yescharacteristic
224C2×D4—71no—
225C2×D4—141no—
226C22×C4—71no—
227C22×C4—141no—
228C22×C4—141no—
229C22×C4—141no—
230C22×C4—141no—
231C22×C4—141no—
232M16—142no—
233C2×C8—71no—
234C2×C8—71no—
235C2×C8—142no—
236C2×C8C22×D711yescharacteristic
237C2×C8C22×D711yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—71no—
241C4⋊C4—71no—
242C4⋊C4—71no—
243C4⋊C4—71no—
244C4⋊C4—71no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4C22×D711yescharacteristic
254C4⋊C4C22×D711yescharacteristic
255C4⋊C4C22×D711yescharacteristic
256C22⋊C4—71no—
257C22⋊C4—71no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C42—21no—
262C42—71no—
263C42—141no—
264C42—141no—
265C42C22×D711yescharacteristic
266C2×C7C2×C22⋊Q811yescharacteristic
267C2×C7#57611yescharacteristic
268C2×C7#57811yescharacteristic
269D7—22no—
270D7—42no—
271C23—71no—
272C23—141no—
273Q8—21no—
274Q8—81no—
275Q8—561no—
276Q8Q8×D712yes—
277D4—72no—
278D4—144no—
279D4—141no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—71no—
285C2×C4—72no—
286C2×C4—72no—
287C2×C4—71no—
288C2×C4—71no—
289C2×C4—144no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—561no—
294C2×C4—561no—
295C2×C4—561no—
296C2×C4—561no—
297C2×C4D4×D711yescharacteristic
298C2×C4D4×D711yescharacteristic
299C2×C4C23×D711yescharacteristic
300C8—21no—
301C8—21no—
302C8—141no—
303C8—141no—
304C7#308311yescharacteristic, complemented, Sylow
305C22—72no—
306C22—141no—
307C22—141no—
308C22—142no—
309C22—281no—
310C22—281no—
311C22C2×D4×D711yescharacteristic
312C4—21no—
313C4—21no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—72no—
319C4—141no—
320C4—141no—
321C4—281no—
322C4—281no—
323C4—281no—
324C4C2×Q8×D711yescharacteristic
325C4D7×#4111yescharacteristic
326C2—142no—
327C2—282no—
328C2D7×C22⋊Q811yescharacteristic
329C2#9045711yescharacteristic
330C2#9046511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#477C2
socleC22×C7C22×D4
prime radicalC2×#1161C22
hypercenter#477D7
hypocenterC7#3083
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #90420C2semidirect
3. C7×#477C2semidirect
4. #89859C2semidirect
5. #89794C2semidirect
6. #89663C2semidirect
7. #89662C2semidirect
8. #89653C2semidirect
9. D7×#107C2semidirect
10. #89619C2semidirect
11. #89611C2semidirect
14. #89571C2semidirect
15. #89566C2semidirect
63. #1161Q8semidirect
114. #477D7semidirect
304. C7#3083semidirect

Maximal subgroups

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

subgroupindex
#904202
#902652
C7×#4772
#898592
#897942
#896632
#896622
#896532
D7×#1072
#896192
#896112
#895762
#895752
#895712
#895662
#30837