Finite Group Databasebeta

← #11738413

Subgroups — #11738413

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#90237C211yesmaximal, characteristic
3C7×#474C211yesmaximal, characteristic, complemented
4#89863C211yesmaximal, characteristic, complemented
5#89781C211yesmaximal, characteristic, complemented
6#89653C211yesmaximal, characteristic, complemented
7D7×#107C211yesmaximal, characteristic, complemented
8#89649C211yesmaximal, characteristic, complemented
9#89647C211yesmaximal, characteristic, complemented
10#89611C211yesmaximal, characteristic, complemented
11#89601C211yesmaximal, characteristic, complemented
12#89544C211yesmaximal, characteristic
13#89543C211yesmaximal, characteristic
14#89535C211yesmaximal, characteristic, complemented
15#89531C211yesmaximal, characteristic, complemented
16C2×#1159C2211yescharacteristic
17C7×C4⋊Q8C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19#6509—21no—
20#6507—21no—
21D4×Dic7C2211yescharacteristic
22C4×C22⋊D7—21no—
23C2×Dic7⋊C4—21no—
24C2×#1131C2211yescharacteristic
25C2×C8×D7C2211yescharacteristic
26#6459—21no—
27#6455C2211yescharacteristic
28D7×C4⋊C4C2211yescharacteristic
29D7×C4⋊C4C2211yescharacteristic
30#6453C2211yescharacteristic
31#6452—21no—
32#6451C2211yescharacteristic
33#6450C2211yescharacteristic
34(C22⋊D7)⋊C4—21no—
35#6440—21no—
36#6435C2211yescharacteristic
37#6433C2211yescharacteristic
38C4×Dic14C2211yescharacteristic
39C7×#107C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×C4⋊C8C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×#102C2211yescharacteristic
44#6406C2211yescharacteristic
45#6406C2211yescharacteristic
46#6392C2211yescharacteristic
47#6391C2211yescharacteristic
48#6390C2211yescharacteristic
49#6390C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6382C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3083—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C2×C22⋊D7—21no—
64C22×Dic7—21no—
65C22×Dic7—21no—
66#1159—22no—
67#1159Q814yescomplemented
68C2×C4×D7—21no—
69C2×C4×D7—21no—
70C2×C4×D7C2311yescharacteristic
71C2×Dic14—21no—
72C2×Dic14C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C22⋊C4—21no—
80C42×C7C2311yescharacteristic
81C22⋊Dic7—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4D411yescharacteristic
84(C2×D7)⋊C4D411yescharacteristic
85C4⋊Dic7—21no—
86C4⋊Dic7C2311yescharacteristic
87C4⋊Dic7C2311yescharacteristic
88C4⋊Dic7C2311yescharacteristic
89Dic7⋊C4—21no—
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4D411yescharacteristic
98Dic7⋊C4D411yescharacteristic
99C4×Dic7—21no—
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#477—71no—
115#474—71no—
116#474D711yescharacteristic, complemented
117#428—71no—
118C2×#107—71no—
119#424—71no—
120#417—71no—
121#416—71no—
122C23×C7—21no—
123C22×D7C2×D411yescharacteristic
124C7×Q8—81no—
125C7×D4—21no—
126C7×D4C2×Q812yes—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—81no—
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2411yescharacteristic
134C22⋊D7—24no—
135C2×Dic7—21no—
136C2×Dic7—24no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—81no—
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144C4×D7—81no—
145C4×D7—81no—
146C4×D7C2×Q812yes—
147C4×D7#4112yes—
148Dic14—21no—
149Dic14—81no—
150Dic14C2×Q812yes—
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×Q8—71no—
168C4×D4—71no—
169C4×D4—141no—
170C4×D4—141no—
171C4×D4C2×D711yescharacteristic
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#103—71no—
193#103—71no—
194#102—71no—
195#102—71no—
196#102C2×D711yescharacteristic
197#102C2×D711yescharacteristic
198C22×C7—22no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7C22×D411yescharacteristic
202C2×D7—21no—
203C2×D7—21no—
204C2×D7C22⋊Q812yes—
205C4×C7—21no—
206C4×C7—21no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7C22×Q811yescharacteristic
211C4×C7C2×#4111yescharacteristic
212Dic7—22no—
213Dic7—21no—
214Dic7—21no—
215Dic7—41no—
216Dic7—41no—
217Dic7—41no—
218Dic7C22⋊Q812yes—
219#41—74no—
220#41—142no—
221C2×Q8—21no—
222C2×Q8—71no—
223C2×Q8—141no—
224C2×D4—141no—
225C2×D4C22×D711yescharacteristic
226C22×C4—21no—
227C22×C4—71no—
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—71no—
240C4⋊C4—71no—
241C4⋊C4—71no—
242C4⋊C4—71no—
243C4⋊C4—71no—
244C4⋊C4—141no—
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—21no—
257C22⋊C4—71no—
258C22⋊C4—71no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C42—71no—
262C42—141no—
263C42—141no—
264C42—141no—
265C42C22×D711yescharacteristic
266C2×C7—42no—
267C2×C7C2×C22⋊Q811yescharacteristic
268C2×C7#57611yescharacteristic
269C2×C7#57811yescharacteristic
270D7—22no—
271C23—21no—
272C23—71no—
273Q8—72no—
274Q8—81no—
275Q8—141no—
276Q8—561no—
277D4—21no—
278D4—144no—
279D4Q8×D712yes—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—71no—
284C2×C4—72no—
285C2×C4—72no—
286C2×C4—71no—
287C2×C4—71no—
288C2×C4—81no—
289C2×C4—144no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
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—22no—
306C22—41no—
307C22—41no—
308C22—72no—
309C22—141no—
310C22—141no—
311C22C2×D4×D711yescharacteristic
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—72no—
318C4—141no—
319C4—141no—
320C4—142no—
321C4—281no—
322C4—281no—
323C4—281no—
324C4C2×Q8×D711yescharacteristic
325C4D7×#4111yescharacteristic
326C2—42no—
327C2—142no—
328C2D7×C22⋊Q811yescharacteristic
329C2#9044811yescharacteristic
330C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#474C2
socleC22×C7C22×D4
prime radicalC2×#1159C22
hypercenter#474D7
hypocenterC7#3083
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #90420C2semidirect
3. C7×#474C2semidirect
4. #89863C2semidirect
5. #89781C2semidirect
6. #89653C2semidirect
7. D7×#107C2semidirect
8. #89649C2semidirect
9. #89647C2semidirect
10. #89611C2semidirect
11. #89601C2semidirect
14. #89535C2semidirect
15. #89531C2semidirect
67. #1159Q8semidirect
116. #474D7semidirect
304. C7#3083semidirect

Maximal subgroups

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

subgroupindex
#904202
#902372
C7×#4742
#898632
#897812
#896532
D7×#1072
#896492
#896472
#896112
#896012
#895442
#895432
#895352
#895312
#30837