Finite Group Databasebeta

← #11737933

Subgroups — #11737933

Order 896.

322 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90422C211yesmaximal, characteristic
2#90274C211yesmaximal, characteristic
3C7×#462C211yesmaximal, characteristic, complemented
4#89961C211yesmaximal, characteristic
5#89960C211yesmaximal, characteristic
6#89954C211yesmaximal, characteristic, complemented
7#89954C211yesmaximal, characteristic, complemented
8#89862C211yesmaximal, characteristic, complemented
9#89794C211yesmaximal, characteristic, complemented
10#89611C211yesmaximal, characteristic
11#89601C211yesmaximal, characteristic, complemented
12#89588C211yesmaximal, characteristic
13#89585C211yesmaximal, characteristic
14#89571C211yesmaximal, characteristic, complemented
15#89568C211yesmaximal, characteristic, complemented
16C2×C7×Q16C2211yescharacteristic
17C2×C7×Q16C2211yescharacteristic
18C7×C4⋊Q8C2211yescharacteristic
19C4×C7×Q8C2211yescharacteristic
20#6509C2211yescharacteristic
21#6509C2211yescharacteristic
22#6509C2211yescharacteristic
23Q8×Dic7C2211yescharacteristic
24Q8×Dic7C2211yescharacteristic
25#6507C2211yescharacteristic
26C2×#1144C2211yescharacteristic
27C2×#1144C2211yescharacteristic
28C2×#1131C2211yescharacteristic
29C2×C8×D7C2211yescharacteristic
30#6461—21no—
31#6461—21no—
32#6461—21no—
33#6460C2211yescharacteristic
34#6453C2211yescharacteristic
35#6452—21no—
36#6452—21no—
37#6452—21no—
38#6439—21no—
39#6435C2211yescharacteristic
40#6433—21no—
41#6433C2211yescharacteristic
42C4×Dic14C2211yescharacteristic
43C7×C4⋊C8C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#103C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6409C2211yescharacteristic
49#6409C2211yescharacteristic
50#6392C2211yescharacteristic
51#6391C2211yescharacteristic
52#6390C2211yescharacteristic
53#6390C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55Dic7⋊C8C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3261—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×C7×Q8C2311yescharacteristic
63C2×C4×D7C2311yescharacteristic
64C2×Dic14C2311yescharacteristic
65C7×Q16—41no—
66C7×Q16—41no—
67C2×C7×C8C2311yescharacteristic
68C2×C7×C8C2311yescharacteristic
69C7×C4⋊C4—21no—
70C7×C4⋊C4—21no—
71C7×C4⋊C4—21no—
72C7×C4⋊C4C2311yescharacteristic
73C42×C7—21no—
74C42×C7C2311yescharacteristic
75#1144—41no—
76#1144—41no—
77(C2×D7)⋊C4—21no—
78(C2×D7)⋊C4—21no—
79(C2×D7)⋊C4—21no—
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4D411yescharacteristic
82(C2×D7)⋊C4D411yescharacteristic
83C4⋊Dic7—21no—
84C4⋊Dic7—21no—
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7C2311yescharacteristic
88C4⋊Dic7C2311yescharacteristic
89C4⋊Dic7C2311yescharacteristic
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4D411yescharacteristic
100Dic7⋊C4D411yescharacteristic
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#1131—41no—
108C8×D7—41no—
109#541—71no—
110#541—71no—
111#479—71no—
112#479—71no—
113#479—71no—
114#479—71no—
115#470—71no—
116#470—71no—
117#467—71no—
118#467—71no—
119#462—71no—
120#462—71no—
121#462—71no—
122#462D711yescharacteristic, complemented
123#424—71no—
124C22×D7C2×D411yescharacteristic
125C7×Q8—21no—
126C7×Q8—21no—
127C7×Q8—21no—
128C7×Q8—21no—
129C7×Q8—21no—
130C7×Q8—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2411yescharacteristic
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145C4×D7—21no—
146C4×D7—21no—
147Dic14—21no—
148Dic14—21no—
149C7×C8—21no—
150C7×C8—21no—
151C7⋊C8—21no—
152C7⋊C8—21no—
153C2×Q16—71no—
154C2×Q16—71no—
155C2×Q16C2×D711yescharacteristic
156C2×Q16C2×D711yescharacteristic
157C2×M16—71no—
158C22×C8—71no—
159C4⋊Q8—71no—
160C4⋊Q8C2×D711yescharacteristic
161#126—141no—
162#126—141no—
163#126—141no—
164#126—141no—
165#125—71no—
166#125—71no—
167#125—141no—
168#125—141no—
169#125—141no—
170#125—141no—
171C22⋊Q8—71no—
172C22⋊Q8—71no—
173C22⋊Q8—71no—
174C22⋊Q8—71no—
175C4×Q8—71no—
176C4×Q8—71no—
177C4×Q8—71no—
178C4×Q8C2×D711yescharacteristic
179#117—71no—
180#107—71no—
181#107—71no—
182#106—71no—
183#106—71no—
184C4⋊C8—71no—
185C4⋊C8—71no—
186C4⋊C8—71no—
187C4⋊C8C2×D711yescharacteristic
188#103—71no—
189#103—71no—
190#103—71no—
191#103—71no—
192#103—71no—
193#103—71no—
194#103C2×D711yescharacteristic
195#103C2×D711yescharacteristic
196C22×C7C22×D411yescharacteristic
197C2×D7—21no—
198C2×D7—21no—
199C2×D7—21no—
200C4×C7—21no—
201C4×C7—21no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7H(5,𝔽2)11yescharacteristic
207C4×C72−1+411yescharacteristic
208Dic7—21no—
209Dic7—21no—
210Dic7—21no—
211Dic7—41no—
212Dic7—41no—
213Dic7—41no—
214Dic7—41no—
215C2×Q8—71no—
216C2×Q8C22×D711yescharacteristic
217C2×Q8C22×D711yescharacteristic
218C2×Q8C22×D711yescharacteristic
219C22×C4—71no—
220Q16—41no—
221Q16—41no—
222Q16—281no—
223Q16—281no—
224M16—281no—
225C2×C8—71no—
226C2×C8—71no—
227C2×C8—281no—
228C2×C8C22×D711yescharacteristic
229C2×C8C22×D711yescharacteristic
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—71no—
234C4⋊C4—71no—
235C4⋊C4—71no—
236C4⋊C4—71no—
237C4⋊C4—71no—
238C4⋊C4—141no—
239C4⋊C4—141no—
240C4⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4C22×D711yescharacteristic
252C22⋊C4—71no—
253C22⋊C4—71no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C42—21no—
259C42—71no—
260C42—141no—
261C42—141no—
262C42—141no—
263C42C22×D711yescharacteristic
264C2×C7#53711yescharacteristic
265C2×C7#57811yescharacteristic
266C2×C7#57811yescharacteristic
267D7—41no—
268C23—71no—
269Q8—21no—
270Q8—21no—
271Q8—21no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—141no—
276Q8—141no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—71no—
282C2×C4—71no—
283C2×C4—71no—
284C2×C4—141no—
285C2×C4—141no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4D4×D711yescharacteristic
291C2×C4D4×D711yescharacteristic
292C2×C4C23×D711yescharacteristic
293C8—21no—
294C8—21no—
295C8—141no—
296C8—141no—
297C7#326111yescharacteristic, complemented, Sylow
298C22—141no—
299C22—141no—
300C22—141no—
301C22C2×D4×D711yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—141no—
309C4—141no—
310C4—141no—
311C4—281no—
312C4—281no—
313C4—281no—
314C4—281no—
315C4#654811yescharacteristic
316C4#655411yescharacteristic
317C2—281no—
318C2#9029911yescharacteristic
319C2#9045711yescharacteristic
320C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#462C2
socleC22×C7C22×D4
prime radicalC22×D7C2×D4
hypercenter#462D7
hypocenterC7#3261
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. C7×#462C2semidirect
6. #89954C2semidirect
7. #89954C2semidirect
8. #89862C2semidirect
9. #89794C2semidirect
11. #89601C2semidirect
14. #89571C2semidirect
15. #89568C2semidirect
122. #462D7semidirect
297. C7#3261semidirect

Maximal subgroups

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

subgroupindex
#904222
#902742
C7×#4622
#899612
#899602
#899542
#898622
#897942
#896112
#896012
#895882
#895852
#895712
#895682
#32617