Finite Group Databasebeta

← #11736808

Subgroups — #11736808

Order 896.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90210C211yesmaximal, characteristic
2#90209C211yesmaximal, characteristic
3C7×#433C211yesmaximal, characteristic
4#89623C211yesmaximal, characteristic, complemented
5#89622C211yesmaximal, characteristic, complemented
6#89601C211yesmaximal, characteristic, complemented
7#89601C211yesmaximal, characteristic, complemented
8#89495C211yesmaximal, characteristic
9#89495C211yesmaximal, characteristic
10#89494C211yesmaximal, characteristic
11#89493C211yesmaximal, characteristic
12#89475C211yesmaximal, characteristic, complemented
13#89474C211yesmaximal, characteristic, complemented
14#89455C211yesmaximal, characteristic, complemented
15#89453C211yesmaximal, characteristic, complemented
16C7×#117C2211yescharacteristic
17#6489C2211yescharacteristic
18C2×C4⋊Dic7C2211yescharacteristic
19#6487C2211yescharacteristic
20#6487C2211yescharacteristic
21#6453—21no—
22#6452—21no—
23#6452—21no—
24#6451—21no—
25#6442—21no—
26#6442—21no—
27#6441—21no—
28#6441—21no—
29#6433C2211yescharacteristic
30#6433C2211yescharacteristic
31#6433C2211yescharacteristic
32#6432C2211yescharacteristic
33#6432C2211yescharacteristic
34#6432C2211yescharacteristic
35C4×Dic14C2211yescharacteristic
36C4×Dic14C2211yescharacteristic
37C7×C4⋊C8C2211yescharacteristic
38C7×C4⋊C8C2211yescharacteristic
39C7×C22⋊C8C2211yescharacteristic
40C7×C22⋊C8C2211yescharacteristic
41C7×#97C2211yescharacteristic
42C4×C7×C8C2211yescharacteristic
43#6392C2211yescharacteristic
44#6392C2211yescharacteristic
45#6392C2211yescharacteristic
46#6392C2211yescharacteristic
47#6391C2211yescharacteristic
48#6391C2211yescharacteristic
49#6391C2211yescharacteristic
50#6391C2211yescharacteristic
51#6390C2211yescharacteristic
52#6390C2211yescharacteristic
53#6390C2211yescharacteristic
54#6390C2211yescharacteristic
55#6390C2211yescharacteristic
56#6390C2211yescharacteristic
57#6390C2211yescharacteristic
58#6390C2211yescharacteristic
59#3242—71nomaximal, Sylow
60C22×C4×C7C2311yescharacteristic
61C22×Dic7—21no—
62C2×Dic14—21no—
63C2×Dic14—21no—
64C2×Dic14C2311yescharacteristic
65C2×Dic14C2311yescharacteristic
66C2×C7×C8C2311yescharacteristic
67C2×C7×C8C2311yescharacteristic
68C2×C7×C8C2311yescharacteristic
69C2×C7×C8C2311yescharacteristic
70C7×C4⋊C4D411yescharacteristic
71C7×C4⋊C4D411yescharacteristic
72C7×C22⋊C4D411yescharacteristic
73C7×C22⋊C4D411yescharacteristic
74C42×C7C2311yescharacteristic
75C42×C7C2311yescharacteristic
76C22⋊Dic7—21no—
77C22⋊Dic7—21no—
78C22⋊Dic7—21no—
79C4⋊Dic7—21no—
80C4⋊Dic7—21no—
81C4⋊Dic7—21no—
82C4⋊Dic7—21no—
83C4⋊Dic7—21no—
84C4⋊Dic7C2311yescharacteristic
85C4⋊Dic7C2311yescharacteristic
86C4⋊Dic7C2311yescharacteristic
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⋊C4—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7—21no—
103#544—71no—
104#541—71no—
105#501—71no—
106#499—71no—
107#491—71no—
108#487—71no—
109#485—71no—
110#485—71no—
111#484—71no—
112#483—71no—
113#479—71no—
114#479—71no—
115#477—71no—
116#475—71no—
117#433D711yescharacteristic
118C23×C7C2×D411yescharacteristic
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7C2×D411yescharacteristic
122C2×C4×C7C2×D411yescharacteristic
123C2×C4×C7C2×D411yescharacteristic
124C2×C4×C7C2×D411yescharacteristic
125C2×C4×C7C2×D411yescharacteristic
126C2×C4×C7C2411yescharacteristic
127C2×Dic7—21no—
128C2×Dic7—21no—
129C2×Dic7—21no—
130C2×Dic7—21no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—81no—
136Dic14—21no—
137Dic14—21no—
138Dic14—21no—
139Dic14—21no—
140Dic14—81no—
141Dic14—81no—
142C7×C8—21no—
143C7×C8—21no—
144C7×C8—21no—
145C7×C8—21no—
146C4⋊Q8—71no—
147C4⋊Q8—71no—
148C4⋊Q8—71no—
149C4⋊Q8—141no—
150#126—141no—
151#126—141no—
152#125—71no—
153#125—71no—
154#125—71no—
155#125—141no—
156#125—141no—
157#125—141no—
158C22⋊Q8—71no—
159C22⋊Q8—71no—
160C22⋊Q8—141no—
161C22⋊Q8—141no—
162C4×Q8—71no—
163C4×Q8—71no—
164#117—71no—
165#117C2×D711yescharacteristic
166C2×C4⋊C4—71no—
167#107—71no—
168#107—71no—
169#107—71no—
170#107—71no—
171#106—71no—
172#106—71no—
173#106—71no—
174#106—71no—
175C4⋊C8C2×D711yescharacteristic
176C4⋊C8C2×D711yescharacteristic
177#103—71no—
178#103—71no—
179#103—71no—
180#103—71no—
181#103—71no—
182#103—71no—
183#103—71no—
184#103—71no—
185C22⋊C8C2×D711yescharacteristic
186C22⋊C8C2×D711yescharacteristic
187#97C2×D711yescharacteristic
188C4×C8C2×D711yescharacteristic
189C22×C7—21no—
190C22×C7—21no—
191C22×C7—21no—
192C22×C7C22×D411yescharacteristic
193C4×C7—21no—
194C4×C7—21no—
195C4×C7—21no—
196C4×C7—21no—
197C4×C7—21no—
198C4×C72−1+411yescharacteristic
199C4×C72−1+411yescharacteristic
200Dic7—41no—
201Dic7—41no—
202Dic7—41no—
203Dic7—41no—
204Dic7—41no—
205Dic7—41no—
206Dic7—41no—
207Dic7—41no—
208C2×Q8—71no—
209C2×Q8—71no—
210C2×Q8—141no—
211C2×Q8—141no—
212C22×C4—141no—
213C22×C4C22×D711yescharacteristic
214C2×C8C22×D711yescharacteristic
215C2×C8C22×D711yescharacteristic
216C2×C8C22×D711yescharacteristic
217C2×C8C22×D711yescharacteristic
218C4⋊C4—71no—
219C4⋊C4—71no—
220C4⋊C4—71no—
221C4⋊C4—71no—
222C4⋊C4—71no—
223C4⋊C4—71no—
224C4⋊C4—141no—
225C4⋊C4—141no—
226C4⋊C4—141no—
227C4⋊C4—141no—
228C4⋊C4—141no—
229C4⋊C4—141no—
230C4⋊C4—141no—
231C4⋊C4—141no—
232C4⋊C4—141no—
233C4⋊C4—141no—
234C4⋊C4—141no—
235C4⋊C4—141no—
236C4⋊C4—141no—
237C4⋊C4—141no—
238C4⋊C4—141no—
239C4⋊C4D2811yescharacteristic
240C4⋊C4D2811yescharacteristic
241C22⋊C4—141no—
242C22⋊C4—141no—
243C22⋊C4—141no—
244C22⋊C4D2811yescharacteristic
245C22⋊C4D2811yescharacteristic
246C42—141no—
247C42—141no—
248C42—141no—
249C42C22×D711yescharacteristic
250C42C22×D711yescharacteristic
251C2×C7—41no—
252C2×C7#53611yescharacteristic
253C2×C7#57711yescharacteristic
254C2×C7#57811yescharacteristic
255C23C2×D2811yescharacteristic
256Q8—141no—
257Q8—141no—
258Q8—141no—
259Q8—141no—
260Q8—561no—
261Q8—561no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—141no—
265C2×C4—141no—
266C2×C4—141no—
267C2×C4—141no—
268C2×C4—141no—
269C2×C4—141no—
270C2×C4—141no—
271C2×C4—141no—
272C2×C4—561no—
273C2×C4C2×D2811yescharacteristic
274C2×C4C2×D2811yescharacteristic
275C2×C4C2×D2811yescharacteristic
276C2×C4C2×D2811yescharacteristic
277C2×C4C2×D2811yescharacteristic
278C2×C4C23×D711yescharacteristic
279C8—21no—
280C8—21no—
281C8—21no—
282C8—21no—
283C7#324211yescharacteristic, complemented, Sylow
284C22—21no—
285C22—21no—
286C22—21no—
287C22C22×D2811yescharacteristic
288C4—21no—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—281no—
294C4—281no—
295C4—281no—
296C4—281no—
297C4—281no—
298C4—281no—
299C4—281no—
300C4—281no—
301C4#655411yescharacteristic
302C4#655411yescharacteristic
303C2—41no—
304C2#9021711yescharacteristic
305C2#9044211yescharacteristic
306C2#9044411yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#433C2
socleC22×C7C22×D4
prime radicalC23×C7C2×D4
hypercenter#433D7
hypocenterC7#3242
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
4. #89623C2semidirect
5. #89622C2semidirect
6. #89601C2semidirect
7. #89601C2semidirect
12. #89475C2semidirect
13. #89474C2semidirect
14. #89455C2semidirect
15. #89453C2semidirect
283. C7#3242semidirect

Maximal subgroups

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

subgroupindex
#902102
#902092
C7×#4332
#896232
#896222
#896012
#894952
#894942
#894932
#894752
#894742
#894552
#894532
#32427