Finite Group Databasebeta

← #11737200

Subgroups — #11737200

Order 896.

340 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#90258C211yesmaximal, characteristic, complemented
2#90230C211yesmaximal, characteristic
3C7×#440C211yesmaximal, characteristic
4#89934C211yesmaximal, characteristic
5#89934C211yesmaximal, characteristic
6#89794C211yesmaximal, characteristic, complemented
7#89791C211yesmaximal, characteristic
8#89649C211yesmaximal, characteristic, complemented
9#89647C211yesmaximal, characteristic, complemented
10#89569C211yesmaximal, characteristic, complemented
11#89564C211yesmaximal, characteristic, complemented
12#89530C211yesmaximal, characteristic
13#89529C211yesmaximal, characteristic
14#89475C211yesmaximal, characteristic, complemented
15#89473C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19#6507C2211yescharacteristic
20#6507C2211yescharacteristic
21D4×Dic7C2211yescharacteristic
22D4×Dic7C2211yescharacteristic
23C2×#1142C2211yescharacteristic
24C2×#1142C2211yescharacteristic
25#6487—21no—
26C2×Dic7⋊C4—21no—
27#6453C2211yescharacteristic
28#6452—21no—
29#6452—21no—
30#6451C2211yescharacteristic
31#6450C2211yescharacteristic
32#6450C2211yescharacteristic
33#6441—21no—
34#6441—21no—
35#6433—21no—
36#6433C2211yescharacteristic
37#6432C2211yescharacteristic
38C4×Dic14—21no—
39C4×Dic14C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#103C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×#97C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6390C2211yescharacteristic
47#6390C2211yescharacteristic
48#6390C2211yescharacteristic
49#6390C2211yescharacteristic
50#6389C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53C8×Dic7C2211yescharacteristic
54#6384C2211yescharacteristic
55#6384C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3372—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C22×Dic7—21no—
64C22×Dic7—21no—
65C2×Dic14—21no—
66C2×Dic14D411yescharacteristic
67C2×Dic14C2311yescharacteristic
68C2×Dic14C2311yescharacteristic
69C7×SD16—22no—
70C2×C7×C8C2311yescharacteristic
71C2×C7×C8C2311yescharacteristic
72C7×C4⋊C4—21no—
73C7×C4⋊C4C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C7×C22⋊C4—21no—
76C42×C7—21no—
77C42×C7C2311yescharacteristic
78C22⋊Dic7—21no—
79C22⋊Dic7—21no—
80#1142—22no—
81#1142—22no—
82C4⋊Dic7—21no—
83C4⋊Dic7—21no—
84C4⋊Dic7—21no—
85C4⋊Dic7D411yescharacteristic
86C4⋊Dic7C2311yescharacteristic
87C4⋊Dic7C2311yescharacteristic
88Dic7⋊C4—21no—
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⋊C4—21no—
98Dic7⋊C4D411yescharacteristic
99Dic7⋊C4D411yescharacteristic
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C4×Dic7C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#557—71no—
107#554—71no—
108#491—71no—
109#491—71no—
110#487—71no—
111#479—71no—
112#479—71no—
113#477—71no—
114#461—71no—
115#461—71no—
116#461—71no—
117#446—71no—
118#440—71no—
119#440D711yescharacteristic
120C4×SD16—71no—
121C23×C7—21no—
122C7×Q8—21no—
123C7×Q8—21no—
124C7×D4—21no—
125C7×D4#4112yes—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—81no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2411yescharacteristic
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—42no—
138C2×Dic7—81no—
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143Dic14—22no—
144Dic14—21no—
145Dic14—21no—
146Dic14—21no—
147Dic14—21no—
148Dic14—81no—
149C7×C8—21no—
150C7×C8—21no—
151C7⋊C8—21no—
152C7⋊C8—21no—
153C2×SD16—71no—
154C2×SD16—71no—
155C2×SD16C2×D711yescharacteristic
156C4⋊Q8—71no—
157C4⋊Q8—71no—
158C4⋊Q8—71no—
159C4⋊Q8—71no—
160#125—71no—
161#125—71no—
162#125—141no—
163#125—141no—
164#125—141no—
165C22⋊Q8—141no—
166C22⋊Q8—141no—
167C22⋊Q8—141no—
168C4×Q8—71no—
169C4×Q8—71no—
170C4×Q8—71no—
171C4×Q8—141no—
172C4×Q8C2×D711yescharacteristic
173C4×D4—71no—
174C4×D4—71no—
175C4×D4C2×D711yescharacteristic
176C2×C4⋊C4—141no—
177#107—71no—
178#107C2×D711yescharacteristic
179#106—71no—
180C4⋊C8—71no—
181C4⋊C8—71no—
182C4⋊C8—71no—
183#103—71no—
184#103—71no—
185#103—71no—
186#103—71no—
187#103—71no—
188#103—71no—
189#103—71no—
190#103—71no—
191#103C2×D711yescharacteristic
192#102C2×D711yescharacteristic
193#97—71no—
194#97C2×D711yescharacteristic
195C4×C8—71no—
196C22×C7—22no—
197C22×C7—41no—
198C22×C7—41no—
199C22×C7C22×D411yescharacteristic
200C4×C7—21no—
201C4×C7—21no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7C2×#4111yescharacteristic
206C4×C72−1+411yescharacteristic
207Dic7—21no—
208Dic7—21no—
209Dic7—21no—
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213Dic7—41no—
214Dic7#13712yes—
215C2×Q8—71no—
216C2×Q8—71no—
217C2×Q8—71no—
218C2×Q8—141no—
219C2×Q8C22×D711yescharacteristic
220C2×D4C22×D711yescharacteristic
221C22×C4—21no—
222C22×C4—141no—
223C22×C4—141no—
224SD16—22no—
225SD16—142no—
226SD16—142no—
227C2×C8—71no—
228C2×C8—71no—
229C2×C8C22×D711yescharacteristic
230C2×C8C22×D711yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—71no—
233C4⋊C4—71no—
234C4⋊C4—71no—
235C4⋊C4—71no—
236C4⋊C4—71no—
237C4⋊C4—141no—
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⋊C4C22×D711yescharacteristic
251C4⋊C4C22×D711yescharacteristic
252C22⋊C4—21no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C42—21no—
256C42—71no—
257C42—71no—
258C42—141no—
259C42—141no—
260C42C22×D711yescharacteristic
261C2×C7—42no—
262C2×C7#54811yescharacteristic
263C2×C7C2×#13711yescharacteristic
264C2×C7#57711yescharacteristic
265C23—21no—
266Q8—21no—
267Q8—21no—
268Q8—142no—
269Q8—141no—
270Q8—141no—
271Q8—141no—
272Q8—141no—
273Q8—561no—
274D4—21no—
275D4#115712yes—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—71no—
280C2×C4—71no—
281C2×C4—71no—
282C2×C4—71no—
283C2×C4—81no—
284C2×C4—141no—
285C2×C4—141no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—282no—
289C2×C4—561no—
290C2×C4D4×D711yescharacteristic
291C2×C4D4×D711yescharacteristic
292C2×C4C23×D711yescharacteristic
293C8—21no—
294C8—21no—
295C8—141no—
296C8—141no—
297C7#337211yescharacteristic, complemented, Sylow
298C22—22no—
299C22—41no—
300C22—41no—
301C22C2×D4×D711yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—72no—
308C4—141no—
309C4—141no—
310C4—141no—
311C4—281no—
312C4—281no—
313C4—281no—
314C4—281no—
315C4C2×#115711yescharacteristic
316C4#655411yescharacteristic
317C2—42no—
318C2#9024311yescharacteristic
319C2#9046611yescharacteristic
320C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#440C2
socleC22×C7C22×D4
prime radicalC2×C7×D4C23
hypercenter#440D7
hypocenterC7#3372
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #90258C2semidirect
6. #89794C2semidirect
8. #89649C2semidirect
9. #89647C2semidirect
10. #89569C2semidirect
11. #89564C2semidirect
14. #89475C2semidirect
15. #89473C2semidirect
297. C7#3372semidirect

Maximal subgroups

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

subgroupindex
#902582
#902302
C7×#4402
#899342
#897942
#897912
#896492
#896472
#895692
#895642
#895302
#895292
#894752
#894732
#33727