Finite Group Databasebeta

← #11737912

Subgroups — #11737912

Order 896.

332 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90423C211yesmaximal, characteristic
2#90260C211yesmaximal, characteristic, complemented
3C7×#462C211yesmaximal, characteristic, complemented
4#89957C211yesmaximal, characteristic
5#89957C211yesmaximal, characteristic
6#89956C211yesmaximal, characteristic, complemented
7Q16×Dic7C211yesmaximal, characteristic, complemented
8#89854C211yesmaximal, characteristic, complemented
9#89798C211yesmaximal, characteristic
10#89619C211yesmaximal, characteristic
11#89600C211yesmaximal, characteristic, complemented
12#89572C211yesmaximal, characteristic
13#89571C211yesmaximal, characteristic, complemented
14#89567C211yesmaximal, characteristic
15#89566C211yesmaximal, characteristic, complemented
16C2×C7×Q16C2211yescharacteristic
17C2×C7×Q16C2211yescharacteristic
18C7×C4⋊Q8C2211yescharacteristic
19C4×C7×Q8C2211yescharacteristic
20#6510C2211yescharacteristic
21#6510C2211yescharacteristic
22Q8×Dic7C2211yescharacteristic
23Q8×Dic7C2211yescharacteristic
24Q8×Dic7C2211yescharacteristic
25Q8×Dic7C2211yescharacteristic
26C2×#1143C2211yescharacteristic
27#6461—21no—
28#6461—21no—
29#6460—21no—
30#6455—21no—
31#6453—21no—
32#6453C2211yescharacteristic
33#6451—21no—
34#6451C2211yescharacteristic
35C4×D28C2211yescharacteristic
36#6433—21no—
37#6433C2211yescharacteristic
38C4×Dic14—21no—
39C4×Dic14C2211yescharacteristic
40C7×C4⋊C8C2211yescharacteristic
41C7×#103C2211yescharacteristic
42C7×#103C2211yescharacteristic
43#6409C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6392C2211yescharacteristic
51#6391C2211yescharacteristic
52#6389C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55C8×Dic7C2211yescharacteristic
56#6383C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3377—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×C7×Q8C2311yescharacteristic
63C2×D28C2311yescharacteristic
64C2×C4×D7—21no—
65C2×Dic14—21no—
66C2×Dic14D411yescharacteristic
67C7×Q16—22no—
68C7×Q16—22no—
69C2×C7×C8C2311yescharacteristic
70C2×C7×C8C2311yescharacteristic
71C7×C4⋊C4—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4C2311yescharacteristic
75C42×C7—21no—
76C42×C7C2311yescharacteristic
77#1143—22no—
78(C2×D7)⋊C4—21no—
79(C2×D7)⋊C4—21no—
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83C4⋊Dic7—21no—
84C4⋊Dic7—21no—
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7D411yescharacteristic
89C4⋊Dic7C2311yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4D411yescharacteristic
97Dic7⋊C4D411yescharacteristic
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C4×Dic7C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#557—71no—
107#556—71no—
108#491—71no—
109#489—71no—
110#488—71no—
111#479—71no—
112#478—71no—
113#477—71no—
114#464—71no—
115#464—71no—
116#462D711yescharacteristic, complemented
117#446—71no—
118#441—71no—
119#440—71no—
120C4×Q16—71no—
121C22×D7—21no—
122C7×Q8—21no—
123C7×Q8—21no—
124C7×Q8—21no—
125C7×Q8—21no—
126C7×Q8—21no—
127C7×Q8#4112yes—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2411yescharacteristic
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7C2×D411yescharacteristic
139C2×Dic7C2×D411yescharacteristic
140C2×Dic7C2×D411yescharacteristic
141C2×Dic7C2×D411yescharacteristic
142D28—21no—
143D28—21no—
144C4×D7—42no—
145Dic14—41no—
146Dic14—42no—
147C7×C8—21no—
148C7×C8—21no—
149C7⋊C8—21no—
150C7⋊C8—21no—
151C2×Q16C2×D711yescharacteristic
152C2×Q16C2×D711yescharacteristic
153C2×SD16—71no—
154C4⋊Q8—71no—
155C4⋊Q8—141no—
156C4⋊Q8C2×D711yescharacteristic
157#126—141no—
158#126—141no—
159#125—71no—
160#125—71no—
161#125—141no—
162#125—141no—
163#124—71no—
164#124—71no—
165C22⋊Q8—141no—
166C4×Q8—71no—
167C4×Q8—71no—
168C4×Q8—71no—
169C4×Q8—71no—
170C4×Q8—71no—
171C4×Q8—141no—
172C4×Q8C2×D711yescharacteristic
173C4×D4—71no—
174#117—141no—
175#107—71no—
176#107—71no—
177#106—71no—
178C4⋊C8—71no—
179C4⋊C8—71no—
180C4⋊C8C2×D711yescharacteristic
181#103—71no—
182#103—71no—
183#103—71no—
184#103—71no—
185#103—71no—
186#103C2×D711yescharacteristic
187#103C2×D711yescharacteristic
188#102—71no—
189#102—71no—
190#102—71no—
191#97—71no—
192#97—71no—
193C4×C8—71no—
194C22×C7C22×D411yescharacteristic
195C2×D7—41no—
196C2×D7—41no—
197C2×D7—41no—
198C4×C7—22no—
199C4×C7—21no—
200C4×C7—21no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7C2×#4111yescharacteristic
205C4×C72−1+411yescharacteristic
206Dic7—21no—
207Dic7—21no—
208Dic7—21no—
209Dic7—21no—
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213C2×Q8—71no—
214C2×Q8—141no—
215C2×Q8C22×D711yescharacteristic
216C2×Q8C22×D711yescharacteristic
217C2×Q8C22×D711yescharacteristic
218C2×D4—71no—
219C22×C4—141no—
220Q16—22no—
221Q16—22no—
222SD16—142no—
223C2×C8—71no—
224C2×C8—71no—
225C2×C8C22×D711yescharacteristic
226C2×C8C22×D711yescharacteristic
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—71no—
231C4⋊C4—71no—
232C4⋊C4—71no—
233C4⋊C4—71no—
234C4⋊C4—71no—
235C4⋊C4—71no—
236C4⋊C4—141no—
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⋊C4C22×D711yescharacteristic
246C22⋊C4—141no—
247C22⋊C4—141no—
248C22⋊C4—141no—
249C22⋊C4—141no—
250C22⋊C4—141no—
251C42—21no—
252C42—71no—
253C42—71no—
254C42—141no—
255C42—141no—
256C42—141no—
257C42—141no—
258C42C22×D711yescharacteristic
259C2×C7#54811yescharacteristic
260C2×C7#57511yescharacteristic
261C2×C7#57811yescharacteristic
262D7—81no—
263C23—141no—
264Q8—21no—
265Q8—21no—
266Q8—21no—
267Q8—21no—
268Q8—21no—
269Q8—281no—
270Q8—282no—
271Q8#115912yes—
272D4—141no—
273D4—141no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—71no—
279C2×C4—71no—
280C2×C4—71no—
281C2×C4—71no—
282C2×C4—141no—
283C2×C4—141no—
284C2×C4—141no—
285C2×C4—282no—
286C2×C4D4×D711yescharacteristic
287C2×C4D4×D711yescharacteristic
288C2×C4C23×D711yescharacteristic
289C8—21no—
290C8—21no—
291C8—141no—
292C8—141no—
293C7#337711yescharacteristic, complemented, Sylow
294C22—281no—
295C22—281no—
296C22—281no—
297C22C2×D4×D711yescharacteristic
298C4—21no—
299C4—21no—
300C4—22no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—141no—
305C4—141no—
306C4—141no—
307C4—141no—
308C4—281no—
309C4—281no—
310C4—281no—
311C4C2×#115911yescharacteristic
312C4#655411yescharacteristic
313C2—561no—
314C2#9028911yescharacteristic
315C2#9045711yescharacteristic
316C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#462C2
socleC22×C7C22×D4
prime radicalC2×D28C23
hypercenter#462D7
hypocenterC7#3377
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
2. #90260C2semidirect
3. C7×#462C2semidirect
6. #89956C2semidirect
7. Q16×Dic7C2semidirect
8. #89854C2semidirect
11. #89600C2semidirect
13. #89571C2semidirect
15. #89566C2semidirect
116. #462D7semidirect
293. C7#3377semidirect

Maximal subgroups

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

subgroupindex
#904232
#902602
C7×#4622
#899572
#899562
Q16×Dic72
#898542
#897982
#896192
#896002
#895722
#895712
#895672
#895662
#33777