Finite Group Databasebeta

← #11737818

Subgroups — #11737818

Order 896.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90404C211yesmaximal, characteristic
2#90260C211yesmaximal, characteristic, complemented
3C7×#460C211yesmaximal, characteristic
4#89938C211yesmaximal, characteristic, complemented
5#89938C211yesmaximal, characteristic, complemented
6#89936C211yesmaximal, characteristic, complemented
7SD16×Dic7C211yesmaximal, characteristic, complemented
8#89843C211yesmaximal, characteristic
9#89802C211yesmaximal, characteristic, complemented
10#89601C211yesmaximal, characteristic, complemented
11#89600C211yesmaximal, characteristic, complemented
12#89540C211yesmaximal, characteristic, complemented
13#89538C211yesmaximal, characteristic, complemented
14#89537C211yesmaximal, characteristic, complemented
15#89533C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C2×C7×SD16C2211yescharacteristic
18C7×#127C2211yescharacteristic
19C4×C7×Q8C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22C2×#1144C2211yescharacteristic
23#6499—21no—
24#6499C2211yescharacteristic
25#6499C2211yescharacteristic
26#6498—21no—
27#6498—21no—
28D4×Dic7—21no—
29D4×Dic7C2211yescharacteristic
30D4×Dic7C2211yescharacteristic
31#6453—21no—
32#6453C2211yescharacteristic
33#6451—21no—
34#6451C2211yescharacteristic
35#6433—21no—
36#6433C2211yescharacteristic
37C4×Dic14—21no—
38C4×Dic14C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C7×C4⋊C8C2211yescharacteristic
41C7×#102C2211yescharacteristic
42C7×#102C2211yescharacteristic
43#6409C2211yescharacteristic
44#6406C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47#6406C2211yescharacteristic
48#6392C2211yescharacteristic
49#6391C2211yescharacteristic
50#6390C2211yescharacteristic
51#6390C2211yescharacteristic
52#6389C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55C8×Dic7C2211yescharacteristic
56#6384C2211yescharacteristic
57#6382C2211yescharacteristic
58#6377C2211yescharacteristic
59#3371—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4—21no—
62C2×C7×D4—21no—
63C2×C7×D4C2311yescharacteristic
64C2×C7×D4C2311yescharacteristic
65C22×Dic7—21no—
66C22×Dic7—21no—
67C2×Dic14—21no—
68C2×Dic14D411yescharacteristic
69C2×Dic14C2311yescharacteristic
70C7×SD16—22no—
71C7×SD16—22no—
72C2×C7×C8C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C7×C4⋊C4—21no—
75C7×C4⋊C4C2311yescharacteristic
76C42×C7—21no—
77C42×C7C2311yescharacteristic
78C22⋊Dic7—21no—
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84#1144—22no—
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⋊C4—21no—
97Dic7⋊C4D411yescharacteristic
98Dic7⋊C4D411yescharacteristic
99C4×Dic7—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C4×Dic7C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#559—71no—
107#557—71no—
108#490—71no—
109#489—71no—
110#488—71no—
111#479—71no—
112#478—71no—
113#476—71no—
114#464—71no—
115#464—71no—
116#460D711yescharacteristic
117#446—71no—
118#441—71no—
119#440—71no—
120C4×SD16—71no—
121C23×C7—21no—
122C23×C7—21no—
123C7×Q8—21no—
124C7×Q8#4112yes—
125C7×D4—21no—
126C7×D4—21no—
127C7×D4—21no—
128C7×D4—21no—
129C7×D4—42no—
130C7×D4—81no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2411yescharacteristic
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—81no—
141C2×Dic7—81no—
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146Dic14—21no—
147Dic14—21no—
148Dic14—41no—
149Dic14—42no—
150C7×C8—21no—
151C7×C8—21no—
152C7⋊C8—21no—
153C7⋊C8—21no—
154C2×Q16—71no—
155C2×SD16C2×D711yescharacteristic
156C2×SD16C2×D711yescharacteristic
157C4⋊Q8—71no—
158C4⋊Q8—141no—
159#127C2×D711yescharacteristic
160#125—71no—
161#125—71no—
162#125—141no—
163#125—141no—
164#124—71no—
165#124—71no—
166#124—141no—
167#123—141no—
168#123—141no—
169C4×Q8—71no—
170C4×Q8—71no—
171C4×Q8—71no—
172C4×Q8—71no—
173C4×Q8—141no—
174C4×Q8C2×D711yescharacteristic
175C4×D4—71no—
176C4×D4—71no—
177C4×D4—141no—
178#107—71no—
179#106—71no—
180#106—71no—
181C4⋊C8—71no—
182C4⋊C8—71no—
183C4⋊C8C2×D711yescharacteristic
184#103—71no—
185#103—71no—
186#103—71no—
187#103—71no—
188#102—71no—
189#102—71no—
190#102—71no—
191#102—71no—
192#102C2×D711yescharacteristic
193#102C2×D711yescharacteristic
194#97—71no—
195#97—71no—
196C4×C8—71no—
197C22×C7—41no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7C22×D411yescharacteristic
204C4×C7—22no—
205C4×C7—21no—
206C4×C7—21no—
207C4×C7—41no—
208C4×C7C2×#4111yescharacteristic
209C4×C72−1+411yescharacteristic
210Dic7—21no—
211Dic7—21no—
212Dic7—21no—
213Dic7—21no—
214Dic7—41no—
215Dic7—41no—
216Dic7—41no—
217Dic7—41no—
218C2×Q8—71no—
219C2×Q8—71no—
220C2×Q8—141no—
221C2×Q8C22×D711yescharacteristic
222C2×D4—21no—
223C2×D4—21no—
224C2×D4C22×D711yescharacteristic
225C2×D4C22×D711yescharacteristic
226C22×C4—141no—
227C22×C4—141no—
228Q16—142no—
229SD16—22no—
230SD16—22no—
231C2×C8—71no—
232C2×C8—71no—
233C2×C8C22×D711yescharacteristic
234C2×C8C22×D711yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—71no—
237C4⋊C4—71no—
238C4⋊C4—71no—
239C4⋊C4—71no—
240C4⋊C4—71no—
241C4⋊C4—71no—
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
251C22⋊C4—141no—
252C22⋊C4—141no—
253C22⋊C4—141no—
254C22⋊C4—141no—
255C22⋊C4—141no—
256C22⋊C4—141no—
257C42—21no—
258C42—71no—
259C42—71no—
260C42—141no—
261C42—141no—
262C42—141no—
263C42C22×D711yescharacteristic
264C2×C7—81no—
265C2×C7—81no—
266C2×C7#54811yescharacteristic
267C2×C7#57511yescharacteristic
268C2×C7#57711yescharacteristic
269C23—21no—
270C23—21no—
271Q8—21no—
272Q8—141no—
273Q8—141no—
274Q8—281no—
275Q8—282no—
276Q8#115912yes—
277D4—21no—
278D4—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—81no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—71no—
286C2×C4—71no—
287C2×C4—71no—
288C2×C4—71no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—561no—
294C2×C4—561no—
295C2×C4D4×D711yescharacteristic
296C2×C4D4×D711yescharacteristic
297C2×C4C23×D711yescharacteristic
298C8—21no—
299C8—21no—
300C8—141no—
301C8—141no—
302C7#337111yescharacteristic, complemented, Sylow
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C2×D4×D711yescharacteristic
310C4—21no—
311C4—21no—
312C4—22no—
313C4—41no—
314C4—141no—
315C4—141no—
316C4—141no—
317C4—141no—
318C4—281no—
319C4—281no—
320C4—281no—
321C4—281no—
322C4C2×#115911yescharacteristic
323C4#655411yescharacteristic
324C2—81no—
325C2—81no—
326C2#9028911yescharacteristic
327C2#9045311yescharacteristic
328C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#460C2
socleC22×C7C22×D4
prime radicalC7×#127C22
hypercenter#460D7
hypocenterC7#3371
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #90260C2semidirect
4. #89938C2semidirect
5. #89938C2semidirect
6. #89936C2semidirect
7. SD16×Dic7C2semidirect
9. #89802C2semidirect
10. #89601C2semidirect
11. #89600C2semidirect
12. #89540C2semidirect
13. #89538C2semidirect
14. #89537C2semidirect
15. #89533C2semidirect
302. C7#3371semidirect

Maximal subgroups

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

subgroupindex
#904042
#902602
C7×#4602
#899382
#899362
SD16×Dic72
#898432
#898022
#896012
#896002
#895402
#895382
#895372
#895332
#33717