Finite Group Databasebeta

← #11741094

Subgroups — #11741094

Order 896.

348 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#555C211yesmaximal, characteristic
2#90232C211yesmaximal, characteristic, complemented
3#89866C211yesmaximal, characteristic, complemented
4#89852C211yesmaximal, characteristic, complemented
5#89814C211yesmaximal, characteristic, complemented
6#89813C211yesmaximal, characteristic, complemented
7#89807C211yesmaximal, characteristic, complemented
8#89806C211yesmaximal, characteristic, complemented
9#89790C211yesmaximal, characteristic, complemented
10#89783C211yesmaximal, characteristic, complemented
11#89781C211yesmaximal, characteristic, complemented
12#89770C211yesmaximal, characteristic, complemented
13#89768C211yesmaximal, characteristic, complemented
14#89768C211yesmaximal, characteristic, complemented
15#89767C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C7×#124—21no—
18C7×#124—21no—
19C7×#121—21no—
20C7×#121C2211yescharacteristic
21C7×#121C2211yescharacteristic
22C4×C7×D4C2211yescharacteristic
23C4×C7×D4C2211yescharacteristic
24C7×#117—21no—
25C7×#117C2211yescharacteristic
26C7×#117C2211yescharacteristic
27#6498—21no—
28C2×#1142C2211yescharacteristic
29#6489C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31#6487C2211yescharacteristic
32#6487C2211yescharacteristic
33C2×#1136C2211yescharacteristic
34C22×C7⋊C8C2211yescharacteristic
35#6452—21no—
36#6442—21no—
37#6440—21no—
38#6433C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40#6406C2211yescharacteristic
41#6406C2211yescharacteristic
42#6406C2211yescharacteristic
43C22⋊(C7⋊C8)C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6384C2211yescharacteristic
46#6384C2211yescharacteristic
47#6384C2211yescharacteristic
48#6382C2211yescharacteristic
49#6382C2211yescharacteristic
50#6382C2211yescharacteristic
51#6382C2211yescharacteristic
52#6382C2211yescharacteristic
53#6381C2211yescharacteristic
54#6381C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3348—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×D4—21no—
62C2×C7×D4—21no—
63C2×C7×D4D411yescharacteristic
64C2×C7×D4C2311yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×Dic7—21no—
69C2×Dic14C2311yescharacteristic
70C7×C4⋊C4—21no—
71C7×C4⋊C4D411yescharacteristic
72C7×C4⋊C4C2311yescharacteristic
73C7×C4⋊C4C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C7×C22⋊C4—21no—
76C7×C22⋊C4—21no—
77C7×C22⋊C4—21no—
78C7×C22⋊C4—21no—
79C7×C22⋊C4—21no—
80C7×C22⋊C4D411yescharacteristic
81C7×C22⋊C4D411yescharacteristic
82C42×C7—21no—
83C42×C7—21no—
84C42×C7C2311yescharacteristic
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88#1142—22no—
89C4⋊Dic7—21no—
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98C4×Dic7—21no—
99C4×Dic7—21no—
100#1136—22no—
101C2×C7⋊C8—22no—
102C2×C7⋊C8C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106#555D711yescharacteristic
107#552—71no—
108#501—71no—
109#485—71no—
110#485—71no—
111#482—71no—
112#481—71no—
113#478—71no—
114#475—71no—
115#468—71no—
116#465—71no—
117#440—71no—
118#435—71no—
119#428—71no—
120#427—71no—
121C23×C7—21no—
122C23×C7C2×D411yescharacteristic
123C23×C7C2×D411yescharacteristic
124C7×Q8—81no—
125C7×D4—21no—
126C7×D4—21no—
127C7×D4—41no—
128C7×D4—81no—
129C7×D4—81no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—42no—
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2411yescharacteristic
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—42no—
148Dic14—21no—
149Dic14—21no—
150C7⋊C8—21no—
151C7⋊C8—21no—
152C7⋊C8—21no—
153C7⋊C8#4112yes—
154C2×SD16—71no—
155C2×M16—71no—
156C22×C8—71no—
157C4⋊Q8C2×D711yescharacteristic
158#126—141no—
159#125—71no—
160#125—141no—
161#124—21no—
162#124—21no—
163#123—141no—
164C22⋊Q8—71no—
165C22⋊Q8—71no—
166#121—21no—
167#121C2×D711yescharacteristic
168#121C2×D711yescharacteristic
169C4×Q8—71no—
170C4×D4C2×D711yescharacteristic
171C4×D4C2×D711yescharacteristic
172#117—21no—
173#117—71no—
174#117—141no—
175#117C2×D711yescharacteristic
176#117C2×D711yescharacteristic
177C2×C4⋊C4—71no—
178#107—71no—
179#107—71no—
180#107—71no—
181#107—71no—
182#106—71no—
183#106—71no—
184#106—71no—
185#106—71no—
186#106—71no—
187C4⋊C8—71no—
188#103—71no—
189#103—71no—
190#103—71no—
191#102—71no—
192#102—71no—
193#102—71no—
194C22⋊C8—71no—
195C22⋊C8—71no—
196#97—71no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7C22×D411yescharacteristic
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7C2×#4111yescharacteristic
215C4×C72−1+411yescharacteristic
216Dic7—41no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220C2×Q8—21no—
221C2×Q8—71no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4C22⋊D711yescharacteristic
225C2×D4C22×D711yescharacteristic
226C22×C4—21no—
227C22×C4—141no—
228C22×C4C22×D711yescharacteristic
229C22×C4C22×D711yescharacteristic
230SD16—142no—
231M16—142no—
232C2×C8—71no—
233C2×C8—71no—
234C2×C8—71no—
235C2×C8—71no—
236C2×C8—142no—
237C4⋊C4—21no—
238C4⋊C4—71no—
239C4⋊C4—71no—
240C4⋊C4—71no—
241C4⋊C4—141no—
242C4⋊C4—141no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
247C4⋊C4C22⋊D711yescharacteristic
248C4⋊C4C22×D711yescharacteristic
249C4⋊C4C22×D711yescharacteristic
250C4⋊C4C22×D711yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—141no—
257C22⋊C4—141no—
258C22⋊C4—141no—
259C22⋊C4C22⋊D711yescharacteristic
260C22⋊C4C22⋊D711yescharacteristic
261C42—21no—
262C42—21no—
263C42—141no—
264C42—141no—
265C42C22×D711yescharacteristic
266C2×C7—41no—
267C2×C7—41no—
268C2×C7—81no—
269C2×C7#54711yescharacteristic
270C2×C7#57511yescharacteristic
271C2×C7#57711yescharacteristic
272C23—21no—
273C23C2×C22⋊D711yescharacteristic
274C23C2×C22⋊D711yescharacteristic
275Q8—81no—
276Q8—141no—
277Q8—141no—
278D4—21no—
279D4—21no—
280D4—41no—
281D4—81no—
282D4—81no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—282no—
296C2×C4C2×C22⋊D711yescharacteristic
297C2×C4C2×C22⋊D711yescharacteristic
298C2×C4C2×C22⋊D711yescharacteristic
299C2×C4C2×C22⋊D711yescharacteristic
300C2×C4C23×D711yescharacteristic
301C8—72no—
302C8—141no—
303C8—141no—
304C8—141no—
305C7#334811yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22C22×C22⋊D711yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—281no—
324C4—281no—
325C4—281no—
326C4—281no—
327C4D7×#4111yescharacteristic
328C4#655411yescharacteristic
329C2—41no—
330C2—41no—
331C2—81no—
332C2#9051311yescharacteristic
333C2#9051511yescharacteristic
334C2#9052711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
Fitting, prime radicalC7×#555C2
socleC22×C7C22×D4
hypercenter#555D7
hypocenterC7#3348
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
2. #90232C2semidirect
3. #89866C2semidirect
4. #89852C2semidirect
5. #89814C2semidirect
6. #89813C2semidirect
7. #89807C2semidirect
8. #89806C2semidirect
9. #89790C2semidirect
10. #89783C2semidirect
11. #89781C2semidirect
12. #89770C2semidirect
13. #89768C2semidirect
14. #89768C2semidirect
15. #89767C2semidirect
305. C7#3348semidirect

Maximal subgroups

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

subgroupindex
C7×#5552
#902322
#898662
#898522
#898142
#898132
#898072
#898062
#897902
#897832
#897812
#897702
#897682
#897672
#33487