Finite Group Databasebeta

← #11737662

Subgroups — #11737662

Order 896.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90518C211yesmaximal, characteristic
2#90316C211yesmaximal, characteristic, complemented
3C7×#452C211yesmaximal, characteristic
4#90011C211yesmaximal, characteristic
5Q16×Dic7C211yesmaximal, characteristic, complemented
6#89954C211yesmaximal, characteristic, complemented
7#89936C211yesmaximal, characteristic, complemented
8#89934C211yesmaximal, characteristic, complemented
9#89815C211yesmaximal, characteristic, complemented
10#89568C211yesmaximal, characteristic, complemented
11#89561C211yesmaximal, characteristic, complemented
12#89538C211yesmaximal, characteristic, complemented
13#89530C211yesmaximal, characteristic, complemented
14#89494C211yesmaximal, characteristic, complemented
15#89492C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×C7×Q16C2211yescharacteristic
18C2×C7×SD16C2211yescharacteristic
19C7×C22⋊Q8C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22Q8×Dic7C2211yescharacteristic
23#6507C2211yescharacteristic
24C2×#1144C2211yescharacteristic
25D4×Dic7—21no—
26D4×Dic7—21no—
27D4×Dic7C2211yescharacteristic
28D4×Dic7C2211yescharacteristic
29C2×#1142C2211yescharacteristic
30#6489—21no—
31#6489C2211yescharacteristic
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7C2211yescharacteristic
34#6487C2211yescharacteristic
35#6453—21no—
36#6453C2211yescharacteristic
37#6452—21no—
38#6450C2211yescharacteristic
39#6442—21no—
40#6442—21no—
41C7×#103C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×C22⋊C8C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6406C2211yescharacteristic
48C22⋊(C7⋊C8)C2211yescharacteristic
49#6392C2211yescharacteristic
50#6392C2211yescharacteristic
51#6390C2211yescharacteristic
52#6390C2211yescharacteristic
53#6389C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55Dic7⋊C8C2211yescharacteristic
56C8×Dic7C2211yescharacteristic
57#6382C2211yescharacteristic
58#6382C2211yescharacteristic
59#3120—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—22no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×Q8C2311yescharacteristic
64C2×C7×D4—21no—
65C2×C7×D4C2311yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7C2311yescharacteristic
68C22×Dic7—21no—
69C22×Dic7—21no—
70C22×Dic7—21no—
71C2×Dic14C2311yescharacteristic
72C7×Q16—22no—
73C7×SD16—22no—
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C7×C4⋊C4—21no—
77C7×C4⋊C4C2311yescharacteristic
78C7×C22⋊C4—21no—
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7D411yescharacteristic
83C22⋊Dic7D411yescharacteristic
84#1144—22no—
85#1142—22no—
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7D411yescharacteristic
92C4⋊Dic7D411yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7C2311yescharacteristic
106C4×Dic7C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#541—71no—
110#520—71no—
111#484—71no—
112#483—71no—
113#479—71no—
114#478—71no—
115#462—71no—
116#461—71no—
117#452—71no—
118#452D711yescharacteristic
119#441—71no—
120#440—71no—
121C4×Q16—71no—
122C4×SD16—71no—
123#433—71no—
124C23×C7—21no—
125C23×C7C2×D411yescharacteristic
126C7×Q8—21no—
127C7×Q8—21no—
128C7×Q8—21no—
129C7×Q8#4112yes—
130C7×D4—24no—
131C7×D4—21no—
132C7×D4#4112yes—
133C2×C4×C7—24no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2411yescharacteristic
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—81no—
146C2×Dic7—81no—
147C2×Dic7—81no—
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151C2×Dic7C2×D411yescharacteristic
152Dic14—21no—
153Dic14—21no—
154C7×C8—21no—
155C7×C8—21no—
156C7⋊C8—21no—
157C7⋊C8—21no—
158C2×#41C2×D711yescharacteristic
159C2×Q16—71no—
160C2×Q16C2×D711yescharacteristic
161C2×SD16—71no—
162C2×SD16C2×D711yescharacteristic
163C4⋊Q8—71no—
164#126—141no—
165#126—141no—
166#125—71no—
167#125—141no—
168#125—141no—
169C22⋊Q8—71no—
170C22⋊Q8C2×D711yescharacteristic
171C4×Q8—71no—
172C4×Q8—71no—
173C4×Q8—71no—
174C4×Q8—71no—
175C4×D4—71no—
176C4×D4—71no—
177C4×D4—141no—
178C4×D4—141no—
179#117—71no—
180#117—141no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—141no—
183#107—71no—
184#107—71no—
185#106—71no—
186#106—71no—
187C4⋊C8—71no—
188C4⋊C8—71no—
189#103—71no—
190#103—71no—
191#103—71no—
192#103—71no—
193#103—71no—
194#103C2×D711yescharacteristic
195#102—71no—
196#102C2×D711yescharacteristic
197C22⋊C8—71no—
198C22⋊C8C2×D711yescharacteristic
199#97—71no—
200C4×C8—71no—
201C22×C7—21no—
202C22×C7—22no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7C22×D411yescharacteristic
208C4×C7—21no—
209C4×C7—22no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7C2×#4111yescharacteristic
213C4×C7C2×#4111yescharacteristic
214Dic7—21no—
215Dic7—21no—
216Dic7—21no—
217Dic7—21no—
218Dic7—41no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222#41—22no—
223#41—22no—
224C2×Q8—71no—
225C2×Q8C22×D711yescharacteristic
226C2×Q8C22×D711yescharacteristic
227C2×D4—21no—
228C2×D4C22×D711yescharacteristic
229C22×C4—21no—
230C22×C4—141no—
231C22×C4—141no—
232C22×C4—141no—
233C22×C4C22×D711yescharacteristic
234Q16—22no—
235Q16—142no—
236SD16—22no—
237SD16—142no—
238C2×C8—71no—
239C2×C8—71no—
240C2×C8C22×D711yescharacteristic
241C2×C8C22×D711yescharacteristic
242C4⋊C4—21no—
243C4⋊C4—71no—
244C4⋊C4—71no—
245C4⋊C4—71no—
246C4⋊C4—71no—
247C4⋊C4—71no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4C22×D711yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—71no—
261C22⋊C4—71no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C42—71no—
266C42—71no—
267C42—141no—
268C42—141no—
269C42—141no—
270C42—141no—
271C2×C7—41no—
272C2×C7—42no—
273C2×C7#52511yescharacteristic
274C2×C7#57711yescharacteristic
275C2×C7#57811yescharacteristic
276C23—21no—
277C23D4×D711yescharacteristic
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—141no—
282Q8—141no—
283Q8#115912yes—
284D4—24no—
285D4—21no—
286D4#115912yes—
287C2×C4—24no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—71no—
294C2×C4—71no—
295C2×C4—71no—
296C2×C4—71no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—561no—
302C2×C4—561no—
303C2×C4—561no—
304C2×C4D4×D711yescharacteristic
305C2×C4C23×D711yescharacteristic
306C8—21no—
307C8—21no—
308C8—141no—
309C8—141no—
310C7#312011yescharacteristic, complemented, Sylow
311C22—21no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—41no—
316C22—41no—
317C22C2×D4×D711yescharacteristic
318C4—21no—
319C4—22no—
320C4—41no—
321C4—41no—
322C4—141no—
323C4—141no—
324C4—141no—
325C4—141no—
326C4—281no—
327C4—281no—
328C4—281no—
329C4—281no—
330C4C2×#115911yescharacteristic
331C4C2×#115911yescharacteristic
332C2—41no—
333C2—42no—
334C2#9027711yescharacteristic
335C2#9046211yescharacteristic
336C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#452C2
socleC22×C7C22×D4
prime radicalC2×C7×#41C22
hypercenter#452D7
hypocenterC7#3120
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #90316C2semidirect
5. Q16×Dic7C2semidirect
6. #89954C2semidirect
7. #89936C2semidirect
8. #89934C2semidirect
9. #89815C2semidirect
10. #89568C2semidirect
11. #89561C2semidirect
12. #89538C2semidirect
13. #89530C2semidirect
14. #89494C2semidirect
15. #89492C2semidirect
310. C7#3120semidirect

Maximal subgroups

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

subgroupindex
#905182
#903162
C7×#4522
#900112
Q16×Dic72
#899542
#899362
#899342
#898152
#895682
#895612
#895382
#895302
#894942
#894922
#31207