Finite Group Databasebeta

← #11738067

Subgroups — #11738067

Order 896.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90368C211yesmaximal, characteristic
2#90272C211yesmaximal, characteristic, complemented
3C7×#464C211yesmaximal, characteristic, complemented
4#89961C211yesmaximal, characteristic, complemented
5#89958C211yesmaximal, characteristic, complemented
6#89946C211yesmaximal, characteristic, complemented
7#89945C211yesmaximal, characteristic, complemented
8#89828C211yesmaximal, characteristic, complemented
9C4×#1144C211yesmaximal, characteristic, complemented
10#89610C211yesmaximal, characteristic, complemented
11#89601C211yesmaximal, characteristic, complemented
12#89581C211yesmaximal, characteristic, complemented
13#89574C211yesmaximal, characteristic, complemented
14#89551C211yesmaximal, characteristic, complemented
15#89543C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×#1159C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C7×#124C2211yescharacteristic
21C4×C7×Q8C2211yescharacteristic
22#6509C2211yescharacteristic
23#6509C2211yescharacteristic
24#6509C2211yescharacteristic
25C2×#1144C2211yescharacteristic
26#6501C2211yescharacteristic
27C2×#1131C2211yescharacteristic
28C2×#1131C2211yescharacteristic
29#6457—21no—
30#6455C2211yescharacteristic
31D7×C4⋊C4C2211yescharacteristic
32#6452—21no—
33#6448—21no—
34#6447—21no—
35(C22⋊D7)⋊C4—21no—
36#6441—21no—
37#6439—21no—
38C4×D28C2211yescharacteristic
39#6435—21no—
40#6433C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#102C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6406C2211yescharacteristic
49#6406C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51(C2×D7)⋊C8C2211yescharacteristic
52#6392C2211yescharacteristic
53#6391C2211yescharacteristic
54#6390C2211yescharacteristic
55#6390C2211yescharacteristic
56#6384C2211yescharacteristic
57#6381C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3327—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C22×Dic7—21no—
66Q8×D7—22no—
67Q8×D7—41no—
68#1159—22no—
69#1159—41no—
70C2×D28D411yescharacteristic
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×C4×D7C2311yescharacteristic
74C2×C4×D7C2311yescharacteristic
75C2×Dic14—21no—
76C2×Dic14C2311yescharacteristic
77C7×Q16—41no—
78C7×SD16—41no—
79C2×C7×C8C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C7×C4⋊C4—21no—
82C7×C4⋊C4C2311yescharacteristic
83C7×C22⋊C4—21no—
84C7×C22⋊C4—21no—
85C42×C7—21no—
86C42×C7C2311yescharacteristic
87C22⋊Dic7—21no—
88#1144—22no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4D411yescharacteristic
94(C2×D7)⋊C4D411yescharacteristic
95C4⋊Dic7D411yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106C4×Dic7—21no—
107C4×Dic7—21no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110#1131—41no—
111#1131—41no—
112#552—71no—
113#548—71no—
114#488—71no—
115#484—71no—
116#482—71no—
117#479—71no—
118#470—71no—
119#468—71no—
120#464D711yescharacteristic, complemented
121#452—71no—
122#451—71no—
123C4×Q16—71no—
124#436—71no—
125#419—71no—
126#417—71no—
127C23×C7—21no—
128C22×D7C2×D411yescharacteristic
129C22×D7C2×D411yescharacteristic
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—21no—
134C7×D4—21no—
135C7×D4—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2411yescharacteristic
142C22⋊D7—42no—
143C22⋊D7—42no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—42no—
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151D28—41no—
152C4×D7—21no—
153C4×D7—21no—
154C4×D7—21no—
155C4×D7—21no—
156C4×D7—42no—
157C4×D7—42no—
158Dic14—21no—
159Dic14—42no—
160Dic14#4112yes—
161C7×C8—21no—
162C7×C8—21no—
163C7⋊C8—21no—
164C7⋊C8—21no—
165C2×#41—71no—
166C22×Q8—71no—
167C2×Q16—71no—
168C2×Q16C2×D711yescharacteristic
169C2×SD16C2×D711yescharacteristic
170C2×M16—71no—
171C2×M16—71no—
172#126—141no—
173#125—71no—
174#125—141no—
175#124—141no—
176#124C2×D711yescharacteristic
177#123—141no—
178C22⋊Q8—71no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182#121—71no—
183#121—141no—
184C4×Q8—71no—
185C4×Q8C2×D711yescharacteristic
186C4×D4—71no—
187C4×D4—141no—
188#117—71no—
189#117—141no—
190C2×C4⋊C4—71no—
191#107—71no—
192#107—71no—
193#106—71no—
194C4⋊C8C2×D711yescharacteristic
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103C2×D711yescharacteristic
202#102—71no—
203#102—71no—
204#102C2×D711yescharacteristic
205C22⋊C8—71no—
206C22⋊C8—71no—
207C4×C8—71no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7C22×D411yescharacteristic
212C2×D7—21no—
213C2×D7—21no—
214C2×D7—21no—
215C2×D7—21no—
216C2×D7—21no—
217C2×D7—21no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7C2×#4111yescharacteristic
224C4×C7H(5,𝔽2)11yescharacteristic
225Dic7—22no—
226Dic7—21no—
227Dic7—21no—
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231#41—142no—
232#41—281no—
233C2×Q8—71no—
234C2×Q8—141no—
235C2×Q8—142no—
236C2×Q8—281no—
237C2×Q8C22×D711yescharacteristic
238C2×Q8C22×D711yescharacteristic
239C2×D4—71no—
240C2×D4—141no—
241C2×D4—141no—
242C2×D4C22×D711yescharacteristic
243C22×C4—71no—
244C22×C4—71no—
245C22×C4—141no—
246C22×C4—141no—
247C22×C4—141no—
248Q16—41no—
249Q16—142no—
250SD16—41no—
251M16—281no—
252M16—281no—
253C2×C8—71no—
254C2×C8—71no—
255C2×C8C22×D711yescharacteristic
256C2×C8C22×D711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—71no—
259C4⋊C4—71no—
260C4⋊C4—71no—
261C4⋊C4—71no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4C22×D711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—71no—
273C22⋊C4—71no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C42—21no—
280C42—141no—
281C42—141no—
282C42C22×D711yescharacteristic
283C2×C7—81no—
284C2×C7#54611yescharacteristic
285C2×C7#57511yescharacteristic
286C2×C7#57811yescharacteristic
287D7—41no—
288D7—41no—
289C23—21no—
290C23—71no—
291C23—71no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—72no—
297Q8—141no—
298Q8—282no—
299D4—21no—
300D4—21no—
301D4—281no—
302D4—282no—
303D4—282no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—71no—
308C2×C4—71no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—282no—
318C2×C4—282no—
319C2×C4—282no—
320C2×C4D4×D711yescharacteristic
321C2×C4D4×D711yescharacteristic
322C2×C4C23×D711yescharacteristic
323C8—21no—
324C8—21no—
325C8—141no—
326C8—141no—
327C7#332711yescharacteristic, complemented, Sylow
328C22—41no—
329C22—41no—
330C22—41no—
331C22—141no—
332C22—141no—
333C22—141no—
334C22—141no—
335C22—141no—
336C22—141no—
337C22C2×D4×D711yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—141no—
344C4—141no—
345C4—142no—
346C4—281no—
347C4—281no—
348C4—281no—
349C4#654811yescharacteristic
350C4D7×#4111yescharacteristic
351C2—81no—
352C2—281no—
353C2—281no—
354C2#9030311yescharacteristic
355C2#9045911yescharacteristic
356C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#464C2
socleC22×C7C22×D4
prime radical#90368C2
hypercenter#464D7
hypocenterC7#3327
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #90272C2semidirect
3. C7×#464C2semidirect
4. #89961C2semidirect
5. #89958C2semidirect
6. #89946C2semidirect
7. #89945C2semidirect
8. #89828C2semidirect
9. C4×#1144C2semidirect
10. #89610C2semidirect
11. #89601C2semidirect
12. #89581C2semidirect
13. #89574C2semidirect
14. #89551C2semidirect
15. #89543C2semidirect
120. #464D7semidirect
327. C7#3327semidirect

Maximal subgroups

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

subgroupindex
#903682
#902722
C7×#4642
#899612
#899582
#899462
#899452
#898282
C4×#11442
#896102
#896012
#895812
#895742
#895512
#895432
#33277