Finite Group Databasebeta

← #11737259

Subgroups — #11737259

Order 896.

370 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#90263C211yesmaximal, characteristic, complemented
2#90252C211yesmaximal, characteristic
3C7×#440C211yesmaximal, characteristic, complemented
4#89942C211yesmaximal, characteristic
5#89938C211yesmaximal, characteristic, complemented
6C4×#1144C211yesmaximal, characteristic, complemented
7C4×#1142C211yesmaximal, characteristic, complemented
8#89660C211yesmaximal, characteristic, complemented
9#89647C211yesmaximal, characteristic, complemented
10#89580C211yesmaximal, characteristic, complemented
11#89563C211yesmaximal, characteristic, complemented
12#89551C211yesmaximal, characteristic
13#89539C211yesmaximal, characteristic, complemented
14#89488C211yesmaximal, characteristic, complemented
15#89481C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×C7×SD16C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19C4×C7×D4C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21C2×#1144C2211yescharacteristic
22#6501C2211yescharacteristic
23#6499C2211yescharacteristic
24C2×#1142C2211yescharacteristic
25#6492—21no—
26#6487—21no—
27C2×Dic28C2211yescharacteristic
28#6460C2211yescharacteristic
29#6455—21no—
30#6455C2211yescharacteristic
31#6451C2211yescharacteristic
32#6450—21no—
33#6450C2211yescharacteristic
34#6448—21no—
35#6442—21no—
36#6435—21no—
37#6435C2211yescharacteristic
38C4×Dic14—21no—
39C4×Dic14C2211yescharacteristic
40C4×Dic14C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×#97C2211yescharacteristic
45#6409C2211yescharacteristic
46#6406C2211yescharacteristic
47(C2×D7)⋊C8C2211yescharacteristic
48(C2×D7)⋊C8C2211yescharacteristic
49#6391C2211yescharacteristic
50#6390C2211yescharacteristic
51#6390C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54#6384C2211yescharacteristic
55#6384C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3121—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C2×C22⋊D7—21no—
64Q8×D7—22no—
65Q8×D7—22no—
66C2×C4×D7—21no—
67C2×C4×D7C2311yescharacteristic
68C2×Dic14—21no—
69C2×Dic14—21no—
70C2×Dic14C2311yescharacteristic
71C2×Dic14C2311yescharacteristic
72C7×SD16—22no—
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C22⋊C4—21no—
79C42×C7—21no—
80C42×C7C2311yescharacteristic
81C22⋊Dic7—21no—
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84#1144—22no—
85#1142—22no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4D411yescharacteristic
89(C2×D7)⋊C4D411yescharacteristic
90C4⋊Dic7—21no—
91C4⋊Dic7—21no—
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4D411yescharacteristic
99Dic7⋊C4D411yescharacteristic
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108Dic28—22no—
109#542—71no—
110#519—71no—
111#485—71no—
112#482—71no—
113#477—71no—
114#475—71no—
115#464—71no—
116#464—71no—
117#451—71no—
118#448—71no—
119#441—71no—
120#440D711yescharacteristic, complemented
121C4×Q16—71no—
122C4×SD16—71no—
123#433—71no—
124C23×C7—21no—
125C22×D7C2×D411yescharacteristic
126C7×Q8—21no—
127C7×Q8#4112yes—
128C7×D4—21no—
129C7×D4—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—81no—
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2411yescharacteristic
137C22⋊D7—81no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145C4×D7—24no—
146C4×D7—21no—
147C4×D7—21no—
148Dic14—24no—
149Dic14—21no—
150Dic14—21no—
151Dic14—21no—
152Dic14—81no—
153Dic14#4112yes—
154C7×C8—21no—
155C7×C8—21no—
156C7⋊C8—21no—
157C7⋊C8—21no—
158C22×Q8—71no—
159C2×Q16—71no—
160C2×Q16—71no—
161C2×SD16—71no—
162C2×SD16C2×D711yescharacteristic
163C4⋊Q8—71no—
164#126—141no—
165#124—71no—
166#124—141no—
167#123—141no—
168C22⋊Q8—71no—
169C22⋊Q8—141no—
170#121—71no—
171C4×Q8—71no—
172C4×Q8—71no—
173C4×Q8—71no—
174C4×Q8—71no—
175C4×Q8—141no—
176C4×Q8—141no—
177C4×Q8C2×D711yescharacteristic
178C4×D4C2×D711yescharacteristic
179#117—71no—
180#117—71no—
181#117—141no—
182#117—141no—
183#107—71no—
184#107C2×D711yescharacteristic
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⋊C8—71no—
199#97C2×D711yescharacteristic
200C4×C8—71no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7C22×D411yescharacteristic
205C2×D7—21no—
206C2×D7—21no—
207C2×D7—21no—
208C4×C7—22no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7C2×#4111yescharacteristic
214C4×C7C2×#4111yescharacteristic
215Dic7—21no—
216Dic7—22no—
217Dic7—21no—
218Dic7—21no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222C2×Q8—71no—
223C2×Q8—71no—
224C2×Q8—141no—
225C2×Q8—141no—
226C2×Q8—142no—
227C2×Q8—142no—
228C2×Q8C22×D711yescharacteristic
229C2×D4—141no—
230C2×D4C22×D711yescharacteristic
231C22×C4—21no—
232C22×C4—71no—
233C22×C4—141no—
234Q16—142no—
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—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4C22×D711yescharacteristic
254C4⋊C4C22×D711yescharacteristic
255C22⋊C4—21no—
256C22⋊C4—71no—
257C22⋊C4—71no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C42—21no—
264C42—71no—
265C42—141no—
266C42—141no—
267C42—141no—
268C42—141no—
269C42—141no—
270C42C22×D711yescharacteristic
271C2×C7—81no—
272C2×C7#52511yescharacteristic
273C2×C7#57711yescharacteristic
274C2×C7#57811yescharacteristic
275D7—41no—
276C23—21no—
277C23—71no—
278Q8—21no—
279Q8—72no—
280Q8—144no—
281Q8—141no—
282Q8—141no—
283Q8—141no—
284Q8—561no—
285Q8#115712yes—
286D4—21no—
287D4—21no—
288D4—561no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—71no—
293C2×C4—71no—
294C2×C4—71no—
295C2×C4—81no—
296C2×C4—144no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4D4×D711yescharacteristic
304C2×C4D4×D711yescharacteristic
305C2×C4C23×D711yescharacteristic
306C8—21no—
307C8—21no—
308C8—141no—
309C8—141no—
310C7#312111yescharacteristic, complemented, Sylow
311C22—41no—
312C22—41no—
313C22—41no—
314C22—141no—
315C22—141no—
316C22—141no—
317C22C2×D4×D711yescharacteristic
318C4—21no—
319C4—21no—
320C4—22no—
321C4—41no—
322C4—41no—
323C4—141no—
324C4—141no—
325C4—141no—
326C4—142no—
327C4—281no—
328C4—281no—
329C4—281no—
330C4C2×#115711yescharacteristic
331C4D7×#4111yescharacteristic
332C2—81no—
333C2—281no—
334C2#9023811yescharacteristic
335C2#9046611yescharacteristic
336C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#440C2
socleC22×C7C22×D4
prime radical#6501C22
hypercenter#440D7
hypocenterC7#3121
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #90263C2semidirect
3. C7×#440C2semidirect
5. #89938C2semidirect
6. C4×#1144C2semidirect
7. C4×#1142C2semidirect
8. #89660C2semidirect
9. #89647C2semidirect
10. #89580C2semidirect
11. #89563C2semidirect
13. #89539C2semidirect
14. #89488C2semidirect
15. #89481C2semidirect
120. #440D7semidirect
310. C7#3121semidirect

Maximal subgroups

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

subgroupindex
#902632
#902522
C7×#4402
#899422
#899382
C4×#11442
C4×#11422
#896602
#896472
#895802
#895632
#895512
#895392
#894882
#894812
#31217