Finite Group Databasebeta

← #11738507

Subgroups — #11738507

Order 896.

398 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90420C211yesmaximal, characteristic, complemented
2#90236C211yesmaximal, characteristic
3C7×#476C211yesmaximal, characteristic, complemented
4#89859C211yesmaximal, characteristic, complemented
5#89781C211yesmaximal, characteristic, complemented
6#89643C211yesmaximal, characteristic, complemented
7#89642C211yesmaximal, characteristic, complemented
8#89633C211yesmaximal, characteristic, complemented
9#89632C211yesmaximal, characteristic, complemented
10#89619C211yesmaximal, characteristic, complemented
11#89611C211yesmaximal, characteristic, complemented
12#89542C211yesmaximal, characteristic
13D7×#102C211yesmaximal, characteristic
14#89535C211yesmaximal, characteristic, complemented
15#89531C211yesmaximal, characteristic, complemented
16C2×D4×D7C2211yescharacteristic
17C7×C4⋊Q8C2211yescharacteristic, complemented
18C4×C7×D4C2211yescharacteristic
19#6509—21no—
20#6507—21no—
21D4×Dic7C2211yescharacteristic
22C4×C22⋊D7—21no—
23C2×(C2×D7)⋊C4—21no—
24C2×#1131C2211yescharacteristic
25C2×C8×D7C2211yescharacteristic
26#6459—21no—
27#6456C2211yescharacteristic
28#6455C2211yescharacteristic
29#6455C2211yescharacteristic
30D7×C4⋊C4C2211yescharacteristic
31#6453C2211yescharacteristic, complemented
32#6452—21no—
33#6451C2211yescharacteristic, complemented
34(C22⋊D7)⋊C4—21no—
35D7×C22⋊C4—21no—
36C4×D28C2211yescharacteristic
37#6435C2211yescharacteristic
38#6433C2211yescharacteristic, complemented
39C7×#106C2211yescharacteristic, complemented
40C7×#106C2211yescharacteristic, complemented
41C7×C4⋊C8C2211yescharacteristic, complemented
42C7×#102C2211yescharacteristic
43C7×#102C2211yescharacteristic
44#6406C2211yescharacteristic
45#6406C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48#6392C2211yescharacteristic, complemented
49#6391C2211yescharacteristic, complemented
50Dic7⋊C8C2211yescharacteristic, complemented
51Dic7⋊C8C2211yescharacteristic, complemented
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic, complemented
55#6381C2211yescharacteristic, complemented
56#6381C2211yescharacteristic, complemented
57#6381C2211yescharacteristic, complemented
58C4⋊(C7⋊C8)C2211yescharacteristic, complemented
59#3081—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×Q8—21no—
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7—21no—
64C2×C22⋊D7—21no—
65C22×Dic7—21no—
66D4×D7—22no—
67D4×D7Q814yescomplemented
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7—21no—
72C2×C4×D7C2311yescharacteristic
73C2×Dic14—21no—
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C7×C4⋊C4—21no—
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C22⋊C4—21no—
81C42×C7C2311yescharacteristic
82C22⋊Dic7—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4D411yescharacteristic
88(C2×D7)⋊C4D411yescharacteristic
89C4⋊Dic7—21no—
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4D411yescharacteristic
99Dic7⋊C4D411yescharacteristic
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105#1131—22no—
106C8×D7—22no—
107#544—71no—
108#518—71no—
109#478—71no—
110#478—71no—
111#478—71no—
112#478—71no—
113#476—71no—
114#476D711yescharacteristic, complemented
115#474—71no—
116#474—71no—
117#428—71no—
118#427—71no—
119#424—71no—
120#418—71no—
121C2×#102—71no—
122C23×C7—21no—
123C22×D7—21no—
124C22×D7—24no—
125C22×D7—81no—
126C22×D7—81no—
127C22×D7C2×D411yescharacteristic
128C7×Q8—81no—
129C7×D4—21no—
130C7×D4C2×Q812yes—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—81no—
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2411yescharacteristic
138C22⋊D7—24no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—81no—
143C2×Dic7C2×D411yescharacteristic
144C2×Dic7C2×D411yescharacteristic
145C2×Dic7C2×D411yescharacteristic
146D28—21no—
147D28C2×Q812yes—
148C4×D7—81no—
149C4×D7—81no—
150C4×D7—81no—
151C4×D7C2×Q812yes—
152C4×D7#4112yes—
153Dic14—81no—
154C7×C8—21no—
155C7×C8—21no—
156C7⋊C8—21no—
157C7⋊C8—21no—
158C22×D4—71no—
159C2×M16—71no—
160C22×C8—71no—
161C4⋊Q8—71no—
162C4⋊Q8—141no—
163C4⋊Q8C2×D711yescharacteristic, complemented
164#125—71no—
165#125—71no—
166#125—141no—
167C22⋊Q8—141no—
168C22⋊Q8—141no—
169C4×D4—71no—
170C4×D4—71no—
171C4×D4—71no—
172C4×D4—141no—
173C4×D4—141no—
174C4×D4C2×D711yescharacteristic
175#117—71no—
176#117—71no—
177#117—71no—
178C2×C4⋊C4—71no—
179C2×C22⋊C4—141no—
180C2×C22⋊C4—141no—
181#107—71no—
182#107—71no—
183#107—71no—
184#107—71no—
185#106—71no—
186#106—71no—
187#106C2×D711yescharacteristic, complemented
188#106C2×D711yescharacteristic, complemented
189C4⋊C8—71no—
190C4⋊C8—71no—
191C4⋊C8—71no—
192C4⋊C8C2×D711yescharacteristic, complemented
193#102—71no—
194#102—71no—
195#102—71no—
196#102—71no—
197#102—71no—
198#102—71no—
199#102C2×D711yescharacteristic
200#102C2×D711yescharacteristic
201C22×C7—22no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7C22×D411yescharacteristic
205C2×D7—21no—
206C2×D7—21no—
207C2×D7—22no—
208C2×D7—41no—
209C2×D7—41no—
210C2×D7—84no—
211C2×D7C22⋊Q812yes—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7C22×Q811yescharacteristic
218C4×C7C2×#4111yescharacteristic
219Dic7—21no—
220Dic7—21no—
221Dic7—41no—
222Dic7—41no—
223Dic7—41no—
224Dic7C22⋊Q812yes—
225C24—141no—
226C2×Q8—21no—
227C2×Q8—141no—
228C2×D4—74no—
229C2×D4—71no—
230C2×D4—142no—
231C2×D4—141no—
232C2×D4C22×D711yescharacteristic
233C22×C4—21no—
234C22×C4—71no—
235C22×C4—141no—
236C22×C4—141no—
237C22×C4—141no—
238C22×C4—141no—
239M16—142no—
240C2×C8—71no—
241C2×C8—71no—
242C2×C8—142no—
243C2×C8C22×D711yescharacteristic
244C2×C8C22×D711yescharacteristic
245C4⋊C4—21no—
246C4⋊C4—71no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—71no—
250C4⋊C4—71no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4C22×D711yescharacteristic
258C4⋊C4C22×D711yescharacteristic
259C4⋊C4C22×D711yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—71no—
262C22⋊C4—71no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C42—71no—
269C42—141no—
270C42—141no—
271C42C22×D711yescharacteristic
272C2×C7—42no—
273C2×C7C2×C22⋊Q811yescharacteristic
274C2×C7#57611yescharacteristic
275C2×C7#57711yescharacteristic
276D7—22no—
277D7—42no—
278C23—21no—
279C23—71no—
280C23—144no—
281C23—141no—
282C23—561no—
283C23—561no—
284Q8—81no—
285Q8—561no—
286D4—21no—
287D4—72no—
288D4—144no—
289D4—141no—
290D4Q8×D712yes—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—71no—
295C2×C4—72no—
296C2×C4—72no—
297C2×C4—71no—
298C2×C4—71no—
299C2×C4—81no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—561no—
304C2×C4—561no—
305C2×C4—561no—
306C2×C4—561no—
307C2×C4D4×D711yescharacteristic
308C2×C4D4×D711yescharacteristic
309C2×C4C23×D711yescharacteristic
310C8—21no—
311C8—21no—
312C8—141no—
313C8—141no—
314C7#308111yescharacteristic, complemented, Sylow
315C22—22no—
316C22—41no—
317C22—41no—
318C22—72no—
319C22—141no—
320C22—141no—
321C22—142no—
322C22—281no—
323C22—281no—
324C22—564no—
325C22C2×D4×D711yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—72no—
332C4—141no—
333C4—141no—
334C4—281no—
335C4—281no—
336C4—281no—
337C4C2×Q8×D711yescharacteristic
338C4D7×#4111yescharacteristic
339C2—42no—
340C2—142no—
341C2—282no—
342C2D7×C22⋊Q811yescharacteristic
343C2#9045311yescharacteristic
344C2#9046511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#476C2
socleC22×C7C22×D4
prime radicalC2×D4×D7C22
hypercenter#476D7
hypocenterC7#3081
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #90420C2semidirect
3. C7×#476C2semidirect
4. #89859C2semidirect
5. #89781C2semidirect
6. #89643C2semidirect
7. #89642C2semidirect
8. #89633C2semidirect
9. #89632C2semidirect
10. #89619C2semidirect
11. #89611C2semidirect
14. #89535C2semidirect
15. #89531C2semidirect
17. C7×C4⋊Q8C22semidirect
31. #6453C22semidirect
33. #6451C22semidirect
38. #6433C22semidirect
39. C7×#106C22semidirect
40. C7×#106C22semidirect
41. C7×C4⋊C8C22semidirect
48. #6392C22semidirect
49. #6391C22semidirect
50. Dic7⋊C8C22semidirect
51. Dic7⋊C8C22semidirect
54. #6382C22semidirect
55. #6381C22semidirect
56. #6381C22semidirect
57. #6381C22semidirect
58. C4⋊(C7⋊C8)C22semidirect
67. D4×D7Q8semidirect
114. #476D7semidirect
163. C4⋊Q8C2×D7semidirect
187. #106C2×D7semidirect
188. #106C2×D7semidirect
192. C4⋊C8C2×D7semidirect
314. C7#3081semidirect

Maximal subgroups

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

subgroupindex
#904202
#902362
C7×#4762
#898592
#897812
#896432
#896422
#896332
#896322
#896192
#896112
#895422
D7×#1022
#895352
#895312
#30817