Finite Group Databasebeta

← #11733439

Subgroups — #11733439

Order 896.

456 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6461C211yesmaximal, characteristic
2C7×#389C211yesmaximal, characteristic, complemented
3#89762C211yesmaximal, characteristic
4#89761C211yesmaximal, characteristic
5#89761C211yesmaximal, characteristic
6#89753C211yesmaximal, characteristic, complemented
7#89748C211yesmaximal, characteristic, complemented
8#89745C211yesmaximal, characteristic, complemented
9#89744C211yesmaximal, characteristic, complemented
10#89712C211yesmaximal, characteristic
11#89705C211yesmaximal, characteristic, complemented
12#89441C211yesmaximal, characteristic
13#89440C211yesmaximal, characteristic
14#89425C211yesmaximal, characteristic, complemented
15#89423C211yesmaximal, characteristic, complemented
16C22×C4×D7C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C42×C7C2211yescharacteristic
22C2×(C2×D7)⋊C4C2211yescharacteristic
23C2×(C2×D7)⋊C4C2211yescharacteristic
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37C2×C4×Dic7C2211yescharacteristic
38C2×C4×Dic7C2211yescharacteristic
39#6461C418yescomplemented
40C7×#95C2211yescharacteristic
41C7×#95C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52#2383—71nomaximal, Sylow
53C23×D7C2311yescharacteristic
54C22×C4×C7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×Dic7C2311yescharacteristic
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C2×C4×D7—41no—
69C2×C4×D7—41no—
70C2×C4×D7—41no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—41no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4C2×C414yescomplemented
75C42×C7—41no—
76(C2×D7)⋊C4—41no—
77(C2×D7)⋊C4—41no—
78(C2×D7)⋊C4—41no—
79(C2×D7)⋊C4C2×C414yes—
80(C2×D7)⋊C4C2×C414yes—
81(C2×D7)⋊C4C2×C414yes—
82C4⋊Dic7—41no—
83C4⋊Dic7C2×C414yescomplemented
84Dic7⋊C4—41no—
85Dic7⋊C4—41no—
86Dic7⋊C4—41no—
87Dic7⋊C4—41no—
88Dic7⋊C4—41no—
89Dic7⋊C4C2×C414yescomplemented
90C4×Dic7—41no—
91C4×Dic7—41no—
92C4×Dic7C2×C414yescomplemented
93C2×#126—71no—
94#389—71no—
95#389—71no—
96#389D711yescharacteristic, complemented
97#388—71no—
98#388—71no—
99#388—71no—
100#387—71no—
101#387—71no—
102#387—71no—
103#385—71no—
104#385—71no—
105#385—71no—
106#384—71no—
107#383—71no—
108C23×C7C2411yescharacteristic
109C22×D7—21no—
110C22×D7—21no—
111C22×D7—21no—
112C22×D7—21no—
113C22×D7—21no—
114C22×D7—21no—
115C22×D7C22×C412yes—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7C22×C412yes—
135C2×C4×C7C22×C412yes—
136C2×C4×C7C22×C412yes—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7C22×C412yes—
156C2×Dic7C22×C412yes—
157C2×Dic7C22×C412yes—
158C4×D7—44no—
159C23×C4—71no—
160#126—78no—
161C2×C4⋊C4—71no—
162C2×C4⋊C4—71no—
163C2×C4⋊C4—71no—
164C2×C4⋊C4—71no—
165C2×C4⋊C4—71no—
166C2×C4⋊C4—71no—
167C2×C4⋊C4—71no—
168C2×C4⋊C4—71no—
169C2×C4⋊C4C2×D711yescharacteristic
170C2×C4⋊C4C2×D711yescharacteristic
171C2×C4⋊C4C2×D711yescharacteristic
172C2×C4⋊C4C2×D711yescharacteristic
173C2×C22⋊C4—71no—
174C2×C22⋊C4—71no—
175C2×C22⋊C4—71no—
176C2×C22⋊C4—71no—
177C2×C22⋊C4—71no—
178C2×C22⋊C4—71no—
179C2×C42—71no—
180C2×C42—71no—
181C2×C42—71no—
182C2×C42C2×D711yescharacteristic
183#95—71no—
184#95—71no—
185#95—71no—
186#95—71no—
187#95—71no—
188#95—71no—
189#95—71no—
190#95—71no—
191#95—71no—
192#95—71no—
193#95C2×D711yescharacteristic
194#95C2×D711yescharacteristic
195C22×C7C23×C411yescharacteristic
196C22×C7H(5,𝔽2)11yescharacteristic
197C22×C7H(5,𝔽2)11yescharacteristic
198C22×C7H(5,𝔽2)11yescharacteristic
199C22×C72−1+411yescharacteristic
200C22×C72−1+411yescharacteristic
201C22×C72−1+411yescharacteristic
202C2×D7—22no—
203C2×D7—22no—
204C2×D7—22no—
205C2×D7—41no—
206C2×D7—41no—
207C2×D7—41no—
208C2×D7—41no—
209C4×C7—22no—
210C4×C7—22no—
211C4×C7—22no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216Dic7—22no—
217Dic7—22no—
218Dic7—22no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222Dic7—41no—
223C24—71no—
224C22×C4—71no—
225C22×C4—71no—
226C22×C4—71no—
227C22×C4—71no—
228C22×C4—71no—
229C22×C4—71no—
230C22×C4—71no—
231C22×C4—281no—
232C22×C4—281no—
233C22×C4—281no—
234C22×C4C22×D711yescharacteristic
235C22×C4C22×D711yescharacteristic
236C22×C4C22×D711yescharacteristic
237C22×C4C22×D711yescharacteristic
238C22×C4C22×D711yescharacteristic
239C22×C4C22×D711yescharacteristic
240C22×C4C22×D711yescharacteristic
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—74no—
245C4⋊C4—74no—
246C4⋊C4—281no—
247C4⋊C4—281no—
248C4⋊C4—281no—
249C4⋊C4—281no—
250C4⋊C4—281no—
251C4⋊C4—281no—
252C4⋊C4C4×D714yescomplemented
253C22⋊C4—74no—
254C22⋊C4—74no—
255C22⋊C4—74no—
256C22⋊C4—281no—
257C22⋊C4—281no—
258C22⋊C4—281no—
259C42—41no—
260C42—74no—
261C42—281no—
262C42—281no—
263C2×C7#52011yescharacteristic
264C2×C7#52011yescharacteristic
265C2×C7#52011yescharacteristic
266C2×C7#56011yescharacteristic
267C2×C7#56311yescharacteristic
268C2×C7#56311yescharacteristic
269C2×C7#56311yescharacteristic
270D7—42no—
271C23—72no—
272C23—141no—
273C23—141no—
274C23—141no—
275C23—141no—
276C23—141no—
277C23—141no—
278C23C23×D711yescharacteristic
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—72no—
298C2×C4—72no—
299C2×C4—72no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
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—141no—
318C2×C4—284no—
319C2×C4C2×C4×D712yes—
320C2×C4C2×C4×D712yes—
321C2×C4C2×C4×D712yes—
322C7#238311yescharacteristic, complemented, Sylow
323C22—142no—
324C22—142no—
325C22—142no—
326C22—281no—
327C22—281no—
328C22—281no—
329C22—281no—
330C22C22×C4×D711yescharacteristic
331C22#654811yescharacteristic
332C22#654811yescharacteristic
333C22#655111yescharacteristic
334C22#655111yescharacteristic
335C22#655311yescharacteristic
336C22#655411yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—142no—
345C4—142no—
346C4—142no—
347C4—281no—
348C4—281no—
349C4—281no—
350C4—281no—
351C2—282no—
352C2#9019511yescharacteristic
353C2#9023711yescharacteristic
354C2#9026511yescharacteristic
355C2#9031111yescharacteristic
356C2#9033711yescharacteristic
357C2#9039011yescharacteristic
358C2#9042611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutatorC22×C7C23×C4
FittingC7×#389C2
socle, cosocleC23×C7C24
prime radicalC23×D7C23
hypercenter#389D7
hypocenterC7#2383
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C7×#389C2semidirect
6. #89753C2semidirect
7. #89748C2semidirect
8. #89745C2semidirect
9. #89744C2semidirect
11. #89705C2semidirect
14. #89425C2semidirect
15. #89423C2semidirect
39. #6461C4semidirect
74. C7×C4⋊C4C2×C4semidirect
83. C4⋊Dic7C2×C4semidirect
89. Dic7⋊C4C2×C4semidirect
92. C4×Dic7C2×C4semidirect
96. #389D7semidirect
252. C4⋊C4C4×D7semidirect
322. C7#2383semidirect

Maximal subgroups

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

subgroupindex
C2×#64612
C7×#3892
#897622
#897612
#897532
#897482
#897452
#897442
#897122
#897052
#894412
#894402
#894252
#894232
#23837