Finite Group Databasebeta

← #11732407

Subgroups — #11732407

Order 896.

548 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D7×C4⋊C4C211yesmaximal, characteristic
2C7×#387C211yesmaximal, characteristic, complemented
3#89759C211yesmaximal, characteristic
4#89753C211yesmaximal, characteristic, complemented
5#89744C211yesmaximal, characteristic, complemented
6C4×(C2×D7)⋊C4C211yesmaximal, characteristic
7C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
8#89439C211yesmaximal, characteristic
9#89439C211yesmaximal, characteristic
10#89438C211yesmaximal, characteristic
11#89438C211yesmaximal, characteristic
12#89426C211yesmaximal, characteristic, complemented
13#89425C211yesmaximal, characteristic, complemented
14#89421C211yesmaximal, characteristic, complemented
15#89419C211yesmaximal, characteristic, complemented
16C22×C4×D7C2211yescharacteristic
17C22×C4×D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
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×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×Dic7⋊C4C2211yescharacteristic
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×C4×Dic7C2211yescharacteristic
35C2×C4×Dic7C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37C2×C4×Dic7C2211yescharacteristic
38C2×C4×Dic7C2211yescharacteristic
39D7×C4⋊C4C418yescomplemented
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#2345—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—22no—
69C2×C4×D7—22no—
70C2×C4×D7—41no—
71C2×C4×D7—41no—
72C2×C4×D7—41no—
73C2×C4×D7—41no—
74C2×C4×D7C2×C414yes—
75C2×C4×D7C2×C414yes—
76C2×C4×D7C2×C414yes—
77C7×C4⋊C4—41no—
78C7×C4⋊C4C2×C414yes—
79C42×C7—41no—
80(C2×D7)⋊C4—22no—
81(C2×D7)⋊C4—22no—
82(C2×D7)⋊C4—22no—
83(C2×D7)⋊C4Q814yes—
84C4⋊Dic7—41no—
85C4⋊Dic7C2×C414yes—
86Dic7⋊C4—22no—
87Dic7⋊C4—22no—
88Dic7⋊C4—41no—
89Dic7⋊C4—41no—
90Dic7⋊C4C2×C414yes—
91Dic7⋊C4C2×C414yes—
92C4×Dic7—22no—
93C4×Dic7—22no—
94C4×Dic7—41no—
95C4×Dic7—41no—
96C4×Dic7—41no—
97C22×C4⋊C4—71no—
98#389—71no—
99#389—71no—
100#387—71no—
101#387—71no—
102#387—71no—
103#387D711yescharacteristic, complemented
104#380—71no—
105#380—71no—
106#380—71no—
107C4×C4⋊C4—71no—
108C4×C4⋊C4—71no—
109C4×C22⋊C4—71no—
110C4×C22⋊C4—71no—
111C4×C22⋊C4—71no—
112C23×C7C2411yescharacteristic
113C22×D7—21no—
114C22×D7—21no—
115C22×D7—21no—
116C22×D7—21no—
117C22×D7C22×C412yes—
118C22×D7C2×Q812yes—
119C22×D7#4112yes—
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×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7C22×C412yes—
138C2×C4×C7C22×C412yes—
139C2×C4×C7C22×C412yes—
140C2×C4×C7C2×Q812yes—
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×Dic7—21no—
156C2×Dic7C22×C412yes—
157C2×Dic7C22×C412yes—
158C2×Dic7C22×C412yes—
159C2×Dic7C2×Q812yes—
160C2×Dic7#4112yes—
161C2×Dic7#4112yes—
162C4×D7—28no—
163C4×D7—44no—
164C4×D7—44no—
165C23×C4—71no—
166C23×C4—71no—
167C23×C4—71no—
168C2×C4⋊C4—71no—
169C2×C4⋊C4—71no—
170C2×C4⋊C4—71no—
171C2×C4⋊C4—78no—
172C2×C4⋊C4—71no—
173C2×C4⋊C4—71no—
174C2×C4⋊C4—71no—
175C2×C4⋊C4—71no—
176C2×C4⋊C4—71no—
177C2×C4⋊C4C2×D711yescharacteristic
178C2×C4⋊C4C2×D711yescharacteristic
179C2×C22⋊C4—71no—
180C2×C22⋊C4—71no—
181C2×C22⋊C4—71no—
182C2×C22⋊C4—71no—
183C2×C42—71no—
184C2×C42—71no—
185C2×C42—71no—
186C2×C42—71no—
187C2×C42—71no—
188C2×C42C2×D711yescharacteristic
189#95—71no—
190#95—71no—
191#95—71no—
192#95—71no—
193#95—71no—
194#95—71no—
195#95—71no—
196#95—71no—
197#95C2×D711yescharacteristic
198#95C2×D711yescharacteristic
199#95C2×D711yescharacteristic
200#95C2×D711yescharacteristic
201C22×C7C23×C411yescharacteristic
202C22×C7C22×Q811yescharacteristic
203C22×C7C2×#4111yescharacteristic
204C22×C7C2×#4111yescharacteristic
205C22×C7C2×#4111yescharacteristic
206C22×C7H(5,𝔽2)11yescharacteristic
207C22×C7H(5,𝔽2)11yescharacteristic
208C2×D7—22no—
209C2×D7—22no—
210C2×D7—22no—
211C2×D7—22no—
212C2×D7—22no—
213C2×D7—22no—
214C2×D7C4×Q814yes—
215C4×C7—22no—
216C4×C7—22no—
217C4×C7—22no—
218C4×C7—22no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—41no—
222Dic7—22no—
223Dic7—22no—
224Dic7—22no—
225Dic7—41no—
226Dic7—41no—
227Dic7—41no—
228Dic7#11714yes—
229C24—71no—
230C22×C4—71no—
231C22×C4—71no—
232C22×C4—71no—
233C22×C4—71no—
234C22×C4—74no—
235C22×C4—74no—
236C22×C4—74no—
237C22×C4—71no—
238C22×C4—71no—
239C22×C4—71no—
240C22×C4—142no—
241C22×C4—142no—
242C22×C4—281no—
243C22×C4—281no—
244C22×C4—281no—
245C22×C4—281no—
246C22×C4C22×D711yescharacteristic
247C22×C4C22×D711yescharacteristic
248C22×C4C22×D711yescharacteristic
249C22×C4C22×D711yescharacteristic
250C22×C4C22×D711yescharacteristic
251C22×C4C22×D711yescharacteristic
252C22×C4C22×D711yescharacteristic
253C4⋊C4—41no—
254C4⋊C4—74no—
255C4⋊C4—74no—
256C4⋊C4—74no—
257C4⋊C4—142no—
258C4⋊C4—142no—
259C4⋊C4—281no—
260C4⋊C4—281no—
261C4⋊C4—281no—
262C4⋊C4C4×D714yes—
263C22⋊C4—74no—
264C22⋊C4—142no—
265C22⋊C4—142no—
266C22⋊C4—142no—
267C42—41no—
268C42—142no—
269C42—142no—
270C42—281no—
271C42—281no—
272C42—281no—
273C2×C7C2×#11711yescharacteristic
274C2×C7C2×C4×Q811yescharacteristic
275C2×C7#51811yescharacteristic
276C2×C7#55311yescharacteristic
277C2×C7#55311yescharacteristic
278C2×C7#55411yescharacteristic
279C2×C7#55411yescharacteristic
280D7—24no—
281C23—72no—
282C23—72no—
283C23—72no—
284C23—141no—
285C23—141no—
286C23—141no—
287C23—141no—
288C23C23×D711yescharacteristic
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—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—72no—
307C2×C4—72no—
308C2×C4—72no—
309C2×C4—72no—
310C2×C4—72no—
311C2×C4—72no—
312C2×C4—148no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—141no—
322C2×C4—141no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—284no—
329C2×C4—284no—
330C2×C4C2×C4×D712yes—
331C2×C4C2×C4×D712yes—
332C2×C4C2×C4×D712yes—
333C2×C4Q8×D712yes—
334C7#234511yescharacteristic, complemented, Sylow
335C22—74no—
336C22—142no—
337C22—142no—
338C22—142no—
339C22—142no—
340C22—142no—
341C22—142no—
342C22C22×C4×D711yescharacteristic
343C22#654811yescharacteristic
344C22C2×Q8×D711yescharacteristic
345C22D7×#4111yescharacteristic
346C22D7×#4111yescharacteristic
347C22D7×#4111yescharacteristic
348C22#655311yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—74no—
357C4—142no—
358C4—142no—
359C4—142no—
360C4—281no—
361C4—281no—
362C4—281no—
363C2—144no—
364C2D7×#11711yescharacteristic
365C2#9023611yescharacteristic
366C2C4×Q8×D711yescharacteristic
367C2#9032711yescharacteristic
368C2#9035211yescharacteristic
369C2#9038111yescharacteristic
370C2#9040011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutatorC22×C7C23×C4
FittingC7×#387C2
socle, cosocleC23×C7C24
prime radicalC23×D7C23
hypercenter#387D7
hypocenterC7#2345
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C7×#387C2semidirect
4. #89753C2semidirect
5. #89744C2semidirect
7. C4×Dic7⋊C4C2semidirect
12. #89426C2semidirect
13. #89425C2semidirect
14. #89421C2semidirect
15. #89419C2semidirect
39. D7×C4⋊C4C4semidirect
103. #387D7semidirect
334. C7#2345semidirect

Maximal subgroups

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

subgroupindex
C2×D7×C4⋊C42
C7×#3872
#897592
#897532
#897442
C4×(C2×D7)⋊C42
C4×Dic7⋊C42
#894392
#894382
#894262
#894252
#894212
#894192
#23457