Finite Group Databasebeta

← #11732992

Subgroups — #11732992

Order 896.

508 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6451C211yesmaximal, characteristic, complemented
2C2×#6441C211yesmaximal, characteristic
3C7×#388C211yesmaximal, characteristic
4#89758C211yesmaximal, characteristic, complemented
5#89723C211yesmaximal, characteristic
6#89719C211yesmaximal, characteristic
7#89719C211yesmaximal, characteristic
8#89714C211yesmaximal, characteristic
9C22⋊C4×Dic7C211yesmaximal, characteristic
10C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
11#89701C211yesmaximal, characteristic, complemented
12#89432C211yesmaximal, characteristic, complemented
13#89418C211yesmaximal, characteristic, complemented
14#89418C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C23×Dic7C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C22×Dic14C2211yescharacteristic
19C22×Dic14C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C42×C7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×C4×Dic7C2211yescharacteristic
35C2×C4×Dic7C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37#6451—24no—
38#6441—24no—
39C7×#95C2211yescharacteristic
40C7×#95C2211yescharacteristic
41C7⋊#95C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2492—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×Dic7—22no—
62C22×Dic7—22no—
63C22×Dic7—22no—
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C2×Dic14—22no—
73C2×Dic14—22no—
74C2×Dic14—22no—
75C2×Dic14—22no—
76C2×Dic14—22no—
77C2×Dic14—22no—
78C2×Dic14—22no—
79C2×Dic14—41no—
80C2×Dic14—41no—
81C7×C4⋊C4—22no—
82C7×C22⋊C4—22no—
83C7×C22⋊C4—41no—
84C7×C22⋊C4Q814yescomplemented
85C42×C7—41no—
86C22⋊Dic7—22no—
87C22⋊Dic7—22no—
88C22⋊Dic7—41no—
89C4⋊Dic7—22no—
90C4⋊Dic7—22no—
91Dic7⋊C4—22no—
92Dic7⋊C4—22no—
93Dic7⋊C4—22no—
94Dic7⋊C4D414yes—
95C4×Dic7—22no—
96C4×Dic7—22no—
97C4×Dic7—41no—
98C2×C4⋊Q8—71no—
99C2×C22⋊Q8—71no—
100#400—71no—
101#400—71no—
102#393—71no—
103#393—71no—
104#391—71no—
105#391—71no—
106#391—71no—
107#391—71no—
108#388—71no—
109#388D711yescharacteristic
110#380—71no—
111C4×C4⋊C4—71no—
112C4×C22⋊C4—71no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7C2×Q812yes—
120C23×C7C2411yescharacteristic
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×C7C2×D412yes—
137C2×C4×C7C2×Q812yes—
138C2×C4×C7C2×Q812yes—
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×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—44no—
159C2×Dic7C2×D412yes—
160C2×Dic7C2×D412yes—
161C2×Dic7#4112yes—
162C2×Dic7#4112yes—
163C2×Dic7#4112yes—
164Dic14—44no—
165Dic14—44no—
166Dic14—44no—
167C22×Q8—71no—
168C22×Q8—71no—
169C22×Q8—71no—
170C23×C4—71no—
171C4⋊Q8—144no—
172C22⋊Q8—144no—
173C2×C4⋊C4—71no—
174C2×C4⋊C4—71no—
175C2×C4⋊C4—71no—
176C2×C4⋊C4—71no—
177C2×C4⋊C4—71no—
178C2×C4⋊C4—71no—
179C2×C4⋊C4C2×D711yescharacteristic
180C2×C22⋊C4—71no—
181C2×C22⋊C4—71no—
182C2×C22⋊C4—71no—
183C2×C22⋊C4C2×D711yescharacteristic
184C2×C22⋊C4C2×D711yescharacteristic
185C2×C22⋊C4C2×D711yescharacteristic
186C2×C42—71no—
187C2×C42—71no—
188C2×C42—71no—
189C2×C42C2×D711yescharacteristic
190#95—71no—
191#95—71no—
192#95—71no—
193#95—71no—
194#95—71no—
195#95—71no—
196#95—71no—
197#95—71no—
198#95—71no—
199#95—71no—
200#95—71no—
201#95—71no—
202#95C2×D711yescharacteristic
203#95C2×D711yescharacteristic
204C22×C7—22no—
205C22×C7—22no—
206C22×C7—22no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7C22×D411yescharacteristic
212C22×C7C22×Q811yescharacteristic
213C22×C7C2×#4111yescharacteristic
214C22×C7C2×#4111yescharacteristic
215C22×C7C2×#4111yescharacteristic
216C22×C7H(5,𝔽2)11yescharacteristic
217C22×C72−1+411yescharacteristic
218C4×C7—22no—
219C4×C7—22no—
220C4×C7—22no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—41no—
224Dic7—22no—
225Dic7—22no—
226Dic7—41no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231Dic7#12414yes—
232C24C22×D711yescharacteristic
233C2×Q8—142no—
234C2×Q8—142no—
235C2×Q8—142no—
236C2×Q8—142no—
237C2×Q8—142no—
238C2×Q8—142no—
239C2×Q8—142no—
240C2×Q8—281no—
241C2×Q8—281no—
242C22×C4—71no—
243C22×C4—71no—
244C22×C4—71no—
245C22×C4—71no—
246C22×C4—71no—
247C22×C4—71no—
248C22×C4—71no—
249C22×C4—71no—
250C22×C4—142no—
251C22×C4—142no—
252C22×C4—142no—
253C22×C4C22×D711yescharacteristic
254C22×C4C22×D711yescharacteristic
255C22×C4C22×D711yescharacteristic
256C22×C4C22×D711yescharacteristic
257C22×C4C22×D711yescharacteristic
258C22×C4C22×D711yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—74no—
261C4⋊C4—142no—
262C4⋊C4—142no—
263C4⋊C4—142no—
264C4⋊C4—142no—
265C4⋊C4—142no—
266C22⋊C4—22no—
267C22⋊C4—41no—
268C22⋊C4—142no—
269C22⋊C4—142no—
270C22⋊C4—281no—
271C22⋊C4Dic1414yescomplemented
272C42—41no—
273C42—142no—
274C42—142no—
275C42—281no—
276C2×C7—42no—
277C2×C7C2×#12411yescharacteristic
278C2×C7#53311yescharacteristic
279C2×C7#54211yescharacteristic
280C2×C7#54611yescharacteristic
281C2×C7D4×Q811yescharacteristic
282C2×C7#55411yescharacteristic
283C2×C7#55611yescharacteristic
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23C2×Dic1412yes—
291C23C23×D711yescharacteristic
292Q8—284no—
293Q8—284no—
294Q8—284no—
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—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—72no—
311C2×C4—72no—
312C2×C4—72no—
313C2×C4—72no—
314C2×C4—72no—
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—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—284no—
335C2×C4C2×Dic1412yes—
336C2×C4C2×Dic1412yes—
337C2×C4D4×D712yes—
338C7#249211yescharacteristic, complemented, Sylow
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22C22×Dic1411yescharacteristic
347C22C2×D4×D711yescharacteristic
348C22#654811yescharacteristic
349C22D7×#4111yescharacteristic
350C22D7×#4111yescharacteristic
351C22D7×#4111yescharacteristic
352C22#655411yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—74no—
360C4—142no—
361C4—142no—
362C4—281no—
363C4—281no—
364C4—281no—
365C4—281no—
366C4—281no—
367C2—42no—
368C2#9020811yescharacteristic
369C2D4×Dic1411yescharacteristic
370C2#9023011yescharacteristic
371C2#9030111yescharacteristic
372C2#9035311yescharacteristic
373C2D7×#12411yescharacteristic
374C2#9039311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#388C2
prime radicalC24×C7C23
hypercenter#388D7
hypocenterC7#2492
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#6451C2semidirect
4. #89758C2semidirect
10. C4×Dic7⋊C4C2semidirect
11. #89701C2semidirect
12. #89432C2semidirect
13. #89418C2semidirect
14. #89418C2semidirect
15. #89418C2semidirect
84. C7×C22⋊C4Q8semidirect
271. C22⋊C4Dic14semidirect
338. C7#2492semidirect

Maximal subgroups

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

subgroupindex
C2×#64512
C2×#64412
C7×#3882
#897582
#897232
#897192
#897142
C22⋊C4×Dic72
C4×Dic7⋊C42
#897012
#894322
#894182
#24927