Finite Group Databasebeta

← #11733043

Subgroups — #11733043

Order 896.

492 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×#6453C211yesmaximal, characteristic, complemented
2C2×#6441C211yesmaximal, characteristic
3C7×#388C211yesmaximal, characteristic
4#89756C211yesmaximal, characteristic, complemented
5#89723C211yesmaximal, characteristic
6#89722C211yesmaximal, characteristic
7#89721C211yesmaximal, characteristic
8#89720C211yesmaximal, characteristic
9C22⋊C4×Dic7C211yesmaximal, characteristic
10C4×C4⋊Dic7C211yesmaximal, characteristic, complemented
11#89700C211yesmaximal, characteristic, complemented
12#89428C211yesmaximal, characteristic, complemented
13#89426C211yesmaximal, characteristic, complemented
14#89419C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C23×Dic7C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C42×C7C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×C4⋊Dic7C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×C4×Dic7C2211yescharacteristic
39C2×C4×Dic7C2211yescharacteristic
40C2×C4×Dic7C2211yescharacteristic
41#6453—24no—
42#6441—24no—
43C7×#95C2211yescharacteristic
44C7×#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2556—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—41no—
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C2×Dic14—22no—
73C2×Dic14—22no—
74C2×Dic14—41no—
75C7×C4⋊C4—22no—
76C7×C22⋊C4—22no—
77C7×C22⋊C4—22no—
78C7×C22⋊C4Q814yescomplemented
79C42×C7—22no—
80C22⋊Dic7—22no—
81C22⋊Dic7—22no—
82C22⋊Dic7—41no—
83C4⋊Dic7—22no—
84C4⋊Dic7—22no—
85C4⋊Dic7—22no—
86C4⋊Dic7—22no—
87C4⋊Dic7—22no—
88C4⋊Dic7D414yes—
89Dic7⋊C4—22no—
90Dic7⋊C4—22no—
91Dic7⋊C4—22no—
92Dic7⋊C4—22no—
93Dic7⋊C4—22no—
94Dic7⋊C4—41no—
95C4×Dic7—22no—
96C4×Dic7—41no—
97C4×Dic7—41no—
98C2×#125—71no—
99C2×C22⋊Q8—71no—
100#399—71no—
101#398—71no—
102#396—71no—
103#395—71no—
104#391—71no—
105#389—71no—
106#389—71no—
107#388—71no—
108#388D711yescharacteristic
109#387—71no—
110#385—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×C7C2×D412yes—
136C2×C4×C7C2×Q812yes—
137C2×C4×C7C2×Q812yes—
138C2×C4×C7#4112yes—
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—21no—
159C2×Dic7—44no—
160C2×Dic7C2×D412yes—
161C2×Dic7C2×D412yes—
162C2×Dic7#4112yes—
163C2×Dic7#4112yes—
164Dic14—44no—
165C22×Q8—71no—
166C23×C4—71no—
167#125—144no—
168C22⋊Q8—144no—
169C2×C4⋊C4—71no—
170C2×C4⋊C4—71no—
171C2×C4⋊C4—71no—
172C2×C4⋊C4—71no—
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⋊C4—71no—
180C2×C4⋊C4—71no—
181C2×C4⋊C4C2×D711yescharacteristic
182C2×C22⋊C4—71no—
183C2×C22⋊C4—71no—
184C2×C22⋊C4—71no—
185C2×C22⋊C4C2×D711yescharacteristic
186C2×C22⋊C4C2×D711yescharacteristic
187C2×C22⋊C4C2×D711yescharacteristic
188C2×C42—71no—
189C2×C42—71no—
190C2×C42—71no—
191C2×C42C2×D711yescharacteristic
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#95C2×D711yescharacteristic
201#95C2×D711yescharacteristic
202C22×C7—22no—
203C22×C7—22no—
204C22×C7—22no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7C22×D411yescharacteristic
210C22×C7C22×Q811yescharacteristic
211C22×C7C2×#4111yescharacteristic
212C22×C7C2×#4111yescharacteristic
213C22×C7C2×#4111yescharacteristic
214C22×C7H(5,𝔽2)11yescharacteristic
215C22×C72−1+411yescharacteristic
216C4×C7—22no—
217C4×C7—22no—
218C4×C7—22no—
219C4×C7—22no—
220C4×C7—41no—
221C4×C7—41no—
222Dic7—22no—
223Dic7—22no—
224Dic7—22no—
225Dic7—22no—
226Dic7—41no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230C24C22×D711yescharacteristic
231C2×Q8—142no—
232C2×Q8—142no—
233C2×Q8—281no—
234C22×C4—71no—
235C22×C4—71no—
236C22×C4—71no—
237C22×C4—71no—
238C22×C4—71no—
239C22×C4—71no—
240C22×C4—71no—
241C22×C4—71no—
242C22×C4—142no—
243C22×C4—142no—
244C22×C4—281no—
245C22×C4C22×D711yescharacteristic
246C22×C4C22×D711yescharacteristic
247C22×C4C22×D711yescharacteristic
248C22×C4C22×D711yescharacteristic
249C22×C4C22×D711yescharacteristic
250C22×C4C22×D711yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—74no—
253C4⋊C4—142no—
254C4⋊C4—142no—
255C4⋊C4—142no—
256C4⋊C4—142no—
257C4⋊C4—142no—
258C4⋊C4—142no—
259C4⋊C4—142no—
260C4⋊C4—142no—
261C4⋊C4—142no—
262C4⋊C4—142no—
263C4⋊C4—281no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—142no—
267C22⋊C4—142no—
268C22⋊C4—281no—
269C22⋊C4Dic1414yescomplemented
270C42—22no—
271C42—142no—
272C42—281no—
273C42—281no—
274C2×C7—42no—
275C2×C7#53211yescharacteristic
276C2×C7#53311yescharacteristic
277C2×C7#53911yescharacteristic
278C2×C7#54611yescharacteristic
279C2×C7D4×Q811yescharacteristic
280C2×C7#55211yescharacteristic
281C2×C7#55411yescharacteristic
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23C2×Dic1412yes—
289C23C23×D711yescharacteristic
290Q8—284no—
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—72no—
306C2×C4—72no—
307C2×C4—72no—
308C2×C4—72no—
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—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—284no—
330C2×C4C2×Dic1412yes—
331C2×C4C2×Dic1412yes—
332C2×C4D4×D712yes—
333C2×C4#115912yes—
334C7#255611yescharacteristic, complemented, Sylow
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22C22×Dic1411yescharacteristic
343C22C2×D4×D711yescharacteristic
344C22C2×#115911yescharacteristic
345C22D7×#4111yescharacteristic
346C22D7×#4111yescharacteristic
347C22#655311yescharacteristic
348C22#655411yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—142no—
356C4—142no—
357C4—142no—
358C4—142no—
359C4—281no—
360C4—281no—
361C4—281no—
362C4—281no—
363C2—42no—
364C2#9020811yescharacteristic
365C2D4×Dic1411yescharacteristic
366C2#9023411yescharacteristic
367C2#9028711yescharacteristic
368C2#9034111yescharacteristic
369C2#9036711yescharacteristic
370C2#9039811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#388C2
prime radicalC24×C7C23
hypercenter#388D7
hypocenterC7#2556
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#6453C2semidirect
4. #89756C2semidirect
10. C4×C4⋊Dic7C2semidirect
11. #89700C2semidirect
12. #89428C2semidirect
13. #89426C2semidirect
14. #89419C2semidirect
15. #89418C2semidirect
78. C7×C22⋊C4Q8semidirect
269. C22⋊C4Dic14semidirect
334. C7#2556semidirect

Maximal subgroups

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

subgroupindex
C2×#64532
C2×#64412
C7×#3882
#897562
#897232
#897222
#897212
#897202
C22⋊C4×Dic72
C4×C4⋊Dic72
#897002
#894282
#894262
#894192
#894182
#25567