Finite Group Databasebeta

← #11735593

Subgroups — #11735593

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×#6459C211yesmaximal, characteristic
2C2×#6451C211yesmaximal, characteristic, complemented
3C7×#400C211yesmaximal, characteristic, complemented
4#89762C211yesmaximal, characteristic
5#89746C211yesmaximal, characteristic, complemented
6#89445C211yesmaximal, characteristic
7#89445C211yesmaximal, characteristic
8#89442C211yesmaximal, characteristic
9#89439C211yesmaximal, characteristic
10#89438C211yesmaximal, characteristic
11#89432C211yesmaximal, characteristic, complemented
12#89421C211yesmaximal, characteristic, complemented
13#89418C211yesmaximal, characteristic, complemented
14#89418C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C22×C4×D7C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C22×Dic14C2211yescharacteristic
19C22×Dic14C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
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×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×C4×Dic7C2211yescharacteristic
34C2×C4×Dic7C2211yescharacteristic
35C2×C4×Dic7C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37#6459—24no—
38#6451—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
54C23×D7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C2×C4×D7—22no—
70C2×C4×D7—22no—
71C2×C4×D7—22no—
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×C4⋊C4—22no—
83(C2×D7)⋊C4—22no—
84(C2×D7)⋊C4—22no—
85(C2×D7)⋊C4—22no—
86(C2×D7)⋊C4—41no—
87(C2×D7)⋊C4—41no—
88(C2×D7)⋊C4Q814yescomplemented
89C4⋊Dic7—22no—
90Dic7⋊C4—22no—
91Dic7⋊C4—22no—
92Dic7⋊C4—22no—
93Dic7⋊C4D414yes—
94C4×Dic7—22no—
95C4×Dic7—22no—
96C4×Dic7—41no—
97C4×Dic7—41no—
98C2×C4⋊Q8—71no—
99C2×C22⋊Q8—71no—
100#400—71no—
101#400D711yescharacteristic, complemented
102#393—71no—
103#393—71no—
104#391—71no—
105#391—71no—
106#391—71no—
107#391—71no—
108#388—71no—
109#388—71no—
110#380—71no—
111C4×C4⋊C4—71no—
112C4×C22⋊C4—71no—
113C23×C7C2411yescharacteristic
114C22×D7—21no—
115C22×D7—21no—
116C22×D7—21no—
117C22×D7—21no—
118C22×D7—21no—
119C22×D7—21no—
120C22×D7C2×Q812yes—
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×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7C2×D412yes—
141C2×C4×C7C2×Q812yes—
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×Dic7C2×D412yes—
158C2×Dic7C2×D412yes—
159C2×Dic7C2×Q812yes—
160C2×Dic7#4112yes—
161C2×Dic7#4112yes—
162C2×Dic7#4112yes—
163C4×D7—44no—
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⋊C4C2×D711yescharacteristic
179C2×C4⋊C4C2×D711yescharacteristic
180C2×C22⋊C4—71no—
181C2×C22⋊C4—71no—
182C2×C22⋊C4—71no—
183C2×C22⋊C4—71no—
184C2×C22⋊C4—71no—
185C2×C22⋊C4—71no—
186C2×C42—71no—
187C2×C42—71no—
188C2×C42—71no—
189C2×C42—71no—
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#95C2×D711yescharacteristic
200#95C2×D711yescharacteristic
201#95C2×D711yescharacteristic
202#95C2×D711yescharacteristic
203#95C2×D711yescharacteristic
204C22×C7C22×D411yescharacteristic
205C22×C7C22×Q811yescharacteristic
206C22×C7C2×#4111yescharacteristic
207C22×C7C2×#4111yescharacteristic
208C22×C7C2×#4111yescharacteristic
209C22×C7H(5,𝔽2)11yescharacteristic
210C22×C72−1+411yescharacteristic
211C2×D7—22no—
212C2×D7—22no—
213C2×D7—22no—
214C2×D7—41no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C4×C7—22no—
219C4×C7—22no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—41no—
225Dic7—22no—
226Dic7—22no—
227Dic7—22no—
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231Dic7#12414yes—
232C24—71no—
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—142no—
250C22×C4—142no—
251C22×C4—142no—
252C22×C4C22×D711yescharacteristic
253C22×C4C22×D711yescharacteristic
254C22×C4C22×D711yescharacteristic
255C22×C4C22×D711yescharacteristic
256C22×C4C22×D711yescharacteristic
257C22×C4C22×D711yescharacteristic
258C22×C4C22×D711yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—74no—
262C4⋊C4—142no—
263C4⋊C4—142no—
264C4⋊C4—142no—
265C4⋊C4—142no—
266C22⋊C4—74no—
267C22⋊C4—142no—
268C22⋊C4—142no—
269C22⋊C4—142no—
270C22⋊C4—281no—
271C22⋊C4—281no—
272C42—142no—
273C42—142no—
274C42—281no—
275C42—281no—
276C2×C7C2×#12411yescharacteristic
277C2×C7#53311yescharacteristic
278C2×C7#54211yescharacteristic
279C2×C7#54611yescharacteristic
280C2×C7D4×Q811yescharacteristic
281C2×C7#55411yescharacteristic
282C2×C7#55611yescharacteristic
283D7—42no—
284C23—72no—
285C23—141no—
286C23—141no—
287C23—141no—
288C23—141no—
289C23—141no—
290C23—141no—
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—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—72no—
315C2×C4—72no—
316C2×C4—72no—
317C2×C4—72no—
318C2×C4—72no—
319C2×C4—72no—
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—141no—
335C2×C4—284no—
336C2×C4D4×D712yes—
337C2×C4Q8×D712yes—
338C7#249211yescharacteristic, complemented, Sylow
339C22—142no—
340C22—142no—
341C22—142no—
342C22—281no—
343C22—281no—
344C22—281no—
345C22—281no—
346C22C2×D4×D711yescharacteristic
347C22C2×Q8×D711yescharacteristic
348C22D7×#4111yescharacteristic
349C22D7×#4111yescharacteristic
350C22D7×#4111yescharacteristic
351C22#655311yescharacteristic
352C22#655411yescharacteristic
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—74no—
361C4—142no—
362C4—142no—
363C4—142no—
364C4—281no—
365C4—281no—
366C4—281no—
367C2—282no—
368C2#9032311yescharacteristic
369C2#9034511yescharacteristic
370C2D7×#12411yescharacteristic
371C2#9037911yescharacteristic
372C2#9038111yescharacteristic
373C2#9039311yescharacteristic
374C2#9040111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#400C2
prime radicalC23×D7C23
hypercenter#400D7
hypocenterC7#2492
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#6451C2semidirect
3. C7×#400C2semidirect
5. #89746C2semidirect
11. #89432C2semidirect
12. #89421C2semidirect
13. #89418C2semidirect
14. #89418C2semidirect
15. #89418C2semidirect
88. (C2×D7)⋊C4Q8semidirect
101. #400D7semidirect
338. C7#2492semidirect

Maximal subgroups

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

subgroupindex
C2×#64592
C2×#64512
C7×#4002
#897622
#897462
#894452
#894422
#894392
#894382
#894322
#894212
#894182
#24927