Finite Group Databasebeta

← #11735787

Subgroups — #11735787

Order 896.

528 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×#6455C211yesmaximal, characteristic
3C7×#400C211yesmaximal, characteristic, complemented
4#89756C211yesmaximal, characteristic, complemented
5#89745C211yesmaximal, characteristic, complemented
6#89440C211yesmaximal, characteristic
7#89440C211yesmaximal, characteristic
8#89439C211yesmaximal, characteristic
9#89436C211yesmaximal, characteristic
10D7×#95C211yesmaximal, characteristic
11#89429C211yesmaximal, characteristic, complemented
12#89428C211yesmaximal, characteristic, complemented
13#89425C211yesmaximal, characteristic, complemented
14#89424C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C22×C4×D7C2211yescharacteristic
17C22×C4×D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
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×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×C4×Dic7C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37#6459—24no—
38#6455—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#2438—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×C4×D7—22no—
73C2×C4×D7—22no—
74C2×C4×D7—22no—
75C2×C4×D7—41no—
76C2×C4×D7—41no—
77C2×C4×D7Q814yescomplemented
78C2×Dic14—22no—
79C2×Dic14—22no—
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)⋊C4D414yes—
87C4⋊Dic7—22no—
88C4⋊Dic7—22no—
89C4⋊Dic7—22no—
90Dic7⋊C4—22no—
91Dic7⋊C4—22no—
92Dic7⋊C4—22no—
93Dic7⋊C4—41no—
94Dic7⋊C4—41no—
95Dic7⋊C4—41no—
96C4×Dic7—22no—
97C4×Dic7—22no—
98C2×C22⋊Q8—71no—
99C2×#117—71no—
100#400D711yescharacteristic, complemented
101#398—71no—
102#398—71no—
103#395—71no—
104#391—71no—
105#389—71no—
106#387—71no—
107#387—71no—
108#385—71no—
109#385—71no—
110#381—71no—
111#380—71no—
112C2×#95—71no—
113C23×C7C2411yescharacteristic
114C22×D7—21no—
115C22×D7—21no—
116C22×D7—21no—
117C22×D7—21no—
118C22×D7C2×D412yes—
119C22×D7C2×Q812yes—
120C22×D7#4112yes—
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×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7C2×D412yes—
160C2×Dic7C2×Q812yes—
161C2×Dic7#4112yes—
162C2×Dic7#4112yes—
163C4×D7—28no—
164C4×D7—44no—
165C4×D7—44no—
166Dic14—44no—
167C22×Q8—71no—
168C23×C4—71no—
169C23×C4—71no—
170C23×C4—71no—
171C22⋊Q8—144no—
172#117—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⋊C4—71no—
180C2×C4⋊C4—71no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4C2×D711yescharacteristic
183C2×C4⋊C4C2×D711yescharacteristic
184C2×C22⋊C4—71no—
185C2×C22⋊C4—71no—
186C2×C22⋊C4—71no—
187C2×C22⋊C4—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—22no—
215C2×D7—22no—
216C2×D7—22no—
217C2×D7C22⋊Q814yes—
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—22no—
229Dic7—41no—
230Dic7—41no—
231Dic7—41no—
232C24—71no—
233C2×Q8—142no—
234C2×Q8—142no—
235C2×Q8—281no—
236C22×C4—71no—
237C22×C4—71no—
238C22×C4—71no—
239C22×C4—71no—
240C22×C4—74no—
241C22×C4—71no—
242C22×C4—71no—
243C22×C4—71no—
244C22×C4—142no—
245C22×C4—142no—
246C22×C4—142no—
247C22×C4—142no—
248C22×C4—142no—
249C22×C4—142no—
250C22×C4—281no—
251C22×C4—281no—
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—142no—
262C4⋊C4—142no—
263C4⋊C4—142no—
264C4⋊C4—142no—
265C4⋊C4—142no—
266C4⋊C4—142no—
267C4⋊C4—281no—
268C4⋊C4—281no—
269C4⋊C4—281no—
270C22⋊C4—74no—
271C22⋊C4—142no—
272C22⋊C4—142no—
273C22⋊C4—142no—
274C42—142no—
275C42—142no—
276C2×C7C2×C22⋊Q811yescharacteristic
277C2×C7#53311yescharacteristic
278C2×C7#54411yescharacteristic
279C2×C7#54611yescharacteristic
280C2×C7#54711yescharacteristic
281C2×C7#55111yescharacteristic
282C2×C7#55211yescharacteristic
283D7—24no—
284C23—72no—
285C23—72no—
286C23—72no—
287C23—141no—
288C23—141no—
289C23—141no—
290C23—141no—
291C23C23×D711yescharacteristic
292Q8—284no—
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—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—72no—
313C2×C4—72no—
314C2×C4—72no—
315C2×C4—72no—
316C2×C4—148no—
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×C4—284no—
336C2×C4D4×D712yes—
337C2×C4Q8×D712yes—
338C7#243811yescharacteristic, complemented, Sylow
339C22—74no—
340C22—142no—
341C22—142no—
342C22—142no—
343C22—142no—
344C22—142no—
345C22—142no—
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—142no—
361C4—142no—
362C4—142no—
363C4—142no—
364C4—281no—
365C4—281no—
366C4—281no—
367C2—144no—
368C2D7×C22⋊Q811yescharacteristic
369C2#9034611yescharacteristic
370C2#9036711yescharacteristic
371C2#9037911yescharacteristic
372C2#9038011yescharacteristic
373C2#9039611yescharacteristic
374C2#9039911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#400C2
prime radicalC23×D7C23
hypercenter#400D7
hypocenterC7#2438
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. C7×#400C2semidirect
4. #89756C2semidirect
5. #89745C2semidirect
11. #89429C2semidirect
12. #89428C2semidirect
13. #89425C2semidirect
14. #89424C2semidirect
15. #89418C2semidirect
77. C2×C4×D7Q8semidirect
100. #400D7semidirect
338. C7#2438semidirect

Maximal subgroups

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

subgroupindex
C2×#64592
C2×#64552
C7×#4002
#897562
#897452
#894402
#894392
#894362
D7×#952
#894292
#894282
#894252
#894242
#894182
#24387