Finite Group Databasebeta

← #11733012

Subgroups — #11733012

Order 896.

428 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#388C211yesmaximal, characteristic
2#89757C211yesmaximal, characteristic, complemented
3#89756C211yesmaximal, characteristic, complemented
4#89723C211yesmaximal, characteristic
5#89722C211yesmaximal, characteristic
6#89721C211yesmaximal, characteristic
7#89720C211yesmaximal, characteristic
8#89714C211yesmaximal, characteristic
9C22⋊C4×Dic7C211yesmaximal, characteristic
10C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
11#89700C211yesmaximal, characteristic, complemented
12#89429C211yesmaximal, characteristic, complemented
13#89427C211yesmaximal, characteristic, complemented
14#89427C211yesmaximal, characteristic, complemented
15#89426C211yesmaximal, characteristic, complemented
16C23×Dic7C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C42×C7C2211yescharacteristic
22C2×C22⋊Dic7C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C4⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37C2×C4×Dic7C2211yescharacteristic
38C2×C4×Dic7C2211yescharacteristic
39C7×#95C2211yescharacteristic
40C7×#95C2211yescharacteristic
41C7⋊#95C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51#2522—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C22×C4×C7C2311yescharacteristic
54C22×C4×C7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×Dic7—22no—
60C22×Dic7—22no—
61C22×Dic7—41no—
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C7×C4⋊C4—22no—
71C7×C22⋊C4—22no—
72C7×C22⋊C4—41no—
73C7×C22⋊C4Q814yes—
74C42×C7—41no—
75C22⋊Dic7—22no—
76C22⋊Dic7—41no—
77C22⋊Dic7—41no—
78C4⋊Dic7—22no—
79C4⋊Dic7—22no—
80C4⋊Dic7—41no—
81C4⋊Dic7—41no—
82C4⋊Dic7—41no—
83Dic7⋊C4—22no—
84Dic7⋊C4—22no—
85Dic7⋊C4—22no—
86Dic7⋊C4—22no—
87Dic7⋊C4—41no—
88Dic7⋊C4—41no—
89C4×Dic7—22no—
90C4×Dic7—41no—
91C4×Dic7—41no—
92#399—71no—
93#398—71no—
94#398—71no—
95#396—71no—
96#389—71no—
97#389—71no—
98#388—71no—
99#388D711yescharacteristic
100#387—71no—
101#387—71no—
102#387—71no—
103#385—71no—
104#380—71no—
105C4×C4⋊C4—71no—
106C4×C22⋊C4—71no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7C2×Q812yes—
114C23×C7C2411yescharacteristic
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
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×C7C2×Q812yes—
131C2×C4×C7C2×Q812yes—
132C2×C4×C7#4112yes—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
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—44no—
155C2×Dic7#4112yes—
156C2×Dic7#4112yes—
157C2×Dic7#4112yes—
158C23×C4—71no—
159C2×C4⋊C4—71no—
160C2×C4⋊C4—71no—
161C2×C4⋊C4—71no—
162C2×C4⋊C4—71no—
163C2×C4⋊C4—71no—
164C2×C4⋊C4—71no—
165C2×C4⋊C4—71no—
166C2×C4⋊C4—71no—
167C2×C4⋊C4—71no—
168C2×C4⋊C4—71no—
169C2×C4⋊C4—71no—
170C2×C4⋊C4C2×D711yescharacteristic
171C2×C22⋊C4—71no—
172C2×C22⋊C4—71no—
173C2×C22⋊C4—71no—
174C2×C22⋊C4C2×D711yescharacteristic
175C2×C22⋊C4C2×D711yescharacteristic
176C2×C22⋊C4C2×D711yescharacteristic
177C2×C42—71no—
178C2×C42—71no—
179C2×C42—71no—
180C2×C42C2×D711yescharacteristic
181#95—71no—
182#95—71no—
183#95—71no—
184#95—71no—
185#95—71no—
186#95—71no—
187#95—71no—
188#95—71no—
189#95—71no—
190#95—71no—
191#95C2×D711yescharacteristic
192#95C2×D711yescharacteristic
193C22×C7—22no—
194C22×C7—22no—
195C22×C7—22no—
196C22×C7—41no—
197C22×C7—41no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7C22×Q811yescharacteristic
201C22×C7C2×#4111yescharacteristic
202C22×C7C2×#4111yescharacteristic
203C22×C7C2×#4111yescharacteristic
204C22×C7C2×#4111yescharacteristic
205C22×C7H(5,𝔽2)11yescharacteristic
206C22×C7H(5,𝔽2)11yescharacteristic
207C4×C7—22no—
208C4×C7—22no—
209C4×C7—22no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—41no—
213Dic7—22no—
214Dic7—22no—
215Dic7—22no—
216Dic7—41no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220Dic7—41no—
221C24C22×D711yescharacteristic
222C22×C4—71no—
223C22×C4—71no—
224C22×C4—71no—
225C22×C4—71no—
226C22×C4—71no—
227C22×C4—71no—
228C22×C4—71no—
229C22×C4—71no—
230C22×C4—142no—
231C22×C4—142no—
232C22×C4—281no—
233C22×C4C22×D711yescharacteristic
234C22×C4C22×D711yescharacteristic
235C22×C4C22×D711yescharacteristic
236C22×C4C22×D711yescharacteristic
237C22×C4C22×D711yescharacteristic
238C22×C4C22×D711yescharacteristic
239C4⋊C4—22no—
240C4⋊C4—142no—
241C4⋊C4—142no—
242C4⋊C4—142no—
243C4⋊C4—142no—
244C4⋊C4—142no—
245C4⋊C4—142no—
246C4⋊C4—281no—
247C4⋊C4—281no—
248C4⋊C4—281no—
249C4⋊C4—281no—
250C4⋊C4—281no—
251C22⋊C4—22no—
252C22⋊C4—41no—
253C22⋊C4—142no—
254C22⋊C4—281no—
255C22⋊C4—281no—
256C22⋊C4Dic1414yes—
257C42—41no—
258C42—142no—
259C42—281no—
260C42—281no—
261C2×C7—42no—
262C2×C7#52911yescharacteristic
263C2×C7#53311yescharacteristic
264C2×C7#54011yescharacteristic
265C2×C7#55111yescharacteristic
266C2×C7#55311yescharacteristic
267C2×C7#55411yescharacteristic
268C2×C7#55411yescharacteristic
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23C2×Dic1412yes—
276C23C23×D711yescharacteristic
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—72no—
293C2×C4—72no—
294C2×C4—72no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—284no—
317C2×C4C2×Dic1412yes—
318C2×C4C2×Dic1412yes—
319C2×C4#115912yes—
320C7#252211yescharacteristic, complemented, Sylow
321C22—22no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22C22×Dic1411yescharacteristic
329C22C2×#115911yescharacteristic
330C22#654811yescharacteristic
331C22D7×#4111yescharacteristic
332C22D7×#4111yescharacteristic
333C22D7×#4111yescharacteristic
334C22#655311yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—142no—
342C4—142no—
343C4—142no—
344C4—281no—
345C4—281no—
346C4—281no—
347C4—281no—
348C4—281no—
349C2—42no—
350C2#9020811yescharacteristic
351C2#9023011yescharacteristic
352C2#9023411yescharacteristic
353C2#9029211yescharacteristic
354C2#9035411yescharacteristic
355C2#9037111yescharacteristic
356C2#9039611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#388C2
prime radicalC24×C7C23
hypercenter#388D7
hypocenterC7#2522
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #89757C2semidirect
3. #89756C2semidirect
10. C4×Dic7⋊C4C2semidirect
11. #89700C2semidirect
12. #89429C2semidirect
13. #89427C2semidirect
14. #89427C2semidirect
15. #89426C2semidirect
320. C7#2522semidirect

Maximal subgroups

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

subgroupindex
C7×#3882
#897572
#897562
#897232
#897222
#897212
#897202
#897142
C22⋊C4×Dic72
C4×Dic7⋊C42
#897002
#894292
#894272
#894262
#25227