Finite Group Databasebeta

← #11739817

Subgroups — #11739817

Order 896.

540 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×D4C211yesmaximal, characteristic
2C2×#6489C211yesmaximal, characteristic, complemented
3#89994C212yesmaximal
4#89984C211yesmaximal, characteristic, complemented
5#89984C211yesmaximal, characteristic, complemented
6#89984C211yesmaximal, characteristic, complemented
7#89750C212yesmaximal, complemented
8#89724C212yesmaximal, complemented
9#89716C212yesmaximal, complemented
10#89700C211yesmaximal, characteristic, complemented
11#89700C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C23×C4×C7C2211yescharacteristic
14C23×C4×C7C2211yescharacteristic
15C4×C7×D4—24no—
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C22⋊C4C2211yescharacteristic
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C42×C7C2211yescharacteristic
20C2×C22⋊Dic7C2212yes—
21C2×C22⋊Dic7C2212yes—
22C2×C22⋊Dic7C2212yes—
23C2×C22⋊Dic7C2212yes—
24#6489—24no—
25C2×C4⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×Dic7⋊C4C2212yes—
30C2×Dic7⋊C4C2212yes—
31C2×C4×Dic7C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2485—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C24×C7C2311yescharacteristic
40C2×C7×D4—22no—
41C2×C7×D4—22no—
42C2×C7×D4—22no—
43C22×C4×C7—22no—
44C22×C4×C7—22no—
45C22×C4×C7—22no—
46C22×C4×C7—22no—
47C22×C4×C7—22no—
48C22×C4×C7D414yes—
49C22×C4×C7C2311yescharacteristic
50C22×C4×C7C2311yescharacteristic
51C22×C4×C7C2311yescharacteristic
52C22×C4×C7C2311yescharacteristic
53C22×C4×C7C2311yescharacteristic
54C22×Dic7C2312yes—
55C22×Dic7C2312yes—
56C22×Dic7C2312yes—
57C22×Dic7C2312yes—
58C7×C4⋊C4—22no—
59C7×C22⋊C4—22no—
60C7×C22⋊C4—22no—
61C42×C7—22no—
62C22⋊Dic7—24no—
63C22⋊Dic7—24no—
64C22⋊Dic7—42no—
65C22⋊Dic7—42no—
66C4⋊Dic7—22no—
67C4⋊Dic7—22no—
68C4⋊Dic7—22no—
69C4⋊Dic7—22no—
70Dic7⋊C4—24no—
71Dic7⋊C4—42no—
72C4×Dic7—24no—
73C2×C4×D4D711yescharacteristic
74C2×#117—71no—
75#397—72no—
76#394—72no—
77#389—71no—
78#389—71no—
79#388—72no—
80#387—72no—
81#380—71no—
82#380—71no—
83#380—71no—
84C23×C7—21no—
85C23×C7—21no—
86C23×C7—21no—
87C23×C7—21no—
88C23×C7—21no—
89C23×C7—21no—
90C23×C7—21no—
91C23×C7—21no—
92C23×C7—21no—
93C23×C7—21no—
94C23×C7—21no—
95C23×C7—21no—
96C23×C7C2×D412yes—
97C23×C7#4112yes—
98C23×C7C2411yescharacteristic
99C7×D4—28no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—28no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—44no—
112C2×C4×C7C2×D412yes—
113C2×C4×C7C2×D412yes—
114C2×C4×C7#4112yes—
115C2×C4×C7#4112yes—
116C2×C4×C7#4112yes—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7—22no—
120C2×Dic7—22no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—22no—
125C2×Dic7—22no—
126C2×Dic7—22no—
127C2×Dic7—22no—
128C2×Dic7—22no—
129C22×D4C2×D711yescharacteristic
130C23×C4C2×D711yescharacteristic
131C23×C4C2×D711yescharacteristic
132C4×D4—24no—
133#117—144no—
134C2×C4⋊C4—71no—
135C2×C4⋊C4—71no—
136C2×C4⋊C4—72no—
137C2×C4⋊C4—71no—
138C2×C4⋊C4—71no—
139C2×C4⋊C4—72no—
140C2×C4⋊C4C2×D711yescharacteristic
141C2×C22⋊C4—72no—
142C2×C22⋊C4—72no—
143C2×C22⋊C4—72no—
144C2×C22⋊C4—72no—
145C2×C22⋊C4C2×D711yescharacteristic
146C2×C22⋊C4C2×D711yescharacteristic
147C2×C42—72no—
148C2×C42C2×D711yescharacteristic
149#95—72no—
150#95—72no—
151#95—72no—
152#95—72no—
153#95—72no—
154C22×C7—22no—
155C22×C7—22no—
156C22×C7—22no—
157C22×C7—22no—
158C22×C7—22no—
159C22×C7—22no—
160C22×C7—41no—
161C22×C7—41no—
162C22×C7—41no—
163C22×C7—41no—
164C22×C7—41no—
165C22×C7—41no—
166C22×C7—41no—
167C22×C7—41no—
168C22×C7C22×D411yescharacteristic
169C22×C7C2×#4111yescharacteristic
170C22×C7C2×#4111yescharacteristic
171C22×C7C2×#4111yescharacteristic
172C22×C7C2×#4111yescharacteristic
173C22×C7H(5,𝔽2)11yescharacteristic
174C22×C72−1+411yescharacteristic
175C4×C7—22no—
176C4×C7—22no—
177C4×C7—41no—
178C4×C7—41no—
179C4×C7#12314yes—
180Dic7—42no—
181Dic7—42no—
182Dic7—42no—
183Dic7—42no—
184C24C22×D711yescharacteristic
185C24C22×D711yescharacteristic
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—72no—
195C22×C4—72no—
196C22×C4—72no—
197C22×C4—72no—
198C22×C4C22⋊D714yes—
199C22×C4C22×D711yescharacteristic
200C22×C4C22×D711yescharacteristic
201C22×C4C22×D711yescharacteristic
202C22×C4C22×D711yescharacteristic
203C22×C4C22×D711yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—144no—
206C4⋊C4—142no—
207C4⋊C4—142no—
208C4⋊C4—142no—
209C4⋊C4—142no—
210C4⋊C4—282no—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—144no—
214C22⋊C4—144no—
215C22⋊C4—282no—
216C22⋊C4—282no—
217C42—22no—
218C42—144no—
219C2×C7—42no—
220C2×C7—42no—
221C2×C7C2×#12311yescharacteristic
222C2×C7#52511yescharacteristic
223C2×C7#53711yescharacteristic
224C2×C7#55211yescharacteristic
225C2×C7#55311yescharacteristic
226C2×C7#55511yescharacteristic
227C2×C7#55611yescharacteristic
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23#115712yes—
241C23C2×C22⋊D712yes—
242C23C23×D711yescharacteristic
243D4—28no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—28no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—44no—
256C2×C4—142no—
257C2×C4—142no—
258C2×C4—142no—
259C2×C4—142no—
260C2×C4—142no—
261C2×C4—142no—
262C2×C4—142no—
263C2×C4—142no—
264C2×C4—142no—
265C2×C4—142no—
266C2×C4—142no—
267C2×C4—142no—
268C2×C4#115712yes—
269C2×C4#115912yes—
270C2×C4#115912yes—
271C2×C4C2×C22⋊D712yes—
272C2×C4C2×C22⋊D712yes—
273C7#248511yescharacteristic, complemented, Sylow
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—22no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22C2×#115711yescharacteristic
289C22C2×#115711yescharacteristic
290C22C2×#115911yescharacteristic
291C22C2×#115911yescharacteristic
292C22#655311yescharacteristic
293C22#655411yescharacteristic
294C22C22×C22⋊D711yescharacteristic
295C4—22no—
296C4—22no—
297C4—41no—
298C4—41no—
299C4—282no—
300C4—282no—
301C4—282no—
302C4—282no—
303C4#649814yes—
304C2—42no—
305C2—42no—
306C2#9023211yescharacteristic
307C2#9023311yescharacteristic
308C2#9025411yescharacteristic
309C2#9025611yescharacteristic
310C2#9048211yescharacteristic
311C2C2×#649811yescharacteristic
312C2#9052411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C4×C7×D4C2
prime radicalC22×C7×D4C22
hypercenterC2×C4×D4D7
hypocenterC7#2485
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#6489C2semidirect
4. #89984C2semidirect
5. #89984C2semidirect
6. #89984C2semidirect
7. #89750C2semidirect
8. #89724C2semidirect
9. #89716C2semidirect
10. #89700C2semidirect
11. #89700C2semidirect
273. C7#2485semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×D42
C2×#64892
#899942
#899842
#897502
#897242
#897162
#897002
#24857