Finite Group Databasebeta

← #11740304

Subgroups — #11740304

Order 896.

708 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#123C211yesmaximal, characteristic, complemented
2C2×#6502C211yesmaximal, characteristic
3C2×#6498C211yesmaximal, characteristic, complemented
4C2×#6492C211yesmaximal, characteristic, complemented
5C22×Dic7⋊C4C211yesmaximal, characteristic, complemented
6C2×#6449C211yesmaximal, characteristic, complemented
7#89764C212yesmaximal, complemented
8#89748C212yesmaximal, complemented
9#89729C212yesmaximal, complemented
10#89721C212yesmaximal, complemented
11C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C23×C4×C7C2211yescharacteristic
14C22×C22⋊D7C2211yescharacteristic
15C22×C22⋊D7C2211yescharacteristic
16C23×Dic7C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C7×#123—24no—
19C2×C7×C4⋊C4C2212yes—
20C2×C7×C22⋊C4C2212yes—
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C22⋊Dic7C2212yes—
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25#6502—24no—
26#6498—24no—
27#6492—24no—
28C2×(C2×D7)⋊C4C2212yes—
29C2×(C2×D7)⋊C4C2212yes—
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×C4⋊Dic7C2212yes—
32C2×Dic7⋊C4—24no—
33C2×Dic7⋊C4C2212yes—
34C2×Dic7⋊C4C2212yes—
35C2×Dic7⋊C4C2212yes—
36C2×Dic7⋊C4C2212yes—
37C2×Dic7⋊C4C2211yescharacteristic
38C2×C4×Dic7C2212yes—
39#6449—24no—
40C7⋊#95C2212yes—
41#2501—71nomaximal, Sylow
42C24×C7C2311yescharacteristic
43C24×C7C2311yescharacteristic
44C23×D7C2311yescharacteristic
45C2×C7×D4—22no—
46C2×C7×D4—22no—
47C2×C7×D4—22no—
48C22×C4×C7—22no—
49C22×C4×C7—22no—
50C22×C4×C7—22no—
51C22×C4×C7C2312yes—
52C22×C4×C7C2311yescharacteristic
53C22×C4×C7C2312yes—
54C2×C22⋊D7—22no—
55C2×C22⋊D7—22no—
56C2×C22⋊D7—22no—
57C2×C22⋊D7—22no—
58C2×C22⋊D7—24no—
59C22×Dic7—24no—
60C22×Dic7—22no—
61C22×Dic7—22no—
62C22×Dic7—22no—
63C22×Dic7D414yes—
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2312yes—
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2312yes—
68C22×Dic7C2311yescharacteristic
69C7×C4⋊C4—24no—
70C7×C22⋊C4—24no—
71C7×C22⋊C4D414yescomplemented
72C22⋊Dic7—24no—
73C22⋊Dic7—22no—
74C22⋊Dic7D414yescomplemented
75(C2×D7)⋊C4—24no—
76(C2×D7)⋊C4—24no—
77(C2×D7)⋊C4—22no—
78C4⋊Dic7—24no—
79Dic7⋊C4—24no—
80Dic7⋊C4—24no—
81Dic7⋊C4—24no—
82Dic7⋊C4—24no—
83Dic7⋊C4—22no—
84C4×Dic7—24no—
85C2×#123—71no—
86C2×#123—71no—
87C2×#123—71no—
88C2×#123D711yescharacteristic, complemented
89C2×#121—71no—
90C22×C4⋊C4—71no—
91#399—72no—
92#399—72no—
93#390—72no—
94#384—72no—
95C4×C22⋊C4—71no—
96C23×C7—22no—
97C23×C7—22no—
98C23×C7—21no—
99C23×C7—21no—
100C23×C7—21no—
101C23×C7—22no—
102C23×C7—22no—
103C23×C7C2×D412yes—
104C23×C7C2×D412yes—
105C23×C7C2×D412yes—
106C23×C7C2411yescharacteristic
107C22×D7—21no—
108C22×D7—21no—
109C22×D7—21no—
110C22×D7—22no—
111C22×D7—22no—
112C7×D4—44no—
113C2×C4×C7—22no—
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—21no—
117C2×C4×C7—22no—
118C2×C4×C7—21no—
119C2×C4×C7—22no—
120C2×C4×C7—21no—
121C2×C4×C7—44no—
122C2×C4×C7C2×D414yes—
123C22⋊D7—44no—
124C22⋊D7—44no—
125C2×Dic7—22no—
126C2×Dic7—22no—
127C2×Dic7—22no—
128C2×Dic7—28no—
129C2×Dic7—22no—
130C2×Dic7—21no—
131C2×Dic7—22no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—22no—
135C2×Dic7—44no—
136C2×Dic7C2×D414yes—
137C2×Dic7C2×D412yes—
138C2×Dic7C2×D412yes—
139C2×Dic7#4114yes—
140C22×D4—71no—
141C22×D4—71no—
142C22×D4C2×D711yescharacteristic
143C23×C4—71no—
144C23×C4—71no—
145C23×C4C2×D711yescharacteristic
146#123—24no—
147#123—144no—
148#123—144no—
149#123—144no—
150#121—144no—
151C2×C4⋊C4—72no—
152C2×C4⋊C4—72no—
153C2×C4⋊C4—71no—
154C2×C4⋊C4—72no—
155C2×C4⋊C4—72no—
156C2×C4⋊C4—72no—
157C2×C4⋊C4—144no—
158C2×C4⋊C4C2×D712yes—
159C2×C22⋊C4—71no—
160C2×C22⋊C4—71no—
161C2×C22⋊C4—72no—
162C2×C22⋊C4—71no—
163C2×C22⋊C4—72no—
164C2×C22⋊C4—72no—
165C2×C22⋊C4C2×D712yes—
166C2×C22⋊C4C2×D711yescharacteristic
167C2×C42—72no—
168#95—72no—
169C22×C7—22no—
170C22×C7—22no—
171C22×C7—24no—
172C22×C7—24no—
173C22×C7—41no—
174C22×C7—41no—
175C22×C7—41no—
176C22×C7—42no—
177C22×C7—42no—
178C22×C7#12714yes—
179C22×C7C22×D411yescharacteristic
180C22×C7C22×D411yescharacteristic
181C22×C7C22×D411yescharacteristic
182C22×C7C2×#4112yes—
183C22×C72−1+412yes—
184C2×D7—41no—
185C2×D7—41no—
186C2×D7—41no—
187C2×D7—42no—
188C2×D7—42no—
189C4×C7—24no—
190C4×C7—42no—
191C4×C7—41no—
192Dic7—22no—
193Dic7—24no—
194Dic7—42no—
195Dic7—41no—
196Dic7#12314yescomplemented
197C24—71no—
198C24C22×D711yescharacteristic
199C24C22×D711yescharacteristic
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—142no—
204C2×D4—144no—
205C2×D4—142no—
206C2×D4—142no—
207C2×D4—142no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—71no—
212C22×C4—74no—
213C22×C4—71no—
214C22×C4—72no—
215C22×C4—72no—
216C22×C4—71no—
217C22×C4—144no—
218C22×C4—142no—
219C22×C4—142no—
220C22×C4—142no—
221C22×C4C22×D712yes—
222C22×C4C22×D711yescharacteristic
223C22×C4C22×D712yes—
224C4⋊C4—24no—
225C4⋊C4—144no—
226C4⋊C4—144no—
227C4⋊C4—142no—
228C4⋊C4—144no—
229C4⋊C4—144no—
230C4⋊C4—144no—
231C22⋊C4—24no—
232C22⋊C4—74no—
233C22⋊C4—142no—
234C22⋊C4—144no—
235C22⋊C4—142no—
236C22⋊C4—144no—
237C22⋊C4—144no—
238C22⋊C4C22⋊D714yescomplemented
239C42—144no—
240C2×C7—24no—
241C2×C7—81no—
242C2×C7C2×#12311yescharacteristic
243C2×C7C2×#12711yescharacteristic
244C2×C7#53611yescharacteristic
245C2×C7#54712yes—
246C2×C7#54712yes—
247D7—81no—
248C23—22no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—141no—
256C23—141no—
257C23—142no—
258C23—142no—
259C23—141no—
260C23D4×D712yes—
261C23D4×D712yes—
262C23C2×C22⋊D712yes—
263C23C23×D711yescharacteristic
264D4—44no—
265D4—284no—
266D4—284no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—44no—
276C2×C4—72no—
277C2×C4—74no—
278C2×C4—74no—
279C2×C4—72no—
280C2×C4—148no—
281C2×C4—142no—
282C2×C4—142no—
283C2×C4—142no—
284C2×C4—142no—
285C2×C4—141no—
286C2×C4—142no—
287C2×C4—141no—
288C2×C4—142no—
289C2×C4—141no—
290C2×C4—284no—
291C2×C4C2×C22⋊D714yes—
292C7#250111yescharacteristic, complemented, Sylow
293C22—24no—
294C22—22no—
295C22—22no—
296C22—24no—
297C22—41no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—42no—
302C22—281no—
303C22—282no—
304C22—281no—
305C22—281no—
306C22—282no—
307C22#650314yes—
308C22C2×D4×D711yescharacteristic
309C22C2×D4×D711yescharacteristic
310C22D7×#4112yes—
311C22#655412yes—
312C22C22×C22⋊D711yescharacteristic
313C4—24no—
314C4—42no—
315C4—41no—
316C4—74no—
317C4—142no—
318C4—144no—
319C4—282no—
320C4—281no—
321C2—24no—
322C2—81no—
323C2—561no—
324C2#9033911yescharacteristic
325C2D7×#12311yescharacteristic
326C2#9034612yes—
327C2C2×#650311yescharacteristic
328C2#9051512yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#123C2
prime radicalC2×#6502C2
hypercenterC2×#123D7
hypocenterC7#2501
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C7×#123C2semidirect
3. C2×#6498C2semidirect
4. C2×#6492C2semidirect
5. C22×Dic7⋊C4C2semidirect
6. C2×#6449C2semidirect
7. #89764C2semidirect
8. #89748C2semidirect
9. #89729C2semidirect
10. #89721C2semidirect
11. C22⋊C4×Dic7C2semidirect
71. C7×C22⋊C4D4semidirect
74. C22⋊Dic7D4semidirect
88. C2×#123D7semidirect
196. Dic7#123semidirect
238. C22⋊C4C22⋊D7semidirect
292. C7#2501semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1232
C2×#65022
C2×#64982
C2×#64922
C22×Dic7⋊C42
C2×#64492
#897642
#897482
#897292
#897212
C22⋊C4×Dic72
#25017