Finite Group Databasebeta

← #11734110

Subgroups — #11734110

Order 896.

944 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#1159C211yesmaximal, characteristic, complemented
2C2×D4×Dic7C212yesmaximal, complemented
3C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
4C7×#392C211yesmaximal, characteristic, complemented
5#89995C212yesmaximal, complemented
6#89994C211yesmaximal, characteristic, complemented
7#89733C212yesmaximal, complemented
8#89719C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic7C212yesmaximal, complemented
10#89438C211yesmaximal, characteristic, complemented
11#89428C211yesmaximal, characteristic, complemented
12C22×C7×D4C2212yes—
13C22×C7×D4C2211yescharacteristic
14C22×C22⋊D7C2211yescharacteristic
15C22×C22⋊D7C2212yes—
16C23×Dic7C2211yescharacteristic
17C23×Dic7C2212yes—
18C2×#1159—28no—
19C2×#1159—24no—
20C22×C4×D7C2211yescharacteristic
21C22×Dic14C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2212yes—
24C2×C22⋊Dic7C2212yes—
25C2×C22⋊Dic7C2212yes—
26C2×C22⋊Dic7C2212yes—
27D4×Dic7—28no—
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2212yes—
30C2×C4⋊Dic7C2212yes—
31C2×Dic7⋊C4C2211yescharacteristic
32C2×C4×Dic7C2211yescharacteristic
33C2×C4×Dic7C2212yes—
34(C22⋊D7)⋊C4—24no—
35C7×#95C2211yescharacteristic
36C7⋊#95C2212yes—
37C7⋊#95C2211yescharacteristic
38C7⋊#95C2212yes—
39#2388—71nomaximal, Sylow
40C24×C7C2312yes—
41C24×C7C2311yescharacteristic
42C23×D7C2311yescharacteristic
43C2×C7×D4—24no—
44C2×C7×D4—24no—
45C2×C7×D4—24no—
46C2×C7×D4—22no—
47C2×C7×D4—24no—
48C22×C4×C7C2311yescharacteristic
49C22×C4×C7C2311yescharacteristic
50C22×C4×C7C2312yes—
51C2×C22⋊D7—24no—
52C2×C22⋊D7—22no—
53C2×C22⋊D7—22no—
54C2×C22⋊D7—24no—
55C2×C22⋊D7—24no—
56C2×C22⋊D7—22no—
57C22×Dic7—24no—
58C22×Dic7—24no—
59C22×Dic7—22no—
60C22×Dic7—24no—
61C22×Dic7D418yescomplemented
62C22×Dic7C2312yes—
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2312yes—
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67#1159—232no—
68C2×C4×D7—24no—
69C2×C4×D7D414yescomplemented
70C2×Dic14—22no—
71C2×Dic14—24no—
72C7×C22⋊C4—22no—
73C7×C22⋊C4—24no—
74C22⋊Dic7—24no—
75C22⋊Dic7—24no—
76C22⋊Dic7—24no—
77(C2×D7)⋊C4—22no—
78(C2×D7)⋊C4—24no—
79C4⋊Dic7—24no—
80Dic7⋊C4—22no—
81C4×Dic7—22no—
82C4×Dic7—24no—
83C22×#41—71no—
84C2×C4×D4—72no—
85C2×C4×D4—71no—
86#395—71no—
87#394—71no—
88#394—72no—
89#393—71no—
90#392—72no—
91#392D711yescharacteristic, complemented
92C4×C22⋊C4—72no—
93C4×C22⋊C4—71no—
94C23×C7—22no—
95C23×C7—21no—
96C23×C7—22no—
97C23×C7—22no—
98C23×C7—22no—
99C23×C7—22no—
100C23×C7—21no—
101C23×C7—22no—
102C23×C7—22no—
103C23×C7—22no—
104C23×C7C2×D414yes—
105C23×C7#4112yescomplemented
106C23×C7C2411yescharacteristic
107C22×D7—22no—
108C22×D7—22no—
109C22×D7—21no—
110C22×D7—21no—
111C22×D7C2×D412yes—
112C7×D4—28no—
113C7×D4—216no—
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—22no—
117C2×C4×C7—21no—
118C2×C4×C7—22no—
119C2×C4×C7C2×D412yes—
120C2×C4×C7#4114yescomplemented
121C22⋊D7—216no—
122C22⋊D7—28no—
123C2×Dic7—21no—
124C2×Dic7—22no—
125C2×Dic7—22no—
126C2×Dic7—21no—
127C2×Dic7—216no—
128C2×Dic7—22no—
129C2×Dic7—21no—
130C2×Dic7—22no—
131C2×Dic7—22no—
132C2×Dic7—22no—
133C2×Dic7—44no—
134C2×Dic7C2×D414yes—
135C2×Dic7C2×D414yes—
136C2×Dic7C2×D412yes—
137C2×Dic7#4112yescomplemented
138C4×D7—28no—
139Dic14—28no—
140C2×#41—148no—
141C2×#41—144no—
142C22×Q8—71no—
143C22×D4—71no—
144C22×D4—72no—
145C22×D4C2×D712yes—
146C22×D4C2×D711yescharacteristic
147C23×C4—72no—
148C23×C4—71no—
149C23×C4—71no—
150C4×D4—144no—
151C4×D4—148no—
152C2×C4⋊C4—71no—
153C2×C4⋊C4—72no—
154C2×C22⋊C4—72no—
155C2×C22⋊C4—72no—
156C2×C22⋊C4—72no—
157C2×C22⋊C4—71no—
158C2×C22⋊C4—72no—
159C2×C22⋊C4C2×D711yescharacteristic
160C2×C22⋊C4C2×D712yes—
161C2×C42—71no—
162C2×C42—72no—
163#95—72no—
164#95—72no—
165#95—71no—
166#95C2×D711yescharacteristic
167C22×C7—24no—
168C22×C7—22no—
169C22×C7—24no—
170C22×C7—24no—
171C22×C7—24no—
172C22×C7—41no—
173C22×C7—42no—
174C22×C7—42no—
175C22×C7—41no—
176C22×C7—42no—
177C22×C7—42no—
178C22×C7—42no—
179C22×C7C22×D411yescharacteristic
180C22×C7C22×D412yes—
181C22×C7C2×#4111yescharacteristic
182C22×C7C2×#4112yes—
183C22×C7C2×#4111yescharacteristic
184C2×D7—22no—
185C2×D7—22no—
186C2×D7—22no—
187C2×D7—42no—
188C2×D7—42no—
189C4×C7—22no—
190C4×C7—24no—
191C4×C7—41no—
192Dic7—24no—
193Dic7—22no—
194Dic7—41no—
195Dic7—42no—
196Dic7#12014yes—
197C24—71no—
198C24C22×D712yes—
199C24C22×D711yescharacteristic
200#41—1432no—
201C2×Q8—144no—
202C2×Q8—142no—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—24no—
208C2×D4—142no—
209C2×D4—144no—
210C2×D4—142no—
211C2×D4—144no—
212C2×D4—144no—
213C2×D4—142no—
214C22×C4—71no—
215C22×C4—78no—
216C22×C4—71no—
217C22×C4—72no—
218C22×C4—74no—
219C22×C4—72no—
220C22×C4—71no—
221C22×C4—144no—
222C22×C4—144no—
223C22×C4—144no—
224C22×C4—142no—
225C22×C4—144no—
226C22×C4C22×D711yescharacteristic
227C22×C4C22×D711yescharacteristic
228C22×C4C22×D712yes—
229C4⋊C4—144no—
230C4⋊C4—142no—
231C22⋊C4—22no—
232C22⋊C4—24no—
233C22⋊C4—142no—
234C22⋊C4—144no—
235C22⋊C4—144no—
236C22⋊C4—144no—
237C22⋊C4—144no—
238C42—144no—
239C42—142no—
240C2×C7—44no—
241C2×C7—42no—
242C2×C7C2×#12011yescharacteristic
243C2×C7#52512yes—
244C2×C7#52511yescharacteristic
245C2×C7#53211yescharacteristic
246C2×C7#53212yes—
247D7—42no—
248C23—21no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—21no—
258C23—72no—
259C23—142no—
260C23—142no—
261C23—141no—
262C23—141no—
263C23D4×D714yes—
264C23#115912yescomplemented
265C23C23×D711yescharacteristic
266Q8—148no—
267D4—216no—
268D4—28no—
269D4—148no—
270D4—1416no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—72no—
277C2×C4—74no—
278C2×C4—74no—
279C2×C4—72no—
280C2×C4—148no—
281C2×C4—1416no—
282C2×C4—141no—
283C2×C4—142no—
284C2×C4—142no—
285C2×C4—141no—
286C2×C4—142no—
287C2×C4—141no—
288C2×C4—142no—
289C2×C4—142no—
290C2×C4—142no—
291C2×C4—284no—
292C2×C4D4×D712yes—
293C2×C4#115914yescomplemented
294C7#238811yescharacteristic, complemented, Sylow
295C22—24no—
296C22—24no—
297C22—24no—
298C22—24no—
299C22—22no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—41no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—142no—
308C22—142no—
309C22—142no—
310C22—282no—
311C22—282no—
312C22C2×D4×D711yescharacteristic
313C22C2×D4×D712yes—
314C22C2×#115912yes—
315C22C2×#115911yescharacteristic
316C22D7×#4111yescharacteristic
317C4—24no—
318C4—22no—
319C4—41no—
320C4—74no—
321C4—144no—
322C4—142no—
323C4—281no—
324C4—282no—
325C2—44no—
326C2—42no—
327C2—282no—
328C2#9027712yes—
329C2D7×#12011yescharacteristic
330C2#9028712yes—
331C2#9029711yescharacteristic
332C2#9040711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#392C2
hypercenter#392D7
hypocenterC7#2388
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#1159C2semidirect
2. C2×D4×Dic7C2semidirect
3. C2×(C22⋊D7)⋊C4C2semidirect
4. C7×#392C2semidirect
5. #89995C2semidirect
6. #89994C2semidirect
7. #89733C2semidirect
8. #89719C2semidirect
9. C22⋊C4×Dic7C2semidirect
10. #89438C2semidirect
11. #89428C2semidirect
61. C22×Dic7D4semidirect
69. C2×C4×D7D4semidirect
91. #392D7semidirect
105. C23×C7#41semidirect
120. C2×C4×C7#41semidirect
137. C2×Dic7#41semidirect
264. C23#1159semidirect
293. C2×C4#1159semidirect
294. C7#2388semidirect

Maximal subgroups

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

subgroupindex
C22×#11592
C2×D4×Dic72
C2×(C22⋊D7)⋊C42
C7×#3922
#899952
#899942
#897332
#897192
C22⋊C4×Dic72
#894382
#894282
#23887