Finite Group Databasebeta

← #11734309

Subgroups — #11734309

Order 896.

914 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 157 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×#1161C211yesmaximal, characteristic, complemented
2C2×#6447C212yesmaximal, complemented
3C2×#6446C211yesmaximal, characteristic, complemented
4C7×#393C211yesmaximal, characteristic, complemented
5#90001C211yesmaximal, characteristic, complemented
6#89732C211yesmaximal, characteristic, complemented
7#89728C212yesmaximal, complemented
8#89444C211yesmaximal, characteristic, complemented
9#89440C212yesmaximal, complemented
10#89437C212yesmaximal, complemented
11D7×#95C211yesmaximal, characteristic, complemented
12C22×C7×Q8C2211yescharacteristic
13C22×C22⋊D7C2211yescharacteristic
14C22×C22⋊D7C2212yes—
15C2×#1161—28no—
16C2×#1161—24no—
17C22×D28C2211yescharacteristic
18C22×D28C2212yes—
19C22×C4×D7C2212yes—
20C22×C4×D7C2212yes—
21C22×C4×D7C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2212yes—
24C2×C22⋊Dic7C2211yescharacteristic
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2212yes—
27C2×(C2×D7)⋊C4C2212yes—
28C2×(C2×D7)⋊C4C2212yes—
29C2×C4⋊Dic7C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2212yes—
32#6447—28no—
33#6446—24no—
34C7×#95C2211yescharacteristic
35C7×#95C2212yes—
36C7⋊#95C2212yes—
37C7⋊#95C2211yescharacteristic
38C7⋊#95C2212yes—
39#2430—71nomaximal, Sylow
40C24×C7C2311yescharacteristic
41C23×D7C2312yes—
42C23×D7C2311yescharacteristic
43C2×C7×Q8—24no—
44C2×C7×Q8—22no—
45C22×C4×C7C2312yes—
46C22×C4×C7C2311yescharacteristic
47C22×C4×C7C2312yes—
48C22×C4×C7C2311yescharacteristic
49C2×C22⋊D7—22no—
50C2×C22⋊D7—24no—
51C2×C22⋊D7—24no—
52C2×C22⋊D7—24no—
53C2×C22⋊D7—42no—
54C22×Dic7C2311yescharacteristic
55C22×Dic7C2311yescharacteristic
56C22×Dic7C2311yescharacteristic
57C22×Dic7C2312yes—
58#1161—232no—
59C2×D28—24no—
60C2×D28—24no—
61C2×D28—24no—
62C2×D28—22no—
63C2×D28D418yescomplemented
64C2×C4×D7—24no—
65C2×C4×D7—24no—
66C2×C4×D7—24no—
67C2×C4×D7—24no—
68C2×C4×D7—24no—
69C2×C4×D7—24no—
70C2×C4×D7—42no—
71C2×C4×D7D414yescomplemented
72C7×C22⋊C4—22no—
73C7×C22⋊C4—24no—
74C22⋊Dic7—22no—
75(C2×D7)⋊C4—22no—
76(C2×D7)⋊C4—24no—
77(C2×D7)⋊C4—24no—
78(C2×D7)⋊C4—42no—
79C4⋊Dic7—22no—
80Dic7⋊C4—22no—
81Dic7⋊C4—24no—
82C22×#41—71no—
83C2×#123—71no—
84C2×#121—72no—
85#395—71no—
86#394—71no—
87#393D711yescharacteristic, complemented
88#392—71no—
89#386—72no—
90#386—72no—
91#385—72no—
92C2×#95—71no—
93C23×C7—21no—
94C23×C7—22no—
95C23×C7—22no—
96C23×C7—21no—
97C23×C7—21no—
98C23×C7C2411yescharacteristic
99C22×D7—22no—
100C22×D7—21no—
101C22×D7—21no—
102C22×D7—22no—
103C22×D7—22no—
104C22×D7—22no—
105C22×D7—22no—
106C22×D7—22no—
107C22×D7—22no—
108C22×D7C2×D414yes—
109C22×D7C2×D412yes—
110C22×D7C2×D414yes—
111C7×Q8—28no—
112C2×C4×C7—22no—
113C2×C4×C7—22no—
114C2×C4×C7—22no—
115C2×C4×C7—21no—
116C2×C4×C7—22no—
117C2×C4×C7—22no—
118C2×C4×C7—22no—
119C2×C4×C7—22no—
120C2×C4×C7C2×D414yes—
121C2×C4×C7C2×D412yes—
122C22⋊D7—48no—
123C22⋊D7—44no—
124C2×Dic7—22no—
125C2×Dic7—22no—
126C2×Dic7—22no—
127C2×Dic7—21no—
128C2×Dic7—21no—
129C2×Dic7—22no—
130C2×Dic7—21no—
131C2×Dic7—21no—
132C2×Dic7C2×D412yes—
133C2×Dic7#4114yescomplemented
134D28—28no—
135D28—216no—
136C4×D7—216no—
137C4×D7—28no—
138C4×D7—48no—
139C2×#41—144no—
140C2×#41—148no—
141C22×Q8C2×D711yescharacteristic
142C22×D4—71no—
143C22×D4—72no—
144C22×D4—72no—
145C22×D4—71no—
146C23×C4—71no—
147C23×C4—72no—
148C23×C4—72no—
149#123—144no—
150#121—148no—
151C2×C4⋊C4—72no—
152C2×C4⋊C4—71no—
153C2×C4⋊C4—71no—
154C2×C22⋊C4—71no—
155C2×C22⋊C4—72no—
156C2×C22⋊C4—71no—
157C2×C22⋊C4—72no—
158C2×C22⋊C4—72no—
159C2×C22⋊C4C2×D712yes—
160C2×C22⋊C4C2×D711yescharacteristic
161#95—72no—
162#95—71no—
163#95—72no—
164#95C2×D711yescharacteristic
165#95C2×D712yes—
166C22×C7—41no—
167C22×C7—41no—
168C22×C7—41no—
169C22×C7—42no—
170C22×C7—42no—
171C22×C7C22×D411yescharacteristic
172C22×C7C22×D412yes—
173C22×C7C2×#4112yes—
174C22×C7H(5,𝔽2)11yescharacteristic
175C22×C72−1+411yescharacteristic
176C2×D7—22no—
177C2×D7—22no—
178C2×D7—24no—
179C2×D7—24no—
180C2×D7—24no—
181C2×D7—24no—
182C2×D7—24no—
183C2×D7—42no—
184C2×D7—42no—
185C2×D7—42no—
186C2×D7—42no—
187C2×D7#12014yes—
188C4×C7—24no—
189C4×C7—22no—
190C4×C7—41no—
191C4×C7—42no—
192Dic7—22no—
193Dic7—24no—
194Dic7—41no—
195Dic7—41no—
196C24—71no—
197C24—72no—
198C24C22×D711yescharacteristic
199#41—1432no—
200C2×Q8—22no—
201C2×Q8—24no—
202C2×D4—78no—
203C2×D4—144no—
204C2×D4—144no—
205C2×D4—144no—
206C2×D4—144no—
207C2×D4—144no—
208C2×D4—144no—
209C2×D4—142no—
210C2×D4—142no—
211C2×D4—282no—
212C22×C4—72no—
213C22×C4—71no—
214C22×C4—74no—
215C22×C4—71no—
216C22×C4—71no—
217C22×C4—144no—
218C22×C4—144no—
219C22×C4—144no—
220C22×C4—144no—
221C22×C4—144no—
222C22×C4—144no—
223C22×C4—282no—
224C22×C4C22×D712yes—
225C22×C4C22×D711yescharacteristic
226C22×C4C22×D711yescharacteristic
227C22×C4C22×D712yes—
228C4⋊C4—142no—
229C4⋊C4—144no—
230C4⋊C4—142no—
231C22⋊C4—24no—
232C22⋊C4—22no—
233C22⋊C4—144no—
234C22⋊C4—144no—
235C22⋊C4—142no—
236C22⋊C4—142no—
237C22⋊C4—282no—
238C2×C7—81no—
239C2×C7C2×#12011yescharacteristic
240C2×C7#53211yescharacteristic
241C2×C7#53711yescharacteristic
242C2×C7#54612yes—
243C2×C7#54712yes—
244D7—24no—
245D7—44no—
246C23—21no—
247C23—22no—
248C23—21no—
249C23—22no—
250C23—21no—
251C23—72no—
252C23—74no—
253C23—74no—
254C23—141no—
255C23—142no—
256C23—141no—
257C23—142no—
258C23—142no—
259C23—142no—
260C23—142no—
261C23—142no—
262C23—142no—
263C23C23×D711yescharacteristic
264Q8—28no—
265D4—1416no—
266D4—148no—
267D4—288no—
268D4—284no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—72no—
278C2×C4—74no—
279C2×C4—148no—
280C2×C4—1416no—
281C2×C4—142no—
282C2×C4—142no—
283C2×C4—142no—
284C2×C4—141no—
285C2×C4—142no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—288no—
290C2×C4D4×D712yes—
291C2×C4D4×D714yes—
292C7#243011yescharacteristic, complemented, Sylow
293C22—41no—
294C22—41no—
295C22—42no—
296C22—41no—
297C22—42no—
298C22—74no—
299C22—142no—
300C22—144no—
301C22—142no—
302C22—144no—
303C22—144no—
304C22—144no—
305C22—144no—
306C22—282no—
307C22—282no—
308C22—282no—
309C22—282no—
310C22C2×D4×D711yescharacteristic
311C22C2×D4×D712yes—
312C22#655111yescharacteristic
313C22D7×#4112yes—
314C22#655311yescharacteristic
315C4—22no—
316C4—24no—
317C4—41no—
318C4—42no—
319C4—142no—
320C4—144no—
321C4—281no—
322C4—281no—
323C2—81no—
324C2—144no—
325C2—284no—
326C2D7×#12011yescharacteristic
327C2#9032011yescharacteristic
328C2#9032212yes—
329C2#9035911yescharacteristic
330C2#9036712yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#393C2
hypercenter#393D7
hypocenterC7#2430
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×#1161C2semidirect
2. C2×#6447C2semidirect
3. C2×#6446C2semidirect
4. C7×#393C2semidirect
5. #90001C2semidirect
6. #89732C2semidirect
7. #89728C2semidirect
8. #89444C2semidirect
9. #89440C2semidirect
10. #89437C2semidirect
11. D7×#95C2semidirect
63. C2×D28D4semidirect
71. C2×C4×D7D4semidirect
87. #393D7semidirect
133. C2×Dic7#41semidirect
292. C7#2430semidirect

Maximal subgroups

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

subgroupindex
C22×#11612
C2×#64472
C2×#64462
C7×#3932
#900012
#897322
#897282
#894442
#894402
#894372
D7×#952
#24307