Finite Group Databasebeta

← #11734416

Subgroups — #11734416

Order 896.

728 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#6503C211yesmaximal, characteristic, complemented
2C2×#6455C211yesmaximal, characteristic, complemented
3C2×#6447C212yesmaximal, complemented
4C7×#394C211yesmaximal, characteristic, complemented
5#89995C211yesmaximal, characteristic, complemented
6#89763C211yesmaximal, characteristic, complemented
7#89733C212yesmaximal, complemented
8#89732C212yesmaximal, complemented
9#89729C212yesmaximal, complemented
10#89443C211yesmaximal, characteristic, complemented
11#89422C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C22×C22⋊D7C2212yes—
15C22×C22⋊D7C2212yes—
16C22×D28C2212yes—
17C22×D28C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2212yes—
21C2×C7×C22⋊C4C2212yes—
22C2×C22⋊Dic7C2212yes—
23#6503—24no—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2212yes—
28C2×(C2×D7)⋊C4C2212yes—
29C2×C4⋊Dic7C2211yescharacteristic
30C2×Dic7⋊C4C2212yes—
31C2×C4×Dic7C2211yescharacteristic
32C2×C4×Dic7C2211yescharacteristic
33#6455—24no—
34#6447—28no—
35C7×#95C2211yescharacteristic
36C7⋊#95C2212yes—
37C7⋊#95C2211yescharacteristic
38#2619—71nomaximal, Sylow
39C24×C7C2312yes—
40C23×D7C2312yes—
41C23×D7C2311yescharacteristic
42C2×C7×D4—22no—
43C2×C7×D4—42no—
44C22×C4×C7C2312yes—
45C22×C4×C7C2311yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C22×C4×C7C2311yescharacteristic
48C2×C22⋊D7—24no—
49C2×C22⋊D7—24no—
50C2×C22⋊D7—24no—
51C2×C22⋊D7—24no—
52C2×C22⋊D7—24no—
53C2×C22⋊D7—42no—
54C2×C22⋊D7—42no—
55C2×C22⋊D7—42no—
56C2×C22⋊D7—42no—
57C22×Dic7C2312yes—
58C22×Dic7C2311yescharacteristic
59C22×Dic7C2311yescharacteristic
60C22×Dic7C2311yescharacteristic
61C2×D28—24no—
62C2×D28—22no—
63C2×D28—42no—
64C2×D28—42no—
65C2×D28—42no—
66C2×C4×D7—22no—
67C2×C4×D7—22no—
68C2×C4×D7D414yescomplemented
69C7×C4⋊C4—22no—
70C7×C22⋊C4—24no—
71C7×C22⋊C4—42no—
72C22⋊Dic7—42no—
73(C2×D7)⋊C4—24no—
74(C2×D7)⋊C4—22no—
75(C2×D7)⋊C4—22no—
76(C2×D7)⋊C4—42no—
77(C2×D7)⋊C4—42no—
78C4⋊Dic7—22no—
79Dic7⋊C4—24no—
80C4×Dic7—22no—
81C4×Dic7D414yescomplemented
82C2×#127—71no—
83C2×#121—72no—
84C2×#117—71no—
85#394—71no—
86#394—72no—
87#394D711yescharacteristic, complemented
88#392—71no—
89#392—72no—
90#392—71no—
91#390—72no—
92#382—71no—
93C23×C7—22no—
94C23×C7—22no—
95C23×C7—22no—
96C23×C7—22no—
97C23×C7—22no—
98C23×C7—22no—
99C23×C7—22no—
100C23×C7C2411yescharacteristic
101C22×D7—21no—
102C22×D7—21no—
103C22×D7—22no—
104C22×D7—22no—
105C22×D7—22no—
106C22×D7—22no—
107C22×D7—22no—
108C22×D7—22no—
109C22×D7—22no—
110C22×D7—22no—
111C22×D7—22no—
112C22×D7C2×D412yes—
113C7×D4—44no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—22no—
117C2×C4×C7—21no—
118C2×C4×C7—22no—
119C2×C4×C7—22no—
120C2×C4×C7—22no—
121C2×C4×C7—22no—
122C2×C4×C7C2×D412yes—
123C2×C4×C7C2×D412yes—
124C22⋊D7—48no—
125C22⋊D7—48no—
126C22⋊D7—48no—
127C2×Dic7—22no—
128C2×Dic7—21no—
129C2×Dic7—22no—
130C2×Dic7—22no—
131C2×Dic7—22no—
132C2×Dic7—22no—
133C2×Dic7C2×D412yes—
134C2×Dic7C2×D412yes—
135C2×Dic7C2×D412yes—
136C2×Dic7#4112yes—
137D28—44no—
138D28—48no—
139C4×D7—28no—
140C22×D4—72no—
141C22×D4—72no—
142C22×D4—72no—
143C22×D4—72no—
144C22×D4—71no—
145C22×D4C2×D711yescharacteristic
146C23×C4—71no—
147#127—144no—
148#121—148no—
149#117—144no—
150C2×C4⋊C4—71no—
151C2×C4⋊C4—72no—
152C2×C4⋊C4C2×D711yescharacteristic
153C2×C22⋊C4—71no—
154C2×C22⋊C4—71no—
155C2×C22⋊C4—72no—
156C2×C22⋊C4—72no—
157C2×C22⋊C4—72no—
158C2×C22⋊C4—72no—
159C2×C22⋊C4C2×D712yes—
160C2×C22⋊C4C2×D712yes—
161C2×C42—71no—
162C2×C42—71no—
163#95—72no—
164#95—71no—
165#95C2×D711yescharacteristic
166C22×C7—42no—
167C22×C7—42no—
168C22×C7—42no—
169C22×C7—42no—
170C22×C7—42no—
171C22×C7—42no—
172C22×C7—42no—
173C22×C7C22×D411yescharacteristic
174C22×C7C22×D411yescharacteristic
175C22×C7C2×#4111yescharacteristic
176C22×C7H(5,𝔽2)12yes—
177C22×C7H(5,𝔽2)12yes—
178C2×D7—22no—
179C2×D7—24no—
180C2×D7—41no—
181C2×D7—41no—
182C2×D7—42no—
183C2×D7—42no—
184C2×D7—42no—
185C2×D7—42no—
186C2×D7—42no—
187C2×D7—42no—
188C2×D7—42no—
189C2×D7—42no—
190C4×C7—22no—
191C4×C7—22no—
192C4×C7—42no—
193C4×C7—41no—
194Dic7—22no—
195Dic7—41no—
196Dic7—42no—
197Dic7#12114yes—
198C24—71no—
199C24—72no—
200C24C22×D712yes—
201C2×D4—22no—
202C2×D4—42no—
203C2×D4—144no—
204C2×D4—144no—
205C2×D4—144no—
206C2×D4—144no—
207C2×D4—144no—
208C2×D4—144no—
209C2×D4—142no—
210C2×D4—282no—
211C2×D4—282no—
212C2×D4—282no—
213C2×D4—282no—
214C2×D4—282no—
215C2×D4—282no—
216C2×D4—282no—
217C22×C4—71no—
218C22×C4—71no—
219C22×C4—74no—
220C22×C4—72no—
221C22×C4—71no—
222C22×C4—142no—
223C22×C4—142no—
224C22×C4C22×D712yes—
225C22×C4C22×D711yescharacteristic
226C22×C4C22×D711yescharacteristic
227C22×C4C22×D711yescharacteristic
228C4⋊C4—22no—
229C4⋊C4—144no—
230C4⋊C4—142no—
231C22⋊C4—24no—
232C22⋊C4—42no—
233C22⋊C4—142no—
234C22⋊C4—142no—
235C22⋊C4—144no—
236C22⋊C4—282no—
237C22⋊C4—282no—
238C22⋊C4—282no—
239C42—74no—
240C42—142no—
241C2×C7—82no—
242C2×C7C2×#12111yescharacteristic
243C2×C7#53511yescharacteristic
244C2×C7#53512yes—
245C2×C7#53511yescharacteristic
246C2×C7#54012yes—
247D7—42no—
248D7—82no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—72no—
257C23—141no—
258C23—142no—
259C23—141no—
260C23—142no—
261C23—142no—
262C23—142no—
263C23—142no—
264C23—142no—
265C23—142no—
266C23—142no—
267C23—142no—
268C23C23×D711yescharacteristic
269D4—44no—
270D4—284no—
271D4—288no—
272D4—288no—
273D4—288no—
274D4—288no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—72no—
284C2×C4—72no—
285C2×C4—72no—
286C2×C4—72no—
287C2×C4—148no—
288C2×C4—142no—
289C2×C4—141no—
290C2×C4—142no—
291C2×C4—142no—
292C2×C4—142no—
293C2×C4—142no—
294C2×C4D4×D712yes—
295C2×C4D4×D712yes—
296C7#261911yescharacteristic, complemented, Sylow
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—142no—
305C22—144no—
306C22—281no—
307C22—282no—
308C22—281no—
309C22—282no—
310C22—282no—
311C22—282no—
312C22—282no—
313C22—282no—
314C22—282no—
315C22—282no—
316C22C2×D4×D711yescharacteristic
317C22C2×D4×D711yescharacteristic
318C22#654812yes—
319C22D7×#4111yescharacteristic
320C22#655312yes—
321C4—22no—
322C4—22no—
323C4—41no—
324C4—42no—
325C4—74no—
326C4—142no—
327C4—282no—
328C4—281no—
329C2—82no—
330C2—282no—
331C2—562no—
332C2#9028311yescharacteristic
333C2D7×#12111yescharacteristic
334C2#9029812yes—
335C2#9035412yes—
336C2#9036511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
hypercenter#394D7
hypocenterC7#2619
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#6503C2semidirect
2. C2×#6455C2semidirect
3. C2×#6447C2semidirect
4. C7×#394C2semidirect
5. #89995C2semidirect
6. #89763C2semidirect
7. #89733C2semidirect
8. #89732C2semidirect
9. #89729C2semidirect
10. #89443C2semidirect
11. #89422C2semidirect
68. C2×C4×D7D4semidirect
81. C4×Dic7D4semidirect
87. #394D7semidirect
296. C7#2619semidirect

Maximal subgroups

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

subgroupindex
C2×#65032
C2×#64552
C2×#64472
C7×#3942
#899952
#897632
#897332
#897322
#897292
#894432
#894222
#26197