Finite Group Databasebeta

← #11734190

Subgroups — #11734190

Order 896.

580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 125 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×#6499C211yesmaximal, characteristic, complemented
2C2×#6448C211yesmaximal, characteristic, complemented
3C7×#392C211yesmaximal, characteristic, complemented
4#89995C211yesmaximal, characteristic, complemented
5#89995C212yesmaximal, complemented
6#89994C212yesmaximal, complemented
7#89731C212yesmaximal, complemented
8#89719C212yesmaximal, complemented
9#89719C211yesmaximal, characteristic, complemented
10#89445C211yesmaximal, characteristic, complemented
11#89422C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C7×D4C2212yes—
14C22×C22⋊D7C2211yescharacteristic
15C22×C22⋊D7C2212yes—
16C22×Dic14C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2212yes—
20C2×C22⋊Dic7C2211yescharacteristic
21C2×C22⋊Dic7C2212yes—
22C2×C22⋊Dic7C2212yes—
23C2×C22⋊Dic7C2212yes—
24C2×C22⋊Dic7C2212yes—
25#6499—24no—
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2212yes—
29C2×Dic7⋊C4C2212yes—
30C2×C4×Dic7C2211yescharacteristic
31#6448—24no—
32C7×#95C2211yescharacteristic
33C7⋊#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2819—71nomaximal, Sylow
38C24×C7C2312yes—
39C24×C7C2311yescharacteristic
40C23×D7C2311yescharacteristic
41C2×C7×D4—22no—
42C2×C7×D4—42no—
43C2×C7×D4—42no—
44C2×C7×D4—42no—
45C2×C7×D4—42no—
46C22×C4×C7C2312yes—
47C22×C4×C7C2311yescharacteristic
48C22×C4×C7C2311yescharacteristic
49C2×C22⋊D7—22no—
50C2×C22⋊D7—42no—
51C2×C22⋊D7—42no—
52C2×C22⋊D7—42no—
53C2×C22⋊D7—42no—
54C22×Dic7C2312yes—
55C22×Dic7C2311yescharacteristic
56C22×Dic7C2312yes—
57C22×Dic7C2311yescharacteristic
58C22×Dic7C2311yescharacteristic
59C2×Dic14—22no—
60C2×Dic14—22no—
61C2×Dic14—42no—
62C2×Dic14—42no—
63C7×C22⋊C4—22no—
64C7×C22⋊C4—42no—
65C22⋊Dic7—22no—
66C22⋊Dic7—24no—
67C22⋊Dic7—24no—
68C22⋊Dic7—42no—
69C22⋊Dic7—42no—
70(C2×D7)⋊C4—22no—
71(C2×D7)⋊C4—22no—
72(C2×D7)⋊C4—42no—
73Dic7⋊C4—42no—
74C4×Dic7D414yescomplemented
75C2×#124—71no—
76C2×#124—71no—
77#394—72no—
78#394—72no—
79#393—72no—
80#393—71no—
81#393—71no—
82#392—72no—
83#392—71no—
84#392D711yescharacteristic, complemented
85#382—71no—
86C23×C7—22no—
87C23×C7—21no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7—22no—
91C23×C7—22no—
92C23×C7—22no—
93C23×C7—22no—
94C23×C7—22no—
95C23×C7—21no—
96C23×C7—22no—
97C23×C7—21no—
98C23×C7C2411yescharacteristic
99C22×D7—21no—
100C22×D7—22no—
101C22×D7—22no—
102C22×D7—21no—
103C22×D7—21no—
104C7×D4—44no—
105C7×D4—48no—
106C2×C4×C7—22no—
107C2×C4×C7—21no—
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7C2×D412yes—
113C22⋊D7—44no—
114C22⋊D7—48no—
115C2×Dic7—21no—
116C2×Dic7—22no—
117C2×Dic7—21no—
118C2×Dic7—21no—
119C2×Dic7—22no—
120C2×Dic7—22no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—22no—
125C2×Dic7—22no—
126C2×Dic7C2×D412yes—
127C2×Dic7C2×D412yes—
128Dic14—44no—
129Dic14—44no—
130C22×Q8—71no—
131C22×Q8—71no—
132C22×D4—71no—
133C22×D4—72no—
134C22×D4C2×D711yescharacteristic
135C22×D4C2×D712yes—
136#124—144no—
137#124—144no—
138C2×C4⋊C4—72no—
139C2×C22⋊C4—72no—
140C2×C22⋊C4—72no—
141C2×C22⋊C4—71no—
142C2×C22⋊C4—71no—
143C2×C22⋊C4—71no—
144C2×C22⋊C4—72no—
145C2×C22⋊C4—72no—
146C2×C22⋊C4—72no—
147C2×C22⋊C4C2×D711yescharacteristic
148C2×C22⋊C4C2×D712yes—
149C2×C42—71no—
150#95—72no—
151#95—71no—
152#95—72no—
153#95—72no—
154#95C2×D711yescharacteristic
155C22×C7—41no—
156C22×C7—42no—
157C22×C7—42no—
158C22×C7—41no—
159C22×C7—42no—
160C22×C7—41no—
161C22×C7—42no—
162C22×C7—42no—
163C22×C7—42no—
164C22×C7—42no—
165C22×C7—42no—
166C22×C7—42no—
167C22×C7C22×D411yescharacteristic
168C22×C7H(5,𝔽2)12yes—
169C22×C7H(5,𝔽2)12yes—
170C22×C7H(5,𝔽2)11yescharacteristic
171C22×C72−1+411yescharacteristic
172C2×D7—41no—
173C2×D7—41no—
174C2×D7—41no—
175C2×D7—42no—
176C2×D7—42no—
177C4×C7—22no—
178C4×C7—41no—
179C4×C7—42no—
180Dic7—22no—
181Dic7—22no—
182Dic7—41no—
183Dic7—42no—
184Dic7—42no—
185C24—71no—
186C24C22×D711yescharacteristic
187C24C22×D712yes—
188C2×Q8—142no—
189C2×Q8—142no—
190C2×Q8—282no—
191C2×Q8—282no—
192C2×D4—22no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—142no—
198C2×D4—282no—
199C2×D4—282no—
200C2×D4—282no—
201C2×D4—282no—
202C22×C4—71no—
203C22×C4—71no—
204C22×C4—72no—
205C22×C4—72no—
206C22×C4—71no—
207C22×C4C22×D712yes—
208C22×C4C22×D711yescharacteristic
209C22×C4C22×D711yescharacteristic
210C4⋊C4—282no—
211C22⋊C4—22no—
212C22⋊C4—42no—
213C22⋊C4—142no—
214C22⋊C4—142no—
215C22⋊C4—144no—
216C22⋊C4—144no—
217C22⋊C4—142no—
218C22⋊C4—282no—
219C22⋊C4—282no—
220C22⋊C4—282no—
221C42—74no—
222C2×C7—81no—
223C2×C7—82no—
224C2×C7#53511yescharacteristic
225C2×C7#53511yescharacteristic
226C2×C7#53711yescharacteristic
227C2×C7H(3,𝔽4)12yes—
228C2×C7#56212yes—
229D7—81no—
230C23—21no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—21no—
241C23—22no—
242C23—141no—
243C23—142no—
244C23—142no—
245C23—141no—
246C23—141no—
247C23C23×D711yescharacteristic
248Q8—284no—
249Q8—284no—
250D4—44no—
251D4—48no—
252D4—284no—
253D4—288no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—72no—
261C2×C4—72no—
262C2×C4—141no—
263C2×C4—142no—
264C2×C4—141no—
265C2×C4—141no—
266C2×C4—142no—
267C2×C4—142no—
268C2×C4—142no—
269C2×C4—142no—
270C2×C4—142no—
271C2×C4—142no—
272C2×C4—142no—
273C2×C4D4×D712yes—
274C7#281911yescharacteristic, complemented, Sylow
275C22—41no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—281no—
288C22—281no—
289C22—282no—
290C22—281no—
291C22—282no—
292C22C2×D4×D711yescharacteristic
293C22#654812yes—
294C22#654812yes—
295C22#654811yescharacteristic
296C22#655411yescharacteristic
297C4—22no—
298C4—41no—
299C4—42no—
300C4—142no—
301C4—142no—
302C4—281no—
303C4—282no—
304C4—282no—
305C2—81no—
306C2—82no—
307C2—561no—
308C2#9028311yescharacteristic
309C2#9028512yes—
310C2#9029911yescharacteristic
311C2#9031212yes—
312C2#9040611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#392C2
hypercenter#392D7
hypocenterC7#2819
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#6499C2semidirect
2. C2×#6448C2semidirect
3. C7×#392C2semidirect
4. #89995C2semidirect
5. #89995C2semidirect
6. #89994C2semidirect
7. #89731C2semidirect
8. #89719C2semidirect
9. #89719C2semidirect
10. #89445C2semidirect
11. #89422C2semidirect
74. C4×Dic7D4semidirect
84. #392D7semidirect
274. C7#2819semidirect

Maximal subgroups

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

subgroupindex
C2×#64992
C2×#64482
C7×#3922
#899952
#899942
#897312
#897192
#894452
#894222
#28197