Finite Group Databasebeta

← #11734987

Subgroups — #11734987

Order 896.

628 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 129 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×#6456C211yesmaximal, characteristic, complemented
2C2×#6448C211yesmaximal, characteristic, complemented
3C7×#397C211yesmaximal, characteristic, complemented
4#89765C211yesmaximal, characteristic, complemented
5#89732C212yesmaximal, complemented
6#89732C211yesmaximal, characteristic, complemented
7#89728C212yesmaximal, complemented
8#89445C212yesmaximal, complemented
9#89443C211yesmaximal, characteristic, complemented
10#89437C212yesmaximal, complemented
11#89423C211yesmaximal, characteristic, complemented
12C22×C22⋊D7C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C22×D28C2211yescharacteristic
15C22×D28C2212yes—
16C22×C4×D7C2212yes—
17C22×Dic14C2211yescharacteristic
18C2×C7×C4⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2212yes—
21C2×C22⋊Dic7C2211yescharacteristic
22C2×(C2×D7)⋊C4C2211yescharacteristic
23C2×(C2×D7)⋊C4C2211yescharacteristic
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2212yes—
27C2×(C2×D7)⋊C4C2212yes—
28C2×C4×Dic7C2211yescharacteristic
29#6456—24no—
30#6448—24no—
31C7×#95C2211yescharacteristic
32C7×#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2211yescharacteristic
36C7⋊#95C2212yes—
37#2631—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C23×D7C2312yes—
41C22×C4×C7C2312yes—
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C2×C22⋊D7—24no—
46C2×C22⋊D7—22no—
47C2×C22⋊D7—42no—
48C2×C22⋊D7—42no—
49C2×C22⋊D7—42no—
50C22×Dic7C2312yes—
51C22×Dic7C2311yescharacteristic
52C22×Dic7C2311yescharacteristic
53C22×Dic7C2311yescharacteristic
54C2×D28—24no—
55C2×D28—22no—
56C2×D28—24no—
57C2×D28—42no—
58C2×D28—42no—
59C2×C4×D7—24no—
60C2×C4×D7—42no—
61C2×C4×D7—42no—
62C2×Dic14—22no—
63C2×Dic14—42no—
64C7×C4⋊C4—22no—
65C7×C22⋊C4—22no—
66C7×C22⋊C4—42no—
67C22⋊Dic7—22no—
68(C2×D7)⋊C4—22no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4—24no—
71(C2×D7)⋊C4—42no—
72(C2×D7)⋊C4—42no—
73(C2×D7)⋊C4—42no—
74C4×Dic7D414yescomplemented
75C2×#124—71no—
76C2×C4×D4—71no—
77#397—71no—
78#397D711yescharacteristic, complemented
79#393—72no—
80#392—72no—
81#392—71no—
82#392—71no—
83#386—72no—
84#386—72no—
85#383—71no—
86C23×C7—21no—
87C23×C7—22no—
88C23×C7—21no—
89C23×C7—22no—
90C23×C7—21no—
91C23×C7C2411yescharacteristic
92C22×D7—22no—
93C22×D7—22no—
94C22×D7—22no—
95C22×D7—22no—
96C22×D7—22no—
97C22×D7—22no—
98C22×D7—21no—
99C22×D7—22no—
100C22×D7—21no—
101C22×D7—21no—
102C22×D7—22no—
103C22×D7#4114yes—
104C2×C4×C7—22no—
105C2×C4×C7—21no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7—22no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7C2×D412yes—
114C22⋊D7—44no—
115C22⋊D7—48no—
116C2×Dic7—22no—
117C2×Dic7—21no—
118C2×Dic7—21no—
119C2×Dic7—21no—
120C2×Dic7—22no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7C2×D412yes—
125C2×Dic7C2×D412yes—
126D28—28no—
127D28—48no—
128C4×D7—48no—
129Dic14—44no—
130C22×Q8—71no—
131C22×D4—71no—
132C22×D4—72no—
133C22×D4—71no—
134C22×D4—72no—
135C23×C4—72no—
136#124—144no—
137C4×D4—144no—
138C2×C4⋊C4C2×D711yescharacteristic
139C2×C22⋊C4—71no—
140C2×C22⋊C4—72no—
141C2×C22⋊C4—71no—
142C2×C22⋊C4—71no—
143C2×C22⋊C4—72no—
144C2×C22⋊C4—72no—
145C2×C22⋊C4—72no—
146C2×C22⋊C4C2×D711yescharacteristic
147C2×C22⋊C4C2×D712yes—
148C2×C42—71no—
149#95—72no—
150#95—72no—
151#95—71no—
152#95—72no—
153#95C2×D711yescharacteristic
154#95C2×D712yes—
155C22×C7—41no—
156C22×C7—41no—
157C22×C7—41no—
158C22×C7—42no—
159C22×C7—42no—
160C22×C7C22×D411yescharacteristic
161C22×C7C2×#4112yes—
162C22×C7H(5,𝔽2)11yescharacteristic
163C22×C7H(5,𝔽2)11yescharacteristic
164C22×C7H(5,𝔽2)12yes—
165C2×D7—24no—
166C2×D7—24no—
167C2×D7—24no—
168C2×D7—42no—
169C2×D7—42no—
170C2×D7—42no—
171C2×D7—42no—
172C2×D7—41no—
173C2×D7—41no—
174C2×D7—41no—
175C2×D7—42no—
176C2×D7—42no—
177C4×C7—22no—
178C4×C7—41no—
179C4×C7—42no—
180C4×C7—42no—
181Dic7—22no—
182Dic7—22no—
183Dic7—41no—
184Dic7—42no—
185C24—72no—
186C24—71no—
187C24C22×D711yescharacteristic
188C2×Q8—142no—
189C2×Q8—282no—
190C2×D4—144no—
191C2×D4—144no—
192C2×D4—142no—
193C2×D4—144no—
194C2×D4—142no—
195C2×D4—282no—
196C2×D4—282no—
197C2×D4—282no—
198C2×D4—282no—
199C2×D4—282no—
200C22×C4—71no—
201C22×C4—71no—
202C22×C4—72no—
203C22×C4—71no—
204C22×C4—144no—
205C22×C4—282no—
206C22×C4—282no—
207C22×C4C22×D712yes—
208C22×C4C22×D712yes—
209C22×C4C22×D711yescharacteristic
210C22×C4C22×D711yescharacteristic
211C4⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—144no—
215C22⋊C4—142no—
216C22⋊C4—142no—
217C22⋊C4—142no—
218C22⋊C4—282no—
219C22⋊C4—282no—
220C22⋊C4—282no—
221C42—74no—
222C2×C7—81no—
223C2×C7#52511yescharacteristic
224C2×C7#53511yescharacteristic
225C2×C7#53511yescharacteristic
226C2×C7#53812yes—
227C2×C7#53812yes—
228D7—44no—
229D7—81no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—21no—
234C23—22no—
235C23—74no—
236C23—142no—
237C23—142no—
238C23—142no—
239C23—142no—
240C23—142no—
241C23—142no—
242C23—141no—
243C23—141no—
244C23—142no—
245C23—141no—
246C23—142no—
247C23C23×D711yescharacteristic
248Q8—284no—
249D4—148no—
250D4—288no—
251D4—284no—
252D4—288no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—72no—
263C2×C4—72no—
264C2×C4—141no—
265C2×C4—141no—
266C2×C4—142no—
267C2×C4—141no—
268C2×C4—142no—
269C2×C4—142no—
270C2×C4—142no—
271C2×C4—142no—
272C2×C4—288no—
273C2×C4D4×D712yes—
274C7#263111yescharacteristic, complemented, Sylow
275C22—42no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—41no—
280C22—144no—
281C22—144no—
282C22—144no—
283C22—282no—
284C22—282no—
285C22—282no—
286C22—282no—
287C22—282no—
288C22—281no—
289C22—281no—
290C22—282no—
291C22—281no—
292C22C2×D4×D711yescharacteristic
293C22#654811yescharacteristic
294C22D7×#4112yes—
295C22#655311yescharacteristic
296C22#655312yes—
297C4—22no—
298C4—42no—
299C4—42no—
300C4—41no—
301C4—142no—
302C4—142no—
303C4—282no—
304C4—281no—
305C2—81no—
306C2—284no—
307C2—561no—
308C2#9029811yescharacteristic
309C2#9034711yescharacteristic
310C2#9034812yes—
311C2#9036511yescharacteristic
312C2#9039512yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
hypercenter#397D7
hypocenterC7#2631
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#6456C2semidirect
2. C2×#6448C2semidirect
3. C7×#397C2semidirect
4. #89765C2semidirect
5. #89732C2semidirect
6. #89732C2semidirect
7. #89728C2semidirect
8. #89445C2semidirect
9. #89443C2semidirect
10. #89437C2semidirect
11. #89423C2semidirect
74. C4×Dic7D4semidirect
78. #397D7semidirect
274. C7#2631semidirect

Maximal subgroups

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

subgroupindex
C2×#64562
C2×#64482
C7×#3972
#897652
#897322
#897282
#894452
#894432
#894372
#894232
#26317