Finite Group Databasebeta

← #11735658

Subgroups — #11735658

Order 896.

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

normal subgroups only

This group has 562 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×#6461C211yesmaximal, characteristic, complemented
2C2×#6456C211yesmaximal, characteristic, complemented
3C7×#400C211yesmaximal, characteristic, complemented
4#89764C211yesmaximal, characteristic, complemented
5#89763C211yesmaximal, characteristic, complemented
6#89448C212yesmaximal, complemented
7#89444C212yesmaximal, complemented
8#89443C211yesmaximal, characteristic
9#89440C212yesmaximal, complemented
10#89437C212yesmaximal, complemented
11#89422C211yesmaximal, characteristic, complemented
12C22×D28C2211yescharacteristic
13C22×D28C2212yes—
14C22×C4×D7C2212yes—
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×(C2×D7)⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2211yescharacteristic
19C2×(C2×D7)⋊C4C2211yescharacteristic
20C2×(C2×D7)⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×C4⋊Dic7C2211yescharacteristic
26C2×Dic7⋊C4C2211yescharacteristic
27C2×Dic7⋊C4C2212yes—
28C2×Dic7⋊C4C2212yes—
29C2×C4×Dic7C2211yescharacteristic
30#6461—24no—
31#6456—24no—
32C7×#95C2211yescharacteristic
33C7×#95C2212yes—
34C7×#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2636—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C23×D7C2312yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C2×D28—24no—
50C2×D28—22no—
51C2×D28—24no—
52C2×D28—42no—
53C2×D28—42no—
54C2×C4×D7—24no—
55C2×C4×D7—42no—
56C2×C4×D7—42no—
57C7×C4⋊C4—22no—
58C7×C4⋊C4—22no—
59(C2×D7)⋊C4—24no—
60(C2×D7)⋊C4—22no—
61(C2×D7)⋊C4—24no—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—42no—
65(C2×D7)⋊C4—42no—
66(C2×D7)⋊C4—42no—
67C4⋊Dic7—22no—
68Dic7⋊C4—22no—
69Dic7⋊C4—42no—
70Dic7⋊C4—42no—
71C4×Dic7D414yescomplemented
72C2×#126—71no—
73C2×C4×D4—71no—
74#400D711yescharacteristic, complemented
75#399—71no—
76#396—72no—
77#394—71no—
78#394—72no—
79#392—71no—
80#386—72no—
81#385—72no—
82#382—71no—
83C23×C7C2411yescharacteristic
84C22×D7—22no—
85C22×D7—22no—
86C22×D7—22no—
87C22×D7—22no—
88C22×D7—22no—
89C22×D7—22no—
90C22×D7—21no—
91C22×D7—22no—
92C22×D7—22no—
93C22×D7—21no—
94C22×D7—21no—
95C22×D7#4114yes—
96C2×C4×C7—22no—
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—21no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—21no—
108C2×C4×C7C2×D412yes—
109C2×Dic7—21no—
110C2×Dic7—22no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7C2×D412yes—
118C2×Dic7C2×D412yes—
119D28—28no—
120D28—48no—
121C4×D7—48no—
122C22×D4—71no—
123C22×D4—72no—
124C23×C4—72no—
125#126—144no—
126C4×D4—144no—
127C2×C4⋊C4—71no—
128C2×C4⋊C4—71no—
129C2×C4⋊C4—72no—
130C2×C4⋊C4—72no—
131C2×C4⋊C4C2×D711yescharacteristic
132C2×C4⋊C4C2×D711yescharacteristic
133C2×C22⋊C4—72no—
134C2×C22⋊C4—71no—
135C2×C22⋊C4—71no—
136C2×C22⋊C4—71no—
137C2×C22⋊C4—72no—
138C2×C22⋊C4—72no—
139C2×C22⋊C4—72no—
140C2×C22⋊C4—72no—
141C2×C42—71no—
142#95—72no—
143#95—71no—
144#95C2×D711yescharacteristic
145#95C2×D712yes—
146#95C2×D712yes—
147C22×C7C22×D411yescharacteristic
148C22×C7C2×#4112yes—
149C22×C7H(5,𝔽2)12yes—
150C22×C7H(5,𝔽2)11yescharacteristic
151C22×C72−1+411yescharacteristic
152C2×D7—24no—
153C2×D7—24no—
154C2×D7—24no—
155C2×D7—42no—
156C2×D7—42no—
157C2×D7—42no—
158C2×D7—42no—
159C2×D7—41no—
160C2×D7—41no—
161C2×D7—41no—
162C2×D7—42no—
163C2×D7—42no—
164C4×C7—22no—
165C4×C7—41no—
166C4×C7—42no—
167C4×C7—41no—
168C4×C7—42no—
169Dic7—22no—
170Dic7—22no—
171Dic7—41no—
172Dic7—42no—
173C24—72no—
174C24—71no—
175C2×D4—144no—
176C2×D4—144no—
177C2×D4—142no—
178C2×D4—282no—
179C2×D4—282no—
180C22×C4—71no—
181C22×C4—71no—
182C22×C4—72no—
183C22×C4—71no—
184C22×C4—144no—
185C22×C4—282no—
186C22×C4—282no—
187C22×C4C22×D712yes—
188C22×C4C22×D711yescharacteristic
189C22×C4C22×D711yescharacteristic
190C22×C4C22×D712yes—
191C22×C4C22×D711yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—142no—
195C4⋊C4—142no—
196C4⋊C4—282no—
197C4⋊C4—282no—
198C22⋊C4—144no—
199C22⋊C4—142no—
200C22⋊C4—142no—
201C22⋊C4—144no—
202C22⋊C4—142no—
203C22⋊C4—282no—
204C22⋊C4—282no—
205C22⋊C4—282no—
206C42—74no—
207C2×C7#52511yescharacteristic
208C2×C7#53511yescharacteristic
209C2×C7#53711yescharacteristic
210C2×C7#53812yes—
211C2×C7#53912yes—
212D7—44no—
213D7—81no—
214C23—74no—
215C23—142no—
216C23—142no—
217C23—142no—
218C23—142no—
219C23—142no—
220C23—142no—
221C23—141no—
222C23—141no—
223C23—142no—
224C23—142no—
225C23—141no—
226C23C23×D711yescharacteristic
227D4—148no—
228D4—288no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—72no—
242C2×C4—72no—
243C2×C4—141no—
244C2×C4—141no—
245C2×C4—142no—
246C2×C4—141no—
247C2×C4—142no—
248C2×C4—142no—
249C2×C4—142no—
250C2×C4—142no—
251C2×C4—288no—
252C2×C4D4×D712yes—
253C7#263611yescharacteristic, complemented, Sylow
254C22—144no—
255C22—144no—
256C22—144no—
257C22—282no—
258C22—282no—
259C22—282no—
260C22—282no—
261C22—282no—
262C22—282no—
263C22—281no—
264C22—281no—
265C22—281no—
266C22C2×D4×D711yescharacteristic
267C22#655111yescharacteristic
268C22D7×#4112yes—
269C22#655312yes—
270C22#655311yescharacteristic
271C4—22no—
272C4—42no—
273C4—41no—
274C4—41no—
275C4—42no—
276C4—142no—
277C4—142no—
278C4—282no—
279C4—281no—
280C2—284no—
281C2—561no—
282C2#9032011yescharacteristic
283C2#9034711yescharacteristic
284C2#9036511yescharacteristic
285C2#9038312yes—
286C2#9039512yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#400C2
prime radical#89443C2
hypercenter#400D7
hypocenterC7#2636
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#6461C2semidirect
2. C2×#6456C2semidirect
3. C7×#400C2semidirect
4. #89764C2semidirect
5. #89763C2semidirect
6. #89448C2semidirect
7. #89444C2semidirect
9. #89440C2semidirect
10. #89437C2semidirect
11. #89422C2semidirect
71. C4×Dic7D4semidirect
74. #400D7semidirect
253. C7#2636semidirect

Maximal subgroups

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

subgroupindex
C2×#64612
C2×#64562
C7×#4002
#897642
#897632
#894482
#894442
#894432
#894402
#894372
#894222
#26367