Finite Group Databasebeta

← #11735261

Subgroups — #11735261

Order 896.

476 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 139 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×#6455C211yesmaximal, characteristic
2C2×#6453C211yesmaximal, characteristic, complemented
3C7×#398C211yesmaximal, characteristic, complemented
4#89765C211yesmaximal, characteristic
5#89761C212yesmaximal
6#89758C211yesmaximal, characteristic, complemented
7#89757C212yesmaximal, complemented
8#89449C211yesmaximal, characteristic
9#89442C212yesmaximal
10#89426C212yesmaximal, complemented
11#89420C211yesmaximal, characteristic, complemented
12C22×C4×D7C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C4⋊C4C2212yes—
16C2×(C2×D7)⋊C4C2211yescharacteristic
17C2×(C2×D7)⋊C4C2211yescharacteristic
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×C4⋊Dic7C2212yes—
21C2×C4⋊Dic7C2211yescharacteristic
22C2×C4⋊Dic7C2211yescharacteristic
23C2×C4⋊Dic7C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×C4×Dic7C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27C2×C4×Dic7C2211yescharacteristic
28#6455—24no—
29#6453—24no—
30C7×#95C2211yescharacteristic
31C7×#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2611—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2312yes—
44C22×Dic7C2312yes—
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C2×C4×D7—22no—
50C2×C4×D7—22no—
51C2×C4×D7—22no—
52C7×C4⋊C4—22no—
53C7×C4⋊C4—22no—
54C7×C4⋊C4—24no—
55(C2×D7)⋊C4—24no—
56(C2×D7)⋊C4—22no—
57(C2×D7)⋊C4—24no—
58(C2×D7)⋊C4—22no—
59C4⋊Dic7—22no—
60C4⋊Dic7—24no—
61C4⋊Dic7—24no—
62C4⋊Dic7—22no—
63Dic7⋊C4—24no—
64C4×Dic7—24no—
65C4×Dic7—22no—
66C4×Dic7D414yes—
67C2×#125—71no—
68C2×#117—71no—
69#400—71no—
70#399—71no—
71#398D711yescharacteristic, complemented
72#397—71no—
73#389—72no—
74#388—72no—
75#387—72no—
76#385—72no—
77#376—71no—
78C23×C7C2411yescharacteristic
79C22×D7—21no—
80C22×D7—22no—
81C22×D7—21no—
82C22×D7—22no—
83C22×D7#4112yes—
84C2×C4×C7—21no—
85C2×C4×C7—22no—
86C2×C4×C7—21no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—21no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7C2×D412yes—
95C2×C4×C7#4112yes—
96C2×C4×C7#4114yes—
97C2×Dic7—22no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—21no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7C2×D412yes—
109C2×Dic7C2×D412yes—
110C4×D7—28no—
111C23×C4—71no—
112#125—144no—
113#117—144no—
114C2×C4⋊C4—71no—
115C2×C4⋊C4—72no—
116C2×C4⋊C4—72no—
117C2×C4⋊C4—71no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4C2×D711yescharacteristic
120C2×C4⋊C4C2×D711yescharacteristic
121C2×C4⋊C4C2×D712yes—
122C2×C22⋊C4—71no—
123C2×C22⋊C4—71no—
124C2×C22⋊C4—72no—
125C2×C22⋊C4—72no—
126C2×C42—71no—
127C2×C42—71no—
128C2×C42—72no—
129#95—72no—
130#95—71no—
131#95—72no—
132#95—72no—
133#95—72no—
134#95C2×D711yescharacteristic
135#95C2×D712yes—
136C22×C7C22×D411yescharacteristic
137C22×C7C2×#4111yescharacteristic
138C22×C7C2×#4112yes—
139C22×C7C2×#4111yescharacteristic
140C22×C72−1+412yes—
141C2×D7—22no—
142C2×D7—24no—
143C2×D7—41no—
144C2×D7—41no—
145C2×D7—42no—
146C4×C7—22no—
147C4×C7—22no—
148C4×C7—24no—
149C4×C7—42no—
150C4×C7—41no—
151Dic7—22no—
152Dic7—22no—
153Dic7—41no—
154Dic7—42no—
155Dic7—42no—
156C24—71no—
157C22×C4—71no—
158C22×C4—71no—
159C22×C4—72no—
160C22×C4—71no—
161C22×C4—72no—
162C22×C4—142no—
163C22×C4—142no—
164C22×C4—142no—
165C22×C4C22×D711yescharacteristic
166C22×C4C22×D712yes—
167C22×C4C22×D711yescharacteristic
168C22×C4C22×D711yescharacteristic
169C22×C4C22×D712yes—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—24no—
173C4⋊C4—144no—
174C4⋊C4—142no—
175C4⋊C4—144no—
176C4⋊C4—142no—
177C4⋊C4—144no—
178C22⋊C4—142no—
179C22⋊C4—144no—
180C22⋊C4—142no—
181C22⋊C4—144no—
182C42—74no—
183C42—142no—
184C42—144no—
185C2×C7#52511yescharacteristic
186C2×C7#53211yescharacteristic
187C2×C7#53611yescharacteristic
188C2×C7#55212yes—
189C2×C7#55212yes—
190D7—42no—
191C23—72no—
192C23—141no—
193C23—141no—
194C23—142no—
195C23—142no—
196C23C23×D711yescharacteristic
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—72no—
208C2×C4—72no—
209C2×C4—148no—
210C2×C4—141no—
211C2×C4—142no—
212C2×C4—141no—
213C2×C4—141no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4D4×D712yes—
222C2×C4#115912yes—
223C2×C4#116114yes—
224C7#261111yescharacteristic, complemented, Sylow
225C22—144no—
226C22—142no—
227C22—281no—
228C22—282no—
229C22—281no—
230C22C2×D4×D711yescharacteristic
231C22C2×#115911yescharacteristic
232C22C2×#116112yes—
233C22D7×#4111yescharacteristic
234C22#655412yes—
235C4—22no—
236C4—24no—
237C4—22no—
238C4—42no—
239C4—41no—
240C4—142no—
241C4—142no—
242C4—282no—
243C4—281no—
244C4—282no—
245C2—282no—
246C2#9029711yescharacteristic
247C2#9033312yes—
248C2#9033911yescharacteristic
249C2#9038512yes—
250C2#9041611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#398C2
prime radicalC23×D7C23
hypercenter#398D7
hypocenterC7#2611
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#6453C2semidirect
3. C7×#398C2semidirect
6. #89758C2semidirect
7. #89757C2semidirect
10. #89426C2semidirect
11. #89420C2semidirect
71. #398D7semidirect
224. C7#2611semidirect

Maximal subgroups

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

subgroupindex
C2×#64552
C2×#64532
C7×#3982
#897652
#897612
#897582
#897572
#894492
#894422
#894262
#894202
#26117