Finite Group Databasebeta

← #11735285

Subgroups — #11735285

Order 896.

434 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 133 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
2C2×#6453C211yesmaximal, characteristic, complemented
3C7×#398C211yesmaximal, characteristic, complemented
4#89766C212yesmaximal
5#89765C211yesmaximal, characteristic
6#89758C211yesmaximal, characteristic, complemented
7#89745C212yesmaximal, complemented
8#89449C211yesmaximal, characteristic
9#89441C212yesmaximal
10#89433C212yesmaximal, complemented
11#89421C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2212yes—
15C2×(C2×D7)⋊C4C2211yescharacteristic
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×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2211yescharacteristic
27C2×C4×Dic7C2212yes—
28C2×C4×Dic7C2211yescharacteristic
29#6461—24no—
30#6453—24no—
31C7×#95C2211yescharacteristic
32C7×#95C2212yes—
33C7⋊#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2664—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2312yes—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C7×C4⋊C4—22no—
50C7×C4⋊C4—22no—
51C7×C4⋊C4—42no—
52(C2×D7)⋊C4—22no—
53(C2×D7)⋊C4—24no—
54(C2×D7)⋊C4—22no—
55(C2×D7)⋊C4—22no—
56(C2×D7)⋊C4—42no—
57C4⋊Dic7—22no—
58C4⋊Dic7—24no—
59C4⋊Dic7—24no—
60C4⋊Dic7—22no—
61Dic7⋊C4—24no—
62Dic7⋊C4—24no—
63Dic7⋊C4—22no—
64C4×Dic7—42no—
65C4×Dic7D414yes—
66C2×#126—71no—
67C2×#125—71no—
68#400—71no—
69#399—71no—
70#398—72no—
71#398D711yescharacteristic, complemented
72#397—71no—
73#396—72no—
74#389—72no—
75#388—72no—
76C4×C4⋊C4—71no—
77C23×C7C2411yescharacteristic
78C22×D7—21no—
79C22×D7—22no—
80C22×D7—21no—
81C22×D7—22no—
82C22×D7—21no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7—21no—
88C2×C4×C7—21no—
89C2×C4×C7—22no—
90C2×C4×C7—21no—
91C2×C4×C7—22no—
92C2×C4×C7—21no—
93C2×C4×C7—22no—
94C2×C4×C7C2×D412yes—
95C2×C4×C7#4114yes—
96C2×Dic7—22no—
97C2×Dic7—21no—
98C2×Dic7—21no—
99C2×Dic7—22no—
100C2×Dic7—21no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7C2×D412yes—
108C2×Dic7C2×D412yes—
109#126—144no—
110#125—144no—
111C2×C4⋊C4—72no—
112C2×C4⋊C4—72no—
113C2×C4⋊C4—71no—
114C2×C4⋊C4—71no—
115C2×C4⋊C4—71no—
116C2×C4⋊C4—72no—
117C2×C4⋊C4—72no—
118C2×C4⋊C4C2×D711yescharacteristic
119C2×C4⋊C4C2×D711yescharacteristic
120C2×C4⋊C4C2×D712yes—
121C2×C22⋊C4—71no—
122C2×C22⋊C4—71no—
123C2×C22⋊C4—71no—
124C2×C22⋊C4—72no—
125C2×C22⋊C4—72no—
126C2×C42—71no—
127C2×C42—72no—
128#95—71no—
129#95—72no—
130#95—72no—
131#95—72no—
132#95C2×D711yescharacteristic
133#95C2×D712yes—
134C22×C7C22×D411yescharacteristic
135C22×C7C2×#4112yes—
136C22×C7H(5,𝔽2)11yescharacteristic
137C22×C72−1+412yes—
138C22×C72−1+411yescharacteristic
139C2×D7—41no—
140C2×D7—41no—
141C2×D7—41no—
142C2×D7—42no—
143C2×D7—42no—
144C4×C7—22no—
145C4×C7—24no—
146C4×C7—41no—
147C4×C7—42no—
148C4×C7—41no—
149Dic7—22no—
150Dic7—22no—
151Dic7—42no—
152Dic7—41no—
153Dic7—42no—
154C24—71no—
155C22×C4—71no—
156C22×C4—71no—
157C22×C4—72no—
158C22×C4—71no—
159C22×C4—72no—
160C22×C4C22×D711yescharacteristic
161C22×C4C22×D711yescharacteristic
162C22×C4C22×D712yes—
163C22×C4C22×D711yescharacteristic
164C22×C4C22×D712yes—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—42no—
168C4⋊C4—144no—
169C4⋊C4—142no—
170C4⋊C4—144no—
171C4⋊C4—142no—
172C4⋊C4—142no—
173C4⋊C4—144no—
174C4⋊C4—144no—
175C22⋊C4—142no—
176C22⋊C4—142no—
177C22⋊C4—142no—
178C22⋊C4—144no—
179C22⋊C4—282no—
180C42—74no—
181C42—282no—
182C2×C7#53211yescharacteristic
183C2×C7#53611yescharacteristic
184C2×C7#53711yescharacteristic
185C2×C7#53912yes—
186C2×C7#54112yes—
187D7—81no—
188C23—141no—
189C23—141no—
190C23—142no—
191C23—141no—
192C23—142no—
193C23C23×D711yescharacteristic
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—72no—
206C2×C4—72no—
207C2×C4—141no—
208C2×C4—142no—
209C2×C4—141no—
210C2×C4—141no—
211C2×C4—142no—
212C2×C4—142no—
213C2×C4—142no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4D4×D712yes—
219C2×C4#116114yes—
220C7#266411yescharacteristic, complemented, Sylow
221C22—282no—
222C22—281no—
223C22—281no—
224C22—282no—
225C22—281no—
226C22C2×D4×D711yescharacteristic
227C22#654811yescharacteristic
228C22C2×#116112yes—
229C22#655411yescharacteristic
230C22#655412yes—
231C4—24no—
232C4—22no—
233C4—41no—
234C4—42no—
235C4—41no—
236C4—142no—
237C4—142no—
238C4—281no—
239C4—282no—
240C4—282no—
241C2—561no—
242C2#9029911yescharacteristic
243C2#9033612yes—
244C2#9033911yescharacteristic
245C2#9038812yes—
246C2#9041611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#398C2
prime radicalC23×D7C23
hypercenter#398D7
hypocenterC7#2664
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. #89745C2semidirect
10. #89433C2semidirect
11. #89421C2semidirect
71. #398D7semidirect
220. C7#2664semidirect

Maximal subgroups

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

subgroupindex
C2×#64612
C2×#64532
C7×#3982
#897662
#897652
#897582
#897452
#894492
#894412
#894332
#894212
#26647