Finite Group Databasebeta

← #11735653

Subgroups — #11735653

Order 896.

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

normal subgroups only

This group has 464 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×#6459C211yesmaximal, characteristic
2C2×#6455C211yesmaximal, characteristic
3C7×#400C211yesmaximal, characteristic, complemented
4#89755C211yesmaximal, characteristic, complemented
5#89754C211yesmaximal, characteristic, complemented
6#89447C212yesmaximal
7#89446C212yesmaximal
8#89439C211yesmaximal, characteristic
9#89433C211yesmaximal, characteristic, complemented
10#89419C212yesmaximal, complemented
11#89418C212yesmaximal, complemented
12C22×C4×D7C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×(C2×D7)⋊C4C2212yes—
17C2×(C2×D7)⋊C4C2211yescharacteristic
18C2×(C2×D7)⋊C4C2211yescharacteristic
19C2×(C2×D7)⋊C4C2212yes—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×C4⋊Dic7C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2211yescharacteristic
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2212yes—
27#6459—24no—
28#6455—24no—
29C7×#95C2212yes—
30C7×#95C2212yes—
31C7×#95C2211yescharacteristic
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2625—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×Dic7C2312yes—
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C2×C4×D7—24no—
50C2×C4×D7D414yes—
51C2×Dic14—22no—
52C2×Dic14—24no—
53C7×C4⋊C4—22no—
54C7×C4⋊C4—22no—
55(C2×D7)⋊C4—24no—
56(C2×D7)⋊C4—22no—
57(C2×D7)⋊C4—22no—
58(C2×D7)⋊C4—42no—
59C4⋊Dic7—22no—
60C4⋊Dic7—42no—
61Dic7⋊C4—22no—
62Dic7⋊C4—22no—
63Dic7⋊C4—24no—
64Dic7⋊C4—42no—
65C4×Dic7—24no—
66C2×C22⋊Q8—71no—
67C2×#117—71no—
68#400—71no—
69#400D711yescharacteristic, complemented
70#398—71no—
71#398—71no—
72#397—72no—
73#397—72no—
74#391—72no—
75#389—72no—
76#380—71no—
77C23×C7C2411yescharacteristic
78C22×D7—22no—
79C22×D7—21no—
80C22×D7—21no—
81C22×D7—22no—
82C22×D7C2×D412yes—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—21no—
90C2×C4×C7—22no—
91C2×C4×C7—21no—
92C2×C4×C7—24no—
93C2×C4×C7—22no—
94C2×C4×C7C2×D412yes—
95C2×Dic7—22no—
96C2×Dic7—21no—
97C2×Dic7—21no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—21no—
103C2×Dic7—22no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7C2×D412yes—
107C2×Dic7#4114yes—
108C4×D7—28no—
109Dic14—44no—
110C22×Q8—71no—
111C23×C4—71no—
112C22⋊Q8—144no—
113#117—144no—
114C2×C4⋊C4—71no—
115C2×C4⋊C4—71no—
116C2×C4⋊C4—71no—
117C2×C4⋊C4—72no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4C2×D711yescharacteristic
121C2×C4⋊C4C2×D711yescharacteristic
122C2×C22⋊C4—72no—
123C2×C22⋊C4—71no—
124C2×C22⋊C4—71no—
125C2×C22⋊C4—72no—
126C2×C42—72no—
127#95—71no—
128#95—72no—
129#95—72no—
130#95—72no—
131#95—72no—
132#95C2×D711yescharacteristic
133#95C2×D712yes—
134#95C2×D712yes—
135C22×C7C22×D411yescharacteristic
136C22×C7C2×#4112yes—
137C22×C7H(5,𝔽2)11yescharacteristic
138C22×C72−1+411yescharacteristic
139C22×C72−1+412yes—
140C2×D7—22no—
141C2×D7—22no—
142C2×D7—22no—
143C2×D7—42no—
144C2×D7—42no—
145C4×C7—22no—
146C4×C7—41no—
147C4×C7—42no—
148C4×C7—41no—
149C4×C7—42no—
150Dic7—41no—
151Dic7—42no—
152Dic7—41no—
153Dic7—42no—
154Dic7#12314yes—
155C24—71no—
156C2×Q8—142no—
157C2×Q8—144no—
158C22×C4—71no—
159C22×C4—74no—
160C22×C4—72no—
161C22×C4—71no—
162C22×C4—72no—
163C22×C4—71no—
164C22×C4—144no—
165C22×C4C22×D712yes—
166C22×C4C22×D711yescharacteristic
167C22×C4C22×D712yes—
168C22×C4C22×D711yescharacteristic
169C22×C4C22×D711yescharacteristic
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—144no—
173C4⋊C4—142no—
174C4⋊C4—142no—
175C4⋊C4—142no—
176C4⋊C4—282no—
177C4⋊C4—282no—
178C22⋊C4—144no—
179C22⋊C4—142no—
180C22⋊C4—142no—
181C22⋊C4—282no—
182C42—144no—
183C2×C7C2×#12311yescharacteristic
184C2×C7#53611yescharacteristic
185C2×C7#53711yescharacteristic
186C2×C7#54112yes—
187C2×C7#54212yes—
188D7—42no—
189C23—72no—
190C23—141no—
191C23—142no—
192C23—141no—
193C23—142no—
194C23C23×D711yescharacteristic
195Q8—284no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—72no—
208C2×C4—74no—
209C2×C4—148no—
210C2×C4—142no—
211C2×C4—141no—
212C2×C4—141no—
213C2×C4—142no—
214C2×C4—141no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—141no—
220C2×C4—142no—
221C2×C4D4×D712yes—
222C7#262511yescharacteristic, complemented, Sylow
223C22—142no—
224C22—142no—
225C22—142no—
226C22—282no—
227C22—282no—
228C22C2×D4×D711yescharacteristic
229C22#655111yescharacteristic
230C22D7×#4112yes—
231C22#655311yescharacteristic
232C22#655412yes—
233C4—22no—
234C4—42no—
235C4—41no—
236C4—41no—
237C4—42no—
238C4—74no—
239C4—282no—
240C4—282no—
241C4—281no—
242C4—281no—
243C2—282no—
244C2#9032011yescharacteristic
245C2D7×#12311yescharacteristic
246C2#9036611yescharacteristic
247C2#9038712yes—
248C2#9040112yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#400C2
prime radicalC23×D7C23
hypercenter#400D7
hypocenterC7#2625
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. C7×#400C2semidirect
4. #89755C2semidirect
5. #89754C2semidirect
9. #89433C2semidirect
10. #89419C2semidirect
11. #89418C2semidirect
69. #400D7semidirect
222. C7#2625semidirect

Maximal subgroups

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

subgroupindex
C2×#64592
C2×#64552
C7×#4002
#897552
#897542
#894472
#894462
#894392
#894332
#894192
#894182
#26257