Finite Group Databasebeta

← #11735805

Subgroups — #11735805

Order 896.

508 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 143 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×#6460C211yesmaximal, characteristic
2C2×D7×C4⋊C4C211yesmaximal, characteristic
3C7×#400C211yesmaximal, characteristic, complemented
4#89754C211yesmaximal, characteristic, complemented
5#89751C211yesmaximal, characteristic, complemented
6#89440C212yesmaximal
7#89439C211yesmaximal, characteristic
8#89436C212yesmaximal
9#89433C212yesmaximal, complemented
10#89429C212yesmaximal, complemented
11#89428C211yesmaximal, characteristic, complemented
12C22×C4×D7C2211yescharacteristic
13C22×C4×D7C2211yescharacteristic
14C22×C4×D7C2211yescharacteristic
15C22×Dic14C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×C4⋊Dic7C2212yes—
22C2×C4⋊Dic7C2212yes—
23C2×Dic7⋊C4C2211yescharacteristic
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27#6460—24no—
28D7×C4⋊C4—24no—
29C7×#95C2211yescharacteristic
30C7×#95C2212yes—
31C7×#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2627—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×Dic7C2312yes—
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C2×C4×D7—22no—
50C2×C4×D7—22no—
51C2×C4×D7—22no—
52C2×C4×D7—42no—
53C2×C4×D7—42no—
54C2×C4×D7D414yes—
55C2×C4×D7Q814yescomplemented
56C2×Dic14—22no—
57C2×Dic14—42no—
58C7×C4⋊C4—22no—
59C7×C4⋊C4—22no—
60(C2×D7)⋊C4—24no—
61(C2×D7)⋊C4—24no—
62C4⋊Dic7—22no—
63C4⋊Dic7—24no—
64C4⋊Dic7—42no—
65Dic7⋊C4—22no—
66Dic7⋊C4—22no—
67Dic7⋊C4—42no—
68Dic7⋊C4—42no—
69C2×C22⋊Q8—71no—
70C22×C4⋊C4—71no—
71#400—71no—
72#400D711yescharacteristic, complemented
73#398—72no—
74#398—72no—
75#395—71no—
76#395—71no—
77#385—72no—
78#381—72no—
79#380—71no—
80C23×C7C2411yescharacteristic
81C22×D7—22no—
82C22×D7—22no—
83C22×D7C2×D412yes—
84C22×D7C2×Q812yes—
85C22×D7#4112yes—
86C2×C4×C7—21no—
87C2×C4×C7—21no—
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×C7—24no—
95C2×C4×C7—22no—
96C2×C4×C7C2×D412yes—
97C2×C4×C7C2×Q812yes—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—22no—
101C2×Dic7—21no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7C2×D412yes—
110C2×Dic7C2×Q812yes—
111C4×D7—28no—
112C4×D7—44no—
113C4×D7—44no—
114Dic14—44no—
115C22×Q8—71no—
116C23×C4—71no—
117C23×C4—71no—
118C23×C4—71no—
119C22⋊Q8—144no—
120C2×C4⋊C4—71no—
121C2×C4⋊C4—71no—
122C2×C4⋊C4—71no—
123C2×C4⋊C4—72no—
124C2×C4⋊C4—72no—
125C2×C4⋊C4—72no—
126C2×C4⋊C4—72no—
127C2×C4⋊C4—144no—
128C2×C4⋊C4C2×D711yescharacteristic
129C2×C4⋊C4C2×D711yescharacteristic
130C2×C22⋊C4—72no—
131C2×C22⋊C4—72no—
132#95—71no—
133#95—72no—
134#95—72no—
135#95—72no—
136#95—72no—
137#95C2×D711yescharacteristic
138#95C2×D712yes—
139#95C2×D712yes—
140C22×C7C22×D411yescharacteristic
141C22×C7C22×Q811yescharacteristic
142C22×C7C2×#4111yescharacteristic
143C22×C7H(5,𝔽2)12yes—
144C22×C72−1+412yes—
145C2×D7—22no—
146C2×D7—22no—
147C2×D7—24no—
148C2×D7—24no—
149C2×D7C22⋊Q814yes—
150C4×C7—22no—
151C4×C7—22no—
152C4×C7—41no—
153C4×C7—42no—
154C4×C7—42no—
155Dic7—22no—
156Dic7—22no—
157Dic7—42no—
158Dic7—41no—
159Dic7—42no—
160C24—71no—
161C2×Q8—142no—
162C2×Q8—282no—
163C22×C4—71no—
164C22×C4—71no—
165C22×C4—74no—
166C22×C4—74no—
167C22×C4—72no—
168C22×C4—72no—
169C22×C4—71no—
170C22×C4—142no—
171C22×C4—142no—
172C22×C4—142no—
173C22×C4—282no—
174C22×C4—282no—
175C22×C4C22×D712yes—
176C22×C4C22×D712yes—
177C22×C4C22×D711yescharacteristic
178C22×C4C22×D711yescharacteristic
179C22×C4C22×D711yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—142no—
183C4⋊C4—142no—
184C4⋊C4—144no—
185C4⋊C4—142no—
186C4⋊C4—282no—
187C4⋊C4—282no—
188C4⋊C4—282no—
189C22⋊C4—144no—
190C22⋊C4—144no—
191C2×C7C2×C22⋊Q811yescharacteristic
192C2×C7#53411yescharacteristic
193C2×C7#53611yescharacteristic
194C2×C7#53912yes—
195C2×C7#54412yes—
196D7—24no—
197C23—72no—
198C23—72no—
199C23—72no—
200C23—142no—
201C23—142no—
202C23C23×D711yescharacteristic
203Q8—284no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—72no—
215C2×C4—72no—
216C2×C4—148no—
217C2×C4—141no—
218C2×C4—141no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—141no—
222C2×C4—142no—
223C2×C4—142no—
224C2×C4—142no—
225C2×C4—142no—
226C2×C4—142no—
227C2×C4—142no—
228C2×C4—284no—
229C2×C4—284no—
230C2×C4D4×D712yes—
231C2×C4Q8×D712yes—
232C7#262711yescharacteristic, complemented, Sylow
233C22—74no—
234C22—144no—
235C22—144no—
236C22—142no—
237C22—142no—
238C22C2×D4×D711yescharacteristic
239C22C2×Q8×D711yescharacteristic
240C22D7×#4111yescharacteristic
241C22#655312yes—
242C22#655412yes—
243C4—22no—
244C4—22no—
245C4—42no—
246C4—41no—
247C4—42no—
248C4—142no—
249C4—142no—
250C4—282no—
251C4—281no—
252C4—282no—
253C2—144no—
254C2D7×C22⋊Q811yescharacteristic
255C2#9034411yescharacteristic
256C2#9036611yescharacteristic
257C2#9038012yes—
258C2#9039812yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#400C2
prime radicalC23×D7C23
hypercenter#400D7
hypocenterC7#2627
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. #89754C2semidirect
5. #89751C2semidirect
9. #89433C2semidirect
10. #89429C2semidirect
11. #89428C2semidirect
55. C2×C4×D7Q8semidirect
72. #400D7semidirect
232. C7#2627semidirect

Maximal subgroups

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

subgroupindex
C2×#64602
C2×D7×C4⋊C42
C7×#4002
#897542
#897512
#894402
#894392
#894362
#894332
#894292
#894282
#26277