Finite Group Databasebeta

← #11734700

Subgroups — #11734700

Order 896.

544 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 159 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×#6509C211yesmaximal, characteristic
2C2×D7×C4⋊C4C211yesmaximal, characteristic
3C7×#395C211yesmaximal, characteristic, complemented
4#90001C211yesmaximal, characteristic, complemented
5#89761C212yesmaximal
6#89755C211yesmaximal, characteristic, complemented
7#89745C212yesmaximal, complemented
8#89440C212yesmaximal
9#89438C211yesmaximal, characteristic
10#89433C211yesmaximal, characteristic, complemented
11#89419C212yesmaximal, complemented
12C22×C7×Q8C2211yescharacteristic
13C22×C4×D7C2211yescharacteristic
14C22×C4×D7C2211yescharacteristic
15C22×C4×D7C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2212yes—
18#6509—24no—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×C4⋊Dic7C2211yescharacteristic
22C2×C4⋊Dic7C2212yes—
23C2×C4⋊Dic7C2211yescharacteristic
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2211yescharacteristic
27C2×Dic7⋊C4C2211yescharacteristic
28C2×Dic7⋊C4C2212yes—
29C2×C4×Dic7C2212yes—
30D7×C4⋊C4—24no—
31C7×#95C2212yes—
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2583—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C2×C7×Q8—22no—
40C2×C7×Q8—42no—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2312yes—
45C22×C4×C7C2311yescharacteristic
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2312yes—
48C22×Dic7C2311yescharacteristic
49C22×Dic7C2312yes—
50C22×Dic7C2311yescharacteristic
51C2×C4×D7—22no—
52C2×C4×D7—22no—
53C2×C4×D7—22no—
54C2×C4×D7—22no—
55C2×C4×D7—42no—
56C2×C4×D7—42no—
57C2×C4×D7D414yes—
58C7×C4⋊C4—22no—
59C7×C4⋊C4—42no—
60(C2×D7)⋊C4—24no—
61(C2×D7)⋊C4Q818yescomplemented
62C4⋊Dic7—22no—
63C4⋊Dic7—22no—
64C4⋊Dic7—24no—
65Dic7⋊C4—22no—
66Dic7⋊C4—24no—
67Dic7⋊C4—22no—
68Dic7⋊C4—24no—
69Dic7⋊C4—42no—
70C4×Dic7—42no—
71C2×C22⋊Q8—71no—
72C22×C4⋊C4—71no—
73#398—71no—
74#398—71no—
75#395—71no—
76#395D711yescharacteristic, complemented
77#389—72no—
78#389—72no—
79#385—72no—
80#385—72no—
81C4×C22⋊C4—71no—
82C23×C7C2411yescharacteristic
83C22×D7—21no—
84C22×D7—21no—
85C22×D7—22no—
86C22×D7C2×D412yes—
87C22×D7C2×Q814yes—
88C7×Q8—44no—
89C2×C4×C7—21no—
90C2×C4×C7—22no—
91C2×C4×C7—21no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—22no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7C2×D412yes—
100C2×C4×C7C2×Q814yes—
101C2×C4×C7#4112yes—
102C2×Dic7—21no—
103C2×Dic7—22no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—21no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7C2×D412yes—
113C2×Dic7C2×Q814yes—
114C2×Dic7#4112yes—
115C4×D7—28no—
116C4×D7—44no—
117C4×D7—44no—
118C22×Q8C2×D711yescharacteristic
119C23×C4—71no—
120C23×C4—71no—
121C23×C4—71no—
122C22⋊Q8—144no—
123C2×C4⋊C4—71no—
124C2×C4⋊C4—71no—
125C2×C4⋊C4—71no—
126C2×C4⋊C4—72no—
127C2×C4⋊C4—71no—
128C2×C4⋊C4—72no—
129C2×C4⋊C4—72no—
130C2×C4⋊C4—72no—
131C2×C4⋊C4—144no—
132C2×C4⋊C4C2×D711yescharacteristic
133C2×C4⋊C4C2×D712yes—
134C2×C22⋊C4—72no—
135C2×C22⋊C4—72no—
136C2×C42—72no—
137#95—71no—
138#95—72no—
139#95—72no—
140#95—72no—
141#95C2×D711yescharacteristic
142#95C2×D712yes—
143C22×C7C22×D411yescharacteristic
144C22×C7C22×Q812yes—
145C22×C7C2×#4111yescharacteristic
146C22×C7C2×#4111yescharacteristic
147C22×C7H(5,𝔽2)12yes—
148C2×D7—22no—
149C2×D7—22no—
150C2×D7—24no—
151C2×D7—24no—
152C2×D7C4⋊Q814yes—
153C4×C7—22no—
154C4×C7—22no—
155C4×C7—24no—
156C4×C7—41no—
157C4×C7—42no—
158Dic7—22no—
159Dic7—22no—
160Dic7—24no—
161Dic7—42no—
162Dic7—41no—
163C24—71no—
164C2×Q8—22no—
165C2×Q8—42no—
166C22×C4—72no—
167C22×C4—71no—
168C22×C4—71no—
169C22×C4—74no—
170C22×C4—72no—
171C22×C4—71no—
172C22×C4—142no—
173C22×C4—142no—
174C22×C4—142no—
175C22×C4—142no—
176C22×C4—282no—
177C22×C4—282no—
178C22×C4C22×D711yescharacteristic
179C22×C4C22×D712yes—
180C22×C4C22×D711yescharacteristic
181C22×C4C22×D712yes—
182C22×C4C22×D711yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—42no—
185C4⋊C4—142no—
186C4⋊C4—144no—
187C4⋊C4—144no—
188C4⋊C4—142no—
189C4⋊C4—142no—
190C4⋊C4—142no—
191C4⋊C4—144no—
192C4⋊C4—282no—
193C22⋊C4—78no—
194C22⋊C4—144no—
195C42—282no—
196C2×C7#52511yescharacteristic
197C2×C7C2×C4⋊Q811yescharacteristic
198C2×C7#53411yescharacteristic
199C2×C7#55412yes—
200C2×C7#55412yes—
201D7—24no—
202C23—72no—
203C23—74no—
204C23—142no—
205C23—141no—
206C23—141no—
207C23C23×D711yescharacteristic
208Q8—44no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—72no—
220C2×C4—74no—
221C2×C4—72no—
222C2×C4—148no—
223C2×C4—141no—
224C2×C4—142no—
225C2×C4—142no—
226C2×C4—142no—
227C2×C4—141no—
228C2×C4—142no—
229C2×C4—141no—
230C2×C4—142no—
231C2×C4—142no—
232C2×C4—142no—
233C2×C4—284no—
234C2×C4—284no—
235C2×C4D4×D712yes—
236C2×C4Q8×D714yes—
237C2×C4#116112yes—
238C7#258311yescharacteristic, complemented, Sylow
239C22—74no—
240C22—144no—
241C22—142no—
242C22—142no—
243C22—144no—
244C22C2×D4×D711yescharacteristic
245C22#654812yes—
246C22C2×Q8×D712yes—
247C22C2×#116111yescharacteristic
248C22D7×#4111yescharacteristic
249C4—22no—
250C4—22no—
251C4—24no—
252C4—41no—
253C4—42no—
254C4—144no—
255C4—142no—
256C4—142no—
257C4—281no—
258C4—282no—
259C2—144no—
260C2#9028011yescharacteristic
261C2#9031811yescharacteristic
262C2#9032712yes—
263C2D7×C4⋊Q811yescharacteristic
264C2#9042112yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#395C2
prime radicalC23×D7C23
hypercenter#395D7
hypocenterC7#2583
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
3. C7×#395C2semidirect
4. #90001C2semidirect
6. #89755C2semidirect
7. #89745C2semidirect
10. #89433C2semidirect
11. #89419C2semidirect
61. (C2×D7)⋊C4Q8semidirect
76. #395D7semidirect
238. C7#2583semidirect

Maximal subgroups

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

subgroupindex
C2×#65092
C2×D7×C4⋊C42
C7×#3952
#900012
#897612
#897552
#897452
#894402
#894382
#894332
#894192
#25837