Finite Group Databasebeta

← #11735210

Subgroups — #11735210

Order 896.

544 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 161 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×D7×C4⋊C4C211yesmaximal, characteristic
3C7×#398C211yesmaximal, characteristic, complemented
4#89761C212yesmaximal
5#89755C211yesmaximal, characteristic, complemented
6#89752C211yesmaximal, characteristic, complemented
7#89748C212yesmaximal, complemented
8#89440C212yesmaximal
9#89438C211yesmaximal, characteristic
10#89433C211yesmaximal, characteristic, complemented
11#89418C212yesmaximal, complemented
12C22×C4×D7C2211yescharacteristic
13C22×C4×D7C2212yes—
14C22×Dic14C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2211yescharacteristic
19C2×(C2×D7)⋊C4C2211yescharacteristic
20C2×(C2×D7)⋊C4C2212yes—
21C2×C4⋊Dic7C2211yescharacteristic
22C2×C4⋊Dic7C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2211yescharacteristic
26C2×Dic7⋊C4C2212yes—
27C2×Dic7⋊C4C2212yes—
28C2×C4×Dic7C2212yes—
29#6459—24no—
30D7×C4⋊C4—24no—
31C7×#95C2212yes—
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2502—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2312yes—
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2312yes—
48C22×Dic7C2311yescharacteristic
49C2×C4×D7—24no—
50C2×C4×D7—24no—
51C2×C4×D7—42no—
52C2×C4×D7—42no—
53C2×C4×D7D414yes—
54C2×Dic14—22no—
55C2×Dic14—24no—
56C7×C4⋊C4—22no—
57C7×C4⋊C4—22no—
58C7×C4⋊C4—42no—
59(C2×D7)⋊C4—22no—
60(C2×D7)⋊C4—22no—
61(C2×D7)⋊C4Q818yescomplemented
62C4⋊Dic7—22no—
63C4⋊Dic7—42no—
64Dic7⋊C4—22no—
65Dic7⋊C4—24no—
66Dic7⋊C4—22no—
67Dic7⋊C4—42no—
68Dic7⋊C4—42no—
69C4×Dic7—24no—
70C2×C22⋊Q8—71no—
71C22×C4⋊C4—71no—
72#398—71no—
73#398—71no—
74#398—71no—
75#398D711yescharacteristic, complemented
76#391—72no—
77#385—72no—
78#385—72no—
79#384—72no—
80C4×C22⋊C4—71no—
81C23×C7C2411yescharacteristic
82C22×D7—22no—
83C22×D7—21no—
84C22×D7—21no—
85C22×D7C2×D412yes—
86C22×D7C2×Q814yes—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—21no—
91C2×C4×C7—22no—
92C2×C4×C7—21no—
93C2×C4×C7—22no—
94C2×C4×C7—21no—
95C2×C4×C7—22no—
96C2×C4×C7—21no—
97C2×C4×C7—22no—
98C2×C4×C7C2×D412yes—
99C2×C4×C7C2×Q814yes—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—21no—
103C2×Dic7—22no—
104C2×Dic7—21no—
105C2×Dic7—22no—
106C2×Dic7—21no—
107C2×Dic7—22no—
108C2×Dic7—21no—
109C2×Dic7—22no—
110C2×Dic7C2×D412yes—
111C2×Dic7C2×Q814yes—
112C2×Dic7#4114yes—
113C4×D7—28no—
114C4×D7—48no—
115Dic14—44no—
116C22×Q8—71no—
117C23×C4—71no—
118C23×C4—72no—
119C22⋊Q8—144no—
120C2×C4⋊C4—72no—
121C2×C4⋊C4—71no—
122C2×C4⋊C4—71no—
123C2×C4⋊C4—71no—
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×C4⋊C4C2×D712yes—
131C2×C22⋊C4—72no—
132C2×C22⋊C4—71no—
133C2×C22⋊C4—71no—
134C2×C42—72no—
135#95—71no—
136#95—72no—
137#95—72no—
138#95—72no—
139#95C2×D711yescharacteristic
140#95C2×D712yes—
141C22×C7C22×D411yescharacteristic
142C22×C7C22×Q812yes—
143C22×C7C2×#4112yes—
144C22×C7H(5,𝔽2)11yescharacteristic
145C22×C7H(5,𝔽2)11yescharacteristic
146C2×D7—22no—
147C2×D7—22no—
148C2×D7—24no—
149C2×D7—24no—
150C2×D7C4⋊Q814yes—
151C4×C7—22no—
152C4×C7—24no—
153C4×C7—41no—
154C4×C7—42no—
155C4×C7—41no—
156Dic7—24no—
157Dic7—41no—
158Dic7—42no—
159Dic7—41no—
160Dic7#12314yes—
161C24—71no—
162C2×Q8—142no—
163C2×Q8—144no—
164C22×C4—71no—
165C22×C4—72no—
166C22×C4—74no—
167C22×C4—71no—
168C22×C4—72no—
169C22×C4—71no—
170C22×C4—144no—
171C22×C4—144no—
172C22×C4—282no—
173C22×C4—282no—
174C22×C4C22×D711yescharacteristic
175C22×C4C22×D712yes—
176C22×C4C22×D711yescharacteristic
177C22×C4C22×D711yescharacteristic
178C22×C4C22×D712yes—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4—144no—
183C4⋊C4—142no—
184C4⋊C4—142no—
185C4⋊C4—142no—
186C4⋊C4—282no—
187C4⋊C4—282no—
188C4⋊C4—282no—
189C22⋊C4—78no—
190C22⋊C4—142no—
191C22⋊C4—142no—
192C42—144no—
193C2×C7C2×#12311yescharacteristic
194C2×C7C2×C4⋊Q811yescharacteristic
195C2×C7#53511yescharacteristic
196C2×C7#55412yes—
197C2×C7#55412yes—
198D7—24no—
199C23—72no—
200C23—74no—
201C23—141no—
202C23—142no—
203C23—141no—
204C23C23×D711yescharacteristic
205Q8—284no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—72no—
218C2×C4—74no—
219C2×C4—74no—
220C2×C4—148no—
221C2×C4—142no—
222C2×C4—142no—
223C2×C4—141no—
224C2×C4—141no—
225C2×C4—142no—
226C2×C4—141no—
227C2×C4—142no—
228C2×C4—142no—
229C2×C4—141no—
230C2×C4—142no—
231C2×C4—288no—
232C2×C4D4×D712yes—
233C2×C4Q8×D714yes—
234C7#250211yescharacteristic, complemented, Sylow
235C22—74no—
236C22—144no—
237C22—142no—
238C22—144no—
239C22—142no—
240C22C2×D4×D711yescharacteristic
241C22#654811yescharacteristic
242C22C2×Q8×D712yes—
243C22D7×#4112yes—
244C22#655311yescharacteristic
245C4—24no—
246C4—22no—
247C4—42no—
248C4—41no—
249C4—41no—
250C4—74no—
251C4—144no—
252C4—282no—
253C4—281no—
254C4—281no—
255C2—144no—
256C2#9029811yescharacteristic
257C2#9032712yes—
258C2D7×#12311yescharacteristic
259C2#9038112yes—
260C2D7×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#398C2
prime radicalC23×D7C23
hypercenter#398D7
hypocenterC7#2502
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. C7×#398C2semidirect
5. #89755C2semidirect
6. #89752C2semidirect
7. #89748C2semidirect
10. #89433C2semidirect
11. #89418C2semidirect
61. (C2×D7)⋊C4Q8semidirect
75. #398D7semidirect
234. C7#2502semidirect

Maximal subgroups

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

subgroupindex
C2×#64592
C2×D7×C4⋊C42
C7×#3982
#897612
#897552
#897522
#897482
#894402
#894382
#894332
#894182
#25027