Finite Group Databasebeta

← #11735263

Subgroups — #11735263

Order 896.

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

normal subgroups only

This group has 528 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×#6459C212yesmaximal
2C2×#6455C211yesmaximal, characteristic
3C2×#6451C211yesmaximal, characteristic, complemented
4C7×#398C211yesmaximal, characteristic, complemented
5#89765C211yesmaximal, characteristic
6#89756C212yesmaximal, complemented
7#89755C211yesmaximal, characteristic, complemented
8#89447C212yesmaximal
9#89446C211yesmaximal, characteristic
10#89422C211yesmaximal, characteristic, complemented
11#89418C212yesmaximal, complemented
12C22×C4×D7C2211yescharacteristic
13C22×Dic14C2212yes—
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C4⋊C4C2212yes—
17C2×(C2×D7)⋊C4C2211yescharacteristic
18C2×(C2×D7)⋊C4C2211yescharacteristic
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×C4⋊Dic7C2211yescharacteristic
22C2×C4⋊Dic7C2211yescharacteristic
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27C2×C4×Dic7C2211yescharacteristic
28#6459—28no—
29#6455—24no—
30#6451—24no—
31C7×#95C2212yes—
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37C7⋊#95C2211yescharacteristic
38#2621—71nomaximal, Sylow
39C23×D7C2311yescharacteristic
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2312yes—
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2312yes—
47C22×Dic7C2312yes—
48C22×Dic7C2311yescharacteristic
49C22×Dic7C2311yescharacteristic
50C2×C4×D7—22no—
51C2×C4×D7—22no—
52C2×C4×D7D414yes—
53C2×Dic14—24no—
54C2×Dic14—24no—
55C2×Dic14—24no—
56C7×C4⋊C4—22no—
57C7×C4⋊C4—22no—
58C7×C4⋊C4—24no—
59(C2×D7)⋊C4—24no—
60(C2×D7)⋊C4—24no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—22no—
63C4⋊Dic7—22no—
64C4⋊Dic7—22no—
65Dic7⋊C4—24no—
66Dic7⋊C4—24no—
67Dic7⋊C4—24no—
68C4×Dic7—22no—
69C4×Dic7D414yes—
70C2×C4⋊Q8—71no—
71C2×C22⋊Q8—72no—
72C2×#117—71no—
73#398—71no—
74#398—72no—
75#398D711yescharacteristic, complemented
76#397—71no—
77#397—71no—
78#397—72no—
79#391—72no—
80#382—71no—
81C23×C7C2411yescharacteristic
82C22×D7—21no—
83C22×D7—21no—
84C22×D7—22no—
85C22×D7—22no—
86C22×D7C2×D412yes—
87C2×C4×C7—21no—
88C2×C4×C7—21no—
89C2×C4×C7—22no—
90C2×C4×C7—21no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—22no—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7C2×D412yes—
99C2×C4×C7C2×D412yes—
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×D412yes—
111C2×Dic7C2×D412yes—
112C2×Dic7#4112yes—
113C4×D7—28no—
114Dic14—48no—
115C22×Q8—72no—
116C23×C4—71no—
117C4⋊Q8—144no—
118C22⋊Q8—148no—
119#117—144no—
120C2×C4⋊C4—71no—
121C2×C4⋊C4—72no—
122C2×C4⋊C4—72no—
123C2×C4⋊C4—72no—
124C2×C4⋊C4—71no—
125C2×C4⋊C4C2×D711yescharacteristic
126C2×C4⋊C4C2×D712yes—
127C2×C4⋊C4C2×D711yescharacteristic
128C2×C22⋊C4—71no—
129C2×C22⋊C4—71no—
130C2×C22⋊C4—72no—
131C2×C22⋊C4—72no—
132C2×C42—71no—
133C2×C42—71no—
134#95—72no—
135#95—71no—
136#95—72no—
137#95—72no—
138#95—72no—
139#95C2×D711yescharacteristic
140#95C2×D712yes—
141C22×C7C22×D411yescharacteristic
142C22×C7C22×D411yescharacteristic
143C22×C7C2×#4111yescharacteristic
144C22×C72−1+412yes—
145C22×C72−1+412yes—
146C2×D7—22no—
147C2×D7—24no—
148C2×D7—41no—
149C2×D7—41no—
150C2×D7—42no—
151C4×C7—22no—
152C4×C7—22no—
153C4×C7—42no—
154C4×C7—42no—
155C4×C7—41no—
156Dic7—22no—
157Dic7—42no—
158Dic7—41no—
159Dic7—42no—
160Dic7#12114yes—
161C24—71no—
162C2×Q8—144no—
163C2×Q8—144no—
164C2×Q8—144no—
165C22×C4—71no—
166C22×C4—71no—
167C22×C4—74no—
168C22×C4—71no—
169C22×C4—72no—
170C22×C4—72no—
171C22×C4—142no—
172C22×C4—142no—
173C22×C4C22×D711yescharacteristic
174C22×C4C22×D712yes—
175C22×C4C22×D712yes—
176C22×C4C22×D711yescharacteristic
177C22×C4C22×D711yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4—144no—
182C4⋊C4—144no—
183C4⋊C4—142no—
184C4⋊C4—144no—
185C4⋊C4—142no—
186C22⋊C4—142no—
187C22⋊C4—144no—
188C22⋊C4—144no—
189C22⋊C4—142no—
190C42—74no—
191C42—142no—
192C2×C7C2×#12111yescharacteristic
193C2×C7#53611yescharacteristic
194C2×C7#53612yes—
195C2×C7#53611yescharacteristic
196C2×C7#54112yes—
197D7—42no—
198C23—72no—
199C23—141no—
200C23—142no—
201C23—141no—
202C23—142no—
203C23C23×D711yescharacteristic
204Q8—288no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—72no—
217C2×C4—72no—
218C2×C4—72no—
219C2×C4—72no—
220C2×C4—148no—
221C2×C4—142no—
222C2×C4—141no—
223C2×C4—142no—
224C2×C4—142no—
225C2×C4—142no—
226C2×C4—142no—
227C2×C4—142no—
228C2×C4—142no—
229C2×C4—142no—
230C2×C4D4×D712yes—
231C2×C4D4×D712yes—
232C7#262111yescharacteristic, complemented, Sylow
233C22—142no—
234C22—144no—
235C22—282no—
236C22—281no—
237C22—281no—
238C22C2×D4×D711yescharacteristic
239C22C2×D4×D711yescharacteristic
240C22#655112yes—
241C22D7×#4111yescharacteristic
242C22#655412yes—
243C4—22no—
244C4—22no—
245C4—42no—
246C4—42no—
247C4—41no—
248C4—74no—
249C4—142no—
250C4—281no—
251C4—282no—
252C4—282no—
253C2—282no—
254C2D7×#12111yescharacteristic
255C2#9031912yes—
256C2#9033911yescharacteristic
257C2#9038712yes—
258C2#9041511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#398C2
prime radicalC23×D7C23
hypercenter#398D7
hypocenterC7#2621
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. C2×#6451C2semidirect
4. C7×#398C2semidirect
6. #89756C2semidirect
7. #89755C2semidirect
10. #89422C2semidirect
11. #89418C2semidirect
75. #398D7semidirect
232. C7#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#64592
C2×#64552
C2×#64512
C7×#3982
#897652
#897562
#897552
#894472
#894462
#894222
#894182
#26217