Finite Group Databasebeta

← #11734978

Subgroups — #11734978

Order 896.

492 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 135 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×#6450C211yesmaximal, characteristic, complemented
2C2×#6440C211yesmaximal, characteristic
3C7×#397C211yesmaximal, characteristic
4#89752C211yesmaximal, characteristic, complemented
5#89723C212yesmaximal
6#89719C211yesmaximal, characteristic
7#89718C212yesmaximal
8#89428C211yesmaximal, characteristic, complemented
9#89426C212yesmaximal, complemented
10#89424C212yesmaximal, complemented
11#89423C211yesmaximal, characteristic, complemented
12C23×Dic7C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C22⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2212yes—
17C2×C22⋊Dic7C2211yescharacteristic
18C2×C22⋊Dic7C2212yes—
19C2×C4⋊Dic7C2212yes—
20C2×Dic7⋊C4C2211yescharacteristic
21C2×Dic7⋊C4C2211yescharacteristic
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×C4×Dic7C2211yescharacteristic
25C2×C4×Dic7C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27#6450—24no—
28#6440—24no—
29C7×#95C2211yescharacteristic
30C7×#95C2212yes—
31C7⋊#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2599—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×Dic7—22no—
44C22×Dic7—22no—
45C22×Dic7—22no—
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C22×Dic7C2311yescharacteristic
50C22×Dic7C2312yes—
51C22×Dic7C2311yescharacteristic
52C2×Dic14—22no—
53C2×Dic14—24no—
54C7×C4⋊C4—22no—
55C7×C22⋊C4—24no—
56C7×C22⋊C4—22no—
57C22⋊Dic7—22no—
58C22⋊Dic7—24no—
59C4⋊Dic7—24no—
60Dic7⋊C4—22no—
61Dic7⋊C4—22no—
62Dic7⋊C4—24no—
63Dic7⋊C4—24no—
64C4×Dic7—22no—
65C4×Dic7—24no—
66C4×Dic7D414yes—
67C2×C4×Q8—71no—
68C2×#117—71no—
69#398—71no—
70#397D711yescharacteristic
71#395—71no—
72#393—71no—
73#388—72no—
74#387—72no—
75#387—72no—
76#385—72no—
77#383—71no—
78C23×C7—21no—
79C23×C7—22no—
80C23×C7—21no—
81C23×C7—22no—
82C23×C7#4112yes—
83C23×C7C2411yescharacteristic
84C2×C4×C7—22no—
85C2×C4×C7—21no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7C2×D412yes—
94C2×Dic7—28no—
95C2×Dic7—22no—
96C2×Dic7—21no—
97C2×Dic7—22no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—21no—
101C2×Dic7—22no—
102C2×Dic7—21no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7C2×D412yes—
108C2×Dic7C2×D412yes—
109C2×Dic7#4112yes—
110C2×Dic7#4114yes—
111Dic14—28no—
112C22×Q8—71no—
113C23×C4—71no—
114C4×Q8—144no—
115#117—144no—
116C2×C4⋊C4—71no—
117C2×C4⋊C4—71no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4—72no—
121C2×C4⋊C4C2×D711yescharacteristic
122C2×C22⋊C4—71no—
123C2×C22⋊C4—72no—
124C2×C22⋊C4C2×D711yescharacteristic
125C2×C22⋊C4C2×D712yes—
126C2×C42—71no—
127C2×C42—71no—
128C2×C42—72no—
129#95—72no—
130#95—72no—
131#95—71no—
132#95—72no—
133#95—72no—
134#95—72no—
135#95C2×D711yescharacteristic
136#95C2×D712yes—
137C22×C7—22no—
138C22×C7—24no—
139C22×C7—41no—
140C22×C7—41no—
141C22×C7—42no—
142C22×C7C22×D411yescharacteristic
143C22×C7C2×#4111yescharacteristic
144C22×C7C2×#4112yes—
145C22×C7C2×#4111yescharacteristic
146C22×C72−1+412yes—
147C4×C7—22no—
148C4×C7—42no—
149C4×C7—41no—
150C4×C7—42no—
151Dic7—24no—
152Dic7—22no—
153Dic7—22no—
154Dic7—22no—
155Dic7—41no—
156Dic7—42no—
157C24C22×D711yescharacteristic
158C2×Q8—144no—
159C2×Q8—142no—
160C22×C4—71no—
161C22×C4—71no—
162C22×C4—72no—
163C22×C4—71no—
164C22×C4—72no—
165C22×C4—71no—
166C22×C4—142no—
167C22×C4—142no—
168C22×C4—142no—
169C22×C4C22×D712yes—
170C22×C4C22×D711yescharacteristic
171C22×C4C22×D712yes—
172C22×C4C22×D711yescharacteristic
173C4⋊C4—22no—
174C4⋊C4—144no—
175C4⋊C4—142no—
176C4⋊C4—144no—
177C4⋊C4—144no—
178C4⋊C4—142no—
179C22⋊C4—24no—
180C22⋊C4—22no—
181C22⋊C4—142no—
182C22⋊C4—144no—
183C42—74no—
184C42—142no—
185C42—144no—
186C2×C7—42no—
187C2×C7#52511yescharacteristic
188C2×C7#52511yescharacteristic
189C2×C7#53611yescharacteristic
190C2×C7#55212yes—
191C2×C7#55612yes—
192C23—21no—
193C23—22no—
194C23—21no—
195C23—22no—
196C23#115912yes—
197C23C23×D711yescharacteristic
198Q8—148no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—72no—
209C2×C4—72no—
210C2×C4—74no—
211C2×C4—72no—
212C2×C4—148no—
213C2×C4—142no—
214C2×C4—141no—
215C2×C4—141no—
216C2×C4—141no—
217C2×C4—141no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—141no—
221C2×C4—142no—
222C2×C4—142no—
223C2×C4—142no—
224C2×C4—142no—
225C2×C4D4×D712yes—
226C7#259911yescharacteristic, complemented, Sylow
227C22—22no—
228C22—24no—
229C22—42no—
230C22—41no—
231C22—41no—
232C22C2×D4×D711yescharacteristic
233C22C2×#115911yescharacteristic
234C22D7×#4111yescharacteristic
235C22D7×#4112yes—
236C22#655412yes—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—41no—
241C4—142no—
242C4—142no—
243C4—144no—
244C4—142no—
245C4—282no—
246C4—281no—
247C2—42no—
248C2#9029711yescharacteristic
249C2#9034112yes—
250C2#9034711yescharacteristic
251C2#9036611yescharacteristic
252C2#9039312yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
prime radicalC24×C7C23
hypercenter#397D7
hypocenterC7#2599
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#6450C2semidirect
4. #89752C2semidirect
8. #89428C2semidirect
9. #89426C2semidirect
10. #89424C2semidirect
11. #89423C2semidirect
226. C7#2599semidirect

Maximal subgroups

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

subgroupindex
C2×#64502
C2×#64402
C7×#3972
#897522
#897232
#897192
#897182
#894282
#894262
#894242
#894232
#25997