Finite Group Databasebeta

← #11735315

Subgroups — #11735315

Order 896.

512 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 129 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, complemented
2C7×#398C211yesmaximal, characteristic, complemented
3#89763C211yesmaximal, characteristic
4#89762C212yesmaximal, complemented
5#89761C212yesmaximal, complemented
6#89746C211yesmaximal, characteristic, complemented
7#89447C212yesmaximal, complemented
8#89437C212yesmaximal
9#89430C212yesmaximal, complemented
10C22×D28C2211yescharacteristic
11C22×C4×D7C2212yes—
12C22×Dic14C2211yescharacteristic
13C2×C7×C4⋊C4C2212yes—
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×(C2×D7)⋊C4C2212yes—
17C2×(C2×D7)⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×C4×Dic7C2211yescharacteristic
24#6459—28no—
25C7×#95C2212yes—
26C7×#95C2211yescharacteristic
27C7⋊#95C2211yescharacteristic
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32#2728—71nomaximal, Sylow
33C23×D7C2312yes—
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2312yes—
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C2×D28—24no—
43C2×D28—41no—
44C2×C4×D7—24no—
45C2×C4×D7—42no—
46C2×C4×D7—42no—
47C2×Dic14—41no—
48C2×Dic14—41no—
49C2×Dic14D414yescomplemented
50C7×C4⋊C4—24no—
51C7×C4⋊C4—41no—
52C7×C4⋊C4—41no—
53(C2×D7)⋊C4—24no—
54(C2×D7)⋊C4—24no—
55(C2×D7)⋊C4—24no—
56(C2×D7)⋊C4—24no—
57(C2×D7)⋊C4—42no—
58Dic7⋊C4—24no—
59Dic7⋊C4—24no—
60C4×Dic7—41no—
61C2×C22⋊Q8—72no—
62#400—72no—
63#398D711yescharacteristic, complemented
64#397—72no—
65#394—71no—
66#391—71no—
67#388—72no—
68#386—72no—
69#385—72no—
70C23×C7C2411yescharacteristic
71C22×D7—22no—
72C22×D7—22no—
73C22×D7—22no—
74C22×D7—22no—
75C22×D7—22no—
76C22×D7—22no—
77C22×D7#4114yes—
78C2×C4×C7—21no—
79C2×C4×C7—21no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7—21no—
87C2×C4×C7—22no—
88C2×C4×C7—21no—
89C2×C4×C7—22no—
90C2×C4×C7C2×D412yes—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7C2×D414yes—
100D28—44no—
101C4×D7—48no—
102Dic14—44no—
103C22×Q8—71no—
104C22×D4—71no—
105C23×C4—72no—
106C22⋊Q8—148no—
107C2×C4⋊C4—72no—
108C2×C4⋊C4—72no—
109C2×C4⋊C4C2×D712yes—
110C2×C4⋊C4C2×D711yescharacteristic
111C2×C4⋊C4C2×D711yescharacteristic
112C2×C22⋊C4—72no—
113C2×C22⋊C4—72no—
114C2×C22⋊C4—72no—
115C2×C22⋊C4—72no—
116C2×C22⋊C4—72no—
117C2×C42—71no—
118#95—72no—
119#95—72no—
120#95—72no—
121#95—72no—
122#95—71no—
123#95C2×D712yes—
124#95C2×D711yescharacteristic
125C22×C7C22×D411yescharacteristic
126C22×C7C2×#4112yes—
127C22×C7H(5,𝔽2)11yescharacteristic
128C22×C7H(5,𝔽2)11yescharacteristic
129C22×C72−1+412yes—
130C2×D7—24no—
131C2×D7—24no—
132C2×D7—24no—
133C2×D7—42no—
134C2×D7—42no—
135C2×D7—42no—
136C2×D7—42no—
137C4×C7—22no—
138C4×C7—42no—
139C4×C7—42no—
140C4×C7—41no—
141C4×C7—41no—
142Dic7—24no—
143Dic7—42no—
144Dic7—42no—
145C24—72no—
146C2×Q8—74no—
147C2×Q8—281no—
148C2×Q8—281no—
149C2×D4—144no—
150C2×D4—281no—
151C22×C4—72no—
152C22×C4—72no—
153C22×C4—72no—
154C22×C4—144no—
155C22×C4—282no—
156C22×C4—282no—
157C22×C4C22×D711yescharacteristic
158C22×C4C22×D711yescharacteristic
159C22×C4C22×D712yes—
160C22×C4C22×D712yes—
161C22×C4C22×D711yescharacteristic
162C4⋊C4—24no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—144no—
166C4⋊C4—144no—
167C22⋊C4—144no—
168C22⋊C4—144no—
169C22⋊C4—144no—
170C22⋊C4—144no—
171C22⋊C4—282no—
172C42—281no—
173C2×C7#53411yescharacteristic
174C2×C7#53912yes—
175C2×C7#54812yes—
176C2×C7#55111yescharacteristic
177C2×C7#56311yescharacteristic
178D7—44no—
179C23—74no—
180C23—142no—
181C23—142no—
182C23—142no—
183C23—142no—
184C23—142no—
185C23—142no—
186C23C23×D711yescharacteristic
187Q8—284no—
188D4—284no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—74no—
202C2×C4—142no—
203C2×C4—142no—
204C2×C4—142no—
205C2×C4—142no—
206C2×C4—142no—
207C2×C4—142no—
208C2×C4—142no—
209C2×C4—142no—
210C2×C4—288no—
211C2×C4D4×D712yes—
212C7#272811yescharacteristic, complemented, Sylow
213C22—144no—
214C22—144no—
215C22—144no—
216C22—282no—
217C22—282no—
218C22—282no—
219C22—282no—
220C22C2×D4×D711yescharacteristic
221C22#654811yescharacteristic
222C22#655112yes—
223C22D7×#4112yes—
224C22#655311yescharacteristic
225C4—22no—
226C4—42no—
227C4—42no—
228C4—41no—
229C4—41no—
230C4—144no—
231C4—282no—
232C4—282no—
233C2—284no—
234C2#9029611yescharacteristic
235C2#9032412yes—
236C2#9035011yescharacteristic
237C2#9038312yes—
238C2#9042611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#398C2
prime radicalC22×D28C22
hypercenter#398D7
hypocenterC7#2728
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#6459C2semidirect
2. C7×#398C2semidirect
4. #89762C2semidirect
5. #89761C2semidirect
6. #89746C2semidirect
7. #89447C2semidirect
9. #89430C2semidirect
49. C2×Dic14D4semidirect
63. #398D7semidirect
212. C7#2728semidirect

Maximal subgroups

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

subgroupindex
C2×#64592
C7×#3982
#897632
#897622
#897612
#897462
#894472
#894372
#894302
#27287