Finite Group Databasebeta

← #11735260

Subgroups — #11735260

Order 896.

540 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 137 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×#6455C212yesmaximal, complemented
2C7×#398C211yesmaximal, characteristic, complemented
3#89766C212yesmaximal, complemented
4#89763C211yesmaximal, characteristic
5#89762C212yesmaximal, complemented
6#89749C211yesmaximal, characteristic, complemented
7#89440C212yesmaximal, complemented
8#89437C212yesmaximal
9#89427C212yesmaximal, complemented
10C22×D28C2211yescharacteristic
11C22×C4×D7C2212yes—
12C2×C7×C4⋊C4C2212yes—
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×(C2×D7)⋊C4C2212yes—
16C2×(C2×D7)⋊C4C2212yes—
17C2×(C2×D7)⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×C4⋊Dic7C2212yes—
21C2×C4⋊Dic7C2211yescharacteristic
22C2×Dic7⋊C4C2212yes—
23C2×C4×Dic7C2212yes—
24C2×C4×Dic7C2211yescharacteristic
25#6455—28no—
26C7×#95C2212yes—
27C7×#95C2211yescharacteristic
28C7⋊#95C2211yescharacteristic
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32#2609—71nomaximal, Sylow
33C23×D7C2312yes—
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2312yes—
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C2×D28—24no—
43C2×D28—41no—
44C2×C4×D7—24no—
45C2×C4×D7—24no—
46C2×C4×D7—24no—
47C7×C4⋊C4—24no—
48C7×C4⋊C4—41no—
49C7×C4⋊C4—41no—
50(C2×D7)⋊C4—24no—
51(C2×D7)⋊C4—24no—
52(C2×D7)⋊C4—24no—
53(C2×D7)⋊C4—24no—
54(C2×D7)⋊C4—42no—
55C4⋊Dic7—24no—
56C4⋊Dic7—41no—
57Dic7⋊C4—24no—
58C4×Dic7—24no—
59C4×Dic7D414yescomplemented
60C2×#117—72no—
61#398D711yescharacteristic, complemented
62#396—72no—
63#394—71no—
64#388—72no—
65#387—72no—
66#386—72no—
67#385—72no—
68#382—71no—
69C23×C7C2411yescharacteristic
70C22×D7—22no—
71C22×D7—22no—
72C22×D7—22no—
73C22×D7—22no—
74C22×D7—22no—
75C22×D7—22no—
76C22×D7#4114yes—
77C2×C4×C7—22no—
78C2×C4×C7—22no—
79C2×C4×C7—21no—
80C2×C4×C7—21no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—22no—
88C2×C4×C7C2×D412yes—
89C2×C4×C7#4114yes—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7C2×D414yes—
99D28—44no—
100C4×D7—216no—
101C22×D4—71no—
102C23×C4—72no—
103#117—148no—
104C2×C4⋊C4—72no—
105C2×C4⋊C4—72no—
106C2×C4⋊C4—71no—
107C2×C4⋊C4C2×D712yes—
108C2×C4⋊C4C2×D711yescharacteristic
109C2×C4⋊C4C2×D711yescharacteristic
110C2×C22⋊C4—72no—
111C2×C22⋊C4—72no—
112C2×C22⋊C4—72no—
113C2×C22⋊C4—72no—
114C2×C22⋊C4—72no—
115C2×C42—71no—
116C2×C42—72no—
117#95—72no—
118#95—72no—
119#95—71no—
120#95—72no—
121#95C2×D711yescharacteristic
122#95C2×D712yes—
123C22×C7C22×D411yescharacteristic
124C22×C7C2×#4112yes—
125C22×C7C2×#4112yes—
126C22×C7H(5,𝔽2)11yescharacteristic
127C22×C72−1+411yescharacteristic
128C2×D7—24no—
129C2×D7—24no—
130C2×D7—24no—
131C2×D7—42no—
132C2×D7—42no—
133C2×D7—42no—
134C2×D7—42no—
135C4×C7—24no—
136C4×C7—22no—
137C4×C7—42no—
138C4×C7—41no—
139C4×C7—41no—
140Dic7—24no—
141Dic7—42no—
142Dic7—42no—
143C24—72no—
144C2×D4—144no—
145C2×D4—281no—
146C22×C4—72no—
147C22×C4—72no—
148C22×C4—72no—
149C22×C4—144no—
150C22×C4—144no—
151C22×C4—144no—
152C22×C4C22×D711yescharacteristic
153C22×C4C22×D711yescharacteristic
154C22×C4C22×D712yes—
155C22×C4C22×D711yescharacteristic
156C22×C4C22×D712yes—
157C4⋊C4—24no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—144no—
161C4⋊C4—144no—
162C4⋊C4—281no—
163C22⋊C4—144no—
164C22⋊C4—144no—
165C22⋊C4—144no—
166C22⋊C4—144no—
167C22⋊C4—282no—
168C42—74no—
169C42—144no—
170C2×C7#52512yes—
171C2×C7#53711yescharacteristic
172C2×C7#55111yescharacteristic
173C2×C7#55212yes—
174C2×C7#55511yescharacteristic
175D7—44no—
176C23—74no—
177C23—142no—
178C23—142no—
179C23—142no—
180C23—142no—
181C23—142no—
182C23—142no—
183C23C23×D711yescharacteristic
184D4—284no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—74no—
197C2×C4—1416no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4—142no—
203C2×C4—142no—
204C2×C4—142no—
205C2×C4—142no—
206C2×C4D4×D712yes—
207C2×C4#116114yes—
208C7#260911yescharacteristic, complemented, Sylow
209C22—144no—
210C22—144no—
211C22—144no—
212C22—282no—
213C22—282no—
214C22—282no—
215C22—282no—
216C22C2×D4×D711yescharacteristic
217C22#654811yescharacteristic
218C22C2×#116112yes—
219C22D7×#4112yes—
220C22#655411yescharacteristic
221C4—22no—
222C4—24no—
223C4—42no—
224C4—41no—
225C4—41no—
226C4—144no—
227C4—282no—
228C4—282no—
229C2—284no—
230C2#9028811yescharacteristic
231C2#9031812yes—
232C2#9035011yescharacteristic
233C2#9038512yes—
234C2#9042511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#398C2
prime radicalC22×D28C22
hypercenter#398D7
hypocenterC7#2609
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#6455C2semidirect
2. C7×#398C2semidirect
3. #89766C2semidirect
5. #89762C2semidirect
6. #89749C2semidirect
7. #89440C2semidirect
9. #89427C2semidirect
59. C4×Dic7D4semidirect
61. #398D7semidirect
208. C7#2609semidirect

Maximal subgroups

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

subgroupindex
C2×#64552
C7×#3982
#897662
#897632
#897622
#897492
#894402
#894372
#894272
#26097