Finite Group Databasebeta

← #13242043

Subgroups — #13242043

Order 1792.

590 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C7×#2296C211yesmaximal, characteristic
2#11729281C212yesmaximal
3#11725444C214yesmaximal, complemented
4C2×C7×#117C411yescharacteristic
5C2×C7×#117C411yescharacteristic
6C22×C42×C7C2211yescharacteristic
7C22×#1136—22no—
8C7×#389C412yes—
9C7×#389C2212yes—
10C7×#385—22no—
11C4×C7×C4⋊C4—22no—
12#89319—22no—
13#89319C2214yes—
14#7632—71nomaximal, Sylow
15C23×C4×C7C2×C411yescharacteristic
16C23×C4×C7D412yes—
17C7×#117—24no—
18C7×#117—42no—
19C2×C7×C4⋊C4—22no—
20C2×C7×C4⋊C4—22no—
21C2×C7×C4⋊C4C2×C412yes—
22C2×C7×C4⋊C4C2×C412yes—
23C2×C7×C22⋊C4—21no—
24C2×C7×C22⋊C4—21no—
25C2×C42×C7—21no—
26C2×C42×C7—21no—
27C2×C42×C7—42no—
28C2×C42×C7C2×C411yescharacteristic
29C2×C42×C7D412yes—
30C2×C42×C7C2311yescharacteristic
31C2×#1136—28no—
32C2×#1136—44no—
33C2×#1136—44no—
34C22×C7⋊C8—24no—
35C7×#95—22no—
36C7×#95—22no—
37#2296D711yescharacteristic
38#1728—72no—
39#1271—74no—
40C24×C7C22⋊C411yescharacteristic
41C22×C4×C7—24no—
42C22×C4×C7—22no—
43C22×C4×C7—22no—
44C22×C4×C7—24no—
45C22×C4×C7—22no—
46C22×C4×C7—22no—
47C22×C4×C7—22no—
48C22×C4×C7—22no—
49C22×C4×C7—22no—
50C22×C4×C7C22⋊C411yescharacteristic
51C22×C4×C7C22⋊C412yes—
52C22×C4×C7C22×C411yescharacteristic
53C22×C4×C7C2×D412yes—
54C7×C4⋊C4—22no—
55C7×C4⋊C4—42no—
56C7×C4⋊C4—44no—
57C7×C4⋊C4—42no—
58C7×C4⋊C4—42no—
59C7×C4⋊C4M1614yescomplemented
60C7×C22⋊C4—24no—
61C7×C22⋊C4—42no—
62C42×C7—24no—
63C42×C7—41no—
64C42×C7—42no—
65C42×C7—41no—
66C42×C7—42no—
67#1136—416no—
68C2×C7⋊C8—28no—
69C2×C7⋊C8—44no—
70C2×C7⋊C8—44no—
71C22×M16—142no—
72C2×#117Dic711yescharacteristic
73C2×#117Dic711yescharacteristic
74C22×C42C2×D711yescharacteristic
75#389Dic712yes—
76#389C2×D712yes—
77#385—22no—
78C4×C4⋊C4—22no—
79#336—74no—
80#336—142no—
81C23×C7—21no—
82C23×C7—21no—
83C23×C7—22no—
84C23×C7—22no—
85C23×C7#9912yes—
86C23×C7C2×C22⋊C411yescharacteristic
87C2×C4×C7—28no—
88C2×C4×C7—28no—
89C2×C4×C7—28no—
90C2×C4×C7—21no—
91C2×C4×C7—22no—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7—22no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—24no—
99C2×C4×C7—42no—
100C2×C4×C7—42no—
101C2×C4×C7—42no—
102C2×C4×C7—42no—
103C2×C4×C7—42no—
104C2×C4×C7#10112yes—
105C2×C4×C7#10414yes—
106C2×C4×C7C2×M1612yes—
107C2×C4×C7AGL(1,ℤ/8)12yes—
108C2×C4×C7#13712yes—
109C7⋊C8—48no—
110C23×C4C2×Dic711yescharacteristic
111C23×C4C22⋊D712yes—
112C2×M16—148no—
113C2×M16—284no—
114C2×M16—284no—
115C22×C8—144no—
116#117—24no—
117#117—42no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4C2×Dic712yes—
121C2×C4⋊C4C2×Dic712yes—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—42no—
127C2×C42C2×Dic711yescharacteristic
128C2×C42C22⋊D712yes—
129C2×C42C22×D711yescharacteristic
130#95—22no—
131#95—22no—
132C22×C7—22no—
133C22×C7—24no—
134C22×C7—41no—
135C22×C7—42no—
136C22×C7—41no—
137C22×C7#40711yescharacteristic
138C22×C7C2×#9911yescharacteristic
139C22×C7C2×#10111yescharacteristic
140C22×C7#41712yes—
141C22×C7C2×#10412yes—
142C4×C7—22no—
143C4×C7—22no—
144C4×C7—22no—
145C4×C7—24no—
146C4×C7—22no—
147C4×C7—44no—
148C4×C7—82no—
149C24C22⋊Dic711yescharacteristic
150C22×C4—22no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4C22⋊Dic711yescharacteristic
160C22×C4C22⋊Dic712yes—
161C22×C4C22×Dic711yescharacteristic
162C22×C4C2×C22⋊D712yes—
163M16—2816no—
164C2×C8—148no—
165C2×C8—284no—
166C2×C8—284no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—44no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4#113614yescomplemented
173C22⋊C4—24no—
174C22⋊C4—42no—
175C42—24no—
176C42—41no—
177C42—42no—
178C42—41no—
179C42—42no—
180C2×C7—42no—
181C2×C7#146311yescharacteristic
182C2×C7#148912yes—
183C2×C7#150412yes—
184C2×C7#152611yescharacteristic
185C2×C7#156111yescharacteristic
186C23—21no—
187C23—21no—
188C23—22no—
189C23—22no—
190C23#640812yes—
191C23C2×C22⋊Dic711yescharacteristic
192C2×C4—28no—
193C2×C4—28no—
194C2×C4—28no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4#641012yes—
210C2×C4#641114yes—
211C2×C4C2×#113612yes—
212C2×C4#651212yes—
213C2×C4#651412yes—
214C8—288no—
215C7#763211yescharacteristic, complemented, Sylow
216C22—24no—
217C22—22no—
218C22—41no—
219C22—41no—
220C22—42no—
221C22#8997911yescharacteristic
222C22C2×#640811yescharacteristic
223C22C2×#641011yescharacteristic
224C22#9000412yes—
225C22C2×#641112yes—
226C4—22no—
227C4—24no—
228C4—22no—
229C4—22no—
230C4—22no—
231C4—44no—
232C4—82no—
233C2—42no—
234C2#1172930611yescharacteristic
235C2#1172939512yes—
236C2#1172946212yes—
237C2#1172957511yescharacteristic
238C2#1172958811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2296C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2296D7
hypocenterC7#7632
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2296

Complements

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

NH ≅ G/Ntype
3. #11725444C2semidirect
59. C7×C4⋊C4M16semidirect
172. C4⋊C4#1136semidirect
215. C7#7632semidirect

Maximal subgroups

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

subgroupindex
C7×#22962
#117292812
#117254442
#76327