Finite Group Databasebeta

← #13251197

Subgroups — #13251197

Order 1792.

466 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 123 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×#2567C211yesmaximal, characteristic
2#11739881C211yesmaximal, characteristic
3#11729528C212yesmaximal
4#11729391C211yesmaximal, characteristic
5#11725426C212yesmaximal
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#395C414yes—
9C7×#391—21no—
10C7×#389—22no—
11C7×#384—21no—
12C4×C7×C4⋊C4C2211yescharacteristic
13C2×#6410—21no—
14#89796—24no—
15#89698—22no—
16#89697C2212yes—
17#89696C2212yes—
18#89341—24no—
19#89325—21no—
20#9400—71nomaximal, Sylow
21C22×C7×Q8C2×C412yes—
22C22×C7×Q8D411yescharacteristic
23C7×C4⋊Q8—22no—
24C7×C4⋊Q8—22no—
25C4×C7×Q8—22no—
26C4×C7×Q8—22no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—24no—
31C2×C7×C4⋊C4C2×C412yes—
32C2×C7×C4⋊C4D411yescharacteristic
33C2×C42×C7—21no—
34C2×C42×C7D412yes—
35C2×C42×C7C2311yescharacteristic
36C2×#1136—22no—
37C7×#95—24no—
38C7×#95C2×C412yes—
39#6410—42no—
40C22⋊(C7⋊C8)—22no—
41C4⋊(C7⋊C8)—24no—
42C4⋊(C7⋊C8)—22no—
43C4⋊(C7⋊C8)—22no—
44C4⋊(C7⋊C8)D414yes—
45#6377—24no—
46C4×C7⋊C8—22no—
47#2983—71no—
48#2567D711yescharacteristic
49#1801—71no—
50#1548—72no—
51#1303—72no—
52C2×C7×Q8—22no—
53C2×C7×Q8—21no—
54C2×C7×Q8—21no—
55C2×C7×Q8—22no—
56C2×C7×Q8—24no—
57C2×C7×Q8—82no—
58C22×C4×C7—22no—
59C22×C4×C7—22no—
60C22×C4×C7C22⋊C412yes—
61C22×C4×C7C22⋊C412yes—
62C22×C4×C7C22×C411yescharacteristic
63C22×C4×C7C2×D411yescharacteristic
64C22×C4×C7C2×D411yescharacteristic
65C7×C4⋊C4—24no—
66C7×C4⋊C4—24no—
67C7×C4⋊C4—22no—
68C7×C4⋊C4—41no—
69C7×C4⋊C4—42no—
70C7×C4⋊C4—41no—
71C7×C4⋊C4—84no—
72C42×C7—22no—
73C42×C7—22no—
74C42×C7—41no—
75C42×C7—41no—
76C42×C7C2×D412yes—
77C42×C7#4112yes—
78#1136—42no—
79#1136—42no—
80C2×C7⋊C8—24no—
81C2×C7⋊C8—42no—
82C2×C4⋊Q8C2×D711yescharacteristic
83C2×C4×Q8C2×D711yescharacteristic
84#446—144no—
85#432—142no—
86#431—72no—
87#423—72no—
88C2×#101—141no—
89#395Dic714yes—
90#391—21no—
91#389—22no—
92#384—21no—
93C4×C4⋊C4C2×D711yescharacteristic
94#343—141no—
95#333—144no—
96C23×C7C2×C22⋊C411yescharacteristic
97C7×Q8—42no—
98C7×Q8—42no—
99C7×Q8—84no—
100C2×C4×C7—22no—
101C2×C4×C7—24no—
102C2×C4×C7—22no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—41no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—44no—
111C2×C4×C7—42no—
112C2×C4×C7C4×D412yes—
113C2×C4×C7#12011yescharacteristic
114C2×C4×C7#12112yes—
115C2×C4×C7#12311yescharacteristic
116C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
117C2×C4×C7#13712yes—
118C2×C4×C7#13711yescharacteristic
119C7⋊C8—44no—
120C7⋊C8—82no—
121C22×Q8C2×Dic712yes—
122C22×Q8C22⋊D711yescharacteristic
123C2×M16—142no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4C2×Dic712yes—
133C2×C4⋊C4C22⋊D711yescharacteristic
134C2×C42—21no—
135C2×C42C22⋊D712yes—
136C2×C42C22×D711yescharacteristic
137C4⋊C8—74no—
138C4⋊C8—144no—
139C4⋊C8—142no—
140C4⋊C8—142no—
141#101—282no—
142C22⋊C8—142no—
143#97—144no—
144C4×C8—142no—
145#95—24no—
146#95C2×Dic712yes—
147C22×C7—21no—
148C22×C7—21no—
149C22×C7#38611yescharacteristic
150C22×C7#41711yescharacteristic
151C22×C7#41911yescharacteristic
152C4×C7—21no—
153C4×C7—22no—
154C4×C7—44no—
155C4×C7—42no—
156C4×C7—42no—
157C4×C7—82no—
158C4×C7#45612yes—
159C4×C7#46812yes—
160C4×C7#47012yes—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—24no—
166C2×Q8—82no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4C22⋊Dic712yes—
170C22×C4C22⋊Dic712yes—
171C22×C4C22×Dic711yescharacteristic
172C22×C4C2×C22⋊D711yescharacteristic
173C22×C4C2×C22⋊D711yescharacteristic
174M16—282no—
175M16—282no—
176C2×C8—144no—
177C2×C8—282no—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—84no—
185C42—22no—
186C42—22no—
187C42—41no—
188C42—41no—
189C42D4×D712yes—
190C42#115912yes—
191C2×C7—21no—
192C2×C7—21no—
193C2×C7#188611yescharacteristic
194C2×C7#189411yescharacteristic
195C2×C7#191411yescharacteristic
196C23C2×C22⋊Dic711yescharacteristic
197Q8—42no—
198Q8—42no—
199Q8—84no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4D4×Dic712yes—
213C2×C4#649811yescharacteristic
214C2×C4#650011yescharacteristic
215C2×C4#650212yes—
216C2×C4#650512yes—
217C2×C4#651211yescharacteristic
218C2×C4#651411yescharacteristic
219C8—284no—
220C8—562no—
221C7#940011yescharacteristic, complemented, Sylow
222C22—21no—
223C22—21no—
224C22#8999211yescharacteristic
225C22#8999711yescharacteristic
226C22#9000411yescharacteristic
227C4—21no—
228C4—22no—
229C4—44no—
230C4—42no—
231C4—42no—
232C4—82no—
233C4#8982112yes—
234C4#8982412yes—
235C4#8986712yes—
236C2—21no—
237C2—21no—
238C2#1172945111yescharacteristic
239C2#1172953811yescharacteristic
240C2#1172960911yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2567C2
socleC22×C7#386
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2567D7
hypocenterC7#9400
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2567

Complements

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

NH ≅ G/Ntype
221. C7#9400semidirect

Maximal subgroups

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

subgroupindex
C7×#25672
#117398812
#117295282
#117293912
#117254262
#94007