Finite Group Databasebeta

← #13251177

Subgroups — #13251177

Order 1792.

470 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 139 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×#2568C211yesmaximal, characteristic
2#11739880C211yesmaximal, characteristic
3#11729529C212yesmaximal
4#11729391C211yesmaximal, characteristic
5#11725426C212yesmaximal
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#400C414yes—
9C7×#391—21no—
10C7×#391—22no—
11C7×#382—21no—
12C4×C7×C4⋊C4C2211yescharacteristic
13C2×#6410—21no—
14Q8×C7⋊C8—24no—
15#89697—22no—
16#89697C2212yes—
17#89696C2212yes—
18#89342—24no—
19#89325—21no—
20#9396—71nomaximal, Sylow
21C22×C7×Q8—21no—
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⋊C4C2×C412yes—
30C2×C7×C4⋊C4C2×C412yes—
31C2×C7×C4⋊C4D411yescharacteristic
32C2×C42×C7—21no—
33C2×C42×C7D412yes—
34C2×C42×C7C2311yescharacteristic
35C2×#1136—22no—
36C7×#95—24no—
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—
45C4×C7⋊C8—24no—
46C4×C7⋊C8—22no—
47#2982—71no—
48#2568D711yescharacteristic
49#1801—71no—
50#1549—72no—
51#1303—72no—
52C2×C7×Q8—22no—
53C2×C7×Q8—22no—
54C2×C7×Q8—44no—
55C2×C7×Q8—41no—
56C2×C7×Q8—41no—
57C2×C7×Q8SD1612yes—
58C2×C7×Q8Q1612yes—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7C22⋊C412yes—
62C22×C4×C7C22⋊C412yes—
63C22×C4×C7C22×C411yescharacteristic
64C22×C4×C7C2×D411yescharacteristic
65C22×C4×C7C2×D411yescharacteristic
66C7×C4⋊C4—24no—
67C7×C4⋊C4—22no—
68C7×C4⋊C4—24no—
69C7×C4⋊C4—42no—
70C7×C4⋊C4—42no—
71C42×C7—22no—
72C42×C7—22no—
73C42×C7—42no—
74C42×C7C2×D412yes—
75C42×C7#4112yes—
76#1136—42no—
77#1136—42no—
78C2×C7⋊C8—24no—
79C2×C7⋊C8—42no—
80C2×C4⋊Q8C2×D711yescharacteristic
81C2×C4×Q8C2×D711yescharacteristic
82C8×Q8—144no—
83#431—72no—
84#431—142no—
85#423—72no—
86C2×#101—141no—
87#400Dic714yes—
88#391—21no—
89#391—22no—
90#382—21no—
91C4×C4⋊C4C2×D711yescharacteristic
92#343—141no—
93#332—144no—
94C23×C7C2×C22⋊C411yescharacteristic
95C7×Q8—24no—
96C7×Q8—24no—
97C7×Q8—82no—
98C7×Q8—82no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—41no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—44no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7#10314yes—
111C2×C4×C7C4×D412yes—
112C2×C4×C7#12011yescharacteristic
113C2×C4×C7#12112yes—
114C2×C4×C7#12311yescharacteristic
115C2×C4×C7C2×SD1611yescharacteristic
116C2×C4×C7C2×Q1611yescharacteristic
117C2×C4×C7#13512yes—
118C7⋊C8—44no—
119C7⋊C8—82no—
120C22×Q8—21no—
121C22×Q8C22⋊D711yescharacteristic
122C2×M16—142no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4C2×Dic712yes—
130C2×C4⋊C4C2×Dic712yes—
131C2×C4⋊C4C22⋊D711yescharacteristic
132C2×C42—21no—
133C2×C42C22⋊D712yes—
134C2×C42C22×D711yescharacteristic
135C4⋊C8—74no—
136C4⋊C8—144no—
137C4⋊C8—142no—
138C4⋊C8—142no—
139#101—282no—
140C22⋊C8—142no—
141C4×C8—144no—
142C4×C8—142no—
143#95—24no—
144#95—24no—
145#95C2×Dic712yes—
146C22×C7—21no—
147C22×C7—21no—
148C22×C7#38611yescharacteristic
149C22×C7C2×#10311yescharacteristic
150C22×C7#41611yescharacteristic
151C4×C7—21no—
152C4×C7—24no—
153C4×C7—22no—
154C4×C7—42no—
155C4×C7—82no—
156C4×C7—82no—
157C4×C7#45412yes—
158C4×C7#46512yes—
159C4×C7#46712yes—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—44no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8#114312yes—
166C2×Q8#114412yes—
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—22no—
180C4⋊C4—24no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C42—22no—
184C42—22no—
185C42—42no—
186C42D4×D712yes—
187C42#115912yes—
188C2×C7—21no—
189C2×C7—21no—
190C2×C7#188811yescharacteristic
191C2×C7#189211yescharacteristic
192C2×C7#191511yescharacteristic
193C23C2×C22⋊Dic711yescharacteristic
194Q8—24no—
195Q8—24no—
196Q8—82no—
197Q8—82no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4#640914yes—
210C2×C4D4×Dic712yes—
211C2×C4#649811yescharacteristic
212C2×C4#650011yescharacteristic
213C2×C4#650212yes—
214C2×C4C2×#114311yescharacteristic
215C2×C4C2×#114411yescharacteristic
216C2×C4#651312yes—
217C8—284no—
218C8—562no—
219C7#939611yescharacteristic, complemented, Sylow
220C22—21no—
221C22—21no—
222C22#8999211yescharacteristic
223C22C2×#640911yescharacteristic
224C22#9000511yescharacteristic
225C4—24no—
226C4—21no—
227C4—22no—
228C4—42no—
229C4—82no—
230C4—82no—
231C4#8982012yes—
232C4#8982312yes—
233C4#8983412yes—
234C2—21no—
235C2—21no—
236C2#1172945011yescharacteristic
237C2#1172947811yescharacteristic
238C2#1172962011yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2568C2
socleC22×C7#386
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2568D7
hypocenterC7#9396
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
219. C7#9396semidirect

Maximal subgroups

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

subgroupindex
C7×#25682
#117398802
#117295292
#117293912
#117254262
#93967