Finite Group Databasebeta

← #13247275

Subgroups — #13247275

Order 1792.

418 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2298C211yesmaximal, characteristic
2#11729348C211yesmaximal, characteristic
3#11729343C211yesmaximal, characteristic
4#11725422C214yesmaximal
5C2×C7×#117C2211yescharacteristic
6C7×#388—22no—
7C7×#387—22no—
8C4×C7×C4⋊C4C412yescomplemented
9C4×C7×C22⋊C4C2212yes—
10C7×#376C411yescharacteristic, complemented
11C43×C7C411yescharacteristic, complemented
12#89983C2212yes—
13#89979C2212yes—
14#89325—22no—
15#89321—22no—
16#8740—71nomaximal, Sylow
17C23×C4×C7C2311yescharacteristic
18C7×#117—24no—
19C2×C7×C4⋊C4—22no—
20C2×C7×C4⋊C4D412yes—
21C2×C7×C22⋊C4—22no—
22C2×C7×C22⋊C4D412yes—
23C2×C42×C7—22no—
24C2×C42×C7C2×C411yescharacteristic
25C2×C42×C7C2×C411yescharacteristic
26C2×C42×C7C2×C412yes—
27C2×C42×C7C2×C412yes—
28C2×C22⋊Dic7—22no—
29C2×#1136—22no—
30C7×#95—22no—
31C7×#95—22no—
32C7×#95—22no—
33C7⋊#95—24no—
34C22⋊(C7⋊C8)—24no—
35#2298D711yescharacteristic
36#1822—71no—
37#1820—71no—
38#1284—74no—
39C24×C7C2×D411yescharacteristic
40C22×C4×C7—22no—
41C22×C4×C7—22no—
42C22×C4×C7—42no—
43C22×C4×C7C22⋊C412yes—
44C22×C4×C7C22⋊C412yes—
45C22×C4×C7C22×C411yescharacteristic
46C22×C4×C7C2×D411yescharacteristic
47C22×C4×C7#4114yes—
48C22×Dic7—42no—
49C7×C4⋊C4—24no—
50C7×C4⋊C4—44no—
51C7×C22⋊C4—24no—
52C7×C22⋊C4—44no—
53C42×C7—22no—
54C42×C7—22no—
55C42×C7—42no—
56C42×C7—42no—
57C42×C7—44no—
58C22⋊Dic7—44no—
59#1136—44no—
60C2×C7⋊C8—42no—
61C2×#117C2×D711yescharacteristic
62#407—72no—
63#388—22no—
64#387—22no—
65#381—72no—
66C4×C4⋊C4Dic712yescomplemented
67C4×C22⋊C4C2×D712yes—
68#376Dic711yescharacteristic, complemented
69C43Dic711yescharacteristic, complemented
70#343—142no—
71#342—142no—
72C23×C7—22no—
73C23×C7—42no—
74C23×C7C2×C22⋊C411yescharacteristic
75C23×C7#12312yes—
76C2×C4×C7—22no—
77C2×C4×C7—22no—
78C2×C4×C7—24no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—44no—
82C2×C4×C7—42no—
83C2×C4×C7—42no—
84C2×C4×C7—42no—
85C2×C4×C7—42no—
86C2×C4×C7—42no—
87C2×C4×C7—42no—
88C2×C4×C7—42no—
89C2×C4×C7#11712yes—
90C2×C4×C7#12312yes—
91C2×Dic7—82no—
92C2×Dic7—84no—
93C7⋊C8—82no—
94C23×C4C22×D711yescharacteristic
95C2×M16—142no—
96#117—24no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4C22⋊D712yes—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—142no—
101C2×C22⋊C4C22⋊D712yes—
102C2×C42—22no—
103C2×C42C2×Dic711yescharacteristic
104C2×C42C2×Dic711yescharacteristic
105C2×C42C2×Dic712yes—
106C2×C42C2×Dic712yes—
107C22⋊C8—144no—
108#95—22no—
109#95—22no—
110#95—22no—
111#95—144no—
112C22×C7—22no—
113C22×C7—22no—
114C22×C7—24no—
115C22×C7—82no—
116C22×C7#38111yescharacteristic
117C22×C7#41012yes—
118C4×C7—22no—
119C4×C7—22no—
120C4×C7—44no—
121C4×C7—42no—
122C4×C7—42no—
123C4×C7—82no—
124Dic7—162no—
125C24C2×C22⋊D711yescharacteristic
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—42no—
129C22×C4—282no—
130C22×C4C22⋊Dic712yes—
131C22×C4C22⋊Dic712yes—
132C22×C4#115714yes—
133C22×C4C22×Dic711yescharacteristic
134C22×C4C2×C22⋊D711yescharacteristic
135M16—284no—
136C2×C8—282no—
137C4⋊C4—24no—
138C4⋊C4—44no—
139C22⋊C4—24no—
140C22⋊C4—44no—
141C22⋊C4—284no—
142C42—22no—
143C42—22no—
144C42—42no—
145C42—42no—
146C42—44no—
147C2×C7—22no—
148C2×C7—42no—
149C2×C7#182011yescharacteristic
150C2×C7#182712yes—
151C23—22no—
152C23—42no—
153C23#649212yes—
154C23C2×C22⋊Dic711yescharacteristic
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—44no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—562no—
169C2×C4—564no—
170C2×C4#648912yes—
171C2×C4#649212yes—
172C8—562no—
173C7#874011yescharacteristic, complemented, Sylow
174C22—22no—
175C22—22no—
176C22—24no—
177C22—82no—
178C22#8998311yescharacteristic
179C22#9001212yes—
180C4—22no—
181C4—22no—
182C4—44no—
183C4—42no—
184C4—42no—
185C4—82no—
186C4—1122no—
187C2—22no—
188C2—42no—
189C2#1172934811yescharacteristic
190C2#1172939812yes—
subgroupG/H
centerC22#89983
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2298C2
socleC22×C7#381
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#2298D7
hypocenterC7#8740
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2298

Complements

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

NH ≅ G/Ntype
8. C4×C7×C4⋊C4C4semidirect
10. C7×#376C4semidirect
11. C43×C7C4semidirect
66. C4×C4⋊C4Dic7semidirect
68. #376Dic7semidirect
69. C43Dic7semidirect
173. C7#8740semidirect

Maximal subgroups

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

subgroupindex
C7×#22982
#117293482
#117293432
#117254222
#87407