Finite Group Databasebeta

← C7×#36640 #13213856

Subgroups — C7×#36640 #13213856

Order 1792.

1990 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2864C214yesmaximal, complemented
2C7×#2859C214yesmaximal, complemented
3C7×#2851C218yesmaximal
4C4×C7×#125C218yesmaximal, complemented
5C7×#2296C214yesmaximal
6C7×#2285C212yesmaximal
7C2×C43×C7C211yesmaximal, characteristic
8C2×C7×#125C2218yes—
9C2×C7×#117C2214yes—
10C22×C42×C7C2212yes—
11C22×C42×C7C2214yes—
12C22×C42×C7C2211yescharacteristic
13C7×#389C22116yes—
14C7×#389C22116yes—
15C7×#387C22132yes—
16C7×#385C22116yes—
17C7×#384C2214yes—
18C7×#382C2214yes—
19C7×#382C2218yes—
20C7×#380C2218yes—
21C4×C7×C4⋊C4C2218yes—
22C4×C7×C4⋊C4C22116yes—
23C43×C7C2214yes—
24C43×C7C2212yes—
25C43×C7C2212yes—
26#36640C711yesmaximal, characteristic, direct product factor, Sylow
27C23×C4×C7C2314yes—
28C23×C4×C7C2312yes—
29C23×C4×C7C2311yescharacteristic
30C7×#125—232no—
31C7×#117—216no—
32C2×C7×C4⋊C4C2318yes—
33C2×C7×C4⋊C4C23116yes—
34C2×C7×C4⋊C4C23132yes—
35C2×C7×C22⋊C4C2318yes—
36C2×C42×C7—28no—
37C2×C42×C7—216no—
38C2×C42×C7Q818yes—
39C2×C42×C7C2318yes—
40C2×C42×C7C2314yes—
41C2×C42×C7C2314yes—
42C2×C42×C7C2314yes—
43C2×C42×C7C2318yes—
44C2×C42×C7C2311yescharacteristic
45C2×C42×C7C2311yescharacteristic
46C2×C42×C7C2312yes—
47C2×C42×C7C2314yes—
48C7×#95C23116yes—
49C7×#95C23132yes—
50#2864C2×C714yescomplemented
51#2859C2×C714yescomplemented
52#2851C2×C718yes—
53C4×#125C2×C718yescomplemented
54#2296C2×C714yes—
55#2285C2×C712yes—
56C2×C43C2×C711yescharacteristic
57C24×C7C2411yescharacteristic
58C22×C4×C7—28no—
59C22×C4×C7—216no—
60C22×C4×C7—28no—
61C22×C4×C7C2×Q818yes—
62C22×C4×C7C2×Q814yes—
63C22×C4×C7#4118yes—
64C22×C4×C7C24116yes—
65C22×C4×C7C2412yes—
66C22×C4×C7C2418yes—
67C22×C4×C7C2411yescharacteristic
68C22×C4×C7C2412yes—
69C22×C4×C7C2411yescharacteristic
70C7×C4⋊C4—216no—
71C7×C4⋊C4—264no—
72C7×C4⋊C4—232no—
73C7×C22⋊C4—216no—
74C42×C7—216no—
75C42×C7—28no—
76C42×C7—28no—
77C42×C7—28no—
78C42×C7—216no—
79C42×C7C2×Q814yes—
80C42×C7C2×Q818yes—
81C42×C7C2×Q814yes—
82C42×C7#41116yes—
83C2×#125C22×C718yes—
84C2×#117C22×C714yes—
85C22×C42C22×C714yes—
86C22×C42C22×C712yes—
87C22×C42C22×C711yescharacteristic
88#389C22×C7116yes—
89#389C22×C7116yes—
90#387C22×C7132yes—
91#385C22×C7116yes—
92#384C22×C714yes—
93#382C22×C718yes—
94#382C22×C714yes—
95#380C22×C718yes—
96C4×C4⋊C4C22×C718yes—
97C4×C4⋊C4C22×C7116yes—
98C43C22×C714yes—
99C43C22×C712yes—
100C43C22×C712yes—
101C23×C7—24no—
102C23×C7C22×Q812yes—
103C23×C7C2×#4114yes—
104C23×C7C2511yescharacteristic
105C2×C4×C7—232no—
106C2×C4×C7—24no—
107C2×C4×C7—28no—
108C2×C4×C7—28no—
109C2×C4×C7—216no—
110C2×C4×C7—216no—
111C2×C4×C7—232no—
112C2×C4×C7#125116yescomplemented
113C2×C4×C7C22×Q814yes—
114C2×C4×C7C22×Q812yes—
115C2×C4×C7C22×Q812yes—
116C2×C4×C7C22×Q814yes—
117C2×C4×C7C2×#41116yes—
118C2×C4×C7C2×#4114yes—
119C2×C4×C7C2×#4114yes—
120C2×C4×C7C2×#4118yes—
121C23×C4C23×C714yes—
122C23×C4C23×C712yes—
123C23×C4C23×C711yescharacteristic
124#125—232no—
125#117—216no—
126C2×C4⋊C4C23×C718yes—
127C2×C4⋊C4C23×C7116yes—
128C2×C4⋊C4C23×C7132yes—
129C2×C22⋊C4C23×C718yes—
130C2×C42—28no—
131C2×C42—216no—
132C2×C42C7×Q818yes—
133C2×C42C23×C718yes—
134C2×C42C23×C714yes—
135C2×C42C23×C718yes—
136C2×C42C23×C714yes—
137C2×C42C23×C714yes—
138C2×C42C23×C712yes—
139C2×C42C23×C711yescharacteristic
140C2×C42C23×C711yescharacteristic
141C2×C42C23×C714yes—
142#95C23×C7116yes—
143#95C23×C7132yes—
144C22×C7—24no—
145C22×C7—28no—
146C22×C7C2×#12514yes—
147C22×C7C23×Q811yescharacteristic
148C22×C7C22×#4112yes—
149C22×C7C22×#4114yes—
150C4×C7—24no—
151C4×C7—216no—
152C4×C7—416no—
153C4×C7C2×#12514yes—
154C4×C7C2×#12514yes—
155C4×C7#53318yes—
156C24C24×C711yescharacteristic
157C22×C4—28no—
158C22×C4—216no—
159C22×C4—28no—
160C22×C4C2×C7×Q818yes—
161C22×C4C2×C7×Q814yes—
162C22×C4C7×#4118yes—
163C22×C4C24×C7116yes—
164C22×C4C24×C712yes—
165C22×C4C24×C718yes—
166C22×C4C24×C711yescharacteristic
167C22×C4C24×C711yescharacteristic
168C22×C4C24×C712yes—
169C4⋊C4—216no—
170C4⋊C4—264no—
171C4⋊C4—232no—
172C22⋊C4—216no—
173C42—216no—
174C42—28no—
175C42—28no—
176C42—28no—
177C42—216no—
178C42C2×C7×Q814yes—
179C42C2×C7×Q818yes—
180C42C2×C7×Q814yes—
181C42C7×#41116yes—
182C2×C7—24no—
183C2×C7C22×#12511yescharacteristic
184C2×C7C2×#53312yes—
185C2×C7#344414yes—
186C23—24no—
187C23C22×C7×Q812yes—
188C23C2×C7×#4114yes—
189C23C25×C711yescharacteristic
190C2×C4—232no—
191C2×C4—24no—
192C2×C4—28no—
193C2×C4—28no—
194C2×C4—216no—
195C2×C4—216no—
196C2×C4—232no—
197C2×C4C7×#125116yescomplemented
198C2×C4C22×C7×Q814yes—
199C2×C4C22×C7×Q812yes—
200C2×C4C22×C7×Q812yes—
201C2×C4C22×C7×Q814yes—
202C2×C4C2×C7×#41116yes—
203C2×C4C2×C7×#4114yes—
204C2×C4C2×C7×#4114yes—
205C2×C4C2×C7×#4118yes—
206C7#3664011yescharacteristic, direct product factor, Sylow
207C22—24no—
208C22—28no—
209C22C2×C7×#12514yes—
210C22C23×C7×Q811yescharacteristic
211C22C22×C7×#4112yes—
212C22C22×C7×#4114yes—
213C4—24no—
214C4—216no—
215C4—416no—
216C4C2×C7×#12514yes—
217C4C2×C7×#12514yes—
218C4C7×#53318yes—
219C2—24no—
220C2C22×C7×#12511yescharacteristic
221C2C2×C7×#53312yes—
222C2C7×#344414yes—
subgroupG/H
centerC22×C4×C7C24
commutator, Frattini, cosocleC23C25×C7
socleC23×C7C25
prime radicalC24×C7C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2864C2semidirect
2. C7×#2859C2semidirect
4. C4×C7×#125C2semidirect
26. #36640C7direct (N × H)
50. #2864C2×C7semidirect
51. #2859C2×C7semidirect
53. C4×#125C2×C7semidirect
112. C2×C4×C7#125semidirect
197. C2×C4C7×#125semidirect
206. C7#36640direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28642
C7×#28592
C7×#28512
C4×C7×#1252
C7×#22962
C7×#22852
C2×C43×C72
#366407