Finite Group Databasebeta

← C7×#34173 #13211389

Subgroups — C7×#34173 #13211389

Order 1792.

5126 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C7×Q8C211yesmaximal, characteristic
2C2×C7×#519C212yesmaximal, complemented
3C7×Q8×C22⋊C4C2116yesmaximal, complemented
4C7×#2310C216yesmaximal, complemented
5C7×#2280C216yesmaximal
6C23×C7×Q8—216no—
7C23×C7×Q8C4116yescomplemented
8C23×C7×Q8C2213yes—
9C23×C7×Q8C2218yes—
10C23×C7×Q8C2211yescharacteristic
11C24×C4×C7C2213yes—
12C7×#519—216no—
13C2×C4×C7×Q8C22116yescomplemented
14C2×C7×#117C22112yes—
15C7×#391C22148yescomplemented
16C7×#381C22112yes—
17C7×#380C22124yes—
18C7×#379C2214yes—
19C4×C7×C22⋊C4C22124yes—
20#34173C711yesmaximal, characteristic, direct product factor, Sylow
21C25×C7C2311yescharacteristic
22C22×C7×Q8—2128no—
23C22×C7×Q8—264no—
24C22×C7×Q8—224no—
25C22×C7×Q8C2×C4164yes—
26C22×C7×Q8C2×C4124yes—
27C22×C7×Q8C2×C418yes—
28C22×C7×Q8D4116yes—
29C22×C7×Q8C23112yes—
30C22×C7×Q8C2314yes—
31C23×C4×C7—224no—
32C23×C4×C7C2×C4124yes—
33C23×C4×C7C23112yes—
34C23×C4×C7C2313yes—
35C23×C4×C7C2313yes—
36C4×C7×Q8—264no—
37C7×#117—248no—
38C2×C7×C4⋊C4C23124yes—
39C2×C7×C22⋊C4C23116yes—
40C2×C7×C22⋊C4C2318yes—
41C2×C42×C7C23124yes—
42C7×#95C23148yes—
43C24×Q8C2×C711yescharacteristic
44C2×#519C2×C712yescomplemented
45Q8×C22⋊C4C2×C7116yescomplemented
46#2310C2×C716yescomplemented
47#2280C2×C716yes—
48C24×C7—24no—
49C24×C7—28no—
50C24×C7C22×C414yes—
51C24×C7C2411yescharacteristic
52C24×C7C2412yes—
53C2×C7×Q8—2256no—
54C2×C7×Q8—296no—
55C2×C7×Q8—248no—
56C2×C7×Q8C22⋊C4164yescomplemented
57C2×C7×Q8C22×C4148yes—
58C2×C7×Q8C22×C4116yes—
59C2×C7×Q8C2×D4132yes—
60C22×C4×C7—212no—
61C22×C4×C7—296no—
62C22×C4×C7—248no—
63C22×C4×C7C22×C4112yes—
64C22×C4×C7C22×C4148yes—
65C22×C4×C7C22×C4112yes—
66C22×C4×C7C2×D4124yes—
67C22×C4×C7C24116yes—
68C22×C4×C7C2416yes—
69C22×C4×C7C2416yes—
70C7×C4⋊C4—248no—
71C7×C22⋊C4—216no—
72C7×C22⋊C4—232no—
73C42×C7—248no—
74C23×Q8—216no—
75C23×Q8C4×C7116yescomplemented
76C23×Q8C22×C713yes—
77C23×Q8C22×C718yes—
78C23×Q8C22×C711yescharacteristic
79C24×C4C22×C713yes—
80#519—216no—
81C2×C4×Q8C22×C7116yescomplemented
82C2×#117C22×C7112yes—
83#391C22×C7148yescomplemented
84#381C22×C7112yes—
85#380C22×C7124yes—
86#379C22×C714yes—
87C4×C22⋊C4C22×C7124yes—
88C23×C7—232no—
89C23×C7—28no—
90C23×C7—216no—
91C23×C7—24no—
92C23×C7—24no—
93C23×C7—24no—
94C23×C7C23×C412yes—
95C23×C7C23×C414yes—
96C23×C7C22×D414yes—
97C23×C72−1+418yes—
98C23×C7C2511yescharacteristic
99C7×Q8—296no—
100C7×Q8C2×C22⋊C4164yes—
101C2×C4×C7—224no—
102C2×C4×C7—212no—
103C2×C4×C7—296no—
104C2×C4×C7—216no—
105C2×C4×C7—232no—
106C2×C4×C7C2×C22⋊C4148yes—
107C2×C4×C7C23×C4112yes—
108C2×C4×C7C23×C4112yes—
109C2×C4×C7C22×D4124yes—
110C25C23×C711yescharacteristic
111C22×Q8—2128no—
112C22×Q8—264no—
113C22×Q8—224no—
114C22×Q8C2×C4×C7164yes—
115C22×Q8C2×C4×C7124yes—
116C22×Q8C2×C4×C718yes—
117C22×Q8C7×D4116yes—
118C22×Q8C23×C7112yes—
119C22×Q8C23×C714yes—
120C23×C4—224no—
121C23×C4C2×C4×C7124yes—
122C23×C4C23×C7112yes—
123C23×C4C23×C713yes—
124C23×C4C23×C713yes—
125C4×Q8—264no—
126#117—248no—
127C2×C4⋊C4C23×C7124yes—
128C2×C22⋊C4C23×C718yes—
129C2×C22⋊C4C23×C7116yes—
130C2×C42C23×C7124yes—
131#95C23×C7148yes—
132C22×C7—232no—
133C22×C7—24no—
134C22×C7—24no—
135C22×C7—24no—
136C22×C7—28no—
137C22×C7—216no—
138C22×C7C22×C22⋊C414yes—
139C22×C7#51918yescomplemented
140C22×C7C24×C411yescharacteristic
141C22×C7C23×D412yes—
142C22×C7C2×2−1+414yes—
143C4×C7—212no—
144C4×C7—416no—
145C4×C7C22×C22⋊C4124yes—
146C24—24no—
147C24—28no—
148C24C22×C4×C714yes—
149C24C24×C711yescharacteristic
150C24C24×C712yes—
151C2×Q8—2256no—
152C2×Q8—296no—
153C2×Q8—248no—
154C2×Q8C7×C22⋊C4164yescomplemented
155C2×Q8C22×C4×C7148yes—
156C2×Q8C22×C4×C7116yes—
157C2×Q8C2×C7×D4132yes—
158C22×C4—212no—
159C22×C4—296no—
160C22×C4—248no—
161C22×C4C22×C4×C7112yes—
162C22×C4C22×C4×C7148yes—
163C22×C4C22×C4×C7112yes—
164C22×C4C2×C7×D4124yes—
165C22×C4C24×C7116yes—
166C22×C4C24×C716yes—
167C22×C4C24×C716yes—
168C4⋊C4—248no—
169C22⋊C4—216no—
170C22⋊C4—232no—
171C42—248no—
172C2×C7—24no—
173C2×C7—28no—
174C2×C7C23×C22⋊C411yescharacteristic
175C2×C7C2×#51912yes—
176C2×C7#348614yes—
177C23—232no—
178C23—24no—
179C23—24no—
180C23—24no—
181C23—28no—
182C23—216no—
183C23C23×C4×C712yes—
184C23C23×C4×C714yes—
185C23C22×C7×D414yes—
186C23C7×2−1+418yes—
187C23C25×C711yescharacteristic
188Q8—296no—
189Q8C2×C7×C22⋊C4164yes—
190C2×C4—212no—
191C2×C4—224no—
192C2×C4—296no—
193C2×C4—216no—
194C2×C4—232no—
195C2×C4C2×C7×C22⋊C4148yes—
196C2×C4C23×C4×C7112yes—
197C2×C4C23×C4×C7112yes—
198C2×C4C22×C7×D4124yes—
199C7#3417311yescharacteristic, direct product factor, Sylow
200C22—232no—
201C22—24no—
202C22—24no—
203C22—24no—
204C22—28no—
205C22—216no—
206C22C22×C7×C22⋊C414yes—
207C22C7×#51918yescomplemented
208C22C24×C4×C711yescharacteristic
209C22C23×C7×D412yes—
210C22C2×C7×2−1+414yes—
211C4—212no—
212C4—416no—
213C4C22×C7×C22⋊C4124yes—
214C2—24no—
215C2—28no—
216C2C23×C7×C22⋊C411yescharacteristic
217C2C2×C7×#51912yes—
218C2C7×#348614yes—
subgroupG/H
center, socleC23×C7C25
commutatorC22C24×C4×C7
Frattini, cosocleC23C25×C7
prime radicalC25×C7C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#519C2semidirect
3. C7×Q8×C22⋊C4C2semidirect
4. C7×#2310C2semidirect
7. C23×C7×Q8C4semidirect
13. C2×C4×C7×Q8C22semidirect
15. C7×#391C22semidirect
20. #34173C7direct (N × H)
44. C2×#519C2×C7semidirect
45. Q8×C22⋊C4C2×C7semidirect
46. #2310C2×C7semidirect
56. C2×C7×Q8C22⋊C4semidirect
75. C23×Q8C4×C7semidirect
81. C2×C4×Q8C22×C7semidirect
83. #391C22×C7semidirect
139. C22×C7#519semidirect
154. C2×Q8C7×C22⋊C4semidirect
199. C7#34173direct (N × H)
207. C22C7×#519semidirect

Maximal subgroups

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

subgroupindex
C24×C7×Q82
C2×C7×#5192
C7×Q8×C22⋊C42
C7×#23102
C7×#22802
#341737