Finite Group Databasebeta

← C7×#9425 #13186641

Subgroups — C7×#9425 #13186641

Order 1792.

534 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 118 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×#3138C211yesmaximal, characteristic
2C7×#2380C211yesmaximal, characteristic
3C7×#1555C211yesmaximal, characteristic
4C7×#1515C212yesmaximal, complemented
5C7×#1288C212yesmaximal, complemented
6C7×#533C2211yescharacteristic
7C2×C7×#127C2211yescharacteristic
8C7×#490—22no—
9C7×#486C414yescomplemented
10C7×#432C2212yes—
11C7×#418—22no—
12C4×C7×M16C2211yescharacteristic
13C7×#386—22no—
14C7×#386—24no—
15C7×#383C2212yes—
16C7×#331—22no—
17C7×#328—24no—
18#9425C711yesmaximal, characteristic, direct product factor, Sylow
19C22×C7×D4—22no—
20C22×C7×D4D412yes—
21C23×C4×C7—22no—
22C2×C7×M16—21no—
23C7×C4⋊Q8C2×C412yescomplemented
24C7×#127—22no—
25C7×#127C2×C412yes—
26C7×#127D412yes—
27C7×#125—21no—
28C7×C22⋊Q8—22no—
29C4×C7×Q8—22no—
30C7×#117—21no—
31C2×C7×C22⋊C4—24no—
32C2×C7×C22⋊C4—22no—
33C2×C42×C7C2311yescharacteristic
34C7×C4⋊C8—22no—
35C7×#102—28no—
36C7×C22⋊C8—22no—
37C7×#97—21no—
38C7×#97—22no—
39C4×C7×C8C2×C412yescomplemented
40C7×#95—22no—
41C7×#95—22no—
42C7×#95D414yes—
43#3138C2×C711yescharacteristic
44#2380C2×C711yescharacteristic
45#1555C2×C711yescharacteristic
46#1515C2×C712yescomplemented
47#1288C2×C712yescomplemented
48C24×C7—22no—
49C2×C7×Q8—22no—
50C2×C7×D4—24no—
51C2×C7×D4—24no—
52C2×C7×D4—22no—
53C2×C7×D4—42no—
54C2×C7×D4—42no—
55C2×C7×D4—82no—
56C2×C7×D4—82no—
57C2×C7×D4C22⋊C414yes—
58C22×C4×C7—24no—
59C22×C4×C7—84no—
60C22×C4×C7—82no—
61C22×C4×C7C2×D411yescharacteristic
62C22×C4×C7C2×D412yes—
63C7×M16—42no—
64C2×C7×C8—22no—
65C2×C7×C8—42no—
66C7×C4⋊C4—22no—
67C7×C4⋊C4—22no—
68C7×C4⋊C4—42no—
69C7×C22⋊C4—41no—
70C7×C22⋊C4—84no—
71C7×C22⋊C4—82no—
72C42×C7—41no—
73C42×C7C22×C411yescharacteristic
74C42×C7C2×D411yescharacteristic
75C42×C7#4112yes—
76#533C22×C711yescharacteristic
77C2×#127C22×C711yescharacteristic
78#490—22no—
79#486C4×C714yescomplemented
80#432C22×C712yes—
81#418—22no—
82C4×M16C22×C711yescharacteristic
83#386—22no—
84#386—24no—
85#383C22×C712yes—
86#331—22no—
87#328—24no—
88C23×C7—24no—
89C23×C7—44no—
90C23×C7—44no—
91C23×C7—42no—
92C23×C7—42no—
93C23×C7#12011yescharacteristic
94C7×Q8—82no—
95C7×D4—44no—
96C7×D4—82no—
97C7×D4—84no—
98C7×D4—84no—
99C2×C4×C7—22no—
100C2×C4×C7—42no—
101C2×C4×C7—48no—
102C2×C4×C7—42no—
103C2×C4×C7—44no—
104C2×C4×C7—88no—
105C2×C4×C7C2×C22⋊C411yescharacteristic
106C2×C4×C7C4×D412yes—
107C2×C4×C7#12112yes—
108C2×C4×C7#12311yescharacteristic
109C7×C8—42no—
110C7×C8—82no—
111C22×D4—22no—
112C22×D4C7×D412yes—
113C23×C4—22no—
114C2×M16—21no—
115C4⋊Q8C2×C4×C712yescomplemented
116#127—22no—
117#127C2×C4×C712yes—
118#127C7×D412yes—
119#125—21no—
120C22⋊Q8—22no—
121C4×Q8—22no—
122#117—21no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—22no—
125C2×C42C23×C711yescharacteristic
126C4⋊C8—22no—
127#102—28no—
128C22⋊C8—22no—
129#97—21no—
130#97—22no—
131C4×C8C2×C4×C712yescomplemented
132#95—22no—
133#95—22no—
134#95C7×D414yes—
135C22×C7—22no—
136C22×C7—44no—
137C22×C7—48no—
138C22×C7—82no—
139C22×C7—82no—
140C22×C7—84no—
141C22×C7#38611yescharacteristic
142C22×C7#45512yes—
143C4×C7—21no—
144C4×C7—21no—
145C4×C7—22no—
146C4×C7—22no—
147C4×C7—84no—
148C4×C7—82no—
149C24—22no—
150C2×Q8—22no—
151C2×D4—24no—
152C2×D4—24no—
153C2×D4—22no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—82no—
157C2×D4—82no—
158C2×D4C7×C22⋊C414yes—
159C22×C4—24no—
160C22×C4—84no—
161C22×C4—82no—
162C22×C4C2×C7×D411yescharacteristic
163C22×C4C2×C7×D412yes—
164M16—42no—
165C2×C8—22no—
166C2×C8—42no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—42no—
170C22⋊C4—41no—
171C22⋊C4—84no—
172C22⋊C4—82no—
173C42—41no—
174C42C22×C4×C711yescharacteristic
175C42C2×C7×D411yescharacteristic
176C42C7×#4112yes—
177C2×C7—22no—
178C2×C7—84no—
179C2×C7#188511yescharacteristic
180C2×C7#190412yes—
181C23—24no—
182C23—44no—
183C23—44no—
184C23—42no—
185C23—42no—
186C23C7×#12011yescharacteristic
187Q8—82no—
188D4—44no—
189D4—82no—
190D4—84no—
191D4—84no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—48no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4—88no—
198C2×C4C2×C7×C22⋊C411yescharacteristic
199C2×C4C4×C7×D412yes—
200C2×C4C7×#12112yes—
201C2×C4C7×#12311yescharacteristic
202C8—42no—
203C8—82no—
204C7#942511yescharacteristic, direct product factor, Sylow
205C22—22no—
206C22—44no—
207C22—48no—
208C22—82no—
209C22—82no—
210C22—84no—
211C22C7×#38611yescharacteristic
212C22C7×#45512yes—
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—84no—
218C4—82no—
219C2—22no—
220C2—84no—
221C2C7×#188511yescharacteristic
222C2C7×#190412yes—
subgroupG/H
center, socleC22×C7#386
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#127C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1515C2semidirect
5. C7×#1288C2semidirect
9. C7×#486C4semidirect
18. #9425C7direct (N × H)
23. C7×C4⋊Q8C2×C4semidirect
39. C4×C7×C8C2×C4semidirect
46. #1515C2×C7semidirect
47. #1288C2×C7semidirect
79. #486C4×C7semidirect
115. C4⋊Q8C2×C4×C7semidirect
131. C4×C8C2×C4×C7semidirect
204. C7#9425direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31382
C7×#23802
C7×#15552
C7×#15152
C7×#12882
#94257