Finite Group Databasebeta

← C7×#10287 #13187503

Subgroups — C7×#10287 #13187503

Order 1792.

566 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 130 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×#3206C212yesmaximal
2C7×#2862C211yesmaximal, characteristic, complemented
3C7×#1962C211yesmaximal, characteristic
4C7×#1548C212yesmaximal, complemented
5C7×#1276C211yesmaximal, characteristic, complemented
6C7×#532C2211yescharacteristic
7C2×C7×C4⋊Q8C2212yes—
8C7×#492—22no—
9C7×#487—22no—
10C7×#461—24no—
11C7×#448—24no—
12C7×#431C2212yes—
13C7×#423C2211yescharacteristic
14C2×C7×#104—22no—
15C7×#391—24no—
16C7×#382—22no—
17C43×C7C2211yescharacteristic
18C7×#333—24no—
19#10287C711yesmaximal, characteristic, direct product factor, Sylow
20C2×C7×#41—21no—
21C22×C7×Q8D414yes—
22C2×C7×SD16—28no—
23C2×C7×M16—21no—
24C7×C4⋊Q8—22no—
25C7×C4⋊Q8—22no—
26C7×C4⋊Q8—22no—
27C7×C4⋊Q8—22no—
28C7×C4⋊Q8D411yescharacteristic
29C7×#127D411yescharacteristic
30C7×#124—21no—
31C7×#121—22no—
32C4×C7×D4—22no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—22no—
35C2×C42×C7—22no—
36C2×C42×C7D412yes—
37C2×C42×C7C2311yescharacteristic
38C7×C4⋊C8—22no—
39C7×C4⋊C8D414yes—
40C7×#104—44no—
41C7×#103—28no—
42C7×C22⋊C8—22no—
43C4×C7×C8—22no—
44C7×#95—28no—
45#3206C2×C712yes—
46#2862C2×C711yescharacteristic, complemented
47#1962C2×C711yescharacteristic
48#1548C2×C712yescomplemented
49#1276C2×C711yescharacteristic, complemented
50C7×#41—42no—
51C7×#41—42no—
52C2×C7×Q8—24no—
53C2×C7×Q8—21no—
54C2×C7×Q8—24no—
55C2×C7×Q8—48no—
56C2×C7×D4—22no—
57C2×C7×D4—21no—
58C2×C7×D4—41no—
59C22×C4×C7—22no—
60C22×C4×C7—24no—
61C22×C4×C7—41no—
62C22×C4×C7C2×D412yes—
63C22×C4×C7C2×D411yescharacteristic
64C7×SD16—88no—
65C7×M16—42no—
66C2×C7×C8—24no—
67C2×C7×C8—41no—
68C7×C4⋊C4—22no—
69C7×C4⋊C4—42no—
70C7×C4⋊C4—42no—
71C7×C4⋊C4—42no—
72C7×C4⋊C4—42no—
73C7×C22⋊C4—42no—
74C42×C7—22no—
75C42×C7—44no—
76C42×C7SD1614yescomplemented
77C42×C7C2×D411yescharacteristic
78C42×C7C2×D412yes—
79C42×C7#4111yescharacteristic
80#532C22×C711yescharacteristic
81C2×C4⋊Q8C22×C712yes—
82#492—22no—
83#487—22no—
84#461—24no—
85#448—24no—
86#431C22×C712yes—
87#423C22×C711yescharacteristic
88C2×#104—22no—
89#391—24no—
90#382—22no—
91C43C22×C711yescharacteristic
92#333—24no—
93C23×C7—41no—
94C23×C7#12011yescharacteristic
95C7×Q8—48no—
96C7×Q8—42no—
97C7×Q8—84no—
98C7×D4—44no—
99C7×D4—42no—
100C7×D4—82no—
101C2×C4×C7—24no—
102C2×C4×C7—24no—
103C2×C4×C7—44no—
104C2×C4×C7—48no—
105C2×C4×C7—41no—
106C2×C4×C7—44no—
107C2×C4×C7—81no—
108C2×C4×C7—81no—
109C2×C4×C7#12012yes—
110C2×C4×C7#12112yes—
111C2×C4×C7#12111yescharacteristic
112C2×C4×C7#12711yescharacteristic
113C2×C4×C7C2×SD1612yes—
114C2×C4×C7#13712yes—
115C7×C8—44no—
116C7×C8—81no—
117C2×#41—21no—
118C22×Q8C7×D414yes—
119C2×SD16—28no—
120C2×M16—21no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8C7×D411yescharacteristic
126#127C7×D411yescharacteristic
127#124—21no—
128#121—22no—
129C4×D4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C42—22no—
133C2×C42C7×D412yes—
134C2×C42C23×C711yescharacteristic
135C4⋊C8—22no—
136C4⋊C8C7×D414yes—
137#104—44no—
138#103—28no—
139C22⋊C8—22no—
140C4×C8—22no—
141#95—28no—
142C22×C7—22no—
143C22×C7—82no—
144C22×C7—81no—
145C22×C7#39211yescharacteristic
146C22×C7#44812yes—
147C4×C7—22no—
148C4×C7—44no—
149C4×C7—84no—
150C4×C7—81no—
151C4×C7#45514yes—
152C4×C7#46112yes—
153C4×C7#49211yescharacteristic
154C4×C7#49711yescharacteristic
155#41—42no—
156#41—42no—
157C2×Q8—24no—
158C2×Q8—21no—
159C2×Q8—24no—
160C2×Q8—48no—
161C2×D4—22no—
162C2×D4—21no—
163C2×D4—41no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—41no—
167C22×C4C2×C7×D412yes—
168C22×C4C2×C7×D411yescharacteristic
169SD16—88no—
170M16—42no—
171C2×C8—24no—
172C2×C8—41no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C22⋊C4—42no—
179C42—22no—
180C42—44no—
181C42C7×SD1614yescomplemented
182C42C2×C7×D411yescharacteristic
183C42C2×C7×D412yes—
184C42C7×#4111yescharacteristic
185C2×C7—22no—
186C2×C7—161no—
187C2×C7#200412yes—
188C2×C7#201311yescharacteristic
189C23—41no—
190C23C7×#12011yescharacteristic
191Q8—48no—
192Q8—42no—
193Q8—84no—
194D4—44no—
195D4—42no—
196D4—82no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—44no—
200C2×C4—48no—
201C2×C4—41no—
202C2×C4—44no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4C7×#12012yes—
206C2×C4C7×#12112yes—
207C2×C4C7×#12111yescharacteristic
208C2×C4C7×#12711yescharacteristic
209C2×C4C2×C7×SD1612yes—
210C2×C4C7×#13712yes—
211C8—44no—
212C8—81no—
213C7#1028711yescharacteristic, direct product factor, Sylow
214C22—22no—
215C22—82no—
216C22—81no—
217C22C7×#39211yescharacteristic
218C22C7×#44812yes—
219C4—22no—
220C4—44no—
221C4—84no—
222C4—81no—
223C4C7×#45514yes—
224C4C7×#46112yes—
225C4C7×#49211yescharacteristic
226C4C7×#49711yescharacteristic
227C2—22no—
228C2—161no—
229C2C7×#200412yes—
230C2C7×#201311yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2862C2semidirect
4. C7×#1548C2semidirect
5. C7×#1276C2semidirect
19. #10287C7direct (N × H)
46. #2862C2×C7semidirect
48. #1548C2×C7semidirect
49. #1276C2×C7semidirect
76. C42×C7SD16semidirect
181. C42C7×SD16semidirect
213. C7#10287direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32062
C7×#28622
C7×#19622
C7×#15482
C7×#12762
#102877