Finite Group Databasebeta

← C7×#10289 #13187505

Subgroups — C7×#10289 #13187505

Order 1792.

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

normal subgroups only

This group has 614 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×#3226C212yesmaximal
2C7×#2868C211yesmaximal, characteristic, complemented
3C7×#1962C211yesmaximal, characteristic
4C7×#1546C212yesmaximal, complemented
5C7×#1276C211yesmaximal, characteristic, complemented
6C7×#532C2211yescharacteristic
7C2×C7×C4⋊Q8—22no—
8C2×C7×C4⋊Q8C2212yes—
9C7×#500—22no—
10C7×#486—22no—
11C7×#480—24no—
12C7×#474—24no—
13C7×#431C2212yes—
14C7×#423C2211yescharacteristic
15C2×C7×#104—22no—
16C7×#384—24no—
17C43×C7C2211yescharacteristic
18C7×#332—24no—
19#10289C711yesmaximal, characteristic, direct product factor, Sylow
20C2×C7×#41—21no—
21C22×C7×Q8—22no—
22C2×C7×M16—21no—
23C7×C4⋊Q8—22no—
24C7×C4⋊Q8—22no—
25C7×C4⋊Q8—22no—
26C7×C4⋊Q8—22no—
27C7×C4⋊Q8—44no—
28C7×C4⋊Q8—44no—
29C7×C4⋊Q8D411yescharacteristic
30C7×#127D411yescharacteristic
31C7×#124—21no—
32C7×#121—22no—
33C4×C7×D4—22no—
34C2×C7×C4⋊C4—28no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4D414yes—
37C2×C42×C7—22no—
38C2×C42×C7D412yes—
39C2×C42×C7C2311yescharacteristic
40C7×#107—28no—
41C7×C4⋊C8—22no—
42C7×C4⋊C8D414yes—
43C7×#104—44no—
44C7×#102—28no—
45C7×C22⋊C8—22no—
46C4×C7×C8—22no—
47#3226C2×C712yes—
48#2868C2×C711yescharacteristic, complemented
49#1962C2×C711yescharacteristic
50#1546C2×C712yescomplemented
51#1276C2×C711yescharacteristic, complemented
52C7×#41—42no—
53C7×#41—42no—
54C2×C7×Q8—21no—
55C2×C7×Q8—42no—
56C2×C7×Q8—42no—
57C2×C7×Q8—44no—
58C2×C7×Q8—44no—
59C2×C7×D4—22no—
60C2×C7×D4—21no—
61C2×C7×D4—41no—
62C22×C4×C7—22no—
63C22×C4×C7—24no—
64C22×C4×C7—41no—
65C22×C4×C7C2×D412yes—
66C22×C4×C7C2×D411yescharacteristic
67C7×M16—42no—
68C2×C7×C8—24no—
69C2×C7×C8—41no—
70C7×C4⋊C4—24no—
71C7×C4⋊C4—22no—
72C7×C4⋊C4—24no—
73C7×C4⋊C4—416no—
74C7×C4⋊C4—42no—
75C7×C4⋊C4—42no—
76C7×C22⋊C4—42no—
77C42×C7—28no—
78C42×C7—22no—
79C42×C7D814yes—
80C42×C7C2×D411yescharacteristic
81C42×C7C2×D412yes—
82C42×C7#4111yescharacteristic
83#532C22×C711yescharacteristic
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C22×C712yes—
86#500—22no—
87#486—22no—
88#480—24no—
89#474—24no—
90#431C22×C712yes—
91#423C22×C711yescharacteristic
92C2×#104—22no—
93#384—24no—
94C43C22×C711yescharacteristic
95#332—24no—
96C23×C7—41no—
97C23×C7#12011yescharacteristic
98C7×Q8—42no—
99C7×Q8—84no—
100C7×Q8—82no—
101C7×Q8—82no—
102C7×D4—44no—
103C7×D4—42no—
104C7×D4—82no—
105C2×C4×C7—28no—
106C2×C4×C7—24no—
107C2×C4×C7—24no—
108C2×C4×C7—41no—
109C2×C4×C7—44no—
110C2×C4×C7—48no—
111C2×C4×C7—81no—
112C2×C4×C7—81no—
113C2×C4×C7#12012yes—
114C2×C4×C7#12112yes—
115C2×C4×C7#12111yescharacteristic
116C2×C4×C7#12711yescharacteristic
117C2×C4×C7C2×D812yes—
118C2×C4×C7AGL(1,ℤ/8)12yes—
119C7×C8—44no—
120C7×C8—81no—
121C2×#41—21no—
122C22×Q8—22no—
123C2×M16—21no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130C4⋊Q8C7×D411yescharacteristic
131#127C7×D411yescharacteristic
132#124—21no—
133#121—22no—
134C4×D4—22no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4C7×D414yes—
138C2×C42—22no—
139C2×C42C7×D412yes—
140C2×C42C23×C711yescharacteristic
141#107—28no—
142C4⋊C8—22no—
143C4⋊C8C7×D414yes—
144#104—44no—
145#102—28no—
146C22⋊C8—22no—
147C4×C8—22no—
148C22×C7—22no—
149C22×C7—82no—
150C22×C7—81no—
151C22×C7#39211yescharacteristic
152C22×C7#44712yes—
153C4×C7—28no—
154C4×C7—22no—
155C4×C7—84no—
156C4×C7—81no—
157C4×C7#45614yes—
158C4×C7#45912yes—
159C4×C7#49311yescharacteristic
160C4×C7#49611yescharacteristic
161#41—42no—
162#41—42no—
163C2×Q8—21no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×D4—22no—
169C2×D4—21no—
170C2×D4—41no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—41no—
174C22×C4C2×C7×D412yes—
175C22×C4C2×C7×D411yescharacteristic
176M16—42no—
177C2×C8—24no—
178C2×C8—41no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4—24no—
182C4⋊C4—416no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C22⋊C4—42no—
186C42—28no—
187C42—22no—
188C42C7×D814yes—
189C42C2×C7×D411yescharacteristic
190C42C2×C7×D412yes—
191C42C7×#4111yescharacteristic
192C2×C7—22no—
193C2×C7—161no—
194C2×C7#200512yes—
195C2×C7#201111yescharacteristic
196C23—41no—
197C23C7×#12011yescharacteristic
198Q8—42no—
199Q8—82no—
200Q8—82no—
201Q8—84no—
202D4—44no—
203D4—42no—
204D4—82no—
205C2×C4—28no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—41no—
209C2×C4—44no—
210C2×C4—48no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4C7×#12012yes—
214C2×C4C7×#12112yes—
215C2×C4C7×#12111yescharacteristic
216C2×C4C7×#12711yescharacteristic
217C2×C4C2×C7×D812yes—
218C2×C4C7×AGL(1,ℤ/8)12yes—
219C8—44no—
220C8—81no—
221C7#1028911yescharacteristic, direct product factor, Sylow
222C22—22no—
223C22—82no—
224C22—81no—
225C22C7×#39211yescharacteristic
226C22C7×#44712yes—
227C4—28no—
228C4—22no—
229C4—84no—
230C4—81no—
231C4C7×#45614yes—
232C4C7×#45912yes—
233C4C7×#49311yescharacteristic
234C4C7×#49611yescharacteristic
235C2—22no—
236C2—161no—
237C2C7×#200512yes—
238C2C7×#201111yescharacteristic
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×#2868C2semidirect
4. C7×#1546C2semidirect
5. C7×#1276C2semidirect
19. #10289C7direct (N × H)
48. #2868C2×C7semidirect
50. #1546C2×C7semidirect
51. #1276C2×C7semidirect
221. C7#10289direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32262
C7×#28682
C7×#19622
C7×#15462
C7×#12762
#102897