Finite Group Databasebeta

← C7×#10014 #13187230

Subgroups — C7×#10014 #13187230

Order 1792.

566 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3103C212yesmaximal
2C7×#2399C211yesmaximal, characteristic, complemented
3C7×#1963C211yesmaximal, characteristic
4C7×#1487C212yesmaximal, complemented
5C7×#1274C211yesmaximal, characteristic, complemented
6C7×#532C2211yescharacteristic
7C2×C4×C7×Q8C2212yes—
8C7×#481—22no—
9C7×#475—22no—
10C7×#460—22no—
11C7×#448—22no—
12C4×C7×SD16—24no—
13C7×#431C2212yes—
14C7×#423C2211yescharacteristic
15C2×C7×#104—22no—
16C7×#398—22no—
17C7×#395—22no—
18C4×C7×C4⋊C4—22no—
19C7×#376C2211yescharacteristic
20C7×Q8⋊C8—24no—
21#10014C711yesmaximal, characteristic, direct product factor, Sylow
22C2×C7×#41—21no—
23C22×C7×Q8D412yes—
24C2×C7×SD16—24no—
25C2×C7×M16—21no—
26C7×C4⋊Q8D411yescharacteristic
27C7×#127D411yescharacteristic
28C7×#124—21no—
29C7×#121—22no—
30C4×C7×Q8—22no—
31C4×C7×Q8—22no—
32C4×C7×Q8—22no—
33C4×C7×Q8—22no—
34C4×C7×D4—22no—
35C2×C7×C4⋊C4—24no—
36C2×C7×C4⋊C4—22no—
37C2×C7×C4⋊C4D412yes—
38C2×C42×C7—22no—
39C2×C42×C7D412yes—
40C2×C42×C7C2311yescharacteristic
41C7×#106—24no—
42C7×C4⋊C8—22no—
43C7×C4⋊C8—22no—
44C7×#104—44no—
45C7×#103—24no—
46C7×#102—24no—
47C7×C22⋊C8—22no—
48C4×C7×C8D414yes—
49C7×#95—22no—
50C7×#95—22no—
51C7×#95—22no—
52#3103C2×C712yes—
53#2399C2×C711yescharacteristic, complemented
54#1963C2×C711yescharacteristic
55#1487C2×C712yescomplemented
56#1274C2×C711yescharacteristic, complemented
57C7×#41—42no—
58C7×#41—42no—
59C2×C7×Q8—22no—
60C2×C7×Q8—22no—
61C2×C7×Q8—21no—
62C2×C7×Q8—44no—
63C2×C7×D4—22no—
64C2×C7×D4—21no—
65C2×C7×D4—41no—
66C22×C4×C7—22no—
67C22×C4×C7—22no—
68C22×C4×C7—22no—
69C22×C4×C7—41no—
70C22×C4×C7C2×D412yes—
71C22×C4×C7C2×D411yescharacteristic
72C7×SD16—48no—
73C7×M16—42no—
74C2×C7×C8—24no—
75C2×C7×C8—41no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—22no—
78C7×C4⋊C4—22no—
79C7×C4⋊C4—48no—
80C7×C4⋊C4—42no—
81C7×C4⋊C4—42no—
82C7×C22⋊C4—42no—
83C42×C7—22no—
84C42×C7—42no—
85C42×C7—42no—
86C42×C7SD1614yescomplemented
87C42×C7C2×D411yescharacteristic
88C42×C7C2×D412yes—
89C42×C7#4111yescharacteristic
90#532C22×C711yescharacteristic
91C2×C4×Q8C22×C712yes—
92#481—22no—
93#475—22no—
94#460—22no—
95#448—22no—
96C4×SD16—24no—
97#431C22×C712yes—
98#423C22×C711yescharacteristic
99C2×#104—22no—
100#398—22no—
101#395—22no—
102C4×C4⋊C4—22no—
103#376C22×C711yescharacteristic
104Q8⋊C8—24no—
105C23×C7—41no—
106C23×C7#12011yescharacteristic
107C7×Q8—24no—
108C7×Q8—24no—
109C7×Q8—42no—
110C7×Q8—42no—
111C7×Q8—42no—
112C7×D4—44no—
113C7×D4—42no—
114C7×D4—82no—
115C2×C4×C7—24no—
116C2×C4×C7—24no—
117C2×C4×C7—42no—
118C2×C4×C7—44no—
119C2×C4×C7—42no—
120C2×C4×C7—42no—
121C2×C4×C7—44no—
122C2×C4×C7—42no—
123C2×C4×C7—41no—
124C2×C4×C7—81no—
125C2×C4×C7—81no—
126C2×C4×C7#12012yes—
127C2×C4×C7#12112yes—
128C2×C4×C7#12111yescharacteristic
129C2×C4×C7#12711yescharacteristic
130C2×C4×C7C2×SD1612yes—
131C2×C4×C7#13712yes—
132C7×C8—44no—
133C7×C8—81no—
134C2×#41—21no—
135C22×Q8C7×D412yes—
136C2×SD16—24no—
137C2×M16—21no—
138C4⋊Q8C7×D411yescharacteristic
139#127C7×D411yescharacteristic
140#124—21no—
141#121—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×D4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4C7×D412yes—
150C2×C42—22no—
151C2×C42C7×D412yes—
152C2×C42C23×C711yescharacteristic
153#106—24no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156#104—44no—
157#103—24no—
158#102—24no—
159C22⋊C8—22no—
160C4×C8C7×D414yes—
161#95—22no—
162#95—22no—
163#95—22no—
164C22×C7—22no—
165C22×C7—82no—
166C22×C7—81no—
167C22×C7#39211yescharacteristic
168C22×C7#44812yes—
169C4×C7—22no—
170C4×C7—22no—
171C4×C7—44no—
172C4×C7—44no—
173C4×C7—82no—
174C4×C7—81no—
175C4×C7#46112yes—
176C4×C7#49211yescharacteristic
177C4×C7#49711yescharacteristic
178#41—42no—
179#41—42no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—44no—
184C2×D4—22no—
185C2×D4—21no—
186C2×D4—41no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—41no—
191C22×C4C2×C7×D412yes—
192C22×C4C2×C7×D411yescharacteristic
193SD16—48no—
194M16—42no—
195C2×C8—24no—
196C2×C8—41no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—48no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C22⋊C4—42no—
204C42—22no—
205C42—42no—
206C42—42no—
207C42C7×SD1614yescomplemented
208C42C2×C7×D411yescharacteristic
209C42C2×C7×D412yes—
210C42C7×#4111yescharacteristic
211C2×C7—22no—
212C2×C7—161no—
213C2×C7#201012yes—
214C2×C7#201311yescharacteristic
215C23—41no—
216C23C7×#12011yescharacteristic
217Q8—24no—
218Q8—24no—
219Q8—42no—
220Q8—42no—
221Q8—42no—
222D4—44no—
223D4—42no—
224D4—82no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4C7×#12012yes—
237C2×C4C7×#12112yes—
238C2×C4C7×#12111yescharacteristic
239C2×C4C7×#12711yescharacteristic
240C2×C4C2×C7×SD1612yes—
241C2×C4C7×#13712yes—
242C8—44no—
243C8—81no—
244C7#1001411yescharacteristic, direct product factor, Sylow
245C22—22no—
246C22—82no—
247C22—81no—
248C22C7×#39211yescharacteristic
249C22C7×#44812yes—
250C4—22no—
251C4—22no—
252C4—44no—
253C4—44no—
254C4—82no—
255C4—81no—
256C4C7×#46112yes—
257C4C7×#49211yescharacteristic
258C4C7×#49711yescharacteristic
259C2—22no—
260C2—161no—
261C2C7×#201012yes—
262C2C7×#201311yescharacteristic
subgroupG/H
centerC2×C4×C7#127
commutator, Frattini, cosocleC2×C42C23×C7
socleC22×C7#392
prime radicalC7×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2399C2semidirect
4. C7×#1487C2semidirect
5. C7×#1274C2semidirect
21. #10014C7direct (N × H)
53. #2399C2×C7semidirect
55. #1487C2×C7semidirect
56. #1274C2×C7semidirect
86. C42×C7SD16semidirect
207. C42C7×SD16semidirect
244. C7#10014direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31032
C7×#23992
C7×#19632
C7×#14872
C7×#12742
#100147