Finite Group Databasebeta

← C7×#10015 #13187231

Subgroups — C7×#10015 #13187231

Order 1792.

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

normal subgroups only

This group has 696 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×#3101C212yesmaximal, complemented
2C7×#2397C211yesmaximal, characteristic
3C7×#1995C211yesmaximal, characteristic, complemented
4C7×#1486C212yesmaximal, complemented
5C7×#1274C211yesmaximal, characteristic, complemented
6C7×#533C2211yescharacteristic
7C2×C4×C7×D4—22no—
8C2×C4×C7×D4C2212yes—
9C7×#483—22no—
10C7×#476—22no—
11C7×#461—22no—
12C7×#450—22no—
13C4×C7×SD16—24no—
14C7×#431C2212yes—
15C7×#423C2211yescharacteristic
16C7×#399—22no—
17C7×#392—22no—
18C7×#376C2211yescharacteristic
19C7×#339—22no—
20C7×D4⋊C8—24no—
21#10015C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×D4—22no—
23C22×C7×D4D412yes—
24C23×C4×C7—22no—
25C2×C7×SD16—24no—
26C2×C7×M16—21no—
27C7×C4⋊Q8D412yescomplemented
28C7×#125—21no—
29C7×C22⋊Q8—22no—
30C4×C7×Q8—22no—
31C4×C7×D4—22no—
32C4×C7×D4—22no—
33C4×C7×D4—22no—
34C4×C7×D4—22no—
35C4×C7×D4—44no—
36C4×C7×D4—44no—
37C7×#117—21no—
38C2×C7×C4⋊C4—22no—
39C2×C7×C4⋊C4D412yes—
40C2×C7×C22⋊C4—24no—
41C2×C7×C22⋊C4—22no—
42C2×C42×C7D412yes—
43C2×C42×C7C2311yescharacteristic
44C7×#106—24no—
45C7×C4⋊C8—22no—
46C7×C4⋊C8—22no—
47C7×#103—24no—
48C7×#102—24no—
49C7×C22⋊C8—22no—
50C4×C7×C8D414yescomplemented
51C7×#95—22no—
52#3101C2×C712yescomplemented
53#2397C2×C711yescharacteristic
54#1995C2×C711yescharacteristic, complemented
55#1486C2×C712yescomplemented
56#1274C2×C711yescharacteristic, complemented
57C24×C7—22no—
58C2×C7×Q8—22no—
59C2×C7×D4—22no—
60C2×C7×D4—22no—
61C2×C7×D4—44no—
62C2×C7×D4—44no—
63C2×C7×D4—48no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—42no—
67C22×C4×C7—42no—
68C22×C4×C7—44no—
69C22×C4×C7—44no—
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—22no—
80C7×C4⋊C4—44no—
81C7×C4⋊C4—42no—
82C7×C22⋊C4—48no—
83C7×C22⋊C4—41no—
84C7×C22⋊C4—42no—
85C7×C22⋊C4—42no—
86C42×C7—22no—
87C42×C7—41no—
88C42×C7SD1614yescomplemented
89C42×C7C2×D411yescharacteristic
90C42×C7C2×D412yes—
91C42×C7#4111yescharacteristic
92#533C22×C711yescharacteristic
93C2×C4×D4—22no—
94C2×C4×D4C22×C712yes—
95#483—22no—
96#476—22no—
97#461—22no—
98#450—22no—
99C4×SD16—24no—
100#431C22×C712yes—
101#423C22×C711yescharacteristic
102#399—22no—
103#392—22no—
104#376C22×C711yescharacteristic
105#339—22no—
106D4⋊C8—24no—
107C23×C7—24no—
108C23×C7—44no—
109C23×C7—42no—
110C23×C7—42no—
111C23×C7—44no—
112C23×C7#12011yescharacteristic
113C7×Q8—44no—
114C7×D4—24no—
115C7×D4—24no—
116C7×D4—42no—
117C7×D4—42no—
118C7×D4—416no—
119C2×C4×C7—24no—
120C2×C4×C7—42no—
121C2×C4×C7—44no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—44no—
125C2×C4×C7—84no—
126C2×C4×C7—82no—
127C2×C4×C7—82no—
128C2×C4×C7#12012yes—
129C2×C4×C7#12112yes—
130C2×C4×C7#12111yescharacteristic
131C2×C4×C7#12711yescharacteristic
132C2×C4×C7C2×SD1612yes—
133C2×C4×C7AGL(1,ℤ/8)12yes—
134C7×C8—44no—
135C7×C8—81no—
136C22×D4—22no—
137C22×D4C7×D412yes—
138C23×C4—22no—
139C2×SD16—24no—
140C2×M16—21no—
141C4⋊Q8C7×D412yescomplemented
142#125—21no—
143C22⋊Q8—22no—
144C4×Q8—22no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—44no—
150C4×D4—44no—
151#117—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C7×D412yes—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—22no—
156C2×C42C7×D412yes—
157C2×C42C23×C711yescharacteristic
158#106—24no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161#103—24no—
162#102—24no—
163C22⋊C8—22no—
164C4×C8C7×D414yescomplemented
165#95—22no—
166C22×C7—22no—
167C22×C7—48no—
168C22×C7—44no—
169C22×C7—84no—
170C22×C7—82no—
171C22×C7—82no—
172C22×C7#39211yescharacteristic
173C22×C7#45012yes—
174C4×C7—22no—
175C4×C7—22no—
176C4×C7—44no—
177C4×C7—82no—
178C4×C7—82no—
179C4×C7#46012yes—
180C4×C7#49211yescharacteristic
181C4×C7#49611yescharacteristic
182C24—22no—
183C2×Q8—22no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4—48no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4C2×C7×D412yes—
196C22×C4C2×C7×D411yescharacteristic
197SD16—48no—
198M16—42no—
199C2×C8—24no—
200C2×C8—41no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—44no—
206C4⋊C4—42no—
207C22⋊C4—48no—
208C22⋊C4—41no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C42—22no—
212C42—41no—
213C42C7×SD1614yescomplemented
214C42C2×C7×D411yescharacteristic
215C42C2×C7×D412yes—
216C42C7×#4111yescharacteristic
217C2×C7—22no—
218C2×C7—84no—
219C2×C7#201012yes—
220C2×C7#201411yescharacteristic
221C23—24no—
222C23—44no—
223C23—42no—
224C23—42no—
225C23—44no—
226C23C7×#12011yescharacteristic
227Q8—44no—
228D4—24no—
229D4—24no—
230D4—42no—
231D4—42no—
232D4—416no—
233C2×C4—24no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—84no—
242C2×C4C7×#12012yes—
243C2×C4C7×#12112yes—
244C2×C4C7×#12111yescharacteristic
245C2×C4C7×#12711yescharacteristic
246C2×C4C2×C7×SD1612yes—
247C2×C4C7×AGL(1,ℤ/8)12yes—
248C8—44no—
249C8—81no—
250C7#1001511yescharacteristic, direct product factor, Sylow
251C22—22no—
252C22—48no—
253C22—44no—
254C22—84no—
255C22—82no—
256C22—82no—
257C22C7×#39211yescharacteristic
258C22C7×#45012yes—
259C4—22no—
260C4—22no—
261C4—44no—
262C4—82no—
263C4—82no—
264C4C7×#46012yes—
265C4C7×#49211yescharacteristic
266C4C7×#49611yescharacteristic
267C2—22no—
268C2—84no—
269C2C7×#201012yes—
270C2C7×#201411yescharacteristic
subgroupG/H
centerC2×C4×C7#127
commutator, Frattini, cosocleC2×C42C23×C7
socleC22×C7#392
prime radicalC7×#2397C2
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
1. C7×#3101C2semidirect
3. C7×#1995C2semidirect
4. C7×#1486C2semidirect
5. C7×#1274C2semidirect
21. #10015C7direct (N × H)
27. C7×C4⋊Q8D4semidirect
50. C4×C7×C8D4semidirect
52. #3101C2×C7semidirect
54. #1995C2×C7semidirect
55. #1486C2×C7semidirect
56. #1274C2×C7semidirect
88. C42×C7SD16semidirect
141. C4⋊Q8C7×D4semidirect
164. C4×C8C7×D4semidirect
213. C42C7×SD16semidirect
250. C7#10015direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31012
C7×#23972
C7×#19952
C7×#14862
C7×#12742
#100157