Finite Group Databasebeta

← C7×#12256 #13189472

Subgroups — C7×#12256 #13189472

Order 1792.

442 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2551C211yesmaximal, characteristic
2C7×#2060C211yesmaximal, characteristic
3C7×#1605C212yesmaximal
4C7×#1354C212yesmaximal
5C7×#1325C211yesmaximal, characteristic
6C7×#544C2211yescharacteristic
7C2×C7×C4⋊Q8—22no—
8C7×#451—22no—
9C7×#428—21no—
10C7×#395—21no—
11C7×#395C2212yes—
12C7×#389—22no—
13C7×#384C2211yescharacteristic
14C4×C7×C4⋊C4—21no—
15C7×#343C2211yescharacteristic
16C7×#339—21no—
17C7×#328—22no—
18C7×#324C2212yes—
19#12256C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×Q8—21no—
21C22×C7×Q8D412yes—
22C2×C7×Q16—42no—
23C2×C7×M16—21no—
24C7×C4⋊Q8—41no—
25C7×C4⋊Q8—48no—
26C7×#125—41no—
27C7×C22⋊Q8—22no—
28C7×#117—21no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C4⋊C4C2311yescharacteristic
36C2×C42×C7—22no—
37C2×C42×C7D411yescharacteristic
38C7×#107—41no—
39C7×#106—41no—
40C7×#103—42no—
41C7×C22⋊C8—22no—
42C7×#95—21no—
43C7×#95—22no—
44C7×#95D412yes—
45#2551C2×C711yescharacteristic
46#2060C2×C711yescharacteristic
47#1605C2×C712yes—
48#1354C2×C712yes—
49#1325C2×C711yescharacteristic
50C2×C7×Q8—22no—
51C2×C7×Q8—44no—
52C2×C7×Q8—44no—
53C2×C7×Q8—42no—
54C2×C7×Q8—81no—
55C22×C4×C7—21no—
56C22×C4×C7—22no—
57C22×C4×C7—22no—
58C22×C4×C7—21no—
59C22×C4×C7C2×D411yescharacteristic
60C22×C4×C7C2×D412yes—
61C7×Q16—84no—
62C7×M16—22no—
63C7×M16—41no—
64C2×C7×C8—41no—
65C2×C7×C8—42no—
66C7×C4⋊C4—24no—
67C7×C4⋊C4—24no—
68C7×C4⋊C4—22no—
69C7×C4⋊C4—21no—
70C7×C4⋊C4—21no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—41no—
73C7×C4⋊C4—44no—
74C7×C4⋊C4—44no—
75C7×C4⋊C4—44no—
76C7×C4⋊C4—42no—
77C7×C22⋊C4—41no—
78C42×C7—21no—
79C42×C7—44no—
80C42×C7—41no—
81C42×C7Q1612yescomplemented
82#544C22×C711yescharacteristic
83C2×C4⋊Q8—22no—
84#451—22no—
85#428—21no—
86#395—21no—
87#395C22×C712yes—
88#389—22no—
89#384C22×C711yescharacteristic
90C4×C4⋊C4—21no—
91#343C22×C711yescharacteristic
92#339—21no—
93#328—22no—
94#324C22×C712yes—
95C23×C7#12011yescharacteristic
96C7×Q8—44no—
97C7×Q8—82no—
98C7×Q8—84no—
99C7×Q8—82no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—24no—
106C2×C4×C7—24no—
107C2×C4×C7—22no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7—41no—
115C2×C4×C7—41no—
116C2×C4×C7—41no—
117C2×C4×C7C2×Q1611yescharacteristic
118C2×C4×C7#13711yescharacteristic
119C7×C8—42no—
120C7×C8—82no—
121C22×Q8—21no—
122C22×Q8C7×D412yes—
123C2×Q16—42no—
124C2×M16—21no—
125C4⋊Q8—41no—
126C4⋊Q8—48no—
127#125—41no—
128C22⋊Q8—22no—
129#117—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4C7×D411yescharacteristic
136C2×C4⋊C4C23×C711yescharacteristic
137C2×C42—22no—
138C2×C42C7×D411yescharacteristic
139#107—41no—
140#106—41no—
141#103—42no—
142C22⋊C8—22no—
143#95—21no—
144#95—22no—
145#95C7×D412yes—
146C22×C7—21no—
147C22×C7—21no—
148C22×C7#45111yescharacteristic
149C22×C7#45611yescharacteristic
150C22×C7B(4,𝔽2)11yescharacteristic
151C4×C7—24no—
152C4×C7—21no—
153C4×C7—21no—
154C4×C7—44no—
155C4×C7—42no—
156C4×C7—44no—
157C4×C7—81no—
158C4×C7—81no—
159C4×C7—81no—
160C2×Q8—22no—
161C2×Q8—44no—
162C2×Q8—44no—
163C2×Q8—42no—
164C2×Q8—81no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—21no—
169C22×C4C2×C7×D411yescharacteristic
170C22×C4C2×C7×D412yes—
171Q16—84no—
172M16—22no—
173M16—41no—
174C2×C8—41no—
175C2×C8—42no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—42no—
187C22⋊C4—41no—
188C42—21no—
189C42—44no—
190C42—41no—
191C42C7×Q1612yescomplemented
192C2×C7—21no—
193C2×C7—21no—
194C2×C7#160511yescharacteristic
195C2×C7#219611yescharacteristic
196C2×C7#220511yescharacteristic
197C23C7×#12011yescharacteristic
198Q8—44no—
199Q8—82no—
200Q8—84no—
201Q8—82no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4C2×C7×Q1611yescharacteristic
220C2×C4C7×#13711yescharacteristic
221C8—42no—
222C8—82no—
223C7#1225611yescharacteristic, direct product factor, Sylow
224C22—21no—
225C22—21no—
226C22C7×#45111yescharacteristic
227C22C7×#45611yescharacteristic
228C22C7×B(4,𝔽2)11yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—44no—
233C4—42no—
234C4—44no—
235C4—81no—
236C4—81no—
237C4—81no—
238C2—21no—
239C2—21no—
240C2C7×#160511yescharacteristic
241C2C7×#219611yescharacteristic
242C2C7×#220511yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC23×C7#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
19. #12256C7direct (N × H)
81. C42×C7Q16semidirect
191. C42C7×Q16semidirect
223. C7#12256direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25512
C7×#20602
C7×#16052
C7×#13542
C7×#13252
#122567