Finite Group Databasebeta

← C7×#12247 #13189463

Subgroups — C7×#12247 #13189463

Order 1792.

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

normal subgroups only

This group has 394 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×#2582C211yesmaximal, characteristic
2C7×#2060C211yesmaximal, characteristic
3C7×#1611C212yesmaximal
4C7×#1346C212yesmaximal
5C7×#1325C211yesmaximal, characteristic
6C7×#544C2211yescharacteristic
7C2×C4×C7×Q8—21no—
8C7×#483—22no—
9C7×#428—21no—
10C7×#398—21no—
11C7×#398C2212yes—
12C7×#391—22no—
13C7×#389—22no—
14C7×#384C2211yescharacteristic
15C7×#343C2211yescharacteristic
16C7×#339—21no—
17C7×#328—22no—
18C7×#324C2212yes—
19#12247C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×Q8—21no—
21C2×C7×M16—21no—
22C7×C4⋊Q8—41no—
23C7×#125—41no—
24C7×C22⋊Q8—22no—
25C4×C7×Q8—42no—
26C4×C7×Q8—42no—
27C7×#117—21no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4D412yes—
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C4⋊C4C2311yescharacteristic
35C2×C42×C7—22no—
36C2×C42×C7D411yescharacteristic
37C7×#107—41no—
38C7×#106—42no—
39C7×#106—41no—
40C7×#103—42no—
41C7×C22⋊C8—22no—
42C7×#95—22no—
43C7×#95—21no—
44C7×#95—22no—
45C7×#95D412yes—
46#2582C2×C711yescharacteristic
47#2060C2×C711yescharacteristic
48#1611C2×C712yes—
49#1346C2×C712yes—
50#1325C2×C711yescharacteristic
51C2×C7×Q8—24no—
52C2×C7×Q8—22no—
53C2×C7×Q8—42no—
54C2×C7×Q8—42no—
55C22×C4×C7—21no—
56C22×C4×C7—22no—
57C22×C4×C7—21no—
58C22×C4×C7—22no—
59C22×C4×C7C2×D411yescharacteristic
60C22×C4×C7C2×D412yes—
61C7×M16—22no—
62C7×M16—41no—
63C2×C7×C8—41no—
64C2×C7×C8—42no—
65C7×C4⋊C4—21no—
66C7×C4⋊C4—22no—
67C7×C4⋊C4—21no—
68C7×C4⋊C4—41no—
69C7×C4⋊C4—42no—
70C7×C4⋊C4—41no—
71C7×C4⋊C4—44no—
72C7×C4⋊C4—42no—
73C7×C4⋊C4—42no—
74C7×C4⋊C4—42no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—81no—
77C7×C22⋊C4—41no—
78C42×C7—21no—
79C42×C7—44no—
80C42×C7—41no—
81C42×C7SD1612yes—
82#544C22×C711yescharacteristic
83C2×C4×Q8—21no—
84#483—22no—
85#428—21no—
86#398—21no—
87#398C22×C712yes—
88#391—22no—
89#389—22no—
90#384C22×C711yescharacteristic
91#343C22×C711yescharacteristic
92#339—21no—
93#328—22no—
94#324C22×C712yes—
95C23×C7#12011yescharacteristic
96C7×Q8—48no—
97C7×Q8—82no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—24no—
104C2×C4×C7—22no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—42no—
111C2×C4×C7—41no—
112C2×C4×C7—41no—
113C2×C4×C7—42no—
114C2×C4×C7—41no—
115C2×C4×C7C2×SD1611yescharacteristic
116C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
117C7×C8—42no—
118C7×C8—82no—
119C22×Q8—21no—
120C2×M16—21no—
121C4⋊Q8—41no—
122#125—41no—
123C22⋊Q8—22no—
124C4×Q8—42no—
125C4×Q8—42no—
126#117—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4C7×D411yescharacteristic
132C2×C4⋊C4C7×D412yes—
133C2×C4⋊C4C23×C711yescharacteristic
134C2×C42—22no—
135C2×C42C7×D411yescharacteristic
136#107—41no—
137#106—41no—
138#106—42no—
139#103—42no—
140C22⋊C8—22no—
141#95—22no—
142#95—21no—
143#95—22no—
144#95C7×D412yes—
145C22×C7—21no—
146C22×C7—21no—
147C22×C7#45011yescharacteristic
148C22×C7#45611yescharacteristic
149C22×C7B(4,𝔽2)11yescharacteristic
150C4×C7—24no—
151C4×C7—21no—
152C4×C7—21no—
153C4×C7—44no—
154C4×C7—42no—
155C4×C7—82no—
156C4×C7—81no—
157C4×C7—81no—
158C4×C7—81no—
159C2×Q8—24no—
160C2×Q8—22no—
161C2×Q8—42no—
162C2×Q8—42no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4C2×C7×D411yescharacteristic
168C22×C4C2×C7×D412yes—
169M16—22no—
170M16—41no—
171C2×C8—41no—
172C2×C8—42no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—21no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C4⋊C4—44no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C22⋊C4—41no—
186C42—21no—
187C42—44no—
188C42—41no—
189C42C7×SD1612yes—
190C2×C7—21no—
191C2×C7—21no—
192C2×C7#160411yescharacteristic
193C2×C7#219811yescharacteristic
194C2×C7#220511yescharacteristic
195C23C7×#12011yescharacteristic
196Q8—48no—
197Q8—82no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4C2×C7×SD1611yescharacteristic
216C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
217C8—42no—
218C8—82no—
219C7#1224711yescharacteristic, direct product factor, Sylow
220C22—21no—
221C22—21no—
222C22C7×#45011yescharacteristic
223C22C7×#45611yescharacteristic
224C22C7×B(4,𝔽2)11yescharacteristic
225C4—24no—
226C4—21no—
227C4—21no—
228C4—44no—
229C4—42no—
230C4—82no—
231C4—81no—
232C4—81no—
233C4—81no—
234C2—21no—
235C2—21no—
236C2C7×#160411yescharacteristic
237C2C7×#219811yescharacteristic
238C2C7×#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. #12247C7direct (N × H)
219. C7#12247direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25822
C7×#20602
C7×#16112
C7×#13462
C7×#13252
#122477