Finite Group Databasebeta

← C7×#12246 #13189462

Subgroups — C7×#12246 #13189462

Order 1792.

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

normal subgroups only

This group has 558 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×#2537C211yesmaximal, characteristic
2C7×#2060C211yesmaximal, characteristic, complemented
3C7×#1599C212yesmaximal, complemented
4C7×#1345C212yesmaximal, complemented
5C7×#1325C211yesmaximal, characteristic, complemented
6C7×#544C2211yescharacteristic
7C2×C7×#121—22no—
8C2×C4×C7×D4—21no—
9C7×#450—22no—
10C7×#428—21no—
11C7×#394—21no—
12C7×#394C2212yes—
13C7×#384C2211yescharacteristic
14C7×#380—22no—
15C7×#343C2211yescharacteristic
16C7×#339—21no—
17C7×#328—22no—
18C7×#324C2212yes—
19#12246C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×D4—21no—
21C22×C7×D4—21no—
22C22×C7×D4D412yes—
23C23×C4×C7—22no—
24C2×C7×SD16—42no—
25C2×C7×M16—21no—
26C7×C4⋊Q8—41no—
27C7×#125—41no—
28C7×C22⋊Q8—22no—
29C7×#121—48no—
30C4×C7×D4—42no—
31C4×C7×D4—42no—
32C7×#117—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C4⋊C4C2311yescharacteristic
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C42×C7D411yescharacteristic
41C7×#107—41no—
42C7×#106—41no—
43C7×#102—42no—
44C7×C22⋊C8—22no—
45C7×#95—21no—
46C7×#95D412yes—
47#2537C2×C711yescharacteristic
48#2060C2×C711yescharacteristic, complemented
49#1599C2×C712yescomplemented
50#1345C2×C712yescomplemented
51#1325C2×C711yescharacteristic, complemented
52C24×C7—22no—
53C2×C7×Q8—22no—
54C2×C7×D4—24no—
55C2×C7×D4—42no—
56C2×C7×D4—42no—
57C2×C7×D4—42no—
58C2×C7×D4—44no—
59C2×C7×D4—44no—
60C2×C7×D4—81no—
61C22×C4×C7—21no—
62C22×C4×C7—22no—
63C22×C4×C7—21no—
64C22×C4×C7—44no—
65C22×C4×C7—44no—
66C22×C4×C7—44no—
67C22×C4×C7C2×D411yescharacteristic
68C22×C4×C7C2×D412yes—
69C7×SD16—84no—
70C7×M16—22no—
71C7×M16—41no—
72C2×C7×C8—41no—
73C2×C7×C8—42no—
74C7×C4⋊C4—22no—
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4—41no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—42no—
80C7×C4⋊C4—44no—
81C7×C4⋊C4—42no—
82C7×C22⋊C4—44no—
83C7×C22⋊C4—44no—
84C7×C22⋊C4—41no—
85C7×C22⋊C4—42no—
86C7×C22⋊C4—42no—
87C42×C7—21no—
88C42×C7—41no—
89C42×C7SD1612yescomplemented
90#544C22×C711yescharacteristic
91C2×#121—22no—
92C2×C4×D4—21no—
93#450—22no—
94#428—21no—
95#394—21no—
96#394C22×C712yes—
97#384C22×C711yescharacteristic
98#380—22no—
99#343C22×C711yescharacteristic
100#339—21no—
101#328—22no—
102#324C22×C712yes—
103C23×C7—24no—
104C23×C7—42no—
105C23×C7—42no—
106C23×C7—42no—
107C23×C7—42no—
108C23×C7—42no—
109C23×C7—42no—
110C23×C7#12011yescharacteristic
111C7×Q8—82no—
112C7×D4—48no—
113C7×D4—44no—
114C7×D4—82no—
115C7×D4—84no—
116C2×C4×C7—22no—
117C2×C4×C7—22no—
118C2×C4×C7—22no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—22no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7—41no—
130C2×C4×C7—88no—
131C2×C4×C7C2×SD1611yescharacteristic
132C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
133C7×C8—42no—
134C7×C8—82no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4C7×D412yes—
138C23×C4—22no—
139C2×SD16—42no—
140C2×M16—21no—
141C4⋊Q8—41no—
142#125—41no—
143C22⋊Q8—22no—
144#121—48no—
145C4×D4—42no—
146C4×D4—42no—
147#117—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4C7×D411yescharacteristic
151C2×C4⋊C4C23×C711yescharacteristic
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C42C7×D411yescharacteristic
156#107—41no—
157#106—41no—
158#102—42no—
159C22⋊C8—22no—
160#95—21no—
161#95C7×D412yes—
162C22×C7—21no—
163C22×C7—21no—
164C22×C7—44no—
165C22×C7—44no—
166C22×C7—44no—
167C22×C7—82no—
168C22×C7—82no—
169C22×C7—82no—
170C22×C7—82no—
171C22×C7#45011yescharacteristic
172C22×C7#45611yescharacteristic
173C22×C7B(4,𝔽2)11yescharacteristic
174C4×C7—24no—
175C4×C7—21no—
176C4×C7—21no—
177C4×C7—42no—
178C4×C7—82no—
179C4×C7—81no—
180C4×C7—81no—
181C4×C7—81no—
182C24—22no—
183C2×Q8—22no—
184C2×D4—24no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—44no—
189C2×D4—44no—
190C2×D4—81no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4C2×C7×D411yescharacteristic
198C22×C4C2×C7×D412yes—
199SD16—84no—
200M16—22no—
201M16—41no—
202C2×C8—41no—
203C2×C8—42no—
204C4⋊C4—22no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—44no—
211C4⋊C4—42no—
212C22⋊C4—44no—
213C22⋊C4—44no—
214C22⋊C4—41no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C42—21no—
218C42—41no—
219C42C7×SD1612yescomplemented
220C2×C7—21no—
221C2×C7—21no—
222C2×C7—84no—
223C2×C7#160411yescharacteristic
224C2×C7#219611yescharacteristic
225C2×C7#220411yescharacteristic
226C23—24no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23C7×#12011yescharacteristic
234Q8—82no—
235D4—48no—
236D4—44no—
237D4—82no—
238D4—84no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—88no—
254C2×C4C2×C7×SD1611yescharacteristic
255C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
256C8—42no—
257C8—82no—
258C7#1224611yescharacteristic, direct product factor, Sylow
259C22—21no—
260C22—21no—
261C22—44no—
262C22—44no—
263C22—44no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22C7×#45011yescharacteristic
269C22C7×#45611yescharacteristic
270C22C7×B(4,𝔽2)11yescharacteristic
271C4—24no—
272C4—21no—
273C4—21no—
274C4—42no—
275C4—82no—
276C4—81no—
277C4—81no—
278C4—81no—
279C2—21no—
280C2—21no—
281C2—84no—
282C2C7×#160411yescharacteristic
283C2C7×#219611yescharacteristic
284C2C7×#220411yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC7×#2537C2
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
2. C7×#2060C2semidirect
3. C7×#1599C2semidirect
4. C7×#1345C2semidirect
5. C7×#1325C2semidirect
19. #12246C7direct (N × H)
48. #2060C2×C7semidirect
49. #1599C2×C7semidirect
50. #1345C2×C7semidirect
51. #1325C2×C7semidirect
89. C42×C7SD16semidirect
219. C42C7×SD16semidirect
258. C7#12246direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25372
C7×#20602
C7×#15992
C7×#13452
C7×#13252
#122467