Finite Group Databasebeta

← C7×#12121 #13189337

Subgroups — C7×#12121 #13189337

Order 1792.

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

normal subgroups only

This group has 470 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×#2607C211yesmaximal, characteristic, complemented
2C7×#2022C211yesmaximal, characteristic
3C7×#1601C212yesmaximal
4C7×#1351C212yesmaximal, complemented
5C7×#1317C211yesmaximal, characteristic, complemented
6C7×#534C2211yescharacteristic
7C2×C7×#123—21no—
8C7×#448—22no—
9C2×C7×#99—21no—
10C7×#399—21no—
11C7×#393C2212yes—
12C7×#391—22no—
13C7×#387—22no—
14C7×#381C2211yescharacteristic
15C4×C7×C22⋊C4—21no—
16C7×#342C2211yescharacteristic
17C7×#327—22no—
18C7×#323C2212yes—
19#12121C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×Q8D412yes—
21C22×C7×D4—21no—
22C23×C4×C7D411yescharacteristic
23C2×C7×SD16—42no—
24C7×#123—42no—
25C7×#123—41no—
26C7×#123—42no—
27C7×#121—22no—
28C7×#120—41no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—41no—
32C2×C7×C22⋊C4—21no—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—41no—
35C2×C7×C22⋊C4D411yescharacteristic
36C2×C7×C22⋊C4C2311yescharacteristic
37C2×C42×C7—21no—
38C2×C42×C7—21no—
39C7×#103—42no—
40C7×#99—42no—
41C7×C22⋊C8—22no—
42C7×#95—22no—
43C7×#95—21no—
44C7×#95—22no—
45C7×#95—22no—
46C7×#95D412yes—
47#2607C2×C711yescharacteristic, complemented
48#2022C2×C711yescharacteristic
49#1601C2×C712yes—
50#1351C2×C712yescomplemented
51#1317C2×C711yescharacteristic, complemented
52C24×C7—21no—
53C24×C7C2×D411yescharacteristic
54C2×C7×Q8—44no—
55C2×C7×Q8—42no—
56C2×C7×D4—22no—
57C2×C7×D4—41no—
58C2×C7×D4—41no—
59C2×C7×D4—41no—
60C2×C7×D4—41no—
61C2×C7×D4—41no—
62C2×C7×D4—41no—
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—22no—
69C22×C4×C7—41no—
70C22×C4×C7—41no—
71C22×C4×C7—41no—
72C22×C4×C7SD1612yescomplemented
73C22×C4×C7C2×D412yes—
74C7×SD16—84no—
75C2×C7×C8—42no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—44no—
78C7×C4⋊C4—44no—
79C7×C4⋊C4—81no—
80C7×C22⋊C4—22no—
81C7×C22⋊C4—41no—
82C7×C22⋊C4—44no—
83C7×C22⋊C4—42no—
84C7×C22⋊C4—44no—
85C7×C22⋊C4—41no—
86C7×C22⋊C4—41no—
87C42×C7—81no—
88C42×C7—81no—
89#534C22×C711yescharacteristic
90C2×#123—21no—
91#448—22no—
92C2×#99—21no—
93#399—21no—
94#393C22×C712yes—
95#391—22no—
96#387—22no—
97#381C22×C711yescharacteristic
98C4×C22⋊C4—21no—
99#342C22×C711yescharacteristic
100#327—22no—
101#323C22×C712yes—
102C23×C7—21no—
103C23×C7—21no—
104C23×C7—42no—
105C23×C7—41no—
106C23×C7—41no—
107C23×C7—41no—
108C23×C7—82no—
109C23×C7#12011yescharacteristic
110C23×C7C2×SD1611yescharacteristic
111C23×C7#13711yescharacteristic
112C7×Q8—44no—
113C7×Q8—82no—
114C7×D4—82no—
115C7×D4—81no—
116C7×D4—81no—
117C7×D4—81no—
118C7×D4—81no—
119C2×C4×C7—24no—
120C2×C4×C7—22no—
121C2×C4×C7—22no—
122C2×C4×C7—42no—
123C2×C4×C7—41no—
124C2×C4×C7—41no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—41no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—41no—
134C2×C4×C7—42no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—81no—
138C2×C4×C7—82no—
139C7×C8—82no—
140C22×Q8C7×D412yes—
141C22×D4—21no—
142C23×C4C7×D411yescharacteristic
143C2×SD16—42no—
144#123—42no—
145#123—41no—
146#123—42no—
147#121—22no—
148#120—41no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—41no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4C7×D411yescharacteristic
156C2×C22⋊C4C23×C711yescharacteristic
157C2×C42—21no—
158C2×C42—21no—
159#103—42no—
160#99—42no—
161C22⋊C8—22no—
162#95—22no—
163#95—21no—
164#95—22no—
165#95—22no—
166#95C7×D412yes—
167C22×C7—22no—
168C22×C7—21no—
169C22×C7—21no—
170C22×C7—41no—
171C22×C7—41no—
172C22×C7—42no—
173C22×C7—81no—
174C22×C7—82no—
175C22×C7—81no—
176C22×C7—82no—
177C22×C7—161no—
178C22×C7#44811yescharacteristic
179C22×C7#45311yescharacteristic
180C22×C7B(4,𝔽2)11yescharacteristic
181C4×C7—42no—
182C4×C7—42no—
183C4×C7—44no—
184C4×C7—82no—
185C4×C7—81no—
186C4×C7—82no—
187C4×C7—81no—
188C24—21no—
189C24C2×C7×D411yescharacteristic
190C2×Q8—44no—
191C2×Q8—42no—
192C2×D4—22no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4C7×SD1612yescomplemented
209C22×C4C2×C7×D412yes—
210SD16—84no—
211C2×C8—42no—
212C4⋊C4—22no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—81no—
216C22⋊C4—22no—
217C22⋊C4—41no—
218C22⋊C4—44no—
219C22⋊C4—42no—
220C22⋊C4—44no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C42—81no—
224C42—81no—
225C2×C7—21no—
226C2×C7—21no—
227C2×C7—41no—
228C2×C7—41no—
229C2×C7—161no—
230C2×C7#159611yescharacteristic
231C2×C7#219712yes—
232C23—21no—
233C23—21no—
234C23—42no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—82no—
239C23C7×#12011yescharacteristic
240C23C2×C7×SD1611yescharacteristic
241C23C7×#13711yescharacteristic
242Q8—44no—
243Q8—82no—
244D4—82no—
245D4—81no—
246D4—81no—
247D4—81no—
248D4—81no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—81no—
268C2×C4—82no—
269C8—82no—
270C7#1212111yescharacteristic, direct product factor, Sylow
271C22—22no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—81no—
278C22—82no—
279C22—81no—
280C22—82no—
281C22—161no—
282C22C7×#44811yescharacteristic
283C22C7×#45311yescharacteristic
284C22C7×B(4,𝔽2)11yescharacteristic
285C4—42no—
286C4—42no—
287C4—44no—
288C4—82no—
289C4—81no—
290C4—82no—
291C4—81no—
292C2—21no—
293C2—21no—
294C2—41no—
295C2—41no—
296C2—161no—
297C2C7×#159611yescharacteristic
298C2C7×#219712yes—
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#534C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2607C2semidirect
4. C7×#1351C2semidirect
5. C7×#1317C2semidirect
19. #12121C7direct (N × H)
47. #2607C2×C7semidirect
50. #1351C2×C7semidirect
51. #1317C2×C7semidirect
72. C22×C4×C7SD16semidirect
208. C22×C4C7×SD16semidirect
270. C7#12121direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26072
C7×#20222
C7×#16012
C7×#13512
C7×#13172
#121217