Finite Group Databasebeta

← C7×#9466 #13186682

Subgroups — C7×#9466 #13186682

Order 1792.

726 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2436C211yesmaximal, characteristic
2C7×#2310C211yesmaximal, characteristic, complemented
3C7×#1784C211yesmaximal, characteristic, complemented
4C7×#1510C212yesmaximal
5C7×#1284C212yesmaximal, complemented
6C23×C7×Q8C2211yescharacteristic
7C2×C7×#124C412yescomplemented
8C2×C7×#121C2211yescharacteristic
9C2×C7×#117—21no—
10C2×C7×#101—22no—
11C7×#407C2212yes—
12C7×#393—22no—
13C7×#391—24no—
14C7×#391C412yescomplemented
15C7×#386—21no—
16C7×#386—21no—
17C7×#381—21no—
18C7×#381C2212yes—
19C4×C7×C22⋊C4C2211yescharacteristic
20C7×#327—24no—
21#9466C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×Q8—22no—
23C22×C7×Q8—22no—
24C22×C7×Q8—44no—
25C22×C7×Q8C2×C412yes—
26C22×C7×Q8D412yes—
27C22×C7×D4—21no—
28C22×C7×D4C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C23×C4×C7C2311yescharacteristic
32C2×C7×M16—22no—
33C7×#124—44no—
34C7×#121—22no—
35C7×#121—22no—
36C7×#117—44no—
37C2×C7×C4⋊C4—22no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4—21no—
44C2×C42×C7—22no—
45C2×C42×C7C2×C412yes—
46C7×#101—44no—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—22no—
49C7×#95—24no—
50C7×#95—22no—
51C7×#95—22no—
52C7×#95—21no—
53C7×#95—22no—
54C7×#95D414yes—
55#2436C2×C711yescharacteristic
56#2310C2×C711yescharacteristic, complemented
57#1784C2×C711yescharacteristic, complemented
58#1510C2×C712yes—
59#1284C2×C712yescomplemented
60C24×C7—21no—
61C24×C7C2×D411yescharacteristic
62C2×C7×Q8—24no—
63C2×C7×Q8—24no—
64C2×C7×Q8—24no—
65C2×C7×Q8—24no—
66C2×C7×Q8—42no—
67C2×C7×Q8—42no—
68C2×C7×Q8—44no—
69C2×C7×Q8—44no—
70C2×C7×Q8—48no—
71C2×C7×D4—22no—
72C2×C7×D4—42no—
73C2×C7×D4—44no—
74C2×C7×D4—41no—
75C2×C7×D4—41no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—24no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—22no—
83C22×C4×C7—24no—
84C22×C4×C7—42no—
85C22×C4×C7—42no—
86C22×C4×C7C22⋊C412yes—
87C22×C4×C7C22⋊C412yes—
88C22×C4×C7C22×C411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D412yes—
91C22×C4×C7#4112yes—
92C7×M16—44no—
93C2×C7×C8—42no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—44no—
96C7×C22⋊C4—44no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—44no—
99C7×C22⋊C4—44no—
100C7×C22⋊C4—42no—
101C42×C7—42no—
102C42×C7—44no—
103C23×Q8C22×C711yescharacteristic
104C2×#124C4×C712yescomplemented
105C2×#121C22×C711yescharacteristic
106C2×#117—21no—
107C2×#101—22no—
108#407C22×C712yes—
109#393—22no—
110#391—24no—
111#391C4×C712yescomplemented
112#386—21no—
113#386—21no—
114#381—21no—
115#381C22×C712yes—
116C4×C22⋊C4C22×C711yescharacteristic
117#327—24no—
118C23×C7—22no—
119C23×C7—22no—
120C23×C7—41no—
121C23×C7—42no—
122C23×C7—42no—
123C23×C7—41no—
124C23×C7—42no—
125C23×C7C2×C22⋊C411yescharacteristic
126C23×C7#12011yescharacteristic
127C23×C7#12311yescharacteristic
128C7×Q8—48no—
129C7×Q8—48no—
130C7×D4—82no—
131C7×D4—82no—
132C7×D4—82no—
133C2×C4×C7—22no—
134C2×C4×C7—22no—
135C2×C4×C7—24no—
136C2×C4×C7—22no—
137C2×C4×C7—22no—
138C2×C4×C7—22no—
139C2×C4×C7—22no—
140C2×C4×C7—22no—
141C2×C4×C7—22no—
142C2×C4×C7—22no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—44no—
148C2×C4×C7—42no—
149C2×C4×C7—44no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—41no—
153C2×C4×C7—42no—
154C2×C4×C7—44no—
155C2×C4×C7#9914yes—
156C2×C4×C7C4×D412yes—
157C2×C4×C7#12112yes—
158C7×C8—82no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—44no—
162C22×Q8C2×C4×C712yes—
163C22×Q8C7×D412yes—
164C22×D4—21no—
165C22×D4C2×C4×C711yescharacteristic
166C23×C4C7×D411yescharacteristic
167C23×C4C7×D411yescharacteristic
168C23×C4C23×C711yescharacteristic
169C2×M16—22no—
170#124—44no—
171#121—22no—
172#121—22no—
173#117—44no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C2×C4×C711yescharacteristic
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—21no—
181C2×C42—22no—
182C2×C42C2×C4×C712yes—
183#101—44no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186#95—24no—
187#95—22no—
188#95—22no—
189#95—21no—
190#95—22no—
191#95C7×D414yes—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—44no—
200C22×C7—84no—
201C22×C7—81no—
202C22×C7—81no—
203C22×C7—81no—
204C22×C7#38611yescharacteristic
205C22×C7C2×#9912yes—
206C22×C7#45412yes—
207C4×C7—22no—
208C4×C7—24no—
209C4×C7—22no—
210C4×C7—42no—
211C4×C7—84no—
212C4×C7—82no—
213C4×C7—81no—
214C24—21no—
215C24C2×C7×D411yescharacteristic
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—24no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—48no—
225C2×D4—22no—
226C2×D4—42no—
227C2×D4—44no—
228C2×D4—41no—
229C2×D4—41no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4C7×C22⋊C412yes—
241C22×C4C7×C22⋊C412yes—
242C22×C4C22×C4×C711yescharacteristic
243C22×C4C2×C7×D411yescharacteristic
244C22×C4C2×C7×D412yes—
245C22×C4C7×#4112yes—
246M16—44no—
247C2×C8—42no—
248C4⋊C4—22no—
249C4⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—42no—
253C22⋊C4—44no—
254C22⋊C4—42no—
255C42—42no—
256C42—44no—
257C2×C7—22no—
258C2×C7—22no—
259C2×C7—41no—
260C2×C7—161no—
261C2×C7#188811yescharacteristic
262C2×C7#189811yescharacteristic
263C2×C7#191211yescharacteristic
264C23—22no—
265C23—22no—
266C23—41no—
267C23—42no—
268C23—42no—
269C23—41no—
270C23—42no—
271C23C2×C7×C22⋊C411yescharacteristic
272C23C7×#12011yescharacteristic
273C23C7×#12311yescharacteristic
274Q8—48no—
275Q8—48no—
276D4—82no—
277D4—82no—
278D4—82no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4C7×#9914yes—
302C2×C4C4×C7×D412yes—
303C2×C4C7×#12112yes—
304C8—82no—
305C7#946611yescharacteristic, direct product factor, Sylow
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—44no—
314C22—84no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22C7×#38611yescharacteristic
319C22C2×C7×#9912yes—
320C22C7×#45412yes—
321C4—22no—
322C4—24no—
323C4—22no—
324C4—42no—
325C4—84no—
326C4—82no—
327C4—81no—
328C2—22no—
329C2—22no—
330C2—41no—
331C2—161no—
332C2C7×#188811yescharacteristic
333C2C7×#189811yescharacteristic
334C2C7×#191211yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC2×C7×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2310C2semidirect
3. C7×#1784C2semidirect
5. C7×#1284C2semidirect
7. C2×C7×#124C4semidirect
14. C7×#391C4semidirect
21. #9466C7direct (N × H)
56. #2310C2×C7semidirect
57. #1784C2×C7semidirect
59. #1284C2×C7semidirect
104. C2×#124C4×C7semidirect
111. #391C4×C7semidirect
305. C7#9466direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24362
C7×#23102
C7×#17842
C7×#15102
C7×#12842
#94667