Finite Group Databasebeta

← C7×#9464 #13186680

Subgroups — C7×#9464 #13186680

Order 1792.

610 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2501C211yesmaximal, characteristic
2C7×#2313C211yesmaximal, characteristic, complemented
3C7×#1821C211yesmaximal, characteristic, complemented
4C7×#1510C212yesmaximal
5C7×#1284C212yesmaximal, complemented
6C2×C7×#123—21no—
7C2×C7×#123—21no—
8C2×C7×#121C2211yescharacteristic
9C22×C7×C4⋊C4C2211yescharacteristic
10C7×#407C2212yes—
11C7×#399—22no—
12C7×#390C412yescomplemented
13C7×#387—24no—
14C7×#384C412yescomplemented
15C7×#381C2212yes—
16C7×#380—21no—
17C4×C7×C22⋊C4—21no—
18C4×C7×C22⋊C4C2211yescharacteristic
19C7×#343—22no—
20C7×#327—24no—
21#9464C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×D4—21no—
23C22×C7×D4C2×C411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C2×C7×M16—22no—
28C7×#123—44no—
29C7×#123—44no—
30C7×#121—22no—
31C7×#121—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4—22no—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4C2×C412yes—
39C2×C7×C4⋊C4D412yes—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4—21no—
45C2×C42×C7—22no—
46C2×C42×C7C2×C412yes—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—22no—
49C7×#95—22no—
50C7×#95—22no—
51C7×#95—21no—
52C7×#95—22no—
53C7×#95D414yes—
54#2501C2×C711yescharacteristic
55#2313C2×C711yescharacteristic, complemented
56#1821C2×C711yescharacteristic, complemented
57#1510C2×C712yes—
58#1284C2×C712yescomplemented
59C24×C7—21no—
60C24×C7C2×D411yescharacteristic
61C2×C7×D4—22no—
62C2×C7×D4—42no—
63C2×C7×D4—42no—
64C2×C7×D4—42no—
65C2×C7×D4—41no—
66C2×C7×D4—41no—
67C22×C4×C7—22no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—24no—
74C22×C4×C7—42no—
75C22×C4×C7—42no—
76C22×C4×C7—42no—
77C22×C4×C7C22⋊C412yes—
78C22×C4×C7C22⋊C412yes—
79C22×C4×C7C22×C411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7C2×D412yes—
82C22×C4×C7#4112yes—
83C7×M16—44no—
84C2×C7×C8—42no—
85C7×C4⋊C4—24no—
86C7×C4⋊C4—24no—
87C7×C4⋊C4—22no—
88C7×C4⋊C4—44no—
89C7×C4⋊C4—44no—
90C7×C4⋊C4—82no—
91C7×C22⋊C4—24no—
92C7×C22⋊C4—44no—
93C7×C22⋊C4—44no—
94C7×C22⋊C4—44no—
95C7×C22⋊C4—42no—
96C42×C7—42no—
97C42×C7—82no—
98C2×#123—21no—
99C2×#123—21no—
100C2×#121C22×C711yescharacteristic
101C22×C4⋊C4C22×C711yescharacteristic
102#407C22×C712yes—
103#399—22no—
104#390C4×C712yescomplemented
105#387—24no—
106#384C4×C712yescomplemented
107#381C22×C712yes—
108#380—21no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4C22×C711yescharacteristic
111#343—22no—
112#327—24no—
113C23×C7—22no—
114C23×C7—22no—
115C23×C7—41no—
116C23×C7—41no—
117C23×C7—41no—
118C23×C7—42no—
119C23×C7—41no—
120C23×C7—42no—
121C23×C7C2×C22⋊C411yescharacteristic
122C23×C7#12011yescharacteristic
123C23×C7#12311yescharacteristic
124C7×D4—82no—
125C7×D4—82no—
126C7×D4—82no—
127C2×C4×C7—22no—
128C2×C4×C7—22no—
129C2×C4×C7—24no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—44no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—44no—
144C2×C4×C7—41no—
145C2×C4×C7—44no—
146C2×C4×C7—81no—
147C2×C4×C7—81no—
148C2×C4×C7C4×D412yes—
149C2×C4×C7#12112yes—
150C7×C8—82no—
151C22×D4—21no—
152C22×D4C2×C4×C711yescharacteristic
153C23×C4C7×D411yescharacteristic
154C23×C4C7×D411yescharacteristic
155C23×C4C23×C711yescharacteristic
156C2×M16—22no—
157#123—44no—
158#123—44no—
159#121—22no—
160#121—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C2×C4×C711yescharacteristic
167C2×C4⋊C4C2×C4×C712yes—
168C2×C4⋊C4C7×D412yes—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—21no—
174C2×C42—22no—
175C2×C42C2×C4×C712yes—
176C22⋊C8—22no—
177C22⋊C8—22no—
178#95—22no—
179#95—22no—
180#95—21no—
181#95—22no—
182#95C7×D414yes—
183C22×C7—21no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—44no—
191C22×C7—82no—
192C22×C7—82no—
193C22×C7—81no—
194C22×C7—81no—
195C22×C7—81no—
196C22×C7#38611yescharacteristic
197C22×C7#41012yes—
198C22×C7#45312yes—
199C4×C7—22no—
200C4×C7—22no—
201C4×C7—44no—
202C4×C7—42no—
203C4×C7—42no—
204C4×C7—84no—
205C4×C7—81no—
206C24—21no—
207C24C2×C7×D411yescharacteristic
208C2×D4—22no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—41no—
213C2×D4—41no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4C7×C22⋊C412yes—
225C22×C4C7×C22⋊C412yes—
226C22×C4C22×C4×C711yescharacteristic
227C22×C4C2×C7×D411yescharacteristic
228C22×C4C2×C7×D412yes—
229C22×C4C7×#4112yes—
230M16—44no—
231C2×C8—42no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—82no—
238C22⋊C4—24no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C22⋊C4—42no—
243C42—42no—
244C42—82no—
245C2×C7—22no—
246C2×C7—22no—
247C2×C7—41no—
248C2×C7—161no—
249C2×C7#188711yescharacteristic
250C2×C7#189911yescharacteristic
251C2×C7#191211yescharacteristic
252C23—22no—
253C23—22no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—42no—
258C23—41no—
259C23—42no—
260C23C2×C7×C22⋊C411yescharacteristic
261C23C7×#12011yescharacteristic
262C23C7×#12311yescharacteristic
263D4—82no—
264D4—82no—
265D4—82no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—44no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4C4×C7×D412yes—
288C2×C4C7×#12112yes—
289C8—82no—
290C7#946411yescharacteristic, direct product factor, Sylow
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—82no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22C7×#38611yescharacteristic
305C22C7×#41012yes—
306C22C7×#45312yes—
307C4—22no—
308C4—22no—
309C4—44no—
310C4—42no—
311C4—42no—
312C4—84no—
313C4—81no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—161no—
318C2C7×#188711yescharacteristic
319C2C7×#189911yescharacteristic
320C2C7×#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×#2313C2semidirect
3. C7×#1821C2semidirect
5. C7×#1284C2semidirect
12. C7×#390C4semidirect
14. C7×#384C4semidirect
21. #9464C7direct (N × H)
55. #2313C2×C7semidirect
56. #1821C2×C7semidirect
58. #1284C2×C7semidirect
104. #390C4×C7semidirect
106. #384C4×C7semidirect
290. C7#9464direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25012
C7×#23132
C7×#18212
C7×#15102
C7×#12842
#94647