Finite Group Databasebeta

← C7×#9465 #13186681

Subgroups — C7×#9465 #13186681

Order 1792.

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

normal subgroups only

This group has 554 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×#2504C211yesmaximal, characteristic
2C7×#2313C211yesmaximal, characteristic, complemented
3C7×#1821C211yesmaximal, characteristic, complemented
4C7×#1510C212yesmaximal
5C7×#1284C212yesmaximal, complemented
6C2×C7×#121C2211yescharacteristic
7C22×C7×C4⋊C4C2211yescharacteristic
8C7×#407C2212yes—
9C7×#397—22no—
10C7×#390C412yescomplemented
11C7×#387—24no—
12C7×#386—21no—
13C7×#386—21no—
14C7×#382C412yescomplemented
15C7×#381C2212yes—
16C7×#380—21no—
17C4×C7×C22⋊C4—21no—
18C4×C7×C22⋊C4C2211yescharacteristic
19C7×#343—22no—
20C7×#327—24no—
21#9465C711yesmaximal, 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×#121—22no—
29C7×#121—22no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4C2×C412yes—
35C2×C7×C4⋊C4D412yes—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C42×C7—22no—
42C2×C42×C7C2×C412yes—
43C7×C22⋊C8—22no—
44C7×C22⋊C8—22no—
45C7×#95—22no—
46C7×#95—22no—
47C7×#95—22no—
48C7×#95—22no—
49C7×#95—21no—
50C7×#95—22no—
51C7×#95D414yes—
52#2504C2×C711yescharacteristic
53#2313C2×C711yescharacteristic, complemented
54#1821C2×C711yescharacteristic, complemented
55#1510C2×C712yes—
56#1284C2×C712yescomplemented
57C24×C7—21no—
58C24×C7C2×D411yescharacteristic
59C2×C7×D4—22no—
60C2×C7×D4—42no—
61C2×C7×D4—44no—
62C2×C7×D4—41no—
63C2×C7×D4—41no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—22no—
70C22×C4×C7—24no—
71C22×C4×C7—42no—
72C22×C4×C7—81no—
73C22×C4×C7—81no—
74C22×C4×C7C22⋊C412yes—
75C22×C4×C7C22⋊C412yes—
76C22×C4×C7C22×C411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7C2×D412yes—
79C22×C4×C7#4112yes—
80C7×M16—44no—
81C2×C7×C8—42no—
82C7×C4⋊C4—24no—
83C7×C4⋊C4—24no—
84C7×C4⋊C4—22no—
85C7×C4⋊C4—82no—
86C7×C22⋊C4—44no—
87C7×C22⋊C4—42no—
88C7×C22⋊C4—42no—
89C7×C22⋊C4—82no—
90C7×C22⋊C4—82no—
91C42×C7—42no—
92C42×C7—82no—
93C2×#121C22×C711yescharacteristic
94C22×C4⋊C4C22×C711yescharacteristic
95#407C22×C712yes—
96#397—22no—
97#390C4×C712yescomplemented
98#387—24no—
99#386—21no—
100#386—21no—
101#382C4×C712yescomplemented
102#381C22×C712yes—
103#380—21no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4C22×C711yescharacteristic
106#343—22no—
107#327—24no—
108C23×C7—22no—
109C23×C7—22no—
110C23×C7—41no—
111C23×C7—42no—
112C23×C7—42no—
113C23×C7—41no—
114C23×C7—42no—
115C23×C7C2×C22⋊C411yescharacteristic
116C23×C7#12011yescharacteristic
117C23×C7#12311yescharacteristic
118C7×D4—82no—
119C7×D4—82no—
120C7×D4—82no—
121C2×C4×C7—22no—
122C2×C4×C7—22no—
123C2×C4×C7—22no—
124C2×C4×C7—22no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—44no—
132C2×C4×C7—42no—
133C2×C4×C7—44no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—41no—
137C2×C4×C7—42no—
138C2×C4×C7—44no—
139C2×C4×C7—81no—
140C2×C4×C7—81no—
141C2×C4×C7C4×D412yes—
142C2×C4×C7#12112yes—
143C7×C8—82no—
144C22×D4—21no—
145C22×D4C2×C4×C711yescharacteristic
146C23×C4C7×D411yescharacteristic
147C23×C4C7×D411yescharacteristic
148C23×C4C23×C711yescharacteristic
149C2×M16—22no—
150#121—22no—
151#121—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C2×C4×C711yescharacteristic
156C2×C4⋊C4C2×C4×C712yes—
157C2×C4⋊C4C7×D412yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C42—22no—
164C2×C42C2×C4×C712yes—
165C22⋊C8—22no—
166C22⋊C8—22no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—21no—
172#95—22no—
173#95C7×D414yes—
174C22×C7—21no—
175C22×C7—21no—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—21no—
179C22×C7—21no—
180C22×C7—21no—
181C22×C7—44no—
182C22×C7—84no—
183C22×C7—81no—
184C22×C7—81no—
185C22×C7—81no—
186C22×C7#38611yescharacteristic
187C22×C7#41012yes—
188C22×C7#45412yes—
189C4×C7—22no—
190C4×C7—22no—
191C4×C7—42no—
192C4×C7—42no—
193C4×C7—84no—
194C4×C7—82no—
195C4×C7—81no—
196C24—21no—
197C24C2×C7×D411yescharacteristic
198C2×D4—22no—
199C2×D4—42no—
200C2×D4—44no—
201C2×D4—41no—
202C2×D4—41no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4C7×C22⋊C412yes—
214C22×C4C7×C22⋊C412yes—
215C22×C4C22×C4×C711yescharacteristic
216C22×C4C2×C7×D411yescharacteristic
217C22×C4C2×C7×D412yes—
218C22×C4C7×#4112yes—
219M16—44no—
220C2×C8—42no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—22no—
224C4⋊C4—82no—
225C22⋊C4—44no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C42—42no—
231C42—82no—
232C2×C7—22no—
233C2×C7—22no—
234C2×C7—41no—
235C2×C7—161no—
236C2×C7#188811yescharacteristic
237C2×C7#189911yescharacteristic
238C2×C7#191111yescharacteristic
239C23—22no—
240C23—22no—
241C23—41no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—42no—
246C23C2×C7×C22⋊C411yescharacteristic
247C23C7×#12011yescharacteristic
248C23C7×#12311yescharacteristic
249D4—82no—
250D4—82no—
251D4—82no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4C4×C7×D412yes—
273C2×C4C7×#12112yes—
274C8—82no—
275C7#946511yescharacteristic, direct product factor, Sylow
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—84no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22C7×#38611yescharacteristic
289C22C7×#41012yes—
290C22C7×#45412yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—84no—
296C4—82no—
297C4—81no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—161no—
302C2C7×#188811yescharacteristic
303C2C7×#189911yescharacteristic
304C2C7×#191111yescharacteristic
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
10. C7×#390C4semidirect
14. C7×#382C4semidirect
21. #9465C7direct (N × H)
53. #2313C2×C7semidirect
54. #1821C2×C7semidirect
56. #1284C2×C7semidirect
97. #390C4×C7semidirect
101. #382C4×C7semidirect
275. C7#9465direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25042
C7×#23132
C7×#18212
C7×#15102
C7×#12842
#94657