Finite Group Databasebeta

← C7×#9408 #13186624

Subgroups — C7×#9408 #13186624

Order 1792.

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

normal subgroups only

This group has 618 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×#2583C211yesmaximal, characteristic
2C7×#2274C211yesmaximal, characteristic
3C7×#1508C212yesmaximal
4C7×#1463C211yesmaximal, characteristic
5C7×#1297C212yesmaximal
6C2×C7×C22⋊Q8C2211yescharacteristic
7C2×C7×#117—21no—
8C22×C7×C4⋊C4C2211yescharacteristic
9C7×#407C2212yes—
10C7×#398C412yescomplemented
11C7×#395C412yescomplemented
12C7×#389—22no—
13C7×#385—21no—
14C7×#385—21no—
15C4×C7×C4⋊C4—24no—
16C4×C7×C22⋊C4—21no—
17C4×C7×C22⋊C4C2211yescharacteristic
18C4×C7×C22⋊C4C2212yes—
19C7×#328—24no—
20C7×#324—22no—
21#9408C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×Q8C2×C411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C2×C7×M16—22no—
27C7×C22⋊Q8—22no—
28C7×C22⋊Q8—22no—
29C7×#117—28no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4C2×C412yes—
38C2×C7×C4⋊C4D412yes—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C42×C7—22no—
43C2×C42×C7—22no—
44C2×C42×C7—22no—
45C2×C42×C7—21no—
46C7×C22⋊C8—22no—
47C7×C22⋊C8—22no—
48C7×#95—22no—
49C7×#95—22no—
50C7×#95—22no—
51C7×#95C2×C412yes—
52C7×#95D414yes—
53#2583C2×C711yescharacteristic
54#2274C2×C711yescharacteristic
55#1508C2×C712yes—
56#1463C2×C711yescharacteristic
57#1297C2×C712yes—
58C24×C7C2×D411yescharacteristic
59C2×C7×Q8—22no—
60C2×C7×Q8—81no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—24no—
64C22×C4×C7—22no—
65C22×C4×C7—21no—
66C22×C4×C7—24no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—81no—
71C22×C4×C7—81no—
72C22×C4×C7C22⋊C412yes—
73C22×C4×C7C22⋊C412yes—
74C22×C4×C7C22×C411yescharacteristic
75C22×C4×C7C2×D412yes—
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7#4112yes—
78C7×M16—82no—
79C2×C7×C8—42no—
80C7×C4⋊C4—28no—
81C7×C4⋊C4—24no—
82C7×C4⋊C4—24no—
83C7×C4⋊C4—22no—
84C7×C4⋊C4—44no—
85C7×C4⋊C4—42no—
86C7×C4⋊C4—82no—
87C7×C22⋊C4—24no—
88C7×C22⋊C4—28no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—41no—
91C42×C7—28no—
92C42×C7—82no—
93C42×C7—81no—
94C42×C7—82no—
95C2×C22⋊Q8C22×C711yescharacteristic
96C2×#117—21no—
97C22×C4⋊C4C22×C711yescharacteristic
98#407C22×C712yes—
99#398C4×C712yescomplemented
100#395C4×C712yescomplemented
101#389—22no—
102#385—21no—
103#385—21no—
104C4×C4⋊C4—24no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4C22×C711yescharacteristic
107C4×C22⋊C4C22×C712yes—
108#328—24no—
109#324—22no—
110C23×C7—22no—
111C23×C7—22no—
112C23×C7—41no—
113C23×C7C2×C22⋊C411yescharacteristic
114C23×C7#12011yescharacteristic
115C23×C7#12311yescharacteristic
116C7×Q8—82no—
117C2×C4×C7—22no—
118C2×C4×C7—22no—
119C2×C4×C7—24no—
120C2×C4×C7—22no—
121C2×C4×C7—22no—
122C2×C4×C7—28no—
123C2×C4×C7—24no—
124C2×C4×C7—22no—
125C2×C4×C7—41no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—41no—
129C2×C4×C7—44no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—44no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—42no—
139C2×C4×C7—81no—
140C2×C4×C7#10414yes—
141C2×C4×C7C4×D412yes—
142C2×C4×C7#12112yes—
143C7×C8—82no—
144C22×Q8C2×C4×C711yescharacteristic
145C23×C4C7×D411yescharacteristic
146C23×C4C7×D411yescharacteristic
147C23×C4C23×C711yescharacteristic
148C2×M16—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151#117—28no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C2×C4×C711yescharacteristic
159C2×C4⋊C4C2×C4×C712yes—
160C2×C4⋊C4C7×D412yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—21no—
168C22⋊C8—22no—
169C22⋊C8—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95C2×C4×C712yes—
174#95C7×D414yes—
175C22×C7—21no—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—21no—
179C22×C7—21no—
180C22×C7—21no—
181C22×C7—21no—
182C22×C7—44no—
183C22×C7#38611yescharacteristic
184C22×C7C2×#10412yes—
185C22×C7#45612yes—
186C4×C7—24no—
187C4×C7—22no—
188C4×C7—22no—
189C4×C7—48no—
190C4×C7—44no—
191C4×C7—42no—
192C4×C7—81no—
193C4×C7—81no—
194C24C2×C7×D411yescharacteristic
195C2×Q8—22no—
196C2×Q8—81no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4C7×C22⋊C412yes—
209C22×C4C7×C22⋊C412yes—
210C22×C4C22×C4×C711yescharacteristic
211C22×C4C2×C7×D412yes—
212C22×C4C2×C7×D411yescharacteristic
213C22×C4C7×#4112yes—
214M16—82no—
215C2×C8—42no—
216C4⋊C4—28no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—22no—
220C4⋊C4—44no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C22⋊C4—28no—
224C22⋊C4—24no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C42—28no—
228C42—82no—
229C42—81no—
230C42—82no—
231C2×C7—22no—
232C2×C7—22no—
233C2×C7—41no—
234C2×C7#188611yescharacteristic
235C2×C7#189611yescharacteristic
236C2×C7#191311yescharacteristic
237C23—22no—
238C23—22no—
239C23—41no—
240C23C2×C7×C22⋊C411yescharacteristic
241C23C7×#12011yescharacteristic
242C23C7×#12311yescharacteristic
243Q8—82no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—28no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—81no—
267C2×C4C7×#10414yes—
268C2×C4C4×C7×D412yes—
269C2×C4C7×#12112yes—
270C8—82no—
271C7#940811yescharacteristic, direct product factor, Sylow
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22C7×#38611yescharacteristic
281C22C2×C7×#10412yes—
282C22C7×#45612yes—
283C4—24no—
284C4—22no—
285C4—22no—
286C4—48no—
287C4—44no—
288C4—42no—
289C4—81no—
290C4—81no—
291C2—22no—
292C2—22no—
293C2—41no—
294C2C7×#188611yescharacteristic
295C2C7×#189611yescharacteristic
296C2C7×#191311yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC24×C7C2×D4
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
10. C7×#398C4semidirect
11. C7×#395C4semidirect
21. #9408C7direct (N × H)
99. #398C4×C7semidirect
100. #395C4×C7semidirect
271. C7#9408direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25832
C7×#22742
C7×#15082
C7×#14632
C7×#12972
#94087