Finite Group Databasebeta

← C7×#9407 #13186623

Subgroups — C7×#9407 #13186623

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×#2561C211yesmaximal, characteristic
2C7×#2273C211yesmaximal, characteristic
3C7×#1508C212yesmaximal, complemented
4C7×#1462C211yesmaximal, characteristic
5C7×#1297C212yesmaximal, complemented
6C2×C7×C22⋊Q8C2211yescharacteristic
7C22×C7×C22⋊C4—21no—
8C22×C7×C22⋊C4C2211yescharacteristic
9C7×#407C2212yes—
10C7×#397C412yescomplemented
11C7×#393C412yescomplemented
12C7×#388—22no—
13C7×#381—21no—
14C7×#379—21no—
15C4×C7×C22⋊C4—21no—
16C4×C7×C22⋊C4—24no—
17C4×C7×C22⋊C4C2211yescharacteristic
18C4×C7×C22⋊C4C2212yes—
19C7×#328—24no—
20C7×#323—22no—
21#9407C711yesmaximal, characteristic, direct product factor, Sylow
22C25×C7D411yescharacteristic
23C22×C7×Q8C2×C411yescharacteristic, complemented
24C23×C4×C7—22no—
25C23×C4×C7D411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C2×C7×M16—22no—
28C7×C22⋊Q8—22no—
29C7×C22⋊Q8—22no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4C2×C411yescharacteristic, complemented
32C2×C7×C22⋊C4—28no—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4C2×C412yes—
42C2×C7×C22⋊C4D412yes—
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×C412yescomplemented
52C7×#95D414yes—
53#2561C2×C711yescharacteristic
54#2273C2×C711yescharacteristic
55#1508C2×C712yescomplemented
56#1462C2×C711yescharacteristic
57#1297C2×C712yescomplemented
58C24×C7—21no—
59C24×C7—24no—
60C24×C7—21no—
61C24×C7—42no—
62C24×C7—81no—
63C24×C7C22⋊C412yes—
64C24×C7C2×D411yescharacteristic
65C2×C7×Q8—22no—
66C2×C7×Q8—81no—
67C22×C4×C7—21no—
68C22×C4×C7—24no—
69C22×C4×C7—28no—
70C22×C4×C7—21no—
71C22×C4×C7—22no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—42no—
75C22×C4×C7—81no—
76C22×C4×C7—82no—
77C22×C4×C7—82no—
78C22×C4×C7C22⋊C412yes—
79C22×C4×C7C22×C411yescharacteristic
80C22×C4×C7C2×D412yes—
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7#4112yes—
83C7×M16—82no—
84C2×C7×C8—42no—
85C7×C4⋊C4—22no—
86C7×C4⋊C4—42no—
87C7×C22⋊C4—28no—
88C7×C22⋊C4—28no—
89C7×C22⋊C4—24no—
90C7×C22⋊C4—24no—
91C7×C22⋊C4—42no—
92C7×C22⋊C4—44no—
93C7×C22⋊C4—41no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—82no—
96C42×C7—82no—
97C42×C7—81no—
98C42×C7—82no—
99C2×C22⋊Q8C22×C711yescharacteristic
100C22×C22⋊C4—21no—
101C22×C22⋊C4C22×C711yescharacteristic
102#407C22×C712yes—
103#397C4×C712yescomplemented
104#393C4×C712yescomplemented
105#388—22no—
106#381—21no—
107#379—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—24no—
110C4×C22⋊C4C22×C711yescharacteristic
111C4×C22⋊C4C22×C712yes—
112#328—24no—
113#323—22no—
114C23×C7—22no—
115C23×C7—22no—
116C23×C7—24no—
117C23×C7—22no—
118C23×C7—22no—
119C23×C7—22no—
120C23×C7—22no—
121C23×C7—41no—
122C23×C7—42no—
123C23×C7—42no—
124C23×C7—42no—
125C23×C7—42no—
126C23×C7—81no—
127C23×C7—81no—
128C23×C7—88no—
129C23×C7#10414yes—
130C23×C7C2×C22⋊C411yescharacteristic
131C23×C7#12011yescharacteristic
132C23×C7#12311yescharacteristic
133C7×Q8—82no—
134C2×C4×C7—28no—
135C2×C4×C7—22no—
136C2×C4×C7—41no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—41no—
140C2×C4×C7—44no—
141C2×C4×C7—42no—
142C2×C4×C7—44no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—81no—
152C2×C4×C7—88no—
153C2×C4×C7C4×D412yes—
154C2×C4×C7#12112yes—
155C7×C8—82no—
156C25C7×D411yescharacteristic
157C22×Q8C2×C4×C711yescharacteristic, complemented
158C23×C4—22no—
159C23×C4C7×D411yescharacteristic
160C23×C4C23×C711yescharacteristic
161C2×M16—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4C2×C4×C711yescharacteristic, complemented
166C2×C22⋊C4—28no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4C2×C4×C712yes—
176C2×C22⋊C4C7×D412yes—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—21no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95C2×C4×C712yescomplemented
186#95C7×D414yes—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—21no—
191C22×C7—21no—
192C22×C7—24no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—44no—
196C22×C7—44no—
197C22×C7—44no—
198C22×C7—44no—
199C22×C7—42no—
200C22×C7—42no—
201C22×C7—82no—
202C22×C7—84no—
203C22×C7#38611yescharacteristic
204C22×C7C2×#10412yes—
205C22×C7#45512yes—
206C4×C7—22no—
207C4×C7—22no—
208C4×C7—48no—
209C4×C7—42no—
210C4×C7—82no—
211C4×C7—81no—
212C4×C7—81no—
213C24—21no—
214C24—24no—
215C24—21no—
216C24—42no—
217C24—81no—
218C24C7×C22⋊C412yes—
219C24C2×C7×D411yescharacteristic
220C2×Q8—22no—
221C2×Q8—81no—
222C22×C4—21no—
223C22×C4—24no—
224C22×C4—28no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—81no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4C7×C22⋊C412yes—
234C22×C4C22×C4×C711yescharacteristic
235C22×C4C2×C7×D412yes—
236C22×C4C2×C7×D411yescharacteristic
237C22×C4C7×#4112yes—
238M16—82no—
239C2×C8—42no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C22⋊C4—28no—
243C22⋊C4—28no—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—42no—
247C22⋊C4—44no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—82no—
251C42—82no—
252C42—81no—
253C42—82no—
254C2×C7—22no—
255C2×C7—22no—
256C2×C7—41no—
257C2×C7—44no—
258C2×C7#188511yescharacteristic
259C2×C7#189611yescharacteristic
260C2×C7#191311yescharacteristic
261C23—22no—
262C23—22no—
263C23—24no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—81no—
274C23—81no—
275C23—88no—
276C23C7×#10414yes—
277C23C2×C7×C22⋊C411yescharacteristic
278C23C7×#12011yescharacteristic
279C23C7×#12311yescharacteristic
280Q8—82no—
281C2×C4—28no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—88no—
300C2×C4C4×C7×D412yes—
301C2×C4C7×#12112yes—
302C8—82no—
303C7#940711yescharacteristic, direct product factor, Sylow
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—24no—
310C22—21no—
311C22—21no—
312C22—44no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22C7×#38611yescharacteristic
321C22C2×C7×#10412yes—
322C22C7×#45512yes—
323C4—22no—
324C4—22no—
325C4—48no—
326C4—42no—
327C4—82no—
328C4—81no—
329C4—81no—
330C2—22no—
331C2—22no—
332C2—41no—
333C2—44no—
334C2C7×#188511yescharacteristic
335C2C7×#189611yescharacteristic
336C2C7×#191311yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC25×C7D4
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
3. C7×#1508C2semidirect
5. C7×#1297C2semidirect
10. C7×#397C4semidirect
11. C7×#393C4semidirect
21. #9407C7direct (N × H)
23. C22×C7×Q8C2×C4semidirect
31. C2×C7×C4⋊C4C2×C4semidirect
51. C7×#95C2×C4semidirect
55. #1508C2×C7semidirect
57. #1297C2×C7semidirect
103. #397C4×C7semidirect
104. #393C4×C7semidirect
157. C22×Q8C2×C4×C7semidirect
165. C2×C4⋊C4C2×C4×C7semidirect
185. #95C2×C4×C7semidirect
303. C7#9407direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25612
C7×#22732
C7×#15082
C7×#14622
C7×#12972
#94077