Finite Group Databasebeta

← C7×#9409 #13186625

Subgroups — C7×#9409 #13186625

Order 1792.

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

normal subgroups only

This group has 840 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×#2539C211yesmaximal, characteristic
2C7×#2273C211yesmaximal, characteristic, complemented
3C7×#1507C212yesmaximal, complemented
4C7×#1462C211yesmaximal, characteristic, complemented
5C7×#1297C212yesmaximal, complemented
6C2×C7×#123—21no—
7C2×C7×#121C2211yescharacteristic
8C2×C7×#120—21no—
9C22×C7×C22⋊C4—21no—
10C22×C7×C22⋊C4C2211yescharacteristic
11C7×#407C2212yes—
12C7×#399C412yescomplemented
13C7×#392C412yescomplemented
14C7×#390—22no—
15C4×C7×C22⋊C4—21no—
16C4×C7×C22⋊C4—24no—
17C4×C7×C22⋊C4C2211yescharacteristic
18C4×C7×C22⋊C4C2212yes—
19C7×#327—24no—
20C7×#323—22no—
21#9409C711yesmaximal, characteristic, direct product factor, Sylow
22C25×C7D411yescharacteristic
23C22×C7×D4—21no—
24C22×C7×D4—22no—
25C22×C7×D4C2×C411yescharacteristic, complemented
26C23×C4×C7—22no—
27C23×C4×C7D411yescharacteristic
28C23×C4×C7C2311yescharacteristic
29C2×C7×M16—22no—
30C7×#123—44no—
31C7×#121—22no—
32C7×#121—22no—
33C7×#120—44no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4C2×C411yescharacteristic, complemented
36C2×C7×C22⋊C4—28no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4C2×C412yes—
46C2×C7×C22⋊C4D412yes—
47C2×C42×C7—22no—
48C2×C42×C7—22no—
49C2×C42×C7—21no—
50C7×C22⋊C8—22no—
51C7×C22⋊C8—22no—
52C7×#95—22no—
53C7×#95C2×C412yescomplemented
54C7×#95D414yescomplemented
55#2539C2×C711yescharacteristic
56#2273C2×C711yescharacteristic, complemented
57#1507C2×C712yescomplemented
58#1462C2×C711yescharacteristic, complemented
59#1297C2×C712yescomplemented
60C24×C7—21no—
61C24×C7—24no—
62C24×C7—21no—
63C24×C7—21no—
64C24×C7—42no—
65C24×C7—81no—
66C24×C7C22⋊C412yes—
67C24×C7C2×D411yescharacteristic
68C2×C7×D4—22no—
69C2×C7×D4—44no—
70C2×C7×D4—42no—
71C2×C7×D4—42no—
72C2×C7×D4—44no—
73C2×C7×D4—41no—
74C2×C7×D4—41no—
75C2×C7×D4—82no—
76C2×C7×D4—81no—
77C22×C4×C7—22no—
78C22×C4×C7—21no—
79C22×C4×C7—24no—
80C22×C4×C7—28no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—42no—
84C22×C4×C7—81no—
85C22×C4×C7—82no—
86C22×C4×C7—82no—
87C22×C4×C7C22⋊C412yes—
88C22×C4×C7C22×C411yescharacteristic
89C22×C4×C7C2×D412yes—
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7#4112yes—
92C7×M16—82no—
93C2×C7×C8—42no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—44no—
96C7×C22⋊C4—28no—
97C7×C22⋊C4—28no—
98C7×C22⋊C4—24no—
99C7×C22⋊C4—24no—
100C7×C22⋊C4—24no—
101C7×C22⋊C4—44no—
102C7×C22⋊C4—44no—
103C7×C22⋊C4—42no—
104C42×C7—82no—
105C42×C7—81no—
106C42×C7—82no—
107C2×#123—21no—
108C2×#121C22×C711yescharacteristic
109C2×#120—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4C22×C711yescharacteristic
112#407C22×C712yes—
113#399C4×C712yescomplemented
114#392C4×C712yescomplemented
115#390—22no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—24no—
118C4×C22⋊C4C22×C711yescharacteristic
119C4×C22⋊C4C22×C712yes—
120#327—24no—
121#323—22no—
122C23×C7—22no—
123C23×C7—22no—
124C23×C7—24no—
125C23×C7—22no—
126C23×C7—22no—
127C23×C7—22no—
128C23×C7—22no—
129C23×C7—41no—
130C23×C7—42no—
131C23×C7—42no—
132C23×C7—42no—
133C23×C7—41no—
134C23×C7—41no—
135C23×C7—42no—
136C23×C7—42no—
137C23×C7—41no—
138C23×C7—42no—
139C23×C7—81no—
140C23×C7—81no—
141C23×C7—88no—
142C23×C7#10414yes—
143C23×C7C2×C22⋊C411yescharacteristic
144C23×C7#12011yescharacteristic
145C23×C7#12311yescharacteristic
146C7×D4—82no—
147C7×D4—88no—
148C7×D4—82no—
149C7×D4—82no—
150C2×C4×C7—28no—
151C2×C4×C7—24no—
152C2×C4×C7—22no—
153C2×C4×C7—41no—
154C2×C4×C7—42no—
155C2×C4×C7—42no—
156C2×C4×C7—41no—
157C2×C4×C7—44no—
158C2×C4×C7—42no—
159C2×C4×C7—42no—
160C2×C4×C7—42no—
161C2×C4×C7—44no—
162C2×C4×C7—42no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—81no—
167C2×C4×C7—88no—
168C2×C4×C7C4×D412yes—
169C2×C4×C7#12112yes—
170C7×C8—82no—
171C25C7×D411yescharacteristic
172C22×D4—21no—
173C22×D4—22no—
174C22×D4C2×C4×C711yescharacteristic, complemented
175C23×C4—22no—
176C23×C4C7×D411yescharacteristic
177C23×C4C23×C711yescharacteristic
178C2×M16—22no—
179#123—44no—
180#121—22no—
181#121—22no—
182#120—44no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C2×C4×C711yescharacteristic, complemented
185C2×C22⋊C4—28no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4C2×C4×C712yes—
195C2×C22⋊C4C7×D412yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201#95—22no—
202#95C2×C4×C712yescomplemented
203#95C7×D414yescomplemented
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—24no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—44no—
213C22×C7—44no—
214C22×C7—44no—
215C22×C7—44no—
216C22×C7—42no—
217C22×C7—42no—
218C22×C7—82no—
219C22×C7—82no—
220C22×C7—82no—
221C22×C7—84no—
222C22×C7—81no—
223C22×C7—81no—
224C22×C7—81no—
225C22×C7#38611yescharacteristic
226C22×C7C2×#10412yes—
227C22×C7#45312yes—
228C4×C7—22no—
229C4×C7—22no—
230C4×C7—48no—
231C4×C7—44no—
232C4×C7—42no—
233C4×C7—81no—
234C24—21no—
235C24—24no—
236C24—21no—
237C24—21no—
238C24—42no—
239C24—81no—
240C24C7×C22⋊C412yes—
241C24C2×C7×D411yescharacteristic
242C2×D4—22no—
243C2×D4—44no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—44no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—82no—
250C2×D4—81no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—28no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—81no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4C7×C22⋊C412yes—
262C22×C4C22×C4×C711yescharacteristic
263C22×C4C2×C7×D412yes—
264C22×C4C2×C7×D411yescharacteristic
265C22×C4C7×#4112yes—
266M16—82no—
267C2×C8—42no—
268C4⋊C4—22no—
269C4⋊C4—44no—
270C22⋊C4—28no—
271C22⋊C4—28no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—42no—
278C42—82no—
279C42—81no—
280C42—82no—
281C2×C7—22no—
282C2×C7—22no—
283C2×C7—41no—
284C2×C7—44no—
285C2×C7—161no—
286C2×C7#188711yescharacteristic
287C2×C7#189611yescharacteristic
288C2×C7#191311yescharacteristic
289C23—22no—
290C23—22no—
291C23—24no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—41no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—41no—
301C23—41no—
302C23—42no—
303C23—42no—
304C23—41no—
305C23—42no—
306C23—81no—
307C23—81no—
308C23—88no—
309C23C7×#10414yes—
310C23C2×C7×C22⋊C411yescharacteristic
311C23C7×#12011yescharacteristic
312C23C7×#12311yescharacteristic
313D4—82no—
314D4—88no—
315D4—82no—
316D4—82no—
317C2×C4—28no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—88no—
335C2×C4C4×C7×D412yes—
336C2×C4C7×#12112yes—
337C8—82no—
338C7#940911yescharacteristic, direct product factor, Sylow
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—24no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—84no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22C7×#38611yescharacteristic
361C22C2×C7×#10412yes—
362C22C7×#45312yes—
363C4—22no—
364C4—22no—
365C4—48no—
366C4—44no—
367C4—42no—
368C4—81no—
369C2—22no—
370C2—22no—
371C2—41no—
372C2—44no—
373C2—161no—
374C2C7×#188711yescharacteristic
375C2C7×#189611yescharacteristic
376C2C7×#191311yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC7×#2539C2
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×#2273C2semidirect
3. C7×#1507C2semidirect
4. C7×#1462C2semidirect
5. C7×#1297C2semidirect
12. C7×#399C4semidirect
13. C7×#392C4semidirect
21. #9409C7direct (N × H)
25. C22×C7×D4C2×C4semidirect
35. C2×C7×C4⋊C4C2×C4semidirect
53. C7×#95C2×C4semidirect
54. C7×#95D4semidirect
56. #2273C2×C7semidirect
57. #1507C2×C7semidirect
58. #1462C2×C7semidirect
59. #1297C2×C7semidirect
113. #399C4×C7semidirect
114. #392C4×C7semidirect
174. C22×D4C2×C4×C7semidirect
184. C2×C4⋊C4C2×C4×C7semidirect
202. #95C2×C4×C7semidirect
203. #95C7×D4semidirect
338. C7#9409direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25392
C7×#22732
C7×#15072
C7×#14622
C7×#12972
#94097