Finite Group Databasebeta

← C7×#11560 #13188776

Subgroups — C7×#11560 #13188776

Order 1792.

534 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2534C211yesmaximal, characteristic
2C7×#1466C211yesmaximal, characteristic, complemented
3C7×#1459C211yesmaximal, characteristic, complemented
4C7×#1317C212yesmaximal, complemented
5C7×#1316C212yesmaximal, complemented
6C2×C7×#123C411yescharacteristic
7C2×C7×#120C411yescharacteristic
8C2×C4×C7×D4C2211yescharacteristic
9C22×C7×C22⋊C4C411yescharacteristic
10C7×#432—21no—
11C2×C7×C22⋊C8—21no—
12C7×#396—21no—
13C7×#394—21no—
14C7×#388—21no—
15C7×#386—21no—
16C7×#381C2211yescharacteristic
17C7×#380C411yescharacteristic
18C7×#379C2211yescharacteristic
19C7×#324—21no—
20C7×#323—21no—
21C7×#323C2212yes—
22C7×#323C2212yes—
23#11560C711yesmaximal, characteristic, direct product factor, Sylow
24C25×C7C2×C411yescharacteristic
25C22×C7×D4—21no—
26C22×C7×D4C2×C411yescharacteristic
27C23×C4×C7C2×C411yescharacteristic
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C22×C7×C8—41no—
31C7×#123—42no—
32C7×#120—42no—
33C4×C7×D4—41no—
34C4×C7×D4—41no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—41no—
45C2×C7×C22⋊C4—41no—
46C2×C7×C22⋊C4C2×C411yescharacteristic
47C2×C7×C22⋊C4C2×C411yescharacteristic
48C2×C7×C22⋊C4D411yescharacteristic
49C2×C7×C22⋊C4C2311yescharacteristic
50C2×C42×C7D411yescharacteristic
51C7×C4⋊C8—41no—
52C7×C22⋊C8—22no—
53C7×C22⋊C8—22no—
54C7×C22⋊C8—41no—
55C7×#97—41no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95—21no—
59#2534C2×C711yescharacteristic
60#1466C2×C711yescharacteristic, complemented
61#1459C2×C711yescharacteristic, complemented
62#1317C2×C712yescomplemented
63#1316C2×C712yescomplemented
64C24×C7—21no—
65C24×C7—22no—
66C24×C7—41no—
67C24×C7—42no—
68C24×C7—41no—
69C24×C7—81no—
70C24×C7C22⋊C411yescharacteristic
71C24×C7C22×C411yescharacteristic
72C2×C7×D4—22no—
73C2×C7×D4—22no—
74C2×C7×D4—41no—
75C2×C7×D4—42no—
76C2×C7×D4—42no—
77C2×C7×D4—81no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—22no—
86C22×C4×C7—41no—
87C22×C4×C7—41no—
88C22×C4×C7—81no—
89C22×C4×C7—81no—
90C22×C4×C7C22⋊C411yescharacteristic
91C22×C4×C7C22⋊C411yescharacteristic
92C22×C4×C7C22⋊C411yescharacteristic
93C22×C4×C7C2×D411yescharacteristic
94C22×C4×C7C2×D411yescharacteristic
95C2×C7×C8—42no—
96C2×C7×C8—42no—
97C2×C7×C8—82no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—42no—
100C7×C4⋊C4—81no—
101C7×C22⋊C4—41no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—42no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—42no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—81no—
112C42×C7—21no—
113C42×C7—21no—
114C2×#123C4×C711yescharacteristic
115C2×#120C4×C711yescharacteristic
116C2×C4×D4C22×C711yescharacteristic
117C22×C22⋊C4C4×C711yescharacteristic
118#432—21no—
119C2×C22⋊C8—21no—
120#396—21no—
121#394—21no—
122#388—21no—
123#386—21no—
124#381C22×C711yescharacteristic
125#380C4×C711yescharacteristic
126#379C22×C711yescharacteristic
127#324—21no—
128#323—21no—
129#323C22×C712yes—
130#323C22×C712yes—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—22no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—42no—
144C23×C7—42no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—82no—
149C23×C7—84no—
150C23×C7—84no—
151C23×C7—82no—
152C23×C7#10012yes—
153C23×C7C2×C22⋊C411yescharacteristic
154C23×C7#13112yes—
155C7×D4—44no—
156C7×D4—84no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—41no—
164C2×C4×C7—42no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—81no—
178C2×C4×C7—81no—
179C2×C4×C7—81no—
180C2×C4×C7—82no—
181C2×C4×C7#9911yescharacteristic
182C2×C4×C7#9911yescharacteristic
183C7×C8—82no—
184C7×C8—82no—
185C25C2×C4×C711yescharacteristic
186C22×D4—21no—
187C22×D4C2×C4×C711yescharacteristic
188C23×C4C2×C4×C711yescharacteristic
189C23×C4C2×C4×C711yescharacteristic
190C23×C4C7×D411yescharacteristic
191C22×C8—41no—
192#123—42no—
193#120—42no—
194C4×D4—41no—
195C4×D4—41no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C7×D411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4C2×C4×C711yescharacteristic
208C2×C22⋊C4C2×C4×C711yescharacteristic
209C2×C22⋊C4C7×D411yescharacteristic
210C2×C22⋊C4C23×C711yescharacteristic
211C2×C42C7×D411yescharacteristic
212C4⋊C8—41no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—41no—
216#97—41no—
217#95—21no—
218#95—21no—
219#95—21no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7—22no—
223C22×C7—22no—
224C22×C7—42no—
225C22×C7—42no—
226C22×C7—42no—
227C22×C7—42no—
228C22×C7—42no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7—41no—
233C22×C7—42no—
234C22×C7—81no—
235C22×C7—81no—
236C22×C7—81no—
237C22×C7—84no—
238C22×C7—84no—
239C22×C7—81no—
240C22×C7—81no—
241C22×C7#40811yescharacteristic
242C22×C7C2×#9911yescharacteristic
243C22×C7C2×#10011yescharacteristic
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—41no—
247C4×C7—41no—
248C4×C7—41no—
249C4×C7—41no—
250C4×C7—81no—
251C4×C7—81no—
252C4×C7—81no—
253C24—21no—
254C24—22no—
255C24—41no—
256C24—42no—
257C24—41no—
258C24—81no—
259C24C7×C22⋊C411yescharacteristic
260C24C22×C4×C711yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—81no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4C7×C22⋊C411yescharacteristic
280C22×C4C7×C22⋊C411yescharacteristic
281C22×C4C7×C22⋊C411yescharacteristic
282C22×C4C2×C7×D411yescharacteristic
283C22×C4C2×C7×D411yescharacteristic
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—82no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—81no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—81no—
301C42—21no—
302C42—21no—
303C2×C7—21no—
304C2×C7—21no—
305C2×C7—42no—
306C2×C7—42no—
307C2×C7—82no—
308C2×C7#147111yescharacteristic
309C2×C7#212011yescharacteristic
310C2×C7#212111yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—22no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—42no—
324C23—42no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—82no—
329C23—84no—
330C23—84no—
331C23—82no—
332C23C7×#10012yes—
333C23C2×C7×C22⋊C411yescharacteristic
334C23C7×#13112yes—
335D4—44no—
336D4—84no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4C7×#9911yescharacteristic
362C2×C4C7×#9911yescharacteristic
363C8—82no—
364C8—82no—
365C7#1156011yescharacteristic, direct product factor, Sylow
366C22—21no—
367C22—21no—
368C22—22no—
369C22—22no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—42no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—84no—
384C22—84no—
385C22—81no—
386C22—81no—
387C22C7×#40811yescharacteristic
388C22C2×C7×#9911yescharacteristic
389C22C2×C7×#10011yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—81no—
397C4—81no—
398C4—81no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—42no—
403C2—82no—
404C2C7×#147111yescharacteristic
405C2C7×#212011yescharacteristic
406C2C7×#212111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC24C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC2×C7×#120C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C7×#1466C2semidirect
3. C7×#1459C2semidirect
4. C7×#1317C2semidirect
5. C7×#1316C2semidirect
23. #11560C7direct (N × H)
60. #1466C2×C7semidirect
61. #1459C2×C7semidirect
62. #1317C2×C7semidirect
63. #1316C2×C7semidirect
365. C7#11560direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25342
C7×#14662
C7×#14592
C7×#13172
C7×#13162
#115607