Finite Group Databasebeta

← C7×#11775 #13188991

Subgroups — C7×#11775 #13188991

Order 1792.

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

normal subgroups only

This group has 532 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×#2543C211yesmaximal, characteristic
2C4×C7×#99C211yesmaximal, characteristic, complemented
3C7×#1466C211yesmaximal, characteristic, complemented
4C7×#1344C212yesmaximal, complemented
5C7×#1343C212yesmaximal, complemented
6C2×C7×#121—21no—
7C2×C7×#121C411yescharacteristic, complemented
8C2×C4×C7×D4C2211yescharacteristic
9C7×#432—21no—
10C2×C7×#99—21no—
11C7×#398C411yescharacteristic, complemented
12C7×#397C2211yescharacteristic
13C7×#394C411yescharacteristic, complemented
14C7×#392C2211yescharacteristic
15C7×#388—21no—
16C7×#386—21no—
17C7×#386—21no—
18C4×C7×C22⋊C4—21no—
19C2×C7×#95C411yescharacteristic, complemented
20C7×#342C2212yes—
21C7×#323—21no—
22C7×#323C2212yes—
23#11775C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4—21no—
25C22×C7×D4—21no—
26C22×C7×D4C2×C411yescharacteristic
27C22×C7×D4D411yescharacteristic
28C23×C4×C7—21no—
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C7×#121—44no—
32C7×#121—41no—
33C7×#121—41no—
34C4×C7×D4—41no—
35C4×C7×D4—41no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4D411yescharacteristic
46C2×C7×C22⋊C4C2311yescharacteristic
47C2×C42×C7—41no—
48C2×C42×C7D411yescharacteristic
49C7×C4⋊C8—41no—
50C7×#99—24no—
51C7×C22⋊C8—22no—
52C7×#97—41no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—41no—
57C7×#95C2×C411yescharacteristic
58C7×#95C2×C411yescharacteristic
59#2543C2×C711yescharacteristic
60C4×#99C2×C711yescharacteristic, complemented
61#1466C2×C711yescharacteristic, complemented
62#1344C2×C712yescomplemented
63#1343C2×C712yescomplemented
64C24×C7—21no—
65C24×C7C22⋊C411yescharacteristic
66C24×C7C2×D411yescharacteristic
67C2×C7×D4—22no—
68C2×C7×D4—22no—
69C2×C7×D4—41no—
70C2×C7×D4—42no—
71C2×C7×D4—41no—
72C2×C7×D4—42no—
73C2×C7×D4—42no—
74C2×C7×D4—42no—
75C2×C7×D4—42no—
76C2×C7×D4—81no—
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—42no—
86C22×C4×C7—42no—
87C22×C4×C7—42no—
88C22×C4×C7—81no—
89C22×C4×C7—81no—
90C22×C4×C7—81no—
91C22×C4×C7C22⋊C411yescharacteristic
92C22×C4×C7C22⋊C411yescharacteristic
93C22×C4×C7C22⋊C411yescharacteristic
94C22×C4×C7C22×C411yescharacteristic
95C22×C4×C7C2×D411yescharacteristic
96C2×C7×C8—42no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—42no—
100C7×C22⋊C4—24no—
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—81no—
110C42×C7—21no—
111C42×C7—21no—
112C42×C7—82no—
113C2×#121—21no—
114C2×#121C4×C711yescharacteristic, complemented
115C2×C4×D4C22×C711yescharacteristic
116#432—21no—
117C2×#99—21no—
118#398C4×C711yescharacteristic, complemented
119#397C22×C711yescharacteristic
120#394C4×C711yescharacteristic, complemented
121#392C22×C711yescharacteristic
122#388—21no—
123#386—21no—
124#386—21no—
125C4×C22⋊C4—21no—
126C2×#95C4×C711yescharacteristic, complemented
127#342C22×C712yes—
128#323—21no—
129#323C22×C712yes—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—41no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7#10412yes—
146C23×C7C2×C22⋊C411yescharacteristic
147C23×C7AGL(1,ℤ/8)12yes—
148C7×D4—44no—
149C7×D4—84no—
150C7×D4—82no—
151C7×D4—84no—
152C2×C4×C7—22no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—41no—
160C2×C4×C7—42no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—44no—
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—81no—
174C2×C4×C7—81no—
175C2×C4×C7—81no—
176C2×C4×C7—84no—
177C2×C4×C7—82no—
178C2×C4×C7—82no—
179C2×C4×C7#9911yescharacteristic
180C2×C4×C7#9911yescharacteristic
181C7×C8—82no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4C2×C4×C711yescharacteristic
185C22×D4C7×D411yescharacteristic
186C23×C4—21no—
187C23×C4C2×C4×C711yescharacteristic
188C23×C4C7×D411yescharacteristic
189#121—44no—
190#121—41no—
191#121—41no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2×C4×C711yescharacteristic
196C2×C4⋊C4C2×C4×C711yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4C7×D411yescharacteristic
204C2×C22⋊C4C23×C711yescharacteristic
205C2×C42—41no—
206C2×C42C7×D411yescharacteristic
207C4⋊C8—41no—
208#99—24no—
209C22⋊C8—22no—
210#97—41no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—41no—
215#95C2×C4×C711yescharacteristic
216#95C2×C4×C711yescharacteristic
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—22no—
220C22×C7—22no—
221C22×C7—42no—
222C22×C7—42no—
223C22×C7—42no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—81no—
229C22×C7—81no—
230C22×C7—81no—
231C22×C7—81no—
232C22×C7—81no—
233C22×C7—81no—
234C22×C7—81no—
235C22×C7—81no—
236C22×C7C2×#9911yescharacteristic
237C22×C7#41811yescharacteristic
238C22×C7C2×#10411yescharacteristic
239C4×C7—21no—
240C4×C7—21no—
241C4×C7—42no—
242C4×C7—41no—
243C4×C7—41no—
244C4×C7—41no—
245C4×C7—41no—
246C4×C7—81no—
247C4×C7—84no—
248C4×C7—81no—
249C24—21no—
250C24C7×C22⋊C411yescharacteristic
251C24C2×C7×D411yescharacteristic
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—81no—
262C2×D4—81no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4C7×C22⋊C411yescharacteristic
277C22×C4C7×C22⋊C411yescharacteristic
278C22×C4C7×C22⋊C411yescharacteristic
279C22×C4C22×C4×C711yescharacteristic
280C22×C4C2×C7×D411yescharacteristic
281C2×C8—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C22⋊C4—24no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—81no—
295C42—21no—
296C42—21no—
297C42—82no—
298C2×C7—21no—
299C2×C7—21no—
300C2×C7—42no—
301C2×C7—42no—
302C2×C7—161no—
303C2×C7#150711yescharacteristic
304C2×C7#212111yescharacteristic
305C2×C7#212611yescharacteristic
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C7×#10412yes—
322C23C2×C7×C22⋊C411yescharacteristic
323C23C7×AGL(1,ℤ/8)12yes—
324D4—44no—
325D4—84no—
326D4—82no—
327D4—84no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—44no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—84no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4C7×#9911yescharacteristic
356C2×C4C7×#9911yescharacteristic
357C8—82no—
358C7#1177511yescharacteristic, direct product factor, Sylow
359C22—21no—
360C22—21no—
361C22—22no—
362C22—22no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22C2×C7×#9911yescharacteristic
379C22C7×#41811yescharacteristic
380C22C2×C7×#10411yescharacteristic
381C4—21no—
382C4—21no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—84no—
390C4—81no—
391C2—21no—
392C2—21no—
393C2—42no—
394C2—42no—
395C2—161no—
396C2C7×#150711yescharacteristic
397C2C7×#212111yescharacteristic
398C2C7×#212611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#2543C2
LCSG ▷ C22×C4 ▷ C23 ▷ 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. C4×C7×#99C2semidirect
3. C7×#1466C2semidirect
4. C7×#1344C2semidirect
5. C7×#1343C2semidirect
7. C2×C7×#121C4semidirect
11. C7×#398C4semidirect
13. C7×#394C4semidirect
19. C2×C7×#95C4semidirect
23. #11775C7direct (N × H)
60. C4×#99C2×C7semidirect
61. #1466C2×C7semidirect
62. #1344C2×C7semidirect
63. #1343C2×C7semidirect
114. C2×#121C4×C7semidirect
118. #398C4×C7semidirect
120. #394C4×C7semidirect
126. C2×#95C4×C7semidirect
358. C7#11775direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25432
C4×C7×#992
C7×#14662
C7×#13442
C7×#13432
#117757