Finite Group Databasebeta

← C7×#11779 #13188995

Subgroups — C7×#11779 #13188995

Order 1792.

768 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2408C211yesmaximal, characteristic
2C4×C7×#100C211yesmaximal, characteristic, complemented
3C7×#1465C211yesmaximal, characteristic, complemented
4C7×#1345C214yesmaximal, complemented
5C23×C7×D4C411yescharacteristic
6C2×C7×#121—21no—
7C2×C7×#120—21no—
8C2×C4×C7×D4C2211yescharacteristic
9C2×C7×#100—21no—
10C7×#407—21no—
11C4×C7×M16—21no—
12C7×#394C2212yes—
13C7×#392C412yes—
14C7×#390—21no—
15C7×#380C411yescharacteristic
16C7×#379—21no—
17C7×#343C2212yescomplemented
18C7×#324C2212yescomplemented
19C7×#323—21no—
20#11779C711yesmaximal, characteristic, direct product factor, Sylow
21C25×C7—21no—
22C22×C7×D4—22no—
23C22×C7×D4—21no—
24C22×C7×D4—41no—
25C22×C7×D4—41no—
26C22×C7×D4—44no—
27C22×C7×D4C2×C412yes—
28C22×C7×D4D411yescharacteristic
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C2×C7×M16—22no—
32C2×C7×M16—41no—
33C7×#121—44no—
34C7×#120—44no—
35C4×C7×D4—41no—
36C4×C7×D4—41no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2311yescharacteristic
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4C2×C411yescharacteristic
44C2×C7×C22⋊C4D411yescharacteristic
45C2×C42×C7D411yescharacteristic
46C7×#100—24no—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—41no—
49C7×#97—41no—
50C4×C7×C8—41no—
51C7×#95C2×C412yes—
52#2408C2×C711yescharacteristic
53C4×#100C2×C711yescharacteristic, complemented
54#1465C2×C711yescharacteristic, complemented
55#1345C2×C714yescomplemented
56C24×C7—22no—
57C24×C7—42no—
58C24×C7—44no—
59C24×C7—42no—
60C24×C7—42no—
61C24×C7—82no—
62C24×C7C22⋊C411yescharacteristic
63C24×C7C22⋊C411yescharacteristic
64C2×C7×D4—24no—
65C2×C7×D4—22no—
66C2×C7×D4—22no—
67C2×C7×D4—41no—
68C2×C7×D4—42no—
69C2×C7×D4—42no—
70C2×C7×D4—48no—
71C2×C7×D4—44no—
72C2×C7×D4—44no—
73C2×C7×D4—44no—
74C2×C7×D4—44no—
75C2×C7×D4—44no—
76C2×C7×D4—41no—
77C2×C7×D4—41no—
78C2×C7×D4—82no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—22no—
84C22×C4×C7—42no—
85C22×C4×C7—42no—
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C22×C411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C7×M16—24no—
92C7×M16—42no—
93C7×M16—82no—
94C2×C7×C8—42no—
95C2×C7×C8—42no—
96C7×C4⋊C4—41no—
97C7×C4⋊C4—42no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—44no—
102C7×C22⋊C4—44no—
103C7×C22⋊C4—82no—
104C42×C7—21no—
105C42×C7—21no—
106C23×D4C4×C711yescharacteristic
107C2×#121—21no—
108C2×#120—21no—
109C2×C4×D4C22×C711yescharacteristic
110C2×#100—21no—
111#407—21no—
112C4×M16—21no—
113#394C22×C712yes—
114#392C4×C712yes—
115#390—21no—
116#380C4×C711yescharacteristic
117#379—21no—
118#343C22×C712yescomplemented
119#324C22×C712yescomplemented
120#323—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—24no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—42no—
132C23×C7—42no—
133C23×C7—42no—
134C23×C7—48no—
135C23×C7—42no—
136C23×C7—42no—
137C23×C7—42no—
138C23×C7—84no—
139C23×C7—88no—
140C23×C7—88no—
141C23×C7—84no—
142C23×C7#10412yes—
143C23×C7C2×C22⋊C411yescharacteristic
144C7×D4—44no—
145C7×D4—44no—
146C7×D4—44no—
147C7×D4—44no—
148C7×D4—44no—
149C7×D4—88no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—22no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—42no—
157C2×C4×C7—42no—
158C2×C4×C7—42no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—42no—
163C2×C4×C7—42no—
164C2×C4×C7—42no—
165C2×C4×C7—81no—
166C2×C4×C7#9911yescharacteristic
167C2×C4×C7#9911yescharacteristic
168C2×C4×C7AGL(1,ℤ/8)12yes—
169C7×C8—44no—
170C7×C8—82no—
171C25—21no—
172C22×D4—22no—
173C22×D4—21no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—44no—
177C22×D4C2×C4×C712yes—
178C22×D4C7×D411yescharacteristic
179C23×C4C2×C4×C711yescharacteristic
180C23×C4C7×D411yescharacteristic
181C2×M16—22no—
182C2×M16—41no—
183#121—44no—
184#120—44no—
185C4×D4—41no—
186C4×D4—41no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C23×C711yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4C2×C4×C711yescharacteristic
194C2×C22⋊C4C7×D411yescharacteristic
195C2×C42C7×D411yescharacteristic
196#100—24no—
197C22⋊C8—22no—
198C22⋊C8—41no—
199#97—41no—
200C4×C8—41no—
201#95C2×C4×C712yes—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—22no—
205C22×C7—42no—
206C22×C7—42no—
207C22×C7—42no—
208C22×C7—44no—
209C22×C7—44no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—44no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—81no—
217C22×C7—82no—
218C22×C7—82no—
219C22×C7—816no—
220C22×C7—82no—
221C22×C7—82no—
222C22×C7C2×#9911yescharacteristic
223C22×C7#41811yescharacteristic
224C22×C7C2×#10411yescharacteristic
225C4×C7—21no—
226C4×C7—22no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—82no—
231C4×C7AΣL(1,𝔽16)12yes—
232C24—22no—
233C24—42no—
234C24—44no—
235C24—42no—
236C24—42no—
237C24—82no—
238C24C7×C22⋊C411yescharacteristic
239C24C7×C22⋊C411yescharacteristic
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—48no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—82no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4C7×C22⋊C411yescharacteristic
263C22×C4C7×C22⋊C411yescharacteristic
264C22×C4C22×C4×C711yescharacteristic
265C22×C4C2×C7×D411yescharacteristic
266C22×C4C2×C7×D411yescharacteristic
267M16—24no—
268M16—42no—
269M16—82no—
270C2×C8—42no—
271C2×C8—42no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—82no—
280C42—21no—
281C42—21no—
282C2×C7—21no—
283C2×C7—21no—
284C2×C7—42no—
285C2×C7—81no—
286C2×C7—84no—
287C2×C7#150711yescharacteristic
288C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
289C2×C7#213411yescharacteristic
290C23—21no—
291C23—21no—
292C23—24no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—48no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—84no—
308C23—88no—
309C23—88no—
310C23—84no—
311C23C7×#10412yes—
312C23C2×C7×C22⋊C411yescharacteristic
313D4—44no—
314D4—44no—
315D4—44no—
316D4—44no—
317D4—44no—
318D4—88no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—81no—
335C2×C4C7×#9911yescharacteristic
336C2×C4C7×#9911yescharacteristic
337C2×C4C7×AGL(1,ℤ/8)12yes—
338C8—44no—
339C8—82no—
340C7#1177911yescharacteristic, direct product factor, Sylow
341C22—21no—
342C22—21no—
343C22—22no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—44no—
348C22—44no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—41no—
355C22—81no—
356C22—82no—
357C22—82no—
358C22—816no—
359C22—82no—
360C22—82no—
361C22C2×C7×#9911yescharacteristic
362C22C7×#41811yescharacteristic
363C22C2×C7×#10411yescharacteristic
364C4—22no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—82no—
370C4C7×AΣL(1,𝔽16)12yes—
371C2—21no—
372C2—21no—
373C2—42no—
374C2—81no—
375C2—84no—
376C2C7×#150711yescharacteristic
377C2C2×C7×AΣL(1,𝔽16)11yescharacteristic
378C2C7×#213411yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC7×#2408C2
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×#100C2semidirect
3. C7×#1465C2semidirect
4. C7×#1345C2semidirect
17. C7×#343C22semidirect
18. C7×#324C22semidirect
20. #11779C7direct (N × H)
53. C4×#100C2×C7semidirect
54. #1465C2×C7semidirect
55. #1345C2×C7semidirect
118. #343C22×C7semidirect
119. #324C22×C7semidirect
340. C7#11779direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24082
C4×C7×#1002
C7×#14652
C7×#13452
#117797