Finite Group Databasebeta

← C7×#11802 #13189018

Subgroups — C7×#11802 #13189018

Order 1792.

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

normal subgroups only

This group has 758 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
1C2×C7×#394C211yesmaximal, characteristic
2C7×#1821C211yesmaximal, characteristic, complemented
3C7×#1463C211yesmaximal, characteristic, complemented
4C7×#1345C214yesmaximal, complemented
5C23×C7×D4C411yescharacteristic
6C22×C7×C4⋊C4C2211yescharacteristic
7C22×C7×C22⋊C4—21no—
8C22×C7×C22⋊C4—21no—
9C7×#407—21no—
10C7×#407—21no—
11C7×#394—21no—
12C7×#394—21no—
13C7×#394C412yes—
14C7×#394C2212yes—
15C2×C7×#95C411yescharacteristic
16C7×#343—21no—
17C7×#343C2212yescomplemented
18C7×#324—21no—
19C7×#324C2212yescomplemented
20#11802C711yesmaximal, characteristic, direct product factor, Sylow
21C25×C7—21no—
22C22×C7×D4—21no—
23C22×C7×D4—41no—
24C22×C7×D4—41no—
25C22×C7×D4—44no—
26C22×C7×D4C2×C412yes—
27C23×C4×C7—21no—
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C2×C7×M16—22no—
32C2×C7×M16—41no—
33C2×C7×C4⋊C4—41no—
34C2×C7×C4⋊C4—41no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C4⋊C4C2311yescharacteristic
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4—44no—
44C2×C7×C22⋊C4—44no—
45C7×C22⋊C8—22no—
46C7×C22⋊C8—41no—
47C7×C22⋊C8—41no—
48C7×C22⋊C8—41no—
49C7×#95—21no—
50C7×#95C2×C412yes—
51C2×#394C2×C711yescharacteristic
52#1821C2×C711yescharacteristic, complemented
53#1463C2×C711yescharacteristic, complemented
54#1345C2×C714yescomplemented
55C24×C7—24no—
56C24×C7—22no—
57C24×C7—42no—
58C24×C7—42no—
59C24×C7—42no—
60C24×C7—42no—
61C24×C7—42no—
62C24×C7—41no—
63C24×C7—41no—
64C24×C7C22⋊C411yescharacteristic
65C2×C7×D4—42no—
66C2×C7×D4—42no—
67C2×C7×D4—44no—
68C2×C7×D4—48no—
69C2×C7×D4—44no—
70C2×C7×D4—44no—
71C2×C7×D4—44no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—22no—
79C22×C4×C7—22no—
80C22×C4×C7—41no—
81C22×C4×C7—42no—
82C22×C4×C7—42no—
83C22×C4×C7—42no—
84C22×C4×C7—42no—
85C22×C4×C7—42no—
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C22⋊C411yescharacteristic
89C22×C4×C7C22×C411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D411yescharacteristic
92C7×M16—42no—
93C7×M16—42no—
94C7×M16—82no—
95C2×C7×C8—42no—
96C2×C7×C8—42no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—41no—
101C7×C22⋊C4—44no—
102C7×C22⋊C4—44no—
103C7×C22⋊C4—44no—
104C7×C22⋊C4—44no—
105C23×D4C4×C711yescharacteristic
106C22×C4⋊C4C22×C711yescharacteristic
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109#407—21no—
110#407—21no—
111#394—21no—
112#394—21no—
113#394C4×C712yes—
114#394C22×C712yes—
115C2×#95C4×C711yescharacteristic
116#343—21no—
117#343C22×C712yescomplemented
118#324—21no—
119#324C22×C712yescomplemented
120C23×C7—22no—
121C23×C7—22no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—24no—
125C23×C7—42no—
126C23×C7—44no—
127C23×C7—48no—
128C23×C7—48no—
129C23×C7—42no—
130C23×C7—42no—
131C23×C7—44no—
132C23×C7—42no—
133C23×C7—42no—
134C23×C7—42no—
135C23×C7—84no—
136C23×C7—84no—
137C23×C7—82no—
138C23×C7—82no—
139C23×C7—82no—
140C23×C7—82no—
141C23×C7#9911yescharacteristic
142C23×C7#9911yescharacteristic
143C23×C7C2×C22⋊C411yescharacteristic
144C7×D4—44no—
145C7×D4—48no—
146C7×D4—44no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
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—42no—
160C2×C4×C7—42no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—42no—
166C2×C4×C7—42no—
167C2×C4×C7—42no—
168C2×C4×C7—81no—
169C2×C4×C7—84no—
170C2×C4×C7AGL(1,ℤ/8)12yes—
171C7×C8—82no—
172C7×C8—82no—
173C25—21no—
174C22×D4—21no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—44no—
178C22×D4C2×C4×C712yes—
179C23×C4—21no—
180C23×C4C2×C4×C711yescharacteristic
181C23×C4C7×D411yescharacteristic
182C23×C4C7×D411yescharacteristic
183C2×M16—22no—
184C2×M16—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4C2×C4×C711yescharacteristic
188C2×C4⋊C4C7×D411yescharacteristic
189C2×C4⋊C4C7×D411yescharacteristic
190C2×C4⋊C4C23×C711yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C22⋊C8—22no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201#95—21no—
202#95C2×C4×C712yes—
203C22×C7—24no—
204C22×C7—24no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—44no—
212C22×C7—44no—
213C22×C7—44no—
214C22×C7—82no—
215C22×C7—82no—
216C22×C7—88no—
217C22×C7—88no—
218C22×C7—82no—
219C22×C7—82no—
220C22×C7AΣL(1,𝔽16)12yes—
221C22×C7#35512yes—
222C22×C7C2×#9911yescharacteristic
223C22×C7#41811yescharacteristic
224C22×C7#42111yescharacteristic
225C4×C7—22no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—82no—
232C24—24no—
233C24—22no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—41no—
240C24—41no—
241C24C7×C22⋊C411yescharacteristic
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—48no—
247C2×D4—44no—
248C2×D4—44no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4C7×C22⋊C411yescharacteristic
264C22×C4C7×C22⋊C411yescharacteristic
265C22×C4C7×C22⋊C411yescharacteristic
266C22×C4C22×C4×C711yescharacteristic
267C22×C4C2×C7×D411yescharacteristic
268C22×C4C2×C7×D411yescharacteristic
269M16—42no—
270M16—42no—
271M16—82no—
272C2×C8—42no—
273C2×C8—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C2×C7—21no—
283C2×C7—21no—
284C2×C7—22no—
285C2×C7—22no—
286C2×C7—84no—
287C2×C7#151011yescharacteristic
288C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
289C2×C7C2×#35511yescharacteristic
290C23—22no—
291C23—22no—
292C23—21no—
293C23—21no—
294C23—24no—
295C23—42no—
296C23—44no—
297C23—48no—
298C23—48no—
299C23—42no—
300C23—42no—
301C23—44no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—84no—
306C23—84no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23C7×#9911yescharacteristic
312C23C7×#9911yescharacteristic
313C23C2×C7×C22⋊C411yescharacteristic
314D4—44no—
315D4—44no—
316D4—48no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—81no—
339C2×C4—84no—
340C2×C4C7×AGL(1,ℤ/8)12yes—
341C8—82no—
342C8—82no—
343C7#1180211yescharacteristic, direct product factor, Sylow
344C22—24no—
345C22—24no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—82no—
356C22—82no—
357C22—88no—
358C22—88no—
359C22—82no—
360C22—82no—
361C22C7×AΣL(1,𝔽16)12yes—
362C22C7×#35512yes—
363C22C2×C7×#9911yescharacteristic
364C22C7×#41811yescharacteristic
365C22C7×#42111yescharacteristic
366C4—22no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—82no—
373C2—21no—
374C2—21no—
375C2—22no—
376C2—22no—
377C2—84no—
378C2C7×#151011yescharacteristic
379C2C2×C7×AΣL(1,𝔽16)11yescharacteristic
380C2C2×C7×#35511yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC23×C7×D4C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C24×C7 ◁ C22×C7×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1821C2semidirect
3. C7×#1463C2semidirect
4. C7×#1345C2semidirect
17. C7×#343C22semidirect
19. C7×#324C22semidirect
20. #11802C7direct (N × H)
52. #1821C2×C7semidirect
53. #1463C2×C7semidirect
54. #1345C2×C7semidirect
117. #343C22×C7semidirect
119. #324C22×C7semidirect
343. C7#11802direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3942
C7×#18212
C7×#14632
C7×#13452
#118027