Finite Group Databasebeta

← C7×#11848 #13189064

Subgroups — C7×#11848 #13189064

Order 1792.

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

normal subgroups only

This group has 572 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×#2406C211yesmaximal, characteristic
2C7×#1821C211yesmaximal, characteristic, complemented
3C2×C7×#324C211yesmaximal, characteristic, complemented
4C7×#1354C212yesmaximal, complemented
5C7×#1345C212yesmaximal, complemented
6C22×C7×#41C411yescharacteristic
7C2×C7×#123—21no—
8C22×C7×C4⋊C4C2211yescharacteristic
9C7×#407—21no—
10C2×C7×C22⋊C8—21no—
11C7×#395C411yescharacteristic
12C7×#395C2211yescharacteristic
13C7×#394C411yescharacteristic
14C7×#394C2211yescharacteristic
15C7×#386—21no—
16C7×#385—21no—
17C7×#385—21no—
18C7×#381C411yescharacteristic
19C7×#343—21no—
20C7×#343C2212yes—
21C7×#324—21no—
22C7×#324C2212yes—
23#11848C711yesmaximal, characteristic, direct product factor, Sylow
24C2×C7×#41—41no—
25C2×C7×#41—41no—
26C2×C7×#41—44no—
27C22×C7×Q8C2×C411yescharacteristic
28C22×C7×D4—21no—
29C22×C7×D4C2×C411yescharacteristic
30C23×C4×C7—21no—
31C23×C4×C7C2×C411yescharacteristic
32C23×C4×C7D411yescharacteristic
33C23×C4×C7D411yescharacteristic
34C2×C7×M16—22no—
35C22×C7×C8—41no—
36C7×#123—44no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—41no—
40C2×C7×C4⋊C4—41no—
41C2×C7×C4⋊C4C2×C411yescharacteristic
42C2×C7×C4⋊C4D411yescharacteristic
43C2×C7×C4⋊C4D411yescharacteristic
44C2×C7×C4⋊C4C2311yescharacteristic
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4—21no—
50C7×C22⋊C8—22no—
51C7×C22⋊C8—41no—
52C7×C22⋊C8—41no—
53C7×C22⋊C8—41no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95C2×C411yescharacteristic
58C7×#95C2×C411yescharacteristic
59#2406C2×C711yescharacteristic
60#1821C2×C711yescharacteristic, complemented
61C2×#324C2×C711yescharacteristic, complemented
62#1354C2×C712yescomplemented
63#1345C2×C712yescomplemented
64C24×C7—21no—
65C24×C7C22⋊C411yescharacteristic
66C7×#41—44no—
67C7×#41—44no—
68C7×#41—44no—
69C7×#41—44no—
70C2×C7×Q8—42no—
71C2×C7×Q8—41no—
72C2×C7×D4—24no—
73C2×C7×D4—41no—
74C2×C7×D4—42no—
75C2×C7×D4—41no—
76C2×C7×D4—41no—
77C2×C7×D4—42no—
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—21no—
87C22×C4×C7—21no—
88C22×C4×C7—41no—
89C22×C4×C7—42no—
90C22×C4×C7—42no—
91C22×C4×C7—42no—
92C22×C4×C7—42no—
93C22×C4×C7—42no—
94C22×C4×C7C22⋊C411yescharacteristic
95C22×C4×C7C22⋊C411yescharacteristic
96C22×C4×C7C22⋊C411yescharacteristic
97C22×C4×C7C22×C411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C7×M16—42no—
101C7×M16—42no—
102C2×C7×C8—42no—
103C2×C7×C8—42no—
104C2×C7×C8—82no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—42no—
110C7×C4⋊C4—42no—
111C7×C22⋊C4—42no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—42no—
115C7×C22⋊C4—42no—
116C22×#41C4×C711yescharacteristic
117C2×#123—21no—
118C22×C4⋊C4C22×C711yescharacteristic
119#407—21no—
120C2×C22⋊C8—21no—
121#395C4×C711yescharacteristic
122#395C22×C711yescharacteristic
123#394C4×C711yescharacteristic
124#394C22×C711yescharacteristic
125#386—21no—
126#385—21no—
127#385—21no—
128#381C4×C711yescharacteristic
129#343—21no—
130#343C22×C712yes—
131#324—21no—
132#324C22×C712yes—
133C23×C7—22no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—22no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7#9911yescharacteristic
145C23×C7#9911yescharacteristic
146C23×C7C2×C22⋊C411yescharacteristic
147C7×Q8—42no—
148C7×Q8—42no—
149C7×D4—44no—
150C7×D4—42no—
151C7×D4—42no—
152C7×D4—44no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—22no—
156C2×C4×C7—21no—
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—42no—
164C2×C4×C7—42no—
165C2×C4×C7—41no—
166C2×C4×C7—42no—
167C2×C4×C7—48no—
168C2×C4×C7—42no—
169C2×C4×C7—42no—
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—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—81no—
182C2×C4×C7#13512yes—
183C7×C8—82no—
184C7×C8—82no—
185C2×#41—41no—
186C2×#41—41no—
187C2×#41—44no—
188C22×Q8C2×C4×C711yescharacteristic
189C22×D4—21no—
190C22×D4C2×C4×C711yescharacteristic
191C23×C4—21no—
192C23×C4C2×C4×C711yescharacteristic
193C23×C4C7×D411yescharacteristic
194C23×C4C7×D411yescharacteristic
195C2×M16—22no—
196C22×C8—41no—
197#123—44no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4C2×C4×C711yescharacteristic
203C2×C4⋊C4C7×D411yescharacteristic
204C2×C4⋊C4C7×D411yescharacteristic
205C2×C4⋊C4C23×C711yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C22⋊C8—22no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95C2×C4×C711yescharacteristic
219#95C2×C4×C711yescharacteristic
220C22×C7—21no—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7—21no—
225C22×C7—21no—
226C22×C7—21no—
227C22×C7—42no—
228C22×C7—42no—
229C22×C7—42no—
230C22×C7—42no—
231C22×C7—42no—
232C22×C7—81no—
233C22×C7—81no—
234C22×C7—81no—
235C22×C7—81no—
236C22×C7AΣL(1,𝔽16)12yes—
237C22×C7C2×#9911yescharacteristic
238C22×C7#41611yescharacteristic
239C22×C7#42111yescharacteristic
240C4×C7—22no—
241C4×C7—41no—
242C4×C7—41no—
243C4×C7—42no—
244C4×C7—41no—
245C4×C7—41no—
246C4×C7—41no—
247C4×C7—81no—
248C4×C7—81no—
249C24—21no—
250C24C7×C22⋊C411yescharacteristic
251#41—44no—
252#41—44no—
253#41—44no—
254#41—44no—
255C2×Q8—42no—
256C2×Q8—41no—
257C2×D4—24no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—42no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4C7×C22⋊C411yescharacteristic
280C22×C4C7×C22⋊C411yescharacteristic
281C22×C4C7×C22⋊C411yescharacteristic
282C22×C4C22×C4×C711yescharacteristic
283C22×C4C2×C7×D411yescharacteristic
284C22×C4C2×C7×D411yescharacteristic
285M16—42no—
286M16—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—82no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C2×C7—21no—
302C2×C7—21no—
303C2×C7—22no—
304C2×C7—41no—
305C2×C7—82no—
306C2×C7#151611yescharacteristic
307C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
308C2×C7#213311yescharacteristic
309C23—22no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23C7×#9911yescharacteristic
321C23C7×#9911yescharacteristic
322C23C2×C7×C22⋊C411yescharacteristic
323Q8—42no—
324Q8—42no—
325D4—44no—
326D4—42no—
327D4—42no—
328D4—44no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—48no—
344C2×C4—42no—
345C2×C4—42no—
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×C4C7×#13512yes—
359C8—82no—
360C8—82no—
361C7#1184811yescharacteristic, direct product factor, Sylow
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22C7×AΣL(1,𝔽16)12yes—
379C22C2×C7×#9911yescharacteristic
380C22C7×#41611yescharacteristic
381C22C7×#42111yescharacteristic
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—81no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2—41no—
395C2—82no—
396C2C7×#151611yescharacteristic
397C2C2×C7×AΣL(1,𝔽16)11yescharacteristic
398C2C7×#213311yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC22×C7×#41C4
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. C2×C7×#324C2semidirect
4. C7×#1354C2semidirect
5. C7×#1345C2semidirect
23. #11848C7direct (N × H)
60. #1821C2×C7semidirect
61. C2×#324C2×C7semidirect
62. #1354C2×C7semidirect
63. #1345C2×C7semidirect
361. C7#11848direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24062
C7×#18212
C2×C7×#3242
C7×#13542
C7×#13452
#118487