Finite Group Databasebeta

← C7×#11563 #13188779

Subgroups — C7×#11563 #13188779

Order 1792.

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

normal subgroups only

This group has 616 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×#2541C211yesmaximal, characteristic
2C7×#1465C211yesmaximal, characteristic, complemented
3C7×#1459C211yesmaximal, characteristic, complemented
4C7×#1318C214yesmaximal, complemented
5C2×C7×#127C411yescharacteristic
6C2×C7×#127C412yes—
7C2×C7×#121—21no—
8C2×C7×#121—21no—
9C2×C4×C7×D4C2211yescharacteristic
10C7×#407—21no—
11C2×C7×C22⋊C8—21no—
12C7×#392—21no—
13C7×#392—21no—
14C7×#390C2212yes—
15C4×C7×C4⋊C4C411yescharacteristic
16C7×#324C2212yes—
17C7×#324C2212yes—
18C7×#323—21no—
19C7×#323—21no—
20#11563C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×D4—22no—
22C22×C7×D4—22no—
23C22×C7×D4—21no—
24C22×C7×D4—21no—
25C22×C7×D4C2×C411yescharacteristic
26C22×C7×D4C2×C412yes—
27C23×C4×C7D411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C2×C7×M16—41no—
30C22×C7×C8—41no—
31C7×#127—41no—
32C7×#127—41no—
33C7×#127—44no—
34C7×#121—44no—
35C7×#121—44no—
36C4×C7×D4—41no—
37C4×C7×D4—41no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4C2311yescharacteristic
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4D411yescharacteristic
43C2×C7×C22⋊C4D411yescharacteristic
44C2×C42×C7C2×C411yescharacteristic
45C2×C42×C7C2×C412yes—
46C7×C22⋊C8—22no—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—41no—
49C7×C22⋊C8—41no—
50C7×#95—21no—
51#2541C2×C711yescharacteristic
52#1465C2×C711yescharacteristic, complemented
53#1459C2×C711yescharacteristic, complemented
54#1318C2×C714yescomplemented
55C24×C7—22no—
56C24×C7C22⋊C411yescharacteristic
57C24×C7C22⋊C411yescharacteristic
58C2×C7×D4—22no—
59C2×C7×D4—22no—
60C2×C7×D4—24no—
61C2×C7×D4—41no—
62C2×C7×D4—42no—
63C2×C7×D4—42no—
64C2×C7×D4—44no—
65C2×C7×D4—44no—
66C2×C7×D4—42no—
67C2×C7×D4—42no—
68C2×C7×D4—42no—
69C2×C7×D4—44no—
70C2×C7×D4—44no—
71C2×C7×D4—41no—
72C2×C7×D4—41no—
73C2×C7×D4—82no—
74C2×C7×D4—82no—
75C2×C7×D4—82no—
76C22×C4×C7—22no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—42no—
82C22×C4×C7—42no—
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22×C411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C7×M16—82no—
89C2×C7×C8—42no—
90C2×C7×C8—42no—
91C2×C7×C8—82no—
92C7×C4⋊C4—24no—
93C7×C4⋊C4—41no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—44no—
97C7×C22⋊C4—44no—
98C42×C7—21no—
99C42×C7—21no—
100C42×C7—42no—
101C2×#127C4×C711yescharacteristic
102C2×#127C4×C712yes—
103C2×#121—21no—
104C2×#121—21no—
105C2×C4×D4C22×C711yescharacteristic
106#407—21no—
107C2×C22⋊C8—21no—
108#392—21no—
109#392—21no—
110#390C22×C712yes—
111C4×C4⋊C4C4×C711yescharacteristic
112#324C22×C712yes—
113#324C22×C712yes—
114#323—21no—
115#323—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—41no—
122C23×C7—41no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—42no—
126C23×C7—42no—
127C23×C7—42no—
128C23×C7—42no—
129C23×C7—42no—
130C23×C7—42no—
131C23×C7—42no—
132C23×C7#10012yes—
133C23×C7C2×C22⋊C411yescharacteristic
134C7×D4—44no—
135C7×D4—84no—
136C7×D4—82no—
137C7×D4—88no—
138C7×D4—88no—
139C7×D4—84no—
140C7×D4—82no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—22no—
144C2×C4×C7—24no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—81no—
156C2×C4×C7—81no—
157C2×C4×C7#9911yescharacteristic
158C2×C4×C7#9911yescharacteristic
159C2×C4×C7#13112yes—
160C7×C8—82no—
161C7×C8—82no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4C2×C4×C711yescharacteristic
167C22×D4C2×C4×C712yes—
168C23×C4C7×D411yescharacteristic
169C23×C4C7×D411yescharacteristic
170C2×M16—41no—
171C22×C8—41no—
172#127—41no—
173#127—41no—
174#127—44no—
175#121—44no—
176#121—44no—
177C4×D4—41no—
178C4×D4—41no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4C23×C711yescharacteristic
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4C7×D411yescharacteristic
184C2×C22⋊C4C7×D411yescharacteristic
185C2×C42C2×C4×C711yescharacteristic
186C2×C42C2×C4×C712yes—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191#95—21no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—22no—
195C22×C7—42no—
196C22×C7—42no—
197C22×C7—41no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—81no—
203C22×C7—81no—
204C22×C7—82no—
205C22×C7—82no—
206C22×C7—82no—
207C22×C7—82no—
208C22×C7—82no—
209C22×C7—82no—
210C22×C7—82no—
211C22×C7#40811yescharacteristic
212C22×C7C2×#9911yescharacteristic
213C22×C7C2×#10011yescharacteristic
214C4×C7—21no—
215C4×C7—22no—
216C4×C7—44no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7#35312yes—
221C24—22no—
222C24C7×C22⋊C411yescharacteristic
223C24C7×C22⋊C411yescharacteristic
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—24no—
227C2×D4—41no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—42no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—82no—
240C2×D4—82no—
241C2×D4—82no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4C7×C22⋊C411yescharacteristic
250C22×C4C7×C22⋊C411yescharacteristic
251C22×C4C22×C4×C711yescharacteristic
252C22×C4C2×C7×D411yescharacteristic
253C22×C4C2×C7×D411yescharacteristic
254M16—82no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—82no—
258C4⋊C4—24no—
259C4⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C42—21no—
265C42—21no—
266C42—42no—
267C2×C7—21no—
268C2×C7—21no—
269C2×C7—42no—
270C2×C7—81no—
271C2×C7—162no—
272C2×C7#147111yescharacteristic
273C2×C7C2×#35311yescharacteristic
274C2×C7#213411yescharacteristic
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23C7×#10012yes—
292C23C2×C7×C22⋊C411yescharacteristic
293D4—44no—
294D4—84no—
295D4—82no—
296D4—88no—
297D4—88no—
298D4—84no—
299D4—82no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4C7×#9911yescharacteristic
317C2×C4C7×#9911yescharacteristic
318C2×C4C7×#13112yes—
319C8—82no—
320C8—82no—
321C7#1156311yescharacteristic, direct product factor, Sylow
322C22—21no—
323C22—21no—
324C22—22no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22—82no—
341C22C7×#40811yescharacteristic
342C22C2×C7×#9911yescharacteristic
343C22C2×C7×#10011yescharacteristic
344C4—22no—
345C4—21no—
346C4—44no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4C7×#35312yes—
351C2—21no—
352C2—21no—
353C2—42no—
354C2—81no—
355C2—162no—
356C2C7×#147111yescharacteristic
357C2C2×C7×#35311yescharacteristic
358C2C7×#213411yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC7×#2541C2
LCSG ▷ C22×C4 ▷ 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×#1465C2semidirect
3. C7×#1459C2semidirect
4. C7×#1318C2semidirect
20. #11563C7direct (N × H)
52. #1465C2×C7semidirect
53. #1459C2×C7semidirect
54. #1318C2×C7semidirect
321. C7#11563direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25412
C7×#14652
C7×#14592
C7×#13182
#115637