Finite Group Databasebeta

← C7×#11577 #13188793

Subgroups — C7×#11577 #13188793

Order 1792.

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

normal subgroups only

This group has 424 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×#2579C211yesmaximal, characteristic
2C7×#1465C211yesmaximal, characteristic
3C7×#1460C211yesmaximal, characteristic
4C7×#1326C212yesmaximal, complemented
5C7×#1326C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C22×C7×C4⋊C4C411yescharacteristic
8C7×#431—21no—
9C7×#407—21no—
10C7×#398—21no—
11C7×#396—21no—
12C7×#387—21no—
13C7×#386C411yescharacteristic
14C7×#386C411yescharacteristic
15C7×#385—21no—
16C7×#385C2211yescharacteristic
17C7×#385C2211yescharacteristic
18C7×#380C411yescharacteristic
19C7×#324—21no—
20C7×#323—21no—
21C7×#323C2212yes—
22C7×#323C2212yes—
23#11577C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4C2×C411yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7C2×C411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C2×C7×M16—41no—
30C4×C7×D4—41no—
31C4×C7×D4—41no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—41no—
38C2×C7×C4⋊C4—41no—
39C2×C7×C4⋊C4D411yescharacteristic
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4C2×C411yescharacteristic
43C2×C7×C22⋊C4C2×C411yescharacteristic
44C2×C7×C22⋊C4D411yescharacteristic
45C2×C7×C22⋊C4C2311yescharacteristic
46C2×C42×C7D411yescharacteristic
47C7×C4⋊C8—41no—
48C7×C22⋊C8—22no—
49C7×C22⋊C8—22no—
50C7×C22⋊C8—41no—
51C4×C7×C8—41no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57#2579C2×C711yescharacteristic
58#1465C2×C711yescharacteristic
59#1460C2×C711yescharacteristic
60#1326C2×C712yescomplemented
61#1326C2×C712yescomplemented
62C24×C7C22⋊C411yescharacteristic
63C24×C7C22×C411yescharacteristic
64C2×C7×D4—22no—
65C2×C7×D4—22no—
66C2×C7×D4—41no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—42no—
76C22×C4×C7—41no—
77C22×C4×C7—41no—
78C22×C4×C7—81no—
79C22×C4×C7—81no—
80C22×C4×C7—81no—
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22⋊C411yescharacteristic
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C22×C4×C7C2×D411yescharacteristic
86C7×M16—82no—
87C2×C7×C8—42no—
88C2×C7×C8—42no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—42no—
91C7×C4⋊C4—42no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—81no—
95C7×C4⋊C4—81no—
96C7×C22⋊C4—22no—
97C7×C22⋊C4—22no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—81no—
102C42×C7—21no—
103C42×C7—21no—
104C2×C4×D4C22×C711yescharacteristic
105C22×C4⋊C4C4×C711yescharacteristic
106#431—21no—
107#407—21no—
108#398—21no—
109#396—21no—
110#387—21no—
111#386C4×C711yescharacteristic
112#386C4×C711yescharacteristic
113#385—21no—
114#385C22×C711yescharacteristic
115#385C22×C711yescharacteristic
116#380C4×C711yescharacteristic
117#324—21no—
118#323—21no—
119#323C22×C712yes—
120#323C22×C712yes—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—41no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7C2×C22⋊C411yescharacteristic
131C23×C7#13112yes—
132C7×D4—44no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—22no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—41no—
140C2×C4×C7—42no—
141C2×C4×C7—41no—
142C2×C4×C7—42no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—42no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—81no—
158C2×C4×C7—81no—
159C2×C4×C7—81no—
160C2×C4×C7—82no—
161C2×C4×C7#9911yescharacteristic
162C2×C4×C7#9911yescharacteristic
163C2×C4×C7#10112yes—
164C7×C8—82no—
165C7×C8—82no—
166C22×D4C2×C4×C711yescharacteristic
167C23×C4C2×C4×C711yescharacteristic
168C23×C4C2×C4×C711yescharacteristic
169C23×C4C2×C4×C711yescharacteristic
170C23×C4C7×D411yescharacteristic
171C2×M16—41no—
172C4×D4—41no—
173C4×D4—41no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4C7×D411yescharacteristic
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4C2×C4×C711yescharacteristic
185C2×C22⋊C4C2×C4×C711yescharacteristic
186C2×C22⋊C4C7×D411yescharacteristic
187C2×C22⋊C4C23×C711yescharacteristic
188C2×C42C7×D411yescharacteristic
189C4⋊C8—41no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—41no—
193C4×C8—41no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—22no—
202C22×C7—42no—
203C22×C7—42no—
204C22×C7—42no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—81no—
211C22×C7#40811yescharacteristic
212C22×C7C2×#9911yescharacteristic
213C22×C7C2×#10111yescharacteristic
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—22no—
217C4×C7—41no—
218C4×C7—42no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—81no—
222C4×C7—81no—
223C4×C7—81no—
224C24C7×C22⋊C411yescharacteristic
225C24C22×C4×C711yescharacteristic
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—81no—
243C22×C4C7×C22⋊C411yescharacteristic
244C22×C4C7×C22⋊C411yescharacteristic
245C22×C4C7×C22⋊C411yescharacteristic
246C22×C4C2×C7×D411yescharacteristic
247C22×C4C2×C7×D411yescharacteristic
248M16—82no—
249C2×C8—42no—
250C2×C8—42no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—81no—
264C42—21no—
265C42—21no—
266C2×C7—21no—
267C2×C7—21no—
268C2×C7—42no—
269C2×C7—81no—
270C2×C7#147211yescharacteristic
271C2×C7#212011yescharacteristic
272C2×C7#212111yescharacteristic
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23C2×C7×C22⋊C411yescharacteristic
283C23C7×#13112yes—
284D4—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4C7×#9911yescharacteristic
314C2×C4C7×#9911yescharacteristic
315C2×C4C7×#10112yes—
316C8—82no—
317C8—82no—
318C7#1157711yescharacteristic, direct product factor, Sylow
319C22—21no—
320C22—21no—
321C22—22no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22C7×#40811yescharacteristic
332C22C2×C7×#9911yescharacteristic
333C22C2×C7×#10111yescharacteristic
334C4—22no—
335C4—21no—
336C4—21no—
337C4—41no—
338C4—42no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—81no—
348C2C7×#147211yescharacteristic
349C2C7×#212011yescharacteristic
350C2C7×#212111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC24C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC22×C7×D4C2×C4
LCSG ▷ C24 ▷ 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
4. C7×#1326C2semidirect
5. C7×#1326C2semidirect
23. #11577C7direct (N × H)
60. #1326C2×C7semidirect
61. #1326C2×C7semidirect
318. C7#11577direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25792
C7×#14652
C7×#14602
C7×#13262
#115777