Finite Group Databasebeta

← C7×#11555 #13188771

Subgroups — C7×#11555 #13188771

Order 1792.

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

normal subgroups only

This group has 414 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×#2502C211yesmaximal, characteristic
2C7×#1463C211yesmaximal, characteristic
3C2×C7×#324C211yesmaximal, characteristic
4C7×#1325C212yesmaximal, complemented
5C7×#1319C212yesmaximal, complemented
6C2×C7×C22⋊Q8C411yescharacteristic
7C22×C7×C4⋊C4C2211yescharacteristic
8C7×#407—21no—
9C2×C7×C22⋊C8—21no—
10C7×#398—21no—
11C7×#398—21no—
12C7×#391C411yescharacteristic
13C7×#391C2211yescharacteristic
14C7×#385—21no—
15C7×#385—21no—
16C7×#384C411yescharacteristic
17C7×#384C2211yescharacteristic
18C4×C7×C22⋊C4C411yescharacteristic
19C7×#324—21no—
20C7×#324—21no—
21C7×#324C2212yes—
22C7×#324C2212yes—
23#11555C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C2×C7×M16—41no—
29C22×C7×C8—41no—
30C7×C22⋊Q8—41no—
31C7×C22⋊Q8—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⋊C4C2×C411yescharacteristic
40C2×C7×C4⋊C4C2×C411yescharacteristic
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C4⋊C4D411yescharacteristic
43C2×C7×C4⋊C4C2311yescharacteristic
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C42×C7C2×C411yescharacteristic
47C2×C42×C7C2×C411yescharacteristic
48C7×C22⋊C8—22no—
49C7×C22⋊C8—22no—
50C7×C22⋊C8—41no—
51C7×C22⋊C8—41no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57#2502C2×C711yescharacteristic
58#1463C2×C711yescharacteristic
59C2×#324C2×C711yescharacteristic
60#1325C2×C712yescomplemented
61#1319C2×C712yescomplemented
62C24×C7C22⋊C411yescharacteristic
63C2×C7×Q8—22no—
64C2×C7×Q8—41no—
65C2×C7×Q8—41no—
66C22×C4×C7—21no—
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—22no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—41no—
78C22×C4×C7—81no—
79C22×C4×C7—81no—
80C22×C4×C7C22⋊C411yescharacteristic
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—
89C2×C7×C8—82no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—42no—
96C7×C4⋊C4—81no—
97C7×C4⋊C4—81no—
98C7×C4⋊C4—81no—
99C7×C4⋊C4—81no—
100C7×C22⋊C4—24no—
101C7×C22⋊C4—41no—
102C7×C22⋊C4—41no—
103C42×C7—22no—
104C42×C7—41no—
105C2×C22⋊Q8C4×C711yescharacteristic
106C22×C4⋊C4C22×C711yescharacteristic
107#407—21no—
108C2×C22⋊C8—21no—
109#398—21no—
110#398—21no—
111#391C4×C711yescharacteristic
112#391C22×C711yescharacteristic
113#385—21no—
114#385—21no—
115#384C4×C711yescharacteristic
116#384C22×C711yescharacteristic
117C4×C22⋊C4C4×C711yescharacteristic
118#324—21no—
119#324—21no—
120#324C22×C712yes—
121#324C22×C712yes—
122C23×C7—22no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—41no—
126C23×C7#9911yescharacteristic
127C23×C7#9911yescharacteristic
128C23×C7C2×C22⋊C411yescharacteristic
129C7×Q8—82no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—22no—
133C2×C4×C7—22no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—41no—
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—41no—
158C2×C4×C7—41no—
159C2×C4×C7—81no—
160C2×C4×C7—81no—
161C2×C4×C7#13112yes—
162C7×C8—82no—
163C7×C8—82no—
164C22×Q8C2×C4×C711yescharacteristic
165C23×C4C2×C4×C711yescharacteristic
166C23×C4C7×D411yescharacteristic
167C23×C4C7×D411yescharacteristic
168C2×M16—41no—
169C22×C8—41no—
170C22⋊Q8—41no—
171C22⋊Q8—41no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4C2×C4×C711yescharacteristic
180C2×C4⋊C4C2×C4×C711yescharacteristic
181C2×C4⋊C4C7×D411yescharacteristic
182C2×C4⋊C4C7×D411yescharacteristic
183C2×C4⋊C4C23×C711yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C42C2×C4×C711yescharacteristic
187C2×C42C2×C4×C711yescharacteristic
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—42no—
205C22×C7—42no—
206C22×C7#35612yes—
207C22×C7#40811yescharacteristic
208C22×C7C2×#9911yescharacteristic
209C22×C7#41311yescharacteristic
210C4×C7—22no—
211C4×C7—42no—
212C4×C7—42no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—81no—
219C4×C7—81no—
220C24C7×C22⋊C411yescharacteristic
221C2×Q8—22no—
222C2×Q8—41no—
223C2×Q8—41no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—41no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4C7×C22⋊C411yescharacteristic
239C22×C4C7×C22⋊C411yescharacteristic
240C22×C4C7×C22⋊C411yescharacteristic
241C22×C4C22×C4×C711yescharacteristic
242C22×C4C2×C7×D411yescharacteristic
243C22×C4C2×C7×D411yescharacteristic
244M16—82no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—82no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—81no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C22⋊C4—24no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C42—22no—
262C42—41no—
263C2×C7—21no—
264C2×C7—21no—
265C2×C7—22no—
266C2×C7—41no—
267C2×C7#146811yescharacteristic
268C2×C7#212711yescharacteristic
269C2×C7C2×#35611yescharacteristic
270C23—22no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23C7×#9911yescharacteristic
275C23C7×#9911yescharacteristic
276C23C2×C7×C22⋊C411yescharacteristic
277Q8—82no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4C7×#13112yes—
310C8—82no—
311C8—82no—
312C7#1155511yescharacteristic, direct product factor, Sylow
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—42no—
321C22—42no—
322C22C7×#35612yes—
323C22C7×#40811yescharacteristic
324C22C2×C7×#9911yescharacteristic
325C22C7×#41311yescharacteristic
326C4—22no—
327C4—42no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C2—21no—
337C2—21no—
338C2—22no—
339C2—41no—
340C2C7×#146811yescharacteristic
341C2C7×#212711yescharacteristic
342C2C2×C7×#35611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC24×C7C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
4. C7×#1325C2semidirect
5. C7×#1319C2semidirect
23. #11555C7direct (N × H)
60. #1325C2×C7semidirect
61. #1319C2×C7semidirect
312. C7#11555direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25022
C7×#14632
C2×C7×#3242
C7×#13252
C7×#13192
#115557