Finite Group Databasebeta

← C7×#9448 #13186664

Subgroups — C7×#9448 #13186664

Order 1792.

486 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2983C211yesmaximal, characteristic, complemented
2C7×#2545C211yesmaximal, characteristic
3C7×#1533C212yesmaximal
4C7×#1467C211yesmaximal, characteristic, complemented
5C7×#1285C212yesmaximal, complemented
6C2×C7×C4⋊Q8C411yescharacteristic
7C2×C7×#124C411yescharacteristic
8C2×C7×#124C2211yescharacteristic
9C2×C4×C7×Q8C2211yescharacteristic
10C7×#446—24no—
11C7×#432—21no—
12C7×#432C2212yes—
13C7×#432C2212yes—
14C7×#431—21no—
15C7×#396—21no—
16C7×#396—21no—
17C7×#391C411yescharacteristic
18C7×#390C411yescharacteristic
19C7×#388—21no—
20C7×#388—21no—
21C4×C7×C4⋊C4C2211yescharacteristic
22C7×#329—24no—
23C7×#324—21no—
24C7×#324—21no—
25#9448C711yesmaximal, characteristic, direct product factor, Sylow
26C22×C7×Q8C2×C411yescharacteristic
27C22×C7×Q8C2×C411yescharacteristic
28C22×C7×D4C2×C411yescharacteristic
29C7×C4⋊Q8—42no—
30C7×#124—22no—
31C7×#124—22no—
32C7×#124—42no—
33C4×C7×Q8—22no—
34C4×C7×Q8—22no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C4⋊C4D411yescharacteristic
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C42×C7C2×C411yescharacteristic
48C2×C42×C7C2×C411yescharacteristic
49C2×C42×C7D411yescharacteristic
50C2×C42×C7D411yescharacteristic
51C2×C42×C7C2311yescharacteristic
52C7×C4⋊C8—22no—
53C7×C4⋊C8—22no—
54C7×C4⋊C8—22no—
55C7×C4⋊C8—22no—
56C7×C22⋊C8—22no—
57C7×C22⋊C8—22no—
58C7×#97—22no—
59C7×#97—22no—
60C7×#97D414yes—
61C4×C7×C8—22no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95—21no—
65#2983C2×C711yescharacteristic, complemented
66#2545C2×C711yescharacteristic
67#1533C2×C712yes—
68#1467C2×C711yescharacteristic, complemented
69#1285C2×C712yescomplemented
70C24×C7—21no—
71C2×C7×Q8—22no—
72C2×C7×Q8—22no—
73C2×C7×Q8—22no—
74C2×C7×Q8—41no—
75C2×C7×Q8—41no—
76C2×C7×Q8—41no—
77C2×C7×D4—22no—
78C2×C7×D4—41no—
79C2×C7×D4—41no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22×C411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C2×C7×C8—24no—
91C2×C7×C8—42no—
92C7×C4⋊C4—22no—
93C7×C4⋊C4—22no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—42no—
96C7×C4⋊C4—42no—
97C7×C4⋊C4—42no—
98C7×C22⋊C4—42no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—41no—
105C42×C7—21no—
106C42×C7—21no—
107C42×C7—21no—
108C42×C7—21no—
109C42×C7—41no—
110C42×C7—41no—
111C42×C7C2×D412yes—
112C42×C7#4112yes—
113C2×C4⋊Q8C4×C711yescharacteristic
114C2×#124C4×C711yescharacteristic
115C2×#124C22×C711yescharacteristic
116C2×C4×Q8C22×C711yescharacteristic
117#446—24no—
118#432—21no—
119#432C22×C712yes—
120#432C22×C712yes—
121#431—21no—
122#396—21no—
123#396—21no—
124#391C4×C711yescharacteristic
125#390C4×C711yescharacteristic
126#388—21no—
127#388—21no—
128C4×C4⋊C4C22×C711yescharacteristic
129#329—24no—
130#324—21no—
131#324—21no—
132C23×C7—41no—
133C23×C7—41no—
134C23×C7—41no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7C2×C22⋊C411yescharacteristic
140C7×Q8—44no—
141C7×Q8—82no—
142C7×D4—82no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—22no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
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—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7#10012yes—
166C2×C4×C7C4×D412yes—
167C2×C4×C7#12011yescharacteristic
168C2×C4×C7#12112yes—
169C2×C4×C7#12311yescharacteristic
170C7×C8—44no—
171C7×C8—82no—
172C22×Q8C2×C4×C711yescharacteristic
173C22×Q8C2×C4×C711yescharacteristic
174C22×D4C2×C4×C711yescharacteristic
175C4⋊Q8—42no—
176#124—22no—
177#124—22no—
178#124—42no—
179C4×Q8—22no—
180C4×Q8—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C2×C4×C711yescharacteristic
185C2×C4⋊C4C7×D411yescharacteristic
186C2×C4⋊C4C7×D411yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C42C2×C4×C711yescharacteristic
194C2×C42C2×C4×C711yescharacteristic
195C2×C42C7×D411yescharacteristic
196C2×C42C7×D411yescharacteristic
197C2×C42C23×C711yescharacteristic
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204#97—22no—
205#97—22no—
206#97C7×D414yes—
207C4×C8—22no—
208#95—21no—
209#95—21no—
210#95—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—81no—
214C22×C7—81no—
215C22×C7—81no—
216C22×C7—81no—
217C22×C7—81no—
218C22×C7—81no—
219C22×C7—81no—
220C22×C7#38611yescharacteristic
221C22×C7C2×#10011yescharacteristic
222C22×C7#41311yescharacteristic
223C4×C7—22no—
224C4×C7—22no—
225C4×C7—22no—
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—42no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—41no—
232C4×C7—81no—
233C4×C7—81no—
234C24—21no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4C7×C22⋊C411yescharacteristic
248C22×C4C7×C22⋊C411yescharacteristic
249C22×C4C7×C22⋊C411yescharacteristic
250C22×C4C7×C22⋊C411yescharacteristic
251C22×C4C22×C4×C711yescharacteristic
252C22×C4C2×C7×D411yescharacteristic
253C22×C4C2×C7×D411yescharacteristic
254C2×C8—24no—
255C2×C8—42no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—41no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—41no—
274C42—41no—
275C42C2×C7×D412yes—
276C42C7×#4112yes—
277C2×C7—21no—
278C2×C7—21no—
279C2×C7—161no—
280C2×C7#188411yescharacteristic
281C2×C7#190011yescharacteristic
282C2×C7#191211yescharacteristic
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23C2×C7×C22⋊C411yescharacteristic
291Q8—44no—
292Q8—82no—
293D4—82no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4C7×#10012yes—
317C2×C4C4×C7×D412yes—
318C2×C4C7×#12011yescharacteristic
319C2×C4C7×#12112yes—
320C2×C4C7×#12311yescharacteristic
321C8—44no—
322C8—82no—
323C7#944811yescharacteristic, direct product factor, Sylow
324C22—21no—
325C22—21no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22C7×#38611yescharacteristic
334C22C2×C7×#10011yescharacteristic
335C22C7×#41311yescharacteristic
336C4—22no—
337C4—22no—
338C4—22no—
339C4—21no—
340C4—21no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C2—21no—
348C2—21no—
349C2—161no—
350C2C7×#188411yescharacteristic
351C2C7×#190011yescharacteristic
352C2C7×#191211yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2983C2semidirect
4. C7×#1467C2semidirect
5. C7×#1285C2semidirect
25. #9448C7direct (N × H)
65. #2983C2×C7semidirect
68. #1467C2×C7semidirect
69. #1285C2×C7semidirect
323. C7#9448direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#29832
C7×#25452
C7×#15332
C7×#14672
C7×#12852
#94487