Finite Group Databasebeta

← C7×#11565 #13188781

Subgroups — C7×#11565 #13188781

Order 1792.

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

normal subgroups only

This group has 412 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×#2545C211yesmaximal, characteristic
2C7×#1467C211yesmaximal, characteristic, complemented
3C7×#1461C211yesmaximal, characteristic, complemented
4C7×#1319C212yesmaximal, complemented
5C7×#1318C212yesmaximal
6C2×C7×C4⋊Q8C411yescharacteristic
7C2×C7×#124C411yescharacteristic
8C2×C7×#124C411yescharacteristic
9C2×C4×C7×Q8C2211yescharacteristic
10C7×#432—21no—
11C7×#431—21no—
12C7×#396—21no—
13C7×#396—21no—
14C7×#391C2211yescharacteristic
15C7×#390C2211yescharacteristic
16C7×#388—21no—
17C7×#388—21no—
18C4×C7×C4⋊C4C411yescharacteristic
19C7×#324—21no—
20C7×#324—21no—
21C7×#324C2212yes—
22C7×#324C2212yes—
23#11565C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×C7×Q8C2×C411yescharacteristic
26C22×C7×D4C2×C411yescharacteristic
27C7×C4⋊Q8—42no—
28C7×#124—41no—
29C7×#124—41no—
30C7×#124—42no—
31C4×C7×Q8—41no—
32C4×C7×Q8—41no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C4⋊C4C2311yescharacteristic
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C42×C7C2×C411yescharacteristic
46C2×C42×C7C2×C411yescharacteristic
47C2×C42×C7C2×C411yescharacteristic
48C2×C42×C7D411yescharacteristic
49C2×C42×C7D411yescharacteristic
50C7×C4⋊C8—41no—
51C7×C4⋊C8—41no—
52C7×C22⋊C8—22no—
53C7×C22⋊C8—22no—
54C7×#97—41no—
55C4×C7×C8—41no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95—21no—
59#2545C2×C711yescharacteristic
60#1467C2×C711yescharacteristic, complemented
61#1461C2×C711yescharacteristic, complemented
62#1319C2×C712yescomplemented
63#1318C2×C712yes—
64C24×C7—21no—
65C2×C7×Q8—22no—
66C2×C7×Q8—22no—
67C2×C7×Q8—22no—
68C2×C7×Q8—41no—
69C2×C7×Q8—41no—
70C2×C7×Q8—41no—
71C2×C7×D4—22no—
72C2×C7×D4—41no—
73C2×C7×D4—41no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22×C411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C2×C7×C8—42no—
85C2×C7×C8—42no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—42no—
90C7×C4⋊C4—42no—
91C7×C4⋊C4—42no—
92C7×C22⋊C4—42no—
93C7×C22⋊C4—42no—
94C7×C22⋊C4—42no—
95C7×C22⋊C4—42no—
96C7×C22⋊C4—41no—
97C7×C22⋊C4—41no—
98C7×C22⋊C4—42no—
99C42×C7—21no—
100C42×C7—21no—
101C42×C7—21no—
102C42×C7—21no—
103C42×C7—21no—
104C42×C7—21no—
105C42×C7—41no—
106C42×C7—41no—
107C2×C4⋊Q8C4×C711yescharacteristic
108C2×#124C4×C711yescharacteristic
109C2×#124C4×C711yescharacteristic
110C2×C4×Q8C22×C711yescharacteristic
111#432—21no—
112#431—21no—
113#396—21no—
114#396—21no—
115#391C22×C711yescharacteristic
116#390C22×C711yescharacteristic
117#388—21no—
118#388—21no—
119C4×C4⋊C4C4×C711yescharacteristic
120#324—21no—
121#324—21no—
122#324C22×C712yes—
123#324C22×C712yes—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7C2×C22⋊C411yescharacteristic
132C7×Q8—44no—
133C7×Q8—82no—
134C7×D4—82no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—22no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—41no—
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#9911yescharacteristic
159C2×C4×C7#9911yescharacteristic
160C2×C4×C7#10012yes—
161C2×C4×C7#13112yes—
162C7×C8—82no—
163C7×C8—82no—
164C22×Q8C2×C4×C711yescharacteristic
165C22×Q8C2×C4×C711yescharacteristic
166C22×D4C2×C4×C711yescharacteristic
167C4⋊Q8—42no—
168#124—41no—
169#124—41no—
170#124—42no—
171C4×Q8—41no—
172C4×Q8—41no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C7×D411yescharacteristic
177C2×C4⋊C4C7×D411yescharacteristic
178C2×C4⋊C4C23×C711yescharacteristic
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C42C2×C4×C711yescharacteristic
186C2×C42C2×C4×C711yescharacteristic
187C2×C42C2×C4×C711yescharacteristic
188C2×C42C7×D411yescharacteristic
189C2×C42C7×D411yescharacteristic
190C4⋊C8—41no—
191C4⋊C8—41no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194#97—41no—
195C4×C8—41no—
196#95—21no—
197#95—21no—
198#95—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—81no—
202C22×C7—81no—
203C22×C7—81no—
204C22×C7—81no—
205C22×C7—81no—
206C22×C7—81no—
207C22×C7—81no—
208C22×C7#40811yescharacteristic
209C22×C7C2×#9911yescharacteristic
210C22×C7C2×#10011yescharacteristic
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—22no—
214C4×C7—22no—
215C4×C7—42no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7—81no—
221C4×C7—81no—
222C24—21no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×D4—22no—
230C2×D4—41no—
231C2×D4—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4C7×C22⋊C411yescharacteristic
236C22×C4C7×C22⋊C411yescharacteristic
237C22×C4C7×C22⋊C411yescharacteristic
238C22×C4C7×C22⋊C411yescharacteristic
239C22×C4C22×C4×C711yescharacteristic
240C22×C4C2×C7×D411yescharacteristic
241C22×C4C2×C7×D411yescharacteristic
242C2×C8—42no—
243C2×C8—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—42no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—41no—
264C42—41no—
265C2×C7—21no—
266C2×C7—21no—
267C2×C7—161no—
268C2×C7#147111yescharacteristic
269C2×C7#212611yescharacteristic
270C2×C7#213311yescharacteristic
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23C2×C7×C22⋊C411yescharacteristic
279Q8—44no—
280Q8—82no—
281D4—82no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
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×C4C7×#9911yescharacteristic
306C2×C4C7×#9911yescharacteristic
307C2×C4C7×#10012yes—
308C2×C4C7×#13112yes—
309C8—82no—
310C8—82no—
311C7#1156511yescharacteristic, direct product factor, Sylow
312C22—21no—
313C22—21no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22C7×#40811yescharacteristic
322C22C2×C7×#9911yescharacteristic
323C22C2×C7×#10011yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—21no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—81no—
334C4—81no—
335C2—21no—
336C2—21no—
337C2—161no—
338C2C7×#147111yescharacteristic
339C2C7×#212611yescharacteristic
340C2C7×#213311yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC22×C7×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1467C2semidirect
3. C7×#1461C2semidirect
4. C7×#1319C2semidirect
23. #11565C7direct (N × H)
60. #1467C2×C7semidirect
61. #1461C2×C7semidirect
62. #1319C2×C7semidirect
311. C7#11565direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25452
C7×#14672
C7×#14612
C7×#13192
C7×#13182
#115657