Finite Group Databasebeta

← C7×#11554 #13188770

Subgroups — C7×#11554 #13188770

Order 1792.

408 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2500C211yesmaximal, characteristic
2C7×#1464C211yesmaximal, characteristic, complemented
3C7×#1458C211yesmaximal, characteristic, complemented
4C7×#1325C212yesmaximal, complemented
5C7×#1318C212yesmaximal, complemented
6C2×C7×#121C411yescharacteristic
7C2×C7×#117C2211yescharacteristic
8C7×#432—21no—
9C7×#431—21no—
10C7×#394—21no—
11C7×#394—21no—
12C7×#390C411yescharacteristic
13C7×#390C2211yescharacteristic
14C7×#388—21no—
15C7×#388—21no—
16C7×#384C2211yescharacteristic
17C7×#382C411yescharacteristic
18C4×C7×C22⋊C4C411yescharacteristic
19C7×#324C2212yes—
20C7×#324C2212yes—
21C7×#323—21no—
22C7×#323—21no—
23#11554C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4—21no—
25C22×C7×D4C2×C411yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C7×#121—41no—
28C7×#121—41no—
29C7×#117—41no—
30C7×#117—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4C2311yescharacteristic
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4D411yescharacteristic
43C2×C7×C22⋊C4D411yescharacteristic
44C2×C42×C7C2×C411yescharacteristic
45C2×C42×C7C2×C411yescharacteristic
46C2×C42×C7D411yescharacteristic
47C2×C42×C7D411yescharacteristic
48C7×C4⋊C8—41no—
49C7×C4⋊C8—41no—
50C7×C22⋊C8—22no—
51C7×C22⋊C8—22no—
52C7×#97—41no—
53C4×C7×C8—41no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57#2500C2×C711yescharacteristic
58#1464C2×C711yescharacteristic, complemented
59#1458C2×C711yescharacteristic, complemented
60#1325C2×C712yescomplemented
61#1318C2×C712yescomplemented
62C24×C7—21no—
63C24×C7C22⋊C411yescharacteristic
64C2×C7×D4—22no—
65C2×C7×D4—41no—
66C2×C7×D4—41no—
67C2×C7×D4—42no—
68C2×C7×D4—81no—
69C2×C7×D4—81no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—22no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—41no—
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22×C411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C2×C7×C8—42no—
84C2×C7×C8—42no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—81no—
89C7×C4⋊C4—81no—
90C7×C22⋊C4—41no—
91C7×C22⋊C4—41no—
92C7×C22⋊C4—42no—
93C7×C22⋊C4—42no—
94C7×C22⋊C4—42no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—41no—
97C7×C22⋊C4—81no—
98C7×C22⋊C4—81no—
99C42×C7—21no—
100C42×C7—21no—
101C42×C7—21no—
102C42×C7—21no—
103C42×C7—22no—
104C42×C7—41no—
105C2×#121C4×C711yescharacteristic
106C2×#117C22×C711yescharacteristic
107#432—21no—
108#431—21no—
109#394—21no—
110#394—21no—
111#390C4×C711yescharacteristic
112#390C22×C711yescharacteristic
113#388—21no—
114#388—21no—
115#384C22×C711yescharacteristic
116#382C4×C711yescharacteristic
117C4×C22⋊C4C4×C711yescharacteristic
118#324C22×C712yes—
119#324C22×C712yes—
120#323—21no—
121#323—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7—41no—
134C23×C7C2×C22⋊C411yescharacteristic
135C7×D4—82no—
136C7×D4—84no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—22no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—42no—
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—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7#9911yescharacteristic
161C2×C4×C7#9911yescharacteristic
162C2×C4×C7#13112yes—
163C7×C8—82no—
164C7×C8—82no—
165C22×D4—21no—
166C22×D4C2×C4×C711yescharacteristic
167C23×C4C2×C4×C711yescharacteristic
168#121—41no—
169#121—41no—
170#117—41no—
171#117—41no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4C2×C4×C711yescharacteristic
175C2×C4⋊C4C2×C4×C711yescharacteristic
176C2×C4⋊C4C23×C711yescharacteristic
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4C7×D411yescharacteristic
184C2×C22⋊C4C7×D411yescharacteristic
185C2×C42C2×C4×C711yescharacteristic
186C2×C42C2×C4×C711yescharacteristic
187C2×C42C7×D411yescharacteristic
188C2×C42C7×D411yescharacteristic
189C4⋊C8—41no—
190C4⋊C8—41no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193#97—41no—
194C4×C8—41no—
195#95—21no—
196#95—21no—
197#95—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—81no—
204C22×C7—81no—
205C22×C7—81no—
206C22×C7—81no—
207C22×C7—81no—
208C22×C7—81no—
209C22×C7—81no—
210C22×C7—81no—
211C22×C7—81no—
212C22×C7#40811yescharacteristic
213C22×C7C2×#9911yescharacteristic
214C22×C7#41311yescharacteristic
215C4×C7—21no—
216C4×C7—21no—
217C4×C7—22no—
218C4×C7—42no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—81no—
224C4×C7—81no—
225C24—21no—
226C24C7×C22⋊C411yescharacteristic
227C2×D4—22no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—42no—
231C2×D4—81no—
232C2×D4—81no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—41no—
240C22×C4C7×C22⋊C411yescharacteristic
241C22×C4C7×C22⋊C411yescharacteristic
242C22×C4C7×C22⋊C411yescharacteristic
243C22×C4C22×C4×C711yescharacteristic
244C22×C4C2×C7×D411yescharacteristic
245C22×C4C2×C7×D411yescharacteristic
246C2×C8—42no—
247C2×C8—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—81no—
261C22⋊C4—81no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—22no—
267C42—41no—
268C2×C7—21no—
269C2×C7—21no—
270C2×C7—81no—
271C2×C7—161no—
272C2×C7#146811yescharacteristic
273C2×C7#212811yescharacteristic
274C2×C7#213211yescharacteristic
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C2×C7×C22⋊C411yescharacteristic
288D4—82no—
289D4—84no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
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—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4C7×#9911yescharacteristic
314C2×C4C7×#9911yescharacteristic
315C2×C4C7×#13112yes—
316C8—82no—
317C8—82no—
318C7#1155411yescharacteristic, direct product factor, Sylow
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22C7×#40811yescharacteristic
334C22C2×C7×#9911yescharacteristic
335C22C7×#41311yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—42no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4—81no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—161no—
350C2C7×#146811yescharacteristic
351C2C7×#212811yescharacteristic
352C2C7×#213211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC2×C7×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1464C2semidirect
3. C7×#1458C2semidirect
4. C7×#1325C2semidirect
5. C7×#1318C2semidirect
23. #11554C7direct (N × H)
58. #1464C2×C7semidirect
59. #1458C2×C7semidirect
60. #1325C2×C7semidirect
61. #1318C2×C7semidirect
318. C7#11554direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25002
C7×#14642
C7×#14582
C7×#13252
C7×#13182
#115547