Finite Group Databasebeta

← C7×#11789 #13189005

Subgroups — C7×#11789 #13189005

Order 1792.

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

normal subgroups only

This group has 462 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×#2564C211yesmaximal, characteristic
2C4×C7×#99C211yesmaximal, characteristic
3C7×#1466C211yesmaximal, characteristic
4C7×#1352C212yesmaximal, complemented
5C7×#1351C212yesmaximal, complemented
6C2×C7×C22⋊Q8C411yescharacteristic, complemented
7C2×C4×C7×D4C2211yescharacteristic
8C7×#432—21no—
9C2×C7×#99—21no—
10C7×#400C411yescharacteristic, complemented
11C7×#399C2211yescharacteristic
12C7×#395C411yescharacteristic, complemented
13C7×#393C2211yescharacteristic
14C7×#387—21no—
15C7×#386—21no—
16C7×#385—21no—
17C7×#380—21no—
18C4×C7×C22⋊C4—21no—
19C2×C7×#95C411yescharacteristic, complemented
20C7×#342C2212yes—
21C7×#323—21no—
22C7×#323C2212yes—
23#11789C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×C7×D4D411yescharacteristic
26C23×C4×C7—21no—
27C23×C4×C7C2×C411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C7×C22⋊Q8—41no—
30C7×C22⋊Q8—41no—
31C4×C7×D4—41no—
32C4×C7×D4—41no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4D411yescharacteristic
43C2×C7×C22⋊C4C2311yescharacteristic
44C2×C42×C7—41no—
45C2×C42×C7D411yescharacteristic
46C7×C4⋊C8—41no—
47C7×#99—24no—
48C7×C22⋊C8—22no—
49C7×#97—41no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—41no—
56C7×#95C2×C411yescharacteristic
57C7×#95C2×C411yescharacteristic
58#2564C2×C711yescharacteristic
59C4×#99C2×C711yescharacteristic
60#1466C2×C711yescharacteristic
61#1352C2×C712yescomplemented
62#1351C2×C712yescomplemented
63C24×C7C22⋊C411yescharacteristic
64C24×C7C2×D411yescharacteristic
65C2×C7×Q8—41no—
66C2×C7×Q8—81no—
67C2×C7×D4—22no—
68C2×C7×D4—22no—
69C2×C7×D4—41no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—42no—
79C22×C4×C7—42no—
80C22×C4×C7—42no—
81C22×C4×C7—81no—
82C22×C4×C7—81no—
83C22×C4×C7—81no—
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22×C411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C2×C7×C8—42no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—42no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—81no—
96C7×C22⋊C4—24no—
97C7×C22⋊C4—41no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C7×C22⋊C4—81no—
103C42×C7—21no—
104C42×C7—21no—
105C42×C7—82no—
106C2×C22⋊Q8C4×C711yescharacteristic, complemented
107C2×C4×D4C22×C711yescharacteristic
108#432—21no—
109C2×#99—21no—
110#400C4×C711yescharacteristic, complemented
111#399C22×C711yescharacteristic
112#395C4×C711yescharacteristic, complemented
113#393C22×C711yescharacteristic
114#387—21no—
115#386—21no—
116#385—21no—
117#380—21no—
118C4×C22⋊C4—21no—
119C2×#95C4×C711yescharacteristic, complemented
120#342C22×C712yes—
121#323—21no—
122#323C22×C712yes—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7#10412yes—
132C23×C7C2×C22⋊C411yescharacteristic
133C23×C7#13712yes—
134C7×Q8—82no—
135C7×D4—44no—
136C2×C4×C7—22no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—41no—
144C2×C4×C7—42no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—44no—
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—81no—
161C2×C4×C7—81no—
162C2×C4×C7—81no—
163C2×C4×C7—84no—
164C2×C4×C7—82no—
165C2×C4×C7—82no—
166C2×C4×C7#9911yescharacteristic
167C2×C4×C7#9911yescharacteristic
168C7×C8—82no—
169C22×Q8C2×C4×C711yescharacteristic
170C22×D4C7×D411yescharacteristic
171C23×C4—21no—
172C23×C4C2×C4×C711yescharacteristic
173C23×C4C7×D411yescharacteristic
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C4×D4—41no—
177C4×D4—41no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C2×C4×C711yescharacteristic
182C2×C4⋊C4C2×C4×C711yescharacteristic
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4C7×D411yescharacteristic
188C2×C22⋊C4C23×C711yescharacteristic
189C2×C42—41no—
190C2×C42C7×D411yescharacteristic
191C4⋊C8—41no—
192#99—24no—
193C22⋊C8—22no—
194#97—41no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—41no—
201#95C2×C4×C711yescharacteristic
202#95C2×C4×C711yescharacteristic
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—22no—
206C22×C7—22no—
207C22×C7—42no—
208C22×C7—42no—
209C22×C7—42no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—81no—
215C22×C7C2×#9911yescharacteristic
216C22×C7#41911yescharacteristic
217C22×C7C2×#10411yescharacteristic
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—42no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—81no—
226C4×C7—84no—
227C4×C7—81no—
228C4×C7—81no—
229C24C7×C22⋊C411yescharacteristic
230C24C2×C7×D411yescharacteristic
231C2×Q8—41no—
232C2×Q8—81no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4C7×C22⋊C411yescharacteristic
251C22×C4C7×C22⋊C411yescharacteristic
252C22×C4C7×C22⋊C411yescharacteristic
253C22×C4C22×C4×C711yescharacteristic
254C22×C4C2×C7×D411yescharacteristic
255C2×C8—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—81no—
262C22⋊C4—24no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—81no—
269C42—21no—
270C42—21no—
271C42—82no—
272C2×C7—21no—
273C2×C7—21no—
274C2×C7—42no—
275C2×C7—42no—
276C2×C7#150811yescharacteristic
277C2×C7#212111yescharacteristic
278C2×C7#212611yescharacteristic
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C7×#10412yes—
288C23C2×C7×C22⋊C411yescharacteristic
289C23C7×#13712yes—
290Q8—82no—
291D4—44no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—44no—
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×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—84no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4C7×#9911yescharacteristic
323C2×C4C7×#9911yescharacteristic
324C8—82no—
325C7#1178911yescharacteristic, direct product factor, Sylow
326C22—21no—
327C22—21no—
328C22—22no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22C2×C7×#9911yescharacteristic
339C22C7×#41911yescharacteristic
340C22C2×C7×#10411yescharacteristic
341C4—21no—
342C4—21no—
343C4—42no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—81no—
349C4—84no—
350C4—81no—
351C4—81no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—42no—
356C2C7×#150811yescharacteristic
357C2C7×#212111yescharacteristic
358C2C7×#212611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ 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×#1352C2semidirect
5. C7×#1351C2semidirect
6. C2×C7×C22⋊Q8C4semidirect
10. C7×#400C4semidirect
12. C7×#395C4semidirect
19. C2×C7×#95C4semidirect
23. #11789C7direct (N × H)
61. #1352C2×C7semidirect
62. #1351C2×C7semidirect
106. C2×C22⋊Q8C4×C7semidirect
110. #400C4×C7semidirect
112. #395C4×C7semidirect
119. C2×#95C4×C7semidirect
325. C7#11789direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25642
C4×C7×#992
C7×#14662
C7×#13522
C7×#13512
#117897