Finite Group Databasebeta

← C7×#8760 #13185976

Subgroups — C7×#8760 #13185976

Order 1792.

530 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2349C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic
3C7×#1459C211yesmaximal, characteristic
4C7×#1294C212yesmaximal, complemented
5C7×#1289C212yesmaximal, complemented
6C2×C4×C7×D4—21no—
7C2×C4×C7×D4C2211yescharacteristic
8C2×C7×#99—21no—
9C2×C7×C22⋊C8C2212yes—
10C7×#389—21no—
11C7×#385—21no—
12C7×#385—21no—
13C7×#382C411yescharacteristic, complemented
14C7×#380C411yescharacteristic, complemented
15C7×#380C2212yes—
16C7×#380C2211yescharacteristic
17C4×C7×C4⋊C4C411yescharacteristic, complemented
18C2×C7×#95C411yescharacteristic, complemented
19C2×C7×#95C2211yescharacteristic
20C7×#342—21no—
21C7×#324—21no—
22C7×#323—21no—
23#8760C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4D411yescharacteristic
25C23×C4×C7—21no—
26C23×C4×C7—21no—
27C23×C4×C7C2×C411yescharacteristic
28C23×C4×C7C2311yescharacteristic
29C22×C7×C8—22no—
30C4×C7×D4—22no—
31C4×C7×D4—22no—
32C4×C7×D4—42no—
33C4×C7×D4—42no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—22no—
39C2×C7×C4⋊C4D411yescharacteristic
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4D411yescharacteristic
44C2×C7×C22⋊C4D411yescharacteristic
45C2×C42×C7—21no—
46C2×C42×C7C2×C411yescharacteristic
47C7×#99—42no—
48C7×C22⋊C8—22no—
49C7×C22⋊C8—22no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—22no—
54C7×#95C2×C411yescharacteristic
55C7×#95C2×C411yescharacteristic
56C7×#95C2×C411yescharacteristic
57C7×#95C2×C411yescharacteristic
58#2349C2×C711yescharacteristic
59#1799C2×C711yescharacteristic
60#1459C2×C711yescharacteristic
61#1294C2×C712yescomplemented
62#1289C2×C712yescomplemented
63C24×C7C22⋊C411yescharacteristic
64C24×C7C2×D411yescharacteristic
65C2×C7×D4—22no—
66C2×C7×D4—22no—
67C2×C7×D4D812yes—
68C2×C7×D4SD1612yes—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—42no—
76C22×C4×C7—42no—
77C22×C4×C7—42no—
78C22×C4×C7—41no—
79C22×C4×C7—41no—
80C22×C4×C7—42no—
81C22×C4×C7—81no—
82C22×C4×C7—81no—
83C22×C4×C7—81no—
84C22×C4×C7—81no—
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C22×C411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7#4112yes—
91C22×C4×C7#4112yes—
92C2×C7×C8—42no—
93C2×C7×C8—82no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—42no—
96C7×C4⋊C4—42no—
97C7×C4⋊C4—81no—
98C7×C4⋊C4—81no—
99C7×C4⋊C4—82no—
100C7×C22⋊C4—22no—
101C7×C22⋊C4—22no—
102C7×C22⋊C4—42no—
103C7×C22⋊C4—41no—
104C7×C22⋊C4—41no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—42no—
107C42×C7—21no—
108C42×C7—21no—
109C42×C7—42no—
110C2×C4×D4—21no—
111C2×C4×D4C22×C711yescharacteristic
112C2×#99—21no—
113C2×C22⋊C8C22×C712yes—
114#389—21no—
115#385—21no—
116#385—21no—
117#382C4×C711yescharacteristic, complemented
118#380C4×C711yescharacteristic, complemented
119#380C22×C712yes—
120#380C22×C711yescharacteristic
121C4×C4⋊C4C4×C711yescharacteristic, complemented
122C2×#95C4×C711yescharacteristic, complemented
123C2×#95C22×C711yescharacteristic
124#342—21no—
125#324—21no—
126#323—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7#10212yes—
134C23×C7C2×C22⋊C411yescharacteristic
135C23×C7#12312yes—
136C23×C7C2×D811yescharacteristic
137C23×C7C2×SD1611yescharacteristic
138C23×C7#13712yes—
139C7×D4—24no—
140C7×D4—24no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—22no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—41no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
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—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—81no—
164C2×C4×C7—81no—
165C2×C4×C7—82no—
166C2×C4×C7—82no—
167C2×C4×C7—84no—
168C2×C4×C7—82no—
169C2×C4×C7—82no—
170C2×C4×C7—82no—
171C2×C4×C7#10212yes—
172C2×C4×C7#11711yescharacteristic
173C2×C4×C7#11711yescharacteristic
174C2×C4×C7#12312yes—
175C7×C8—82no—
176C22×D4C7×D411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4C2×C4×C711yescharacteristic
180C23×C4C23×C711yescharacteristic
181C22×C8—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—42no—
185C4×D4—42no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4C7×D411yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4C7×D411yescharacteristic
196C2×C22⋊C4C7×D411yescharacteristic
197C2×C42—21no—
198C2×C42C2×C4×C711yescharacteristic
199#99—42no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—22no—
206#95C2×C4×C711yescharacteristic
207#95C2×C4×C711yescharacteristic
208#95C2×C4×C711yescharacteristic
209#95C2×C4×C711yescharacteristic
210C22×C7—21no—
211C22×C7—22no—
212C22×C7—21no—
213C22×C7—22no—
214C22×C7—22no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—42no—
218C22×C7—42no—
219C22×C7—42no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7#38111yescharacteristic
223C22×C7C2×#10211yescharacteristic
224C22×C7#41911yescharacteristic
225C22×C7#48012yes—
226C22×C7#48312yes—
227C4×C7—22no—
228C4×C7—22no—
229C4×C7—21no—
230C4×C7—21no—
231C4×C7—42no—
232C4×C7—41no—
233C4×C7—41no—
234C4×C7—81no—
235C4×C7—81no—
236C4×C7—81no—
237C4×C7—162no—
238C24C7×C22⋊C411yescharacteristic
239C24C2×C7×D411yescharacteristic
240C2×D4—22no—
241C2×D4—22no—
242C2×D4C7×D812yes—
243C2×D4C7×SD1612yes—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4C7×C22⋊C411yescharacteristic
261C22×C4C7×C22⋊C411yescharacteristic
262C22×C4C7×C22⋊C411yescharacteristic
263C22×C4C22×C4×C711yescharacteristic
264C22×C4C2×C7×D411yescharacteristic
265C22×C4C7×#4112yes—
266C22×C4C7×#4112yes—
267C2×C8—42no—
268C2×C8—82no—
269C4⋊C4—22no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4—82no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C42—21no—
283C42—21no—
284C42—42no—
285C2×C7—21no—
286C2×C7—22no—
287C2×C7—21no—
288C2×C7—22no—
289C2×C7—42no—
290C2×C7#182311yescharacteristic
291C2×C7#182711yescharacteristic
292C2×C7#182811yescharacteristic
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23C7×#10212yes—
300C23C2×C7×C22⋊C411yescharacteristic
301C23C7×#12312yes—
302C23C2×C7×D811yescharacteristic
303C23C2×C7×SD1611yescharacteristic
304C23C7×#13712yes—
305D4—24no—
306D4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—84no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4C7×#10212yes—
338C2×C4C7×#11711yescharacteristic
339C2×C4C7×#11711yescharacteristic
340C2×C4C7×#12312yes—
341C8—82no—
342C7#876011yescharacteristic, direct product factor, Sylow
343C22—21no—
344C22—22no—
345C22—21no—
346C22—22no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22C7×#38111yescharacteristic
356C22C2×C7×#10211yescharacteristic
357C22C7×#41911yescharacteristic
358C22C7×#48012yes—
359C22C7×#48312yes—
360C4—22no—
361C4—22no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—162no—
371C2—21no—
372C2—22no—
373C2—21no—
374C2—22no—
375C2—42no—
376C2C7×#182311yescharacteristic
377C2C7×#182711yescharacteristic
378C2C7×#182811yescharacteristic
subgroupG/H
center, socleC22×C7#381
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1294C2semidirect
5. C7×#1289C2semidirect
13. C7×#382C4semidirect
14. C7×#380C4semidirect
17. C4×C7×C4⋊C4C4semidirect
18. C2×C7×#95C4semidirect
23. #8760C7direct (N × H)
61. #1294C2×C7semidirect
62. #1289C2×C7semidirect
117. #382C4×C7semidirect
118. #380C4×C7semidirect
121. C4×C4⋊C4C4×C7semidirect
122. C2×#95C4×C7semidirect
342. C7#8760direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23492
C7×#17992
C7×#14592
C7×#12942
C7×#12892
#87607