Finite Group Databasebeta

← C7×#8761 #13185977

Subgroups — C7×#8761 #13185977

Order 1792.

530 conjugacy classes of subgroups.

The subgroup lattice has 372 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#8761C711yesmaximal, 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—44no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—22no—
38C2×C7×C4⋊C4D411yescharacteristic
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4D411yescharacteristic
43C2×C7×C22⋊C4D411yescharacteristic
44C2×C42×C7—21no—
45C2×C42×C7C2×C411yescharacteristic
46C7×#99—42no—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—22no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—22no—
53C7×#95C2×C411yescharacteristic
54C7×#95C2×C411yescharacteristic
55C7×#95C2×C411yescharacteristic
56C7×#95C2×C411yescharacteristic
57#2349C2×C711yescharacteristic
58#1799C2×C711yescharacteristic
59#1459C2×C711yescharacteristic
60#1294C2×C712yescomplemented
61#1289C2×C712yescomplemented
62C24×C7C22⋊C411yescharacteristic
63C24×C7C2×D411yescharacteristic
64C2×C7×D4—22no—
65C2×C7×D4—22no—
66C2×C7×D4SD1612yes—
67C2×C7×D4Q1612yes—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—42no—
75C22×C4×C7—42no—
76C22×C4×C7—42no—
77C22×C4×C7—42no—
78C22×C4×C7—42no—
79C22×C4×C7—81no—
80C22×C4×C7—81no—
81C22×C4×C7—81no—
82C22×C4×C7—81no—
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22×C411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7#4112yes—
89C22×C4×C7#4112yes—
90C2×C7×C8—42no—
91C2×C7×C8—82no—
92C7×C4⋊C4—22no—
93C7×C4⋊C4—42no—
94C7×C4⋊C4—42no—
95C7×C4⋊C4—81no—
96C7×C4⋊C4—81no—
97C7×C4⋊C4—82no—
98C7×C22⋊C4—22no—
99C7×C22⋊C4—22no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C7×C22⋊C4—42no—
103C7×C22⋊C4—42no—
104C42×C7—21no—
105C42×C7—21no—
106C42×C7—42no—
107C2×C4×D4—21no—
108C2×C4×D4C22×C711yescharacteristic
109C2×#99—21no—
110C2×C22⋊C8C22×C712yes—
111#389—21no—
112#385—21no—
113#385—21no—
114#382C4×C711yescharacteristic, complemented
115#380C4×C711yescharacteristic, complemented
116#380C22×C712yes—
117#380C22×C711yescharacteristic
118C4×C4⋊C4C4×C711yescharacteristic, complemented
119C2×#95C4×C711yescharacteristic, complemented
120C2×#95C22×C711yescharacteristic
121#342—21no—
122#324—21no—
123#323—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—41no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7#10312yes—
131C23×C7C2×C22⋊C411yescharacteristic
132C23×C7#12312yes—
133C23×C7C2×SD1611yescharacteristic
134C23×C7C2×Q1611yescharacteristic
135C23×C7AGL(1,ℤ/8)12yes—
136C7×D4—24no—
137C7×D4—24no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—22no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—41no—
144C2×C4×C7—42no—
145C2×C4×C7—42no—
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—41no—
159C2×C4×C7—41no—
160C2×C4×C7—81no—
161C2×C4×C7—81no—
162C2×C4×C7—84no—
163C2×C4×C7—84no—
164C2×C4×C7—82no—
165C2×C4×C7—82no—
166C2×C4×C7—82no—
167C2×C4×C7#10312yes—
168C2×C4×C7#11711yescharacteristic
169C2×C4×C7#11711yescharacteristic
170C2×C4×C7#12312yes—
171C7×C8—82no—
172C22×D4C7×D411yescharacteristic
173C23×C4—21no—
174C23×C4—21no—
175C23×C4C2×C4×C711yescharacteristic
176C23×C4C23×C711yescharacteristic
177C22×C8—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—44no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4C7×D411yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C7×D411yescharacteristic
191C2×C22⋊C4C7×D411yescharacteristic
192C2×C42—21no—
193C2×C42C2×C4×C711yescharacteristic
194#99—42no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—22no—
201#95C2×C4×C711yescharacteristic
202#95C2×C4×C711yescharacteristic
203#95C2×C4×C711yescharacteristic
204#95C2×C4×C711yescharacteristic
205C22×C7—21no—
206C22×C7—22no—
207C22×C7—21no—
208C22×C7—22no—
209C22×C7—22no—
210C22×C7—42no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—42no—
214C22×C7—42no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7#38111yescharacteristic
218C22×C7C2×#10311yescharacteristic
219C22×C7#41811yescharacteristic
220C22×C7#48112yes—
221C22×C7#48412yes—
222C4×C7—22no—
223C4×C7—22no—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—42no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7—81no—
230C4×C7—81no—
231C4×C7—81no—
232C4×C7—162no—
233C24C7×C22⋊C411yescharacteristic
234C24C2×C7×D411yescharacteristic
235C2×D4—22no—
236C2×D4—22no—
237C2×D4C7×SD1612yes—
238C2×D4C7×Q1612yes—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4C7×C22⋊C411yescharacteristic
255C22×C4C7×C22⋊C411yescharacteristic
256C22×C4C7×C22⋊C411yescharacteristic
257C22×C4C22×C4×C711yescharacteristic
258C22×C4C2×C7×D411yescharacteristic
259C22×C4C7×#4112yes—
260C22×C4C7×#4112yes—
261C2×C8—42no—
262C2×C8—82no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C4⋊C4—82no—
269C22⋊C4—22no—
270C22⋊C4—22no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C42—21no—
276C42—21no—
277C42—42no—
278C2×C7—21no—
279C2×C7—22no—
280C2×C7—21no—
281C2×C7—22no—
282C2×C7—42no—
283C2×C7#182411yescharacteristic
284C2×C7#182711yescharacteristic
285C2×C7#182811yescharacteristic
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C7×#10312yes—
293C23C2×C7×C22⋊C411yescharacteristic
294C23C7×#12312yes—
295C23C2×C7×SD1611yescharacteristic
296C23C2×C7×Q1611yescharacteristic
297C23C7×AGL(1,ℤ/8)12yes—
298D4—24no—
299D4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
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—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4C7×#10312yes—
330C2×C4C7×#11711yescharacteristic
331C2×C4C7×#11711yescharacteristic
332C2×C4C7×#12312yes—
333C8—82no—
334C7#876111yescharacteristic, direct product factor, Sylow
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—42no—
345C22—41no—
346C22—41no—
347C22C7×#38111yescharacteristic
348C22C2×C7×#10311yescharacteristic
349C22C7×#41811yescharacteristic
350C22C7×#48112yes—
351C22C7×#48412yes—
352C4—22no—
353C4—22no—
354C4—21no—
355C4—21no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—162no—
363C2—21no—
364C2—22no—
365C2—21no—
366C2—22no—
367C2—42no—
368C2C7×#182411yescharacteristic
369C2C7×#182711yescharacteristic
370C2C7×#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. #8761C7direct (N × H)
60. #1294C2×C7semidirect
61. #1289C2×C7semidirect
114. #382C4×C7semidirect
115. #380C4×C7semidirect
118. C4×C4⋊C4C4×C7semidirect
119. C2×#95C4×C7semidirect
334. C7#8761direct (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
#87617