Finite Group Databasebeta

← C7×#9474 #13186690

Subgroups — C7×#9474 #13186690

Order 1792.

576 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2973C211yesmaximal, characteristic
2C7×#2486C211yesmaximal, characteristic
3C7×#1823C212yesmaximal, complemented
4C7×#1799C211yesmaximal, characteristic
5C7×#1289C212yesmaximal, complemented
6C2×C7×#123—21no—
7C2×C4×C7×D4C2211yescharacteristic
8C22×C7×C4⋊C4C2211yescharacteristic
9C7×#435—24no—
10C7×#432—21no—
11C2×C7×#99—21no—
12C7×#407—21no—
13C2×C7×C22⋊C8C2212yes—
14C7×#398C412yescomplemented
15C7×#397C412yescomplemented
16C7×#389—21no—
17C7×#381—21no—
18C7×#380—21no—
19C7×#380C2212yes—
20C7×#380C2211yescharacteristic
21C7×#342—21no—
22C7×#340—24no—
23#9474C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4D411yescharacteristic
25C23×C4×C7—21no—
26C23×C4×C7D411yescharacteristic
27C23×C4×C7C2311yescharacteristic
28C2×C7×M16—22no—
29C22×C7×C8D414yes—
30C7×#123—44no—
31C4×C7×D4—22no—
32C4×C7×D4—22no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—22no—
39C2×C7×C4⋊C4C2×C412yes—
40C2×C7×C4⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4C2×C411yescharacteristic
45C2×C7×C22⋊C4D411yescharacteristic
46C2×C42×C7D411yescharacteristic
47C7×C4⋊C8—22no—
48C7×#99—42no—
49C7×C22⋊C8—22no—
50C7×C22⋊C8—22no—
51C7×C22⋊C8—22no—
52C7×#97—22no—
53C7×#95—22no—
54C7×#95—22no—
55C7×#95—22no—
56C7×#95C2×C412yes—
57#2973C2×C711yescharacteristic
58#2486C2×C711yescharacteristic
59#1823C2×C712yescomplemented
60#1799C2×C711yescharacteristic
61#1289C2×C712yescomplemented
62C24×C7C22⋊C411yescharacteristic
63C24×C7C2×D411yescharacteristic
64C2×C7×D4—22no—
65C2×C7×D4—21no—
66C2×C7×D4—22no—
67C2×C7×D4—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—22no—
71C22×C4×C7—22no—
72C22×C4×C7—42no—
73C22×C4×C7—42no—
74C22×C4×C7—42no—
75C22×C4×C7—42no—
76C22×C4×C7—41no—
77C22×C4×C7—41no—
78C22×C4×C7—42no—
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22×C411yescharacteristic
83C22×C4×C7C2×D412yes—
84C22×C4×C7C2×D411yescharacteristic
85C22×C4×C7#4112yes—
86C7×M16—82no—
87C2×C7×C8—24no—
88C2×C7×C8—44no—
89C7×C4⋊C4—22no—
90C7×C4⋊C4—24no—
91C7×C4⋊C4—42no—
92C7×C4⋊C4—44no—
93C7×C4⋊C4—44no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C22⋊C4—22no—
97C7×C22⋊C4—22no—
98C7×C22⋊C4—44no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C42×C7—21no—
103C42×C7—21no—
104C2×#123—21no—
105C2×C4×D4C22×C711yescharacteristic
106C22×C4⋊C4C22×C711yescharacteristic
107#435—24no—
108#432—21no—
109C2×#99—21no—
110#407—21no—
111C2×C22⋊C8C22×C712yes—
112#398C4×C712yescomplemented
113#397C4×C712yescomplemented
114#389—21no—
115#381—21no—
116#380—21no—
117#380C22×C712yes—
118#380C22×C711yescharacteristic
119#342—21no—
120#340—24no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7—41no—
128C23×C7C2×C22⋊C411yescharacteristic
129C23×C7#12112yes—
130C23×C7AGL(1,ℤ/8)11yescharacteristic
131C23×C7AGL(1,ℤ/8)12yes—
132C23×C7#13711yescharacteristic
133C7×D4—42no—
134C7×D4—42no—
135C2×C4×C7—24no—
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—41no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
153C2×C4×C7—81no—
154C2×C4×C7—82no—
155C2×C4×C7—82no—
156C2×C4×C7—82no—
157C2×C4×C7—82no—
158C2×C4×C7—82no—
159C2×C4×C7C4×D412yes—
160C2×C4×C7#12011yescharacteristic
161C2×C4×C7#12311yescharacteristic
162C7×C8—44no—
163C22×D4C7×D411yescharacteristic
164C23×C4—21no—
165C23×C4C7×D411yescharacteristic
166C23×C4C23×C711yescharacteristic
167C2×M16—22no—
168C22×C8C7×D414yes—
169#123—44no—
170C4×D4—22no—
171C4×D4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C2×C4×C712yes—
179C2×C4⋊C4C2×C4×C711yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4C2×C4×C711yescharacteristic
184C2×C22⋊C4C7×D411yescharacteristic
185C2×C42C7×D411yescharacteristic
186C4⋊C8—22no—
187#99—42no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191#97—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95C2×C4×C712yes—
196C22×C7—21no—
197C22×C7—22no—
198C22×C7—21no—
199C22×C7—22no—
200C22×C7—42no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—42no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—81no—
208C22×C7#38611yescharacteristic
209C22×C7#41711yescharacteristic
210C22×C7#41811yescharacteristic
211C22×C7#46812yes—
212C22×C7#46912yes—
213C4×C7—21no—
214C4×C7—22no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—82no—
219C4×C7—82no—
220C4×C7—162no—
221C4×C7#45812yes—
222C24C7×C22⋊C411yescharacteristic
223C24C2×C7×D411yescharacteristic
224C2×D4—22no—
225C2×D4—21no—
226C2×D4—22no—
227C2×D4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4C7×C22⋊C411yescharacteristic
240C22×C4C7×C22⋊C411yescharacteristic
241C22×C4C7×C22⋊C411yescharacteristic
242C22×C4C22×C4×C711yescharacteristic
243C22×C4C2×C7×D412yes—
244C22×C4C2×C7×D411yescharacteristic
245C22×C4C7×#4112yes—
246M16—82no—
247C2×C8—24no—
248C2×C8—44no—
249C4⋊C4—22no—
250C4⋊C4—24no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—44no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C42—21no—
263C42—21no—
264C2×C7—21no—
265C2×C7—22no—
266C2×C7—21no—
267C2×C7—22no—
268C2×C7—81no—
269C2×C7#189011yescharacteristic
270C2×C7#189511yescharacteristic
271C2×C7#190911yescharacteristic
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23C2×C7×C22⋊C411yescharacteristic
280C23C7×#12112yes—
281C23C7×AGL(1,ℤ/8)12yes—
282C23C7×AGL(1,ℤ/8)11yescharacteristic
283C23C7×#13711yescharacteristic
284D4—42no—
285D4—42no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—81no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4C4×C7×D412yes—
311C2×C4C7×#12011yescharacteristic
312C2×C4C7×#12311yescharacteristic
313C8—44no—
314C7#947411yescharacteristic, direct product factor, Sylow
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C7×#38611yescharacteristic
328C22C7×#41711yescharacteristic
329C22C7×#41811yescharacteristic
330C22C7×#46812yes—
331C22C7×#46912yes—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—82no—
338C4—82no—
339C4—162no—
340C4C7×#45812yes—
341C2—21no—
342C2—22no—
343C2—21no—
344C2—22no—
345C2—81no—
346C2C7×#189011yescharacteristic
347C2C7×#189511yescharacteristic
348C2C7×#190911yescharacteristic
subgroupG/H
center, socleC22×C7#386
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
3. C7×#1823C2semidirect
5. C7×#1289C2semidirect
14. C7×#398C4semidirect
15. C7×#397C4semidirect
23. #9474C7direct (N × H)
59. #1823C2×C7semidirect
61. #1289C2×C7semidirect
112. #398C4×C7semidirect
113. #397C4×C7semidirect
314. C7#9474direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#29732
C7×#24862
C7×#18232
C7×#17992
C7×#12892
#94747