Finite Group Databasebeta

← C7×#8776 #13185992

Subgroups — C7×#8776 #13185992

Order 1792.

480 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2353C211yesmaximal, characteristic
2C4×C7×#99C211yesmaximal, characteristic
3C7×#1465C211yesmaximal, characteristic
4C7×#1297C212yesmaximal, complemented
5C7×#1284C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×C7×#99—21no—
8C7×#407C2212yes—
9C7×#388—21no—
10C7×#387—21no—
11C7×#386—21no—
12C7×#383C411yescharacteristic, complemented
13C7×#381C2211yescharacteristic
14C7×#380C411yescharacteristic, complemented
15C4×C7×C22⋊C4—21no—
16C4×C7×C22⋊C4C2212yes—
17C4×C7×C22⋊C4C2211yescharacteristic
18C7×#376C411yescharacteristic, complemented
19C2×C7×#95C411yescharacteristic, complemented
20C7×#342—21no—
21C7×#324—21no—
22C7×#323—21no—
23#8776C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4D411yescharacteristic
25C23×C4×C7—21no—
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7C2311yescharacteristic
28C2×C7×M16—22no—
29C4×C7×D4—22no—
30C4×C7×D4—22no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4D411yescharacteristic
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—21no—
35C2×C7×C22⋊C4—21no—
36C2×C7×C22⋊C4—21no—
37C2×C7×C22⋊C4D411yescharacteristic
38C2×C7×C22⋊C4D411yescharacteristic
39C2×C42×C7—21no—
40C2×C42×C7—21no—
41C2×C42×C7—22no—
42C2×C42×C7C2×C411yescharacteristic
43C7×#99—24no—
44C7×C22⋊C8—22no—
45C7×C22⋊C8—22no—
46C7×#95—21no—
47C7×#95—21no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—22no—
52C7×#95C2×C411yescharacteristic
53C7×#95C2×C411yescharacteristic
54C7×#95C2×C411yescharacteristic
55C7×#95C2×C411yescharacteristic
56#2353C2×C711yescharacteristic
57C4×#99C2×C711yescharacteristic
58#1465C2×C711yescharacteristic
59#1297C2×C712yescomplemented
60#1284C2×C712yescomplemented
61C24×C7C22⋊C411yescharacteristic
62C24×C7C2×D411yescharacteristic
63C2×C7×D4—22no—
64C2×C7×D4—22no—
65C2×C7×D4—41no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—42no—
73C22×C4×C7—42no—
74C22×C4×C7—42no—
75C22×C4×C7—81no—
76C22×C4×C7—81no—
77C22×C4×C7—81no—
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22×C411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7#4112yes—
84C22×C4×C7#4112yes—
85C7×M16—82no—
86C2×C7×C8—42no—
87C7×C4⋊C4—22no—
88C7×C4⋊C4—81no—
89C7×C22⋊C4—22no—
90C7×C22⋊C4—22no—
91C7×C22⋊C4—24no—
92C7×C22⋊C4—42no—
93C7×C22⋊C4—42no—
94C7×C22⋊C4—42no—
95C7×C22⋊C4—81no—
96C42×C7—21no—
97C42×C7—21no—
98C42×C7—42no—
99C42×C7—81no—
100C42×C7—82no—
101C2×C4×D4C22×C711yescharacteristic
102C2×#99—21no—
103#407C22×C712yes—
104#388—21no—
105#387—21no—
106#386—21no—
107#383C4×C711yescharacteristic, complemented
108#381C22×C711yescharacteristic
109#380C4×C711yescharacteristic, complemented
110C4×C22⋊C4—21no—
111C4×C22⋊C4C22×C712yes—
112C4×C22⋊C4C22×C711yescharacteristic
113#376C4×C711yescharacteristic, complemented
114C2×#95C4×C711yescharacteristic, complemented
115#342—21no—
116#324—21no—
117#323—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—41no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7#10412yes—
127C23×C7C2×C22⋊C411yescharacteristic
128C23×C7#12312yes—
129C7×D4—44no—
130C2×C4×C7—22no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—41no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—44no—
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—81no—
155C2×C4×C7—81no—
156C2×C4×C7—84no—
157C2×C4×C7—82no—
158C2×C4×C7—82no—
159C2×C4×C7#11711yescharacteristic
160C2×C4×C7#11711yescharacteristic
161C2×C4×C7#12312yes—
162C7×C8—82no—
163C22×D4C7×D411yescharacteristic
164C23×C4—21no—
165C23×C4C2×C4×C711yescharacteristic
166C23×C4C23×C711yescharacteristic
167C2×M16—22no—
168C4×D4—22no—
169C4×D4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4C7×D411yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4C7×D411yescharacteristic
177C2×C22⋊C4C7×D411yescharacteristic
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42C2×C4×C711yescharacteristic
182#99—24no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95—22no—
191#95C2×C4×C711yescharacteristic
192#95C2×C4×C711yescharacteristic
193#95C2×C4×C711yescharacteristic
194#95C2×C4×C711yescharacteristic
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—22no—
198C22×C7—22no—
199C22×C7—22no—
200C22×C7—22no—
201C22×C7—42no—
202C22×C7—42no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—81no—
206C22×C7—81no—
207C22×C7#38111yescharacteristic
208C22×C7C2×#10411yescharacteristic
209C22×C7#42111yescharacteristic
210C4×C7—22no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—42no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—81no—
218C4×C7—81no—
219C4×C7—81no—
220C4×C7—84no—
221C24C7×C22⋊C411yescharacteristic
222C24C2×C7×D411yescharacteristic
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—41no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4C7×C22⋊C411yescharacteristic
239C22×C4C7×C22⋊C411yescharacteristic
240C22×C4C7×C22⋊C411yescharacteristic
241C22×C4C22×C4×C711yescharacteristic
242C22×C4C2×C7×D411yescharacteristic
243C22×C4C7×#4112yes—
244C22×C4C7×#4112yes—
245M16—82no—
246C2×C8—42no—
247C4⋊C4—22no—
248C4⋊C4—81no—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—24no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—81no—
256C42—21no—
257C42—21no—
258C42—42no—
259C42—81no—
260C42—82no—
261C2×C7—21no—
262C2×C7—21no—
263C2×C7—42no—
264C2×C7—42no—
265C2×C7#182611yescharacteristic
266C2×C7#182711yescharacteristic
267C2×C7#182811yescharacteristic
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23C7×#10412yes—
277C23C2×C7×C22⋊C411yescharacteristic
278C23C7×#12312yes—
279D4—44no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—84no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4C7×#11711yescharacteristic
310C2×C4C7×#11711yescharacteristic
311C2×C4C7×#12312yes—
312C8—82no—
313C7#877611yescharacteristic, direct product factor, Sylow
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22C7×#38111yescharacteristic
327C22C2×C7×#10411yescharacteristic
328C22C7×#42111yescharacteristic
329C4—22no—
330C4—21no—
331C4—21no—
332C4—42no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—84no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—42no—
344C2C7×#182611yescharacteristic
345C2C7×#182711yescharacteristic
346C2C7×#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×#1297C2semidirect
5. C7×#1284C2semidirect
12. C7×#383C4semidirect
14. C7×#380C4semidirect
18. C7×#376C4semidirect
19. C2×C7×#95C4semidirect
23. #8776C7direct (N × H)
59. #1297C2×C7semidirect
60. #1284C2×C7semidirect
107. #383C4×C7semidirect
109. #380C4×C7semidirect
113. #376C4×C7semidirect
114. C2×#95C4×C7semidirect
313. C7#8776direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23532
C4×C7×#992
C7×#14652
C7×#12972
C7×#12842
#87767