Finite Group Databasebeta

← C7×#11777 #13188993

Subgroups — C7×#11777 #13188993

Order 1792.

480 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2547C211yesmaximal, characteristic
2C4×C7×#99C211yesmaximal, characteristic
3C7×#1465C211yesmaximal, characteristic
4C7×#1344C214yesmaximal, complemented
5C2×C7×C4⋊Q8C411yescharacteristic, complemented
6C2×C7×#124—21no—
7C2×C4×C7×D4C2211yescharacteristic
8C2×C7×#99—21no—
9C7×#407—21no—
10C7×#397C412yescomplemented
11C7×#397C2212yes—
12C7×#391—21no—
13C7×#380—21no—
14C7×#380—21no—
15C4×C7×C4⋊C4C411yescharacteristic, complemented
16C4×C7×C22⋊C4—21no—
17C7×#342C2212yes—
18C7×#324—21no—
19C7×#323C2212yes—
20#11777C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×D4D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C2×C7×M16—41no—
26C7×C4⋊Q8—41no—
27C7×C4⋊Q8—41no—
28C7×#124—44no—
29C4×C7×D4—41no—
30C4×C7×D4—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C412yes—
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4C2311yescharacteristic
40C2×C42×C7—21no—
41C2×C42×C7—41no—
42C2×C42×C7C2×C411yescharacteristic
43C7×#99—24no—
44C7×C22⋊C8—22no—
45C7×C22⋊C8—41no—
46C7×#95—22no—
47C7×#95—22no—
48C7×#95—41no—
49C7×#95C2×C412yes—
50#2547C2×C711yescharacteristic
51C4×#99C2×C711yescharacteristic
52#1465C2×C711yescharacteristic
53#1344C2×C714yescomplemented
54C24×C7C22⋊C411yescharacteristic
55C24×C7C2×D411yescharacteristic
56C2×C7×Q8—42no—
57C2×C7×Q8—42no—
58C2×C7×Q8—42no—
59C2×C7×D4—22no—
60C2×C7×D4—22no—
61C2×C7×D4—41no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—22no—
67C22×C4×C7—22no—
68C22×C4×C7—42no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22×C411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C7×M16—82no—
77C2×C7×C8—42no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—42no—
80C7×C4⋊C4—42no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C7×C22⋊C4—24no—
84C7×C22⋊C4—41no—
85C7×C22⋊C4—41no—
86C7×C22⋊C4—44no—
87C7×C22⋊C4—42no—
88C7×C22⋊C4—44no—
89C42×C7—21no—
90C42×C7—21no—
91C42×C7—42no—
92C42×C7—82no—
93C2×C4⋊Q8C4×C711yescharacteristic, complemented
94C2×#124—21no—
95C2×C4×D4C22×C711yescharacteristic
96C2×#99—21no—
97#407—21no—
98#397C4×C712yescomplemented
99#397C22×C712yes—
100#391—21no—
101#380—21no—
102#380—21no—
103C4×C4⋊C4C4×C711yescharacteristic, complemented
104C4×C22⋊C4—21no—
105#342C22×C712yes—
106#324—21no—
107#323C22×C712yes—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7—41no—
114C23×C7—41no—
115C23×C7—41no—
116C23×C7—41no—
117C23×C7C2×C22⋊C411yescharacteristic
118C23×C7AGL(1,ℤ/8)12yes—
119C7×Q8—84no—
120C7×D4—44no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—22no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—41no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7—44no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—81no—
140C2×C4×C7—81no—
141C2×C4×C7—82no—
142C2×C4×C7—82no—
143C2×C4×C7#9911yescharacteristic
144C2×C4×C7#9911yescharacteristic
145C2×C4×C7#10412yes—
146C7×C8—82no—
147C22×Q8—21no—
148C22×D4C7×D411yescharacteristic
149C23×C4C7×D411yescharacteristic
150C23×C4C7×D411yescharacteristic
151C2×M16—41no—
152C4⋊Q8—41no—
153C4⋊Q8—41no—
154#124—44no—
155C4×D4—41no—
156C4×D4—41no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C2×C4×C712yes—
160C2×C4⋊C4C7×D411yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4C2×C4×C711yescharacteristic
165C2×C22⋊C4C23×C711yescharacteristic
166C2×C42—21no—
167C2×C42—41no—
168C2×C42C2×C4×C711yescharacteristic
169#99—24no—
170C22⋊C8—22no—
171C22⋊C8—41no—
172#95—22no—
173#95—22no—
174#95—41no—
175#95C2×C4×C712yes—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—22no—
179C22×C7—42no—
180C22×C7—42no—
181C22×C7—41no—
182C22×C7—41no—
183C22×C7—41no—
184C22×C7—41no—
185C22×C7—41no—
186C22×C7—81no—
187C22×C7—81no—
188C22×C7C2×#9911yescharacteristic
189C22×C7#41811yescharacteristic
190C22×C7C2×#10411yescharacteristic
191C4×C7—21no—
192C4×C7—22no—
193C4×C7—41no—
194C4×C7—41no—
195C4×C7—41no—
196C4×C7—82no—
197C4×C7—84no—
198C4×C7—82no—
199C4×C7#35412yes—
200C24C7×C22⋊C411yescharacteristic
201C24C2×C7×D411yescharacteristic
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—41no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4C7×C22⋊C411yescharacteristic
218C22×C4C7×C22⋊C411yescharacteristic
219C22×C4C7×C22⋊C411yescharacteristic
220C22×C4C22×C4×C711yescharacteristic
221C22×C4C2×C7×D411yescharacteristic
222M16—82no—
223C2×C8—42no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C22⋊C4—24no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—44no—
235C42—21no—
236C42—21no—
237C42—42no—
238C42—82no—
239C2×C7—21no—
240C2×C7—21no—
241C2×C7—42no—
242C2×C7—81no—
243C2×C7#150711yescharacteristic
244C2×C7#212311yescharacteristic
245C2×C7C2×#35411yescharacteristic
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23C2×C7×C22⋊C411yescharacteristic
256C23C7×AGL(1,ℤ/8)12yes—
257Q8—84no—
258D4—44no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4C7×#9911yescharacteristic
282C2×C4C7×#9911yescharacteristic
283C2×C4C7×#10412yes—
284C8—82no—
285C7#1177711yescharacteristic, direct product factor, Sylow
286C22—21no—
287C22—21no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—81no—
298C22C2×C7×#9911yescharacteristic
299C22C7×#41811yescharacteristic
300C22C2×C7×#10411yescharacteristic
301C4—22no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—82no—
307C4—84no—
308C4—82no—
309C4C7×#35412yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—81no—
314C2C7×#150711yescharacteristic
315C2C7×#212311yescharacteristic
316C2C2×C7×#35411yescharacteristic
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×#1344C2semidirect
5. C2×C7×C4⋊Q8C4semidirect
10. C7×#397C4semidirect
15. C4×C7×C4⋊C4C4semidirect
20. #11777C7direct (N × H)
53. #1344C2×C7semidirect
93. C2×C4⋊Q8C4×C7semidirect
98. #397C4×C7semidirect
103. C4×C4⋊C4C4×C7semidirect
285. C7#11777direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25472
C4×C7×#992
C7×#14652
C7×#13442
#117777