Finite Group Databasebeta

← C7×#12800 #13190016

Subgroups — C7×#12800 #13190016

Order 1792.

524 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2326C211yesmaximal, characteristic
2C2×C7×#354C212yesmaximal
3C7×#1799C212yesmaximal
4C7×#1393C212yesmaximal
5C2×C7×#124C2211yescharacteristic
6C2×C4×C7×Q8C411yescharacteristic, complemented
7C2×C4×C7×D4C2211yescharacteristic
8C2×C7×#99C2214yes—
9C7×#391C411yescharacteristic, complemented
10C7×#390C2211yescharacteristic
11C7×#388—22no—
12C7×#385—22no—
13C7×#384C411yescharacteristic, complemented
14C7×#380—22no—
15C7×#376C411yescharacteristic, complemented
16C7×#354—24no—
17C7×#342—22no—
18#12800C711yesmaximal, characteristic, direct product factor, Sylow
19C22×C7×Q8C2×C411yescharacteristic
20C22×C7×D4C2311yescharacteristic
21C23×C4×C7D412yes—
22C7×#124—22no—
23C7×#124D412yes—
24C7×#124Q812yes—
25C4×C7×Q8—42no—
26C4×C7×D4—42no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4—42no—
30C2×C7×C4⋊C4C2×C411yescharacteristic
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C22⋊C4—22no—
33C2×C7×C22⋊C4—24no—
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4D412yes—
36C2×C42×C7—21no—
37C2×C42×C7C2×C411yescharacteristic
38C2×C42×C7C2×C411yescharacteristic
39C2×C42×C7C2×C411yescharacteristic
40C7×#99—24no—
41C7×#99—24no—
42C7×#95—21no—
43C7×#95—22no—
44C7×#95—21no—
45C7×#95—42no—
46#2326C2×C711yescharacteristic
47C2×#354C2×C712yes—
48#1799C2×C712yes—
49#1393C2×C712yes—
50C24×C7C2×D412yes—
51C2×C7×Q8—21no—
52C2×C7×Q8—41no—
53C2×C7×Q8—41no—
54C2×C7×Q8C4⋊C412yescomplemented
55C2×C7×D4—41no—
56C2×C7×D4—41no—
57C2×C7×D4C2×D411yescharacteristic
58C2×C7×D4C2×Q811yescharacteristic
59C2×C7×D4#4112yes—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—44no—
67C22×C4×C7—82no—
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22⋊C412yes—
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22×C411yescharacteristic
72C7×C4⋊C4—41no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—42no—
75C7×C4⋊C4—82no—
76C7×C4⋊C4—84no—
77C7×C22⋊C4—24no—
78C7×C22⋊C4—24no—
79C7×C22⋊C4—42no—
80C7×C22⋊C4—44no—
81C7×C22⋊C4—44no—
82C7×C22⋊C4—82no—
83C42×C7—22no—
84C42×C7—21no—
85C42×C7—41no—
86C42×C7—42no—
87C42×C7C4⋊C412yescomplemented
88C2×#124C22×C711yescharacteristic
89C2×C4×Q8C4×C711yescharacteristic, complemented
90C2×C4×D4C22×C711yescharacteristic
91C2×#99C22×C714yes—
92#391C4×C711yescharacteristic, complemented
93#390C22×C711yescharacteristic
94#388—22no—
95#385—22no—
96#384C4×C711yescharacteristic, complemented
97#380—22no—
98#376C4×C711yescharacteristic, complemented
99#354—24no—
100#342—22no—
101C23×C7—22no—
102C23×C7—22no—
103C23×C7—42no—
104C23×C7—42no—
105C23×C7C2×C22⋊C411yescharacteristic
106C23×C7#12112yes—
107C23×C7C22⋊Q812yes—
108C7×Q8—42no—
109C7×Q8—42no—
110C7×D4—41no—
111C7×D4—41no—
112C7×D4—42no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—22no—
116C2×C4×C7—22no—
117C2×C4×C7—21no—
118C2×C4×C7—22no—
119C2×C4×C7—41no—
120C2×C4×C7—44no—
121C2×C4×C7—41no—
122C2×C4×C7—41no—
123C2×C4×C7—41no—
124C2×C4×C7—41no—
125C2×C4×C7—42no—
126C2×C4×C7—41no—
127C2×C4×C7—41no—
128C2×C4×C7—42no—
129C2×C4×C7—41no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7—82no—
133C2×C4×C7—84no—
134C2×C4×C7—84no—
135C2×C4×C7—84no—
136C2×C4×C7#9912yes—
137C2×C4×C7C2×C4⋊C411yescharacteristic
138C2×C4×C7#11711yescharacteristic
139C22×Q8C2×C4×C711yescharacteristic
140C22×D4C23×C711yescharacteristic
141C23×C4C7×D412yes—
142#124—22no—
143#124C7×D412yes—
144#124C7×Q812yes—
145C4×Q8—42no—
146C4×D4—42no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4C2×C4×C711yescharacteristic
151C2×C4⋊C4C2×C4×C711yescharacteristic
152C2×C22⋊C4—22no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4C7×D412yes—
156C2×C42—21no—
157C2×C42C2×C4×C711yescharacteristic
158C2×C42C2×C4×C711yescharacteristic
159C2×C42C2×C4×C711yescharacteristic
160#99—24no—
161#99—24no—
162#95—21no—
163#95—21no—
164#95—22no—
165#95—42no—
166C22×C7—21no—
167C22×C7—21no—
168C22×C7—22no—
169C22×C7—22no—
170C22×C7—42no—
171C22×C7—42no—
172C22×C7—42no—
173C22×C7—42no—
174C22×C7—82no—
175C22×C7#38011yescharacteristic
176C22×C7C2×#9911yescharacteristic
177C22×C7#41011yescharacteristic
178C4×C7—21no—
179C4×C7—21no—
180C4×C7—22no—
181C4×C7—42no—
182C4×C7—42no—
183C4×C7—42no—
184C4×C7—41no—
185C4×C7—81no—
186C4×C7—81no—
187C4×C7—164no—
188C24C2×C7×D412yes—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8C7×C4⋊C412yescomplemented
193C2×D4—41no—
194C2×D4—41no—
195C2×D4C2×C7×D411yescharacteristic
196C2×D4C2×C7×Q811yescharacteristic
197C2×D4C7×#4112yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—44no—
205C22×C4—82no—
206C22×C4C7×C22⋊C411yescharacteristic
207C22×C4C7×C22⋊C412yes—
208C22×C4C7×C22⋊C411yescharacteristic
209C22×C4C22×C4×C711yescharacteristic
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—82no—
214C4⋊C4—84no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—42no—
218C22⋊C4—44no—
219C22⋊C4—44no—
220C22⋊C4—82no—
221C42—22no—
222C42—21no—
223C42—41no—
224C42—42no—
225C42C7×C4⋊C412yescomplemented
226C2×C7—21no—
227C2×C7—21no—
228C2×C7—42no—
229C2×C7—42no—
230C2×C7#179911yescharacteristic
231C2×C7#212711yescharacteristic
232C2×C7#212911yescharacteristic
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23C2×C7×C22⋊C411yescharacteristic
238C23C7×#12112yes—
239C23C7×C22⋊Q812yes—
240Q8—42no—
241Q8—42no—
242D4—41no—
243D4—41no—
244D4—42no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—44no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—82no—
265C2×C4—84no—
266C2×C4—84no—
267C2×C4—84no—
268C2×C4C7×#9912yes—
269C2×C4C2×C7×C4⋊C411yescharacteristic
270C2×C4C7×#11711yescharacteristic
271C7#1280011yescharacteristic, direct product factor, Sylow
272C22—21no—
273C22—21no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—82no—
281C22C7×#38011yescharacteristic
282C22C2×C7×#9911yescharacteristic
283C22C7×#41011yescharacteristic
284C4—21no—
285C4—21no—
286C4—22no—
287C4—42no—
288C4—42no—
289C4—42no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—164no—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—42no—
298C2C7×#179911yescharacteristic
299C2C7×#212711yescharacteristic
300C2C7×#212911yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC22×D4C23×C7
prime radicalC22×C7×D4C23
LCSG ▷ C22×C4 ▷ C22 ▷ 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
6. C2×C4×C7×Q8C4semidirect
9. C7×#391C4semidirect
13. C7×#384C4semidirect
15. C7×#376C4semidirect
18. #12800C7direct (N × H)
54. C2×C7×Q8C4⋊C4semidirect
87. C42×C7C4⋊C4semidirect
89. C2×C4×Q8C4×C7semidirect
92. #391C4×C7semidirect
96. #384C4×C7semidirect
98. #376C4×C7semidirect
192. C2×Q8C7×C4⋊C4semidirect
225. C42C7×C4⋊C4semidirect
271. C7#12800direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23262
C2×C7×#3542
C7×#17992
C7×#13932
#128007