Finite Group Databasebeta

← C7×#9451 #13186667

Subgroups — C7×#9451 #13186667

Order 1792.

518 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2973C211yesmaximal, characteristic
2C7×#2547C211yesmaximal, characteristic
3C7×#1558C212yesmaximal, complemented
4C7×#1466C211yesmaximal, characteristic
5C7×#1285C212yesmaximal, complemented
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C7×#124C412yes—
8C2×C4×C7×D4C2211yescharacteristic
9C7×#435—24no—
10C7×#432C2212yes—
11C7×#432C2212yes—
12C7×#407—21no—
13C2×C7×C22⋊C8—21no—
14C7×#397—21no—
15C7×#397—21no—
16C7×#391C412yes—
17C7×#380—21no—
18C7×#380—21no—
19C4×C7×C4⋊C4C2211yescharacteristic
20C7×#333—24no—
21C7×#323—21no—
22C7×#323—21no—
23#9451C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C412yes—
25C22×C7×D4C2×C411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C2×C7×M16—22no—
29C22×C7×C8—22no—
30C7×C4⋊Q8—22no—
31C7×C4⋊Q8—22no—
32C7×#124—44no—
33C4×C7×D4—22no—
34C4×C7×D4—22no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4D411yescharacteristic
42C2×C7×C22⋊C4D411yescharacteristic
43C2×C42×C7C2×C412yes—
44C2×C42×C7C2311yescharacteristic
45C7×C4⋊C8—22no—
46C7×C4⋊C8D414yes—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—22no—
49C7×C22⋊C8—22no—
50C7×C22⋊C8—22no—
51C7×#97—22no—
52C7×#97—22no—
53C7×#95—22no—
54C7×#95—22no—
55C7×#95—21no—
56#2973C2×C711yescharacteristic
57#2547C2×C711yescharacteristic
58#1558C2×C712yescomplemented
59#1466C2×C711yescharacteristic
60#1285C2×C712yescomplemented
61C24×C7C22⋊C411yescharacteristic
62C24×C7C22⋊C411yescharacteristic
63C2×C7×Q8—24no—
64C2×C7×Q8—42no—
65C2×C7×Q8—42no—
66C2×C7×D4—22no—
67C2×C7×D4—22no—
68C2×C7×D4—41no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—22no—
74C22×C4×C7—22no—
75C22×C4×C7—42no—
76C22×C4×C7—42no—
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22×C411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C7×M16—82no—
83C2×C7×C8—24no—
84C2×C7×C8—42no—
85C2×C7×C8—82no—
86C7×C4⋊C4—22no—
87C7×C4⋊C4—42no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—42no—
90C7×C4⋊C4—41no—
91C7×C22⋊C4—22no—
92C7×C22⋊C4—22no—
93C7×C22⋊C4—44no—
94C7×C22⋊C4—44no—
95C42×C7—42no—
96C42×C7C2×D412yes—
97C42×C7#4112yes—
98C2×C4⋊Q8C22×C711yescharacteristic
99C2×#124C4×C712yes—
100C2×C4×D4C22×C711yescharacteristic
101#435—24no—
102#432C22×C712yes—
103#432C22×C712yes—
104#407—21no—
105C2×C22⋊C8—21no—
106#397—21no—
107#397—21no—
108#391C4×C712yes—
109#380—21no—
110#380—21no—
111C4×C4⋊C4C22×C711yescharacteristic
112#333—24no—
113#323—21no—
114#323—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—41no—
121C23×C7—41no—
122C23×C7—41no—
123C23×C7—41no—
124C23×C7#10012yes—
125C23×C7C2×C22⋊C411yescharacteristic
126C7×Q8—84no—
127C7×D4—44no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—22no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—41no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7—81no—
145C2×C4×C7—81no—
146C2×C4×C7C4×D412yes—
147C2×C4×C7#12011yescharacteristic
148C2×C4×C7#12112yes—
149C2×C4×C7#12311yescharacteristic
150C7×C8—44no—
151C7×C8—82no—
152C22×Q8C2×C4×C712yes—
153C22×D4C2×C4×C711yescharacteristic
154C23×C4C7×D411yescharacteristic
155C23×C4C7×D411yescharacteristic
156C2×M16—22no—
157C22×C8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160#124—44no—
161C4×D4—22no—
162C4×D4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4C2×C4×C711yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4C7×D411yescharacteristic
170C2×C22⋊C4C7×D411yescharacteristic
171C2×C42C2×C4×C712yes—
172C2×C42C23×C711yescharacteristic
173C4⋊C8—22no—
174C4⋊C8C7×D414yes—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179#97—22no—
180#97—22no—
181#95—22no—
182#95—22no—
183#95—21no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—22no—
187C22×C7—42no—
188C22×C7—42no—
189C22×C7—41no—
190C22×C7—41no—
191C22×C7—41no—
192C22×C7—41no—
193C22×C7—41no—
194C22×C7—81no—
195C22×C7—81no—
196C22×C7#38611yescharacteristic
197C22×C7C2×#10011yescharacteristic
198C22×C7#41311yescharacteristic
199C4×C7—21no—
200C4×C7—22no—
201C4×C7—22no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—82no—
205C4×C7—82no—
206C4×C7#45512yes—
207C24C7×C22⋊C411yescharacteristic
208C24C7×C22⋊C411yescharacteristic
209C2×Q8—24no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C7×C22⋊C411yescharacteristic
224C22×C4C7×C22⋊C411yescharacteristic
225C22×C4C22×C4×C711yescharacteristic
226C22×C4C2×C7×D411yescharacteristic
227C22×C4C2×C7×D411yescharacteristic
228M16—82no—
229C2×C8—24no—
230C2×C8—42no—
231C2×C8—82no—
232C4⋊C4—22no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C42—42no—
242C42C2×C7×D412yes—
243C42C7×#4112yes—
244C2×C7—21no—
245C2×C7—21no—
246C2×C7—42no—
247C2×C7—81no—
248C2×C7#188511yescharacteristic
249C2×C7#190011yescharacteristic
250C2×C7#190911yescharacteristic
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23C7×#10012yes—
261C23C2×C7×C22⋊C411yescharacteristic
262Q8—84no—
263D4—44no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4C4×C7×D412yes—
283C2×C4C7×#12011yescharacteristic
284C2×C4C7×#12112yes—
285C2×C4C7×#12311yescharacteristic
286C8—44no—
287C8—82no—
288C7#945111yescharacteristic, direct product factor, Sylow
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22C7×#38611yescharacteristic
302C22C2×C7×#10011yescharacteristic
303C22C7×#41311yescharacteristic
304C4—21no—
305C4—22no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—82no—
310C4—82no—
311C4C7×#45512yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2C7×#188511yescharacteristic
317C2C7×#190011yescharacteristic
318C2C7×#190911yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C7×#1558C2semidirect
5. C7×#1285C2semidirect
23. #9451C7direct (N × H)
58. #1558C2×C7semidirect
60. #1285C2×C7semidirect
288. C7#9451direct (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×#25472
C7×#15582
C7×#14662
C7×#12852
#94517