Finite Group Databasebeta

← C7×#9453 #13186669

Subgroups — C7×#9453 #13186669

Order 1792.

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

normal subgroups only

This group has 498 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×#2563C211yesmaximal, characteristic
3C7×#1556C212yesmaximal, complemented
4C7×#1466C211yesmaximal, characteristic
5C7×#1285C212yesmaximal, complemented
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C7×#435—24no—
9C7×#432C2212yes—
10C7×#432C2212yes—
11C7×#407—21no—
12C2×C7×C22⋊C8—21no—
13C7×#393—21no—
14C7×#393—21no—
15C7×#390C412yes—
16C7×#382C412yes—
17C7×#380—21no—
18C7×#380—21no—
19C4×C7×C4⋊C4C2211yescharacteristic
20C7×#332—24no—
21C7×#323—21no—
22C7×#323—21no—
23#9453C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8—21no—
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—
32C4×C7×D4—22no—
33C4×C7×D4—22no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C412yes—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4D411yescharacteristic
41C2×C7×C22⋊C4D411yescharacteristic
42C2×C42×C7C2×C412yes—
43C2×C42×C7C2311yescharacteristic
44C7×C4⋊C8—22no—
45C7×C4⋊C8D414yes—
46C7×C22⋊C8—22no—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—22no—
49C7×C22⋊C8—22no—
50C7×#97—22no—
51C7×#97—22no—
52C7×#95—22no—
53C7×#95—22no—
54C7×#95—21no—
55#2973C2×C711yescharacteristic
56#2563C2×C711yescharacteristic
57#1556C2×C712yescomplemented
58#1466C2×C711yescharacteristic
59#1285C2×C712yescomplemented
60C24×C7C22⋊C411yescharacteristic
61C24×C7C22⋊C411yescharacteristic
62C2×C7×Q8—42no—
63C2×C7×Q8—82no—
64C2×C7×D4—22no—
65C2×C7×D4—22no—
66C2×C7×D4—41no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—22no—
72C22×C4×C7—22no—
73C22×C4×C7—42no—
74C22×C4×C7—42no—
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C7×M16—82no—
81C2×C7×C8—24no—
82C2×C7×C8—42no—
83C2×C7×C8—82no—
84C7×C4⋊C4—22no—
85C7×C4⋊C4—24no—
86C7×C4⋊C4—42no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C22⋊C4—22no—
90C7×C22⋊C4—22no—
91C7×C22⋊C4—82no—
92C7×C22⋊C4—82no—
93C42×C7—42no—
94C42×C7C2×D412yes—
95C42×C7#4112yes—
96C2×C4⋊Q8C22×C711yescharacteristic
97C2×C4×D4C22×C711yescharacteristic
98#435—24no—
99#432C22×C712yes—
100#432C22×C712yes—
101#407—21no—
102C2×C22⋊C8—21no—
103#393—21no—
104#393—21no—
105#390C4×C712yes—
106#382C4×C712yes—
107#380—21no—
108#380—21no—
109C4×C4⋊C4C22×C711yescharacteristic
110#332—24no—
111#323—21no—
112#323—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—41no—
119C23×C7—41no—
120C23×C7—41no—
121C23×C7—41no—
122C23×C7#10112yes—
123C23×C7C2×C22⋊C411yescharacteristic
124C7×Q8—84no—
125C7×D4—44no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—22no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—41no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—81no—
143C2×C4×C7—81no—
144C2×C4×C7C4×D412yes—
145C2×C4×C7#12011yescharacteristic
146C2×C4×C7#12112yes—
147C2×C4×C7#12311yescharacteristic
148C7×C8—44no—
149C7×C8—82no—
150C22×Q8—21no—
151C22×D4C2×C4×C711yescharacteristic
152C23×C4C7×D411yescharacteristic
153C23×C4C7×D411yescharacteristic
154C2×M16—22no—
155C22×C8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4×D4—22no—
159C4×D4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×C4×C712yes—
163C2×C4⋊C4C2×C4×C711yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4C7×D411yescharacteristic
167C2×C22⋊C4C7×D411yescharacteristic
168C2×C42C2×C4×C712yes—
169C2×C42C23×C711yescharacteristic
170C4⋊C8—22no—
171C4⋊C8C7×D414yes—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176#97—22no—
177#97—22no—
178#95—22no—
179#95—22no—
180#95—21no—
181C22×C7—21no—
182C22×C7—21no—
183C22×C7—22no—
184C22×C7—42no—
185C22×C7—42no—
186C22×C7—41no—
187C22×C7—41no—
188C22×C7—41no—
189C22×C7—41no—
190C22×C7—41no—
191C22×C7—81no—
192C22×C7—81no—
193C22×C7#38611yescharacteristic
194C22×C7C2×#10111yescharacteristic
195C22×C7#41311yescharacteristic
196C4×C7—21no—
197C4×C7—22no—
198C4×C7—22no—
199C4×C7—41no—
200C4×C7—41no—
201C4×C7—82no—
202C4×C7—82no—
203C4×C7#45512yes—
204C24C7×C22⋊C411yescharacteristic
205C24C7×C22⋊C411yescharacteristic
206C2×Q8—42no—
207C2×Q8—82no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4C7×C22⋊C411yescharacteristic
220C22×C4C7×C22⋊C411yescharacteristic
221C22×C4C22×C4×C711yescharacteristic
222C22×C4C2×C7×D411yescharacteristic
223C22×C4C2×C7×D411yescharacteristic
224M16—82no—
225C2×C8—24no—
226C2×C8—42no—
227C2×C8—82no—
228C4⋊C4—22no—
229C4⋊C4—24no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C42—42no—
238C42C2×C7×D412yes—
239C42C7×#4112yes—
240C2×C7—21no—
241C2×C7—21no—
242C2×C7—42no—
243C2×C7—81no—
244C2×C7#188511yescharacteristic
245C2×C7#190111yescharacteristic
246C2×C7#191011yescharacteristic
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23C7×#10112yes—
257C23C2×C7×C22⋊C411yescharacteristic
258Q8—84no—
259D4—44no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4C4×C7×D412yes—
279C2×C4C7×#12011yescharacteristic
280C2×C4C7×#12112yes—
281C2×C4C7×#12311yescharacteristic
282C8—44no—
283C8—82no—
284C7#945311yescharacteristic, direct product factor, Sylow
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22C7×#38611yescharacteristic
298C22C2×C7×#10111yescharacteristic
299C22C7×#41311yescharacteristic
300C4—21no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—82no—
306C4—82no—
307C4C7×#45512yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2C7×#188511yescharacteristic
313C2C7×#190111yescharacteristic
314C2C7×#191011yescharacteristic
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×#1556C2semidirect
5. C7×#1285C2semidirect
23. #9453C7direct (N × H)
57. #1556C2×C7semidirect
59. #1285C2×C7semidirect
284. C7#9453direct (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×#25632
C7×#15562
C7×#14662
C7×#12852
#94537