Finite Group Databasebeta

← C7×#9028 #13186244

Subgroups — C7×#9028 #13186244

Order 1792.

696 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C7×#397C211yesmaximal, characteristic
2C7×#2312C211yesmaximal, characteristic
3C7×#1823C212yesmaximal, complemented
4C2×C7×#323C211yesmaximal, characteristic
5C7×#1294C212yesmaximal, complemented
6C22×C7×C4⋊C4C2211yescharacteristic
7C22×C7×C22⋊C4—21no—
8C22×C7×C22⋊C4C2211yescharacteristic
9C2×C7×C22⋊C8C2212yes—
10C7×#397—22no—
11C7×#397C414yescomplemented
12C7×#388—24no—
13C7×#380C2212yes—
14C7×#379—21no—
15C4×C7×C22⋊C4—21no—
16C2×C7×#95—21no—
17C2×C7×#95C2211yescharacteristic
18C7×#340—24no—
19C7×#323—22no—
20#9028C711yesmaximal, characteristic, direct product factor, Sylow
21C25×C7D411yescharacteristic
22C23×C4×C7—21no—
23C23×C4×C7—21no—
24C23×C4×C7D411yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C22×C7×C8—22no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4C2×C412yes—
31C2×C7×C4⋊C4D414yes—
32C2×C7×C22⋊C4—22no—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—24no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—44no—
39C2×C7×C22⋊C4C2×C412yes—
40C2×C7×C22⋊C4D412yes—
41C2×C42×C7—22no—
42C7×C22⋊C8—22no—
43C7×C22⋊C8—22no—
44C7×#95—22no—
45C7×#95—24no—
46C7×#95—21no—
47C7×#95—22no—
48C7×#95C2×C412yes—
49C2×#397C2×C711yescharacteristic
50#2312C2×C711yescharacteristic
51#1823C2×C712yescomplemented
52C2×#323C2×C711yescharacteristic
53#1294C2×C712yescomplemented
54C24×C7—21no—
55C24×C7—22no—
56C24×C7—21no—
57C24×C7—22no—
58C24×C7—42no—
59C24×C7—42no—
60C24×C7C22⋊C412yes—
61C24×C7C2×D411yescharacteristic
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—24no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7—41no—
72C22×C4×C7—41no—
73C22×C4×C7—42no—
74C22×C4×C7—42no—
75C22×C4×C7—42no—
76C22×C4×C7C22⋊C412yes—
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D412yes—
79C22×C4×C7C2×D411yescharacteristic
80C22×C4×C7#4112yes—
81C2×C7×C8—44no—
82C2×C7×C8—42no—
83C7×C4⋊C4—24no—
84C7×C4⋊C4—44no—
85C7×C4⋊C4—42no—
86C7×C22⋊C4—24no—
87C7×C22⋊C4—24no—
88C7×C22⋊C4—48no—
89C7×C22⋊C4—42no—
90C7×C22⋊C4—42no—
91C7×C22⋊C4—82no—
92C7×C22⋊C4—82no—
93C42×C7—82no—
94C22×C4⋊C4C22×C711yescharacteristic
95C22×C22⋊C4—21no—
96C22×C22⋊C4C22×C711yescharacteristic
97C2×C22⋊C8C22×C712yes—
98#397—22no—
99#397C4×C714yescomplemented
100#388—24no—
101#380C22×C712yes—
102#379—21no—
103C4×C22⋊C4—21no—
104C2×#95—21no—
105C2×#95C22×C711yescharacteristic
106#340—24no—
107#323—22no—
108C23×C7—22no—
109C23×C7—22no—
110C23×C7—22no—
111C23×C7—22no—
112C23×C7—22no—
113C23×C7—42no—
114C23×C7—42no—
115C23×C7—48no—
116C23×C7—42no—
117C23×C7—42no—
118C23×C7—42no—
119C23×C7—42no—
120C23×C7—81no—
121C23×C7—81no—
122C23×C7—84no—
123C23×C7C2×C22⋊C411yescharacteristic
124C23×C7#12011yescharacteristic
125C23×C7#12311yescharacteristic
126C23×C7#13512yes—
127C23×C7AGL(1,ℤ/8)12yes—
128C2×C4×C7—24no—
129C2×C4×C7—22no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7—42no—
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—42no—
140C2×C4×C7—42no—
141C2×C4×C7—44no—
142C2×C4×C7—44no—
143C2×C4×C7—44no—
144C2×C4×C7—81no—
145C2×C4×C7—84no—
146C2×C4×C7—82no—
147C2×C4×C7—82no—
148C2×C4×C7C4×D412yes—
149C2×C4×C7#12112yes—
150C7×C8—82no—
151C25C7×D411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C7×D411yescharacteristic
155C23×C4C23×C711yescharacteristic
156C22×C8—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4C2×C4×C712yes—
161C2×C4⋊C4C7×D414yes—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4C2×C4×C712yes—
170C2×C22⋊C4C7×D412yes—
171C2×C42—22no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174#95—22no—
175#95—24no—
176#95—21no—
177#95—22no—
178#95C2×C4×C712yes—
179C22×C7—24no—
180C22×C7—24no—
181C22×C7—21no—
182C22×C7—21no—
183C22×C7—21no—
184C22×C7—21no—
185C22×C7—44no—
186C22×C7—42no—
187C22×C7—42no—
188C22×C7—42no—
189C22×C7—82no—
190C22×C7—84no—
191C22×C7—84no—
192C22×C7#38611yescharacteristic
193C22×C7#41611yescharacteristic
194C22×C7#41811yescharacteristic
195C22×C7#44912yes—
196C22×C7#45512yes—
197C22×C7#45812yes—
198C22×C7#48212yes—
199C4×C7—22no—
200C4×C7—22no—
201C4×C7—42no—
202C4×C7—82no—
203C4×C7—84no—
204C4×C7—82no—
205C24—21no—
206C24—22no—
207C24—21no—
208C24—22no—
209C24—42no—
210C24—42no—
211C24C7×C22⋊C412yes—
212C24C2×C7×D411yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4C7×C22⋊C412yes—
228C22×C4C22×C4×C711yescharacteristic
229C22×C4C2×C7×D412yes—
230C22×C4C2×C7×D411yescharacteristic
231C22×C4C7×#4112yes—
232C2×C8—44no—
233C2×C8—42no—
234C4⋊C4—24no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—48no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C22⋊C4—82no—
244C42—82no—
245C2×C7—21no—
246C2×C7—21no—
247C2×C7—22no—
248C2×C7—82no—
249C2×C7#161214yes—
250C2×C7#187511yescharacteristic
251C2×C7#188511yescharacteristic
252C2×C7#189011yescharacteristic
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23—48no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—81no—
266C23—81no—
267C23—84no—
268C23C2×C7×C22⋊C411yescharacteristic
269C23C7×#12011yescharacteristic
270C23C7×#12311yescharacteristic
271C23C7×#13512yes—
272C23C7×AGL(1,ℤ/8)12yes—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—81no—
290C2×C4—84no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4C4×C7×D412yes—
294C2×C4C7×#12112yes—
295C8—82no—
296C7#902811yescharacteristic, direct product factor, Sylow
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—44no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—82no—
308C22—84no—
309C22—84no—
310C22C7×#38611yescharacteristic
311C22C7×#41611yescharacteristic
312C22C7×#41811yescharacteristic
313C22C7×#44912yes—
314C22C7×#45512yes—
315C22C7×#45812yes—
316C22C7×#48212yes—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—82no—
321C4—84no—
322C4—82no—
323C2—21no—
324C2—22no—
325C2—21no—
326C2—82no—
327C2C7×#161214yes—
328C2C7×#187511yescharacteristic
329C2C7×#188511yescharacteristic
330C2C7×#189011yescharacteristic
subgroupG/H
center, socleC23×C7#120
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC25×C7D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1823C2semidirect
5. C7×#1294C2semidirect
11. C7×#397C4semidirect
20. #9028C7direct (N × H)
51. #1823C2×C7semidirect
53. #1294C2×C7semidirect
99. #397C4×C7semidirect
296. C7#9028direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3972
C7×#23122
C7×#18232
C2×C7×#3232
C7×#12942
#90287