Finite Group Databasebeta

← C7×#9032 #13186248

Subgroups — C7×#9032 #13186248

Order 1792.

516 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2983C211yesmaximal, characteristic, complemented
2C7×#2398C211yesmaximal, characteristic
3C4×C7×#101C211yesmaximal, characteristic, complemented
4C7×#1490C212yesmaximal
5C7×#1300C212yesmaximal, complemented
6C2×C4×C7×Q8C2211yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C7×#446—24no—
9C7×#432—21no—
10C7×#432C2212yes—
11C7×#431—21no—
12C2×C7×#101—21no—
13C4×C7×M16C2212yes—
14C7×#400C411yescharacteristic
15C7×#400C411yescharacteristic
16C7×#397—21no—
17C7×#394C411yescharacteristic
18C7×#394C411yescharacteristic
19C7×#393—21no—
20C4×C7×C22⋊C4—21no—
21C4×C7×C22⋊C4—21no—
22C7×#376C2211yescharacteristic
23C7×#343—21no—
24C7×D4⋊C8—24no—
25#9032C711yesmaximal, characteristic, direct product factor, Sylow
26C22×C7×Q8D411yescharacteristic
27C22×C7×D4C2×C411yescharacteristic
28C23×C4×C7—21no—
29C2×C7×M16—22no—
30C4×C7×Q8—22no—
31C4×C7×Q8—22no—
32C4×C7×D4—22no—
33C4×C7×D4—22no—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C42×C7—21no—
44C2×C42×C7D411yescharacteristic
45C2×C42×C7D411yescharacteristic
46C2×C42×C7C2311yescharacteristic
47C7×C4⋊C8—22no—
48C7×C4⋊C8—22no—
49C7×C4⋊C8D414yes—
50C7×#101—24no—
51C7×C22⋊C8—22no—
52C7×#97—22no—
53C7×#97—22no—
54C7×#97—22no—
55C4×C7×C8—22no—
56C4×C7×C8—22no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—21no—
60C7×#95—21no—
61C7×#95—21no—
62C7×#95C2×C411yescharacteristic
63C7×#95C2×C411yescharacteristic
64#2983C2×C711yescharacteristic, complemented
65#2398C2×C711yescharacteristic
66C4×#101C2×C711yescharacteristic, complemented
67#1490C2×C712yes—
68#1300C2×C712yescomplemented
69C24×C7—21no—
70C2×C7×Q8—22no—
71C2×C7×Q8—22no—
72C2×C7×Q8—41no—
73C2×C7×D4—22no—
74C2×C7×D4—42no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—42no—
79C22×C4×C7—42no—
80C22×C4×C7—41no—
81C22×C4×C7—41no—
82C22×C4×C7C22⋊C411yescharacteristic
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22×C411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C7×M16—24no—
90C7×M16—42no—
91C2×C7×C8—24no—
92C2×C7×C8—42no—
93C7×C4⋊C4—22no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—22no—
96C7×C4⋊C4—22no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—42no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C42×C7—21no—
103C42×C7—21no—
104C42×C7—21no—
105C42×C7—21no—
106C42×C7—42no—
107C42×C7C2×D412yes—
108C42×C7#4112yes—
109C2×C4×Q8C22×C711yescharacteristic
110C2×C4×D4C22×C711yescharacteristic
111#446—24no—
112#432—21no—
113#432C22×C712yes—
114#431—21no—
115C2×#101—21no—
116C4×M16C22×C712yes—
117#400C4×C711yescharacteristic
118#400C4×C711yescharacteristic
119#397—21no—
120#394C4×C711yescharacteristic
121#394C4×C711yescharacteristic
122#393—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125#376C22×C711yescharacteristic
126#343—21no—
127D4⋊C8—24no—
128C23×C7—22no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7—41no—
134C23×C7—41no—
135C23×C7C2×C22⋊C411yescharacteristic
136C7×Q8—44no—
137C7×D4—24no—
138C7×D4—42no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—22no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—82no—
162C2×C4×C7—82no—
163C2×C4×C7#10412yes—
164C2×C4×C7C4×D412yes—
165C2×C4×C7#12011yescharacteristic
166C2×C4×C7#12112yes—
167C2×C4×C7#12311yescharacteristic
168C7×C8—44no—
169C7×C8—44no—
170C22×Q8C7×D411yescharacteristic
171C22×D4C2×C4×C711yescharacteristic
172C23×C4—21no—
173C2×M16—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×D4—22no—
177C4×D4—22no—
178C2×C4⋊C4C2×C4×C711yescharacteristic
179C2×C4⋊C4C2×C4×C711yescharacteristic
180C2×C4⋊C4C2×C4×C711yescharacteristic
181C2×C4⋊C4C7×D411yescharacteristic
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C42—21no—
188C2×C42C7×D411yescharacteristic
189C2×C42C7×D411yescharacteristic
190C2×C42C23×C711yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8C7×D414yes—
194#101—24no—
195C22⋊C8—22no—
196#97—22no—
197#97—22no—
198#97—22no—
199C4×C8—22no—
200C4×C8—22no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95C2×C4×C711yescharacteristic
207#95C2×C4×C711yescharacteristic
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—42no—
211C22×C7—42no—
212C22×C7—42no—
213C22×C7—81no—
214C22×C7—81no—
215C22×C7—81no—
216C22×C7—81no—
217C22×C7#38611yescharacteristic
218C22×C7C2×#10411yescharacteristic
219C22×C7#42111yescharacteristic
220C4×C7—22no—
221C4×C7—22no—
222C4×C7—21no—
223C4×C7—22no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7—81no—
228C4×C7—81no—
229C4×C7—81no—
230C4×C7#45412yes—
231C24—21no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×D4—22no—
236C2×D4—42no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4C7×C22⋊C411yescharacteristic
245C22×C4C7×C22⋊C411yescharacteristic
246C22×C4C7×C22⋊C411yescharacteristic
247C22×C4C7×C22⋊C411yescharacteristic
248C22×C4C22×C4×C711yescharacteristic
249C22×C4C2×C7×D411yescharacteristic
250C22×C4C2×C7×D411yescharacteristic
251M16—24no—
252M16—42no—
253C2×C8—24no—
254C2×C8—42no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—42no—
269C42C2×C7×D412yes—
270C42C7×#4112yes—
271C2×C7—21no—
272C2×C7—21no—
273C2×C7—82no—
274C2×C7#187111yescharacteristic
275C2×C7#188811yescharacteristic
276C2×C7#189111yescharacteristic
277C23—22no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23C2×C7×C22⋊C411yescharacteristic
285Q8—44no—
286D4—24no—
287D4—42no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4C7×#10412yes—
313C2×C4C4×C7×D412yes—
314C2×C4C7×#12011yescharacteristic
315C2×C4C7×#12112yes—
316C2×C4C7×#12311yescharacteristic
317C8—44no—
318C8—44no—
319C7#903211yescharacteristic, direct product factor, Sylow
320C22—21no—
321C22—21no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22C7×#38611yescharacteristic
330C22C2×C7×#10411yescharacteristic
331C22C7×#42111yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4C7×#45412yes—
343C2—21no—
344C2—21no—
345C2—82no—
346C2C7×#187111yescharacteristic
347C2C7×#188811yescharacteristic
348C2C7×#189111yescharacteristic
subgroupG/H
centerC2×C4×C7#120
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
socleC22×C7#386
prime radicalC22×C7×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2983C2semidirect
3. C4×C7×#101C2semidirect
5. C7×#1300C2semidirect
25. #9032C7direct (N × H)
64. #2983C2×C7semidirect
66. C4×#101C2×C7semidirect
68. #1300C2×C7semidirect
319. C7#9032direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#29832
C7×#23982
C4×C7×#1012
C7×#14902
C7×#13002
#90327