Finite Group Databasebeta

← C7×#11831 #13189047

Subgroups — C7×#11831 #13189047

Order 1792.

384 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2399C211yesmaximal, characteristic
2C7×#1801C211yesmaximal, characteristic
3C7×#1467C211yesmaximal, characteristic
4C7×#1354C212yesmaximal
5C7×#1346C212yesmaximal
6C2×C4×C7×Q8C411yescharacteristic
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#432—21no—
9C7×#423—21no—
10C2×C7×#101—21no—
11C7×#398—21no—
12C7×#398C411yescharacteristic
13C7×#398C2211yescharacteristic
14C7×#395—21no—
15C7×#395C411yescharacteristic
16C7×#395C2211yescharacteristic
17C4×C7×C4⋊C4—21no—
18C4×C7×C4⋊C4—21no—
19C7×#376C411yescharacteristic
20C7×#343C2212yes—
21C7×#324—21no—
22C7×#324C2212yes—
23#11831C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×C7×Q8D411yescharacteristic
26C2×C7×M16—22no—
27C4×C7×Q8—41no—
28C4×C7×Q8—41no—
29C4×C7×Q8—41no—
30C4×C7×Q8—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4D411yescharacteristic
39C2×C7×C4⋊C4C2311yescharacteristic
40C2×C42×C7—21no—
41C2×C42×C7—21no—
42C2×C42×C7C2×C411yescharacteristic
43C2×C42×C7D411yescharacteristic
44C2×C42×C7D411yescharacteristic
45C7×C4⋊C8—41no—
46C7×C4⋊C8—41no—
47C7×C4⋊C8—41no—
48C7×#101—42no—
49C7×C22⋊C8—22no—
50C7×#97—41no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95C2×C411yescharacteristic
57C7×#95C2×C411yescharacteristic
58#2399C2×C711yescharacteristic
59#1801C2×C711yescharacteristic
60#1467C2×C711yescharacteristic
61#1354C2×C712yes—
62#1346C2×C712yes—
63C2×C7×Q8—21no—
64C2×C7×Q8—21no—
65C2×C7×Q8—41no—
66C2×C7×Q8—41no—
67C2×C7×Q8—42no—
68C2×C7×Q8—41no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C7×M16—42no—
81C7×M16—42no—
82C2×C7×C8—42no—
83C2×C7×C8—42no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—42no—
89C7×C4⋊C4—42no—
90C7×C4⋊C4—42no—
91C7×C4⋊C4—42no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—41no—
94C42×C7—24no—
95C42×C7—21no—
96C42×C7—21no—
97C42×C7—21no—
98C42×C7—21no—
99C42×C7—21no—
100C42×C7—21no—
101C42×C7—41no—
102C42×C7—41no—
103C2×C4×Q8C4×C711yescharacteristic
104C2×C4×Q8C22×C711yescharacteristic
105#432—21no—
106#423—21no—
107C2×#101—21no—
108#398—21no—
109#398C4×C711yescharacteristic
110#398C22×C711yescharacteristic
111#395—21no—
112#395C4×C711yescharacteristic
113#395C22×C711yescharacteristic
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116#376C4×C711yescharacteristic
117#343C22×C712yes—
118#324—21no—
119#324C22×C712yes—
120C23×C7C2×C22⋊C411yescharacteristic
121C7×Q8—42no—
122C7×Q8—42no—
123C7×Q8—42no—
124C7×Q8—42no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—22no—
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×C4×C7—22no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7#9911yescharacteristic
153C2×C4×C7#9911yescharacteristic
154C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C7#13711yescharacteristic
156C7×C8—82no—
157C7×C8—82no—
158C22×Q8C2×C4×C711yescharacteristic
159C22×Q8C7×D411yescharacteristic
160C2×M16—22no—
161C4×Q8—41no—
162C4×Q8—41no—
163C4×Q8—41no—
164C4×Q8—41no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4C2×C4×C711yescharacteristic
171C2×C4⋊C4C2×C4×C711yescharacteristic
172C2×C4⋊C4C7×D411yescharacteristic
173C2×C4⋊C4C23×C711yescharacteristic
174C2×C42—21no—
175C2×C42—21no—
176C2×C42C2×C4×C711yescharacteristic
177C2×C42C7×D411yescharacteristic
178C2×C42C7×D411yescharacteristic
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182#101—42no—
183C22⋊C8—22no—
184#97—41no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95C2×C4×C711yescharacteristic
191#95C2×C4×C711yescharacteristic
192C22×C7—21no—
193C22×C7—21no—
194C22×C7C2×#9911yescharacteristic
195C22×C7#41711yescharacteristic
196C22×C7#42111yescharacteristic
197C4×C7—21no—
198C4×C7—21no—
199C4×C7—21no—
200C4×C7—42no—
201C4×C7—42no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—42no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7—81no—
209C4×C7#35612yes—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—42no—
215C2×Q8—41no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4C7×C22⋊C411yescharacteristic
221C22×C4C7×C22⋊C411yescharacteristic
222C22×C4C7×C22⋊C411yescharacteristic
223C22×C4C7×C22⋊C411yescharacteristic
224C22×C4C22×C4×C711yescharacteristic
225C22×C4C2×C7×D411yescharacteristic
226C22×C4C2×C7×D411yescharacteristic
227M16—42no—
228M16—42no—
229C2×C8—42no—
230C2×C8—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C42—24no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—41no—
249C42—41no—
250C2×C7—21no—
251C2×C7—21no—
252C2×C7#151311yescharacteristic
253C2×C7#212011yescharacteristic
254C2×C7C2×#35611yescharacteristic
255C23C2×C7×C22⋊C411yescharacteristic
256Q8—42no—
257Q8—42no—
258Q8—42no—
259Q8—42no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4C7×#9911yescharacteristic
288C2×C4C7×#9911yescharacteristic
289C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
290C2×C4C7×#13711yescharacteristic
291C8—82no—
292C8—82no—
293C7#1183111yescharacteristic, direct product factor, Sylow
294C22—21no—
295C22—21no—
296C22C2×C7×#9911yescharacteristic
297C22C7×#41711yescharacteristic
298C22C7×#42111yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—42no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4C7×#35612yes—
312C2—21no—
313C2—21no—
314C2C7×#151311yescharacteristic
315C2C7×#212011yescharacteristic
316C2C2×C7×#35611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC23×C7C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
23. #11831C7direct (N × H)
293. C7#11831direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#23992
C7×#18012
C7×#14672
C7×#13542
C7×#13462
#118317