Finite Group Databasebeta

← C7×#11850 #13189066

Subgroups — C7×#11850 #13189066

Order 1792.

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

normal subgroups only

This group has 396 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×#2484C211yesmaximal, characteristic
2C7×#1821C211yesmaximal, characteristic
3C2×C7×#324C211yesmaximal, characteristic
4C7×#1353C212yesmaximal, complemented
5C7×#1346C212yesmaximal, complemented
6C2×C7×#117C411yescharacteristic
7C22×C7×C4⋊C4C2211yescharacteristic
8C7×#407—21no—
9C2×C7×C22⋊C8—21no—
10C7×#400C411yescharacteristic
11C7×#400C2211yescharacteristic
12C7×#398C411yescharacteristic
13C7×#398C2211yescharacteristic
14C7×#387—21no—
15C7×#387—21no—
16C7×#385—21no—
17C7×#381—21no—
18C7×#381C411yescharacteristic
19C7×#343—21no—
20C7×#343C2212yes—
21C7×#324—21no—
22C7×#324C2212yes—
23#11850C711yesmaximal, characteristic, direct product factor, Sylow
24C23×C4×C7C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C2×C7×M16—22no—
28C22×C7×C8—41no—
29C7×#117—41no—
30C7×#117—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—41no—
34C2×C7×C4⋊C4—41no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4D411yescharacteristic
39C2×C7×C4⋊C4D411yescharacteristic
40C2×C7×C4⋊C4C2311yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C42×C7—21no—
44C7×C22⋊C8—22no—
45C7×C22⋊C8—41no—
46C7×C22⋊C8—41no—
47C7×C22⋊C8—41no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95C2×C411yescharacteristic
56C7×#95C2×C411yescharacteristic
57#2484C2×C711yescharacteristic
58#1821C2×C711yescharacteristic
59C2×#324C2×C711yescharacteristic
60#1353C2×C712yescomplemented
61#1346C2×C712yescomplemented
62C24×C7C22⋊C411yescharacteristic
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—22no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—41no—
75C22×C4×C7—42no—
76C22×C4×C7—42no—
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22×C411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C7×M16—42no—
84C7×M16—42no—
85C2×C7×C8—42no—
86C2×C7×C8—42no—
87C2×C7×C8—82no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—42no—
95C7×C4⋊C4—42no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—41no—
98C7×C22⋊C4—41no—
99C42×C7—42no—
100C2×#117C4×C711yescharacteristic
101C22×C4⋊C4C22×C711yescharacteristic
102#407—21no—
103C2×C22⋊C8—21no—
104#400C4×C711yescharacteristic
105#400C22×C711yescharacteristic
106#398C4×C711yescharacteristic
107#398C22×C711yescharacteristic
108#387—21no—
109#387—21no—
110#385—21no—
111#381—21no—
112#381C4×C711yescharacteristic
113#343—21no—
114#343C22×C712yes—
115#324—21no—
116#324C22×C712yes—
117C23×C7—22no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—41no—
121C23×C7#9911yescharacteristic
122C23×C7#9911yescharacteristic
123C23×C7C2×C22⋊C411yescharacteristic
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
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—42no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—81no—
154C2×C4×C7#13512yes—
155C7×C8—82no—
156C7×C8—82no—
157C23×C4C2×C4×C711yescharacteristic
158C23×C4C7×D411yescharacteristic
159C23×C4C7×D411yescharacteristic
160C2×M16—22no—
161C22×C8—41no—
162#117—41no—
163#117—41no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4C2×C4×C711yescharacteristic
169C2×C4⋊C4C2×C4×C711yescharacteristic
170C2×C4⋊C4C2×C4×C711yescharacteristic
171C2×C4⋊C4C7×D411yescharacteristic
172C2×C4⋊C4C7×D411yescharacteristic
173C2×C4⋊C4C23×C711yescharacteristic
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C42—21no—
177C22⋊C8—22no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95C2×C4×C711yescharacteristic
189#95C2×C4×C711yescharacteristic
190C22×C7—21no—
191C22×C7—21no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—42no—
198C22×C7—42no—
199C22×C7#35212yes—
200C22×C7C2×#9911yescharacteristic
201C22×C7#41611yescharacteristic
202C22×C7#42111yescharacteristic
203C4×C7—22no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7—41no—
209C4×C7—81no—
210C4×C7—81no—
211C4×C7—81no—
212C4×C7—81no—
213C24C7×C22⋊C411yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C7×C22⋊C411yescharacteristic
229C22×C4C7×C22⋊C411yescharacteristic
230C22×C4C7×C22⋊C411yescharacteristic
231C22×C4C22×C4×C711yescharacteristic
232C22×C4C2×C7×D411yescharacteristic
233C22×C4C2×C7×D411yescharacteristic
234M16—42no—
235M16—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—82no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C42—42no—
251C2×C7—21no—
252C2×C7—21no—
253C2×C7—22no—
254C2×C7—41no—
255C2×C7#151611yescharacteristic
256C2×C7C2×#35211yescharacteristic
257C2×C7#213311yescharacteristic
258C23—22no—
259C23—21no—
260C23—21no—
261C23—41no—
262C23C7×#9911yescharacteristic
263C23C7×#9911yescharacteristic
264C23C2×C7×C22⋊C411yescharacteristic
265C2×C4—21no—
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—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
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×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4C7×#13512yes—
296C8—82no—
297C8—82no—
298C7#1185011yescharacteristic, direct product factor, Sylow
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—42no—
307C22—42no—
308C22C7×#35212yes—
309C22C2×C7×#9911yescharacteristic
310C22C7×#41611yescharacteristic
311C22C7×#42111yescharacteristic
312C4—22no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—41no—
326C2C7×#151611yescharacteristic
327C2C2×C7×#35211yescharacteristic
328C2C7×#213311yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC24×C7C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C24×C7 ◁ C22×C7×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1353C2semidirect
5. C7×#1346C2semidirect
23. #11850C7direct (N × H)
60. #1353C2×C7semidirect
61. #1346C2×C7semidirect
298. C7#11850direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24842
C7×#18212
C2×C7×#3242
C7×#13532
C7×#13462
#118507