Finite Group Databasebeta

← C7×#11849 #13189065

Subgroups — C7×#11849 #13189065

Order 1792.

376 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2599C211yesmaximal, characteristic
2C7×#1822C211yesmaximal, characteristic
3C7×#1458C211yesmaximal, characteristic
4C7×#1354C212yesmaximal, complemented
5C7×#1346C212yesmaximal, complemented
6C2×C4×C7×Q8C411yescharacteristic
7C2×C7×#117C2211yescharacteristic
8C7×#431—21no—
9C7×#407—21no—
10C7×#398C2211yescharacteristic
11C7×#397C411yescharacteristic
12C7×#395C2211yescharacteristic
13C7×#393C411yescharacteristic
14C7×#388—21no—
15C7×#387—21no—
16C7×#387—21no—
17C7×#385—21no—
18C7×#383C411yescharacteristic
19C7×#343—21no—
20C7×#343C2212yes—
21C7×#324C2212yes—
22C7×#323—21no—
23#11849C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C2×C7×M16—22no—
27C4×C7×Q8—41no—
28C4×C7×Q8—41no—
29C7×#117—41no—
30C7×#117—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C4⋊C4C2311yescharacteristic
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4C2×C411yescharacteristic
40C2×C7×C22⋊C4D411yescharacteristic
41C2×C42×C7—21no—
42C2×C42×C7C2×C411yescharacteristic
43C2×C42×C7D411yescharacteristic
44C7×C4⋊C8—41no—
45C7×C22⋊C8—22no—
46C7×C22⋊C8—41no—
47C7×C22⋊C8—41no—
48C4×C7×C8—41no—
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#2599C2×C711yescharacteristic
58#1822C2×C711yescharacteristic
59#1458C2×C711yescharacteristic
60#1354C2×C712yescomplemented
61#1346C2×C712yescomplemented
62C24×C7C22⋊C411yescharacteristic
63C2×C7×Q8—42no—
64C2×C7×Q8—41no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—22no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—41no—
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22×C411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C7×M16—42no—
80C7×M16—42no—
81C2×C7×C8—42no—
82C2×C7×C8—42no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—42no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—42no—
90C7×C22⋊C4—41no—
91C7×C22⋊C4—41no—
92C7×C22⋊C4—42no—
93C7×C22⋊C4—42no—
94C42×C7—21no—
95C42×C7—21no—
96C42×C7—21no—
97C42×C7—21no—
98C42×C7—42no—
99C2×C4×Q8C4×C711yescharacteristic
100C2×#117C22×C711yescharacteristic
101#431—21no—
102#407—21no—
103#398C22×C711yescharacteristic
104#397C4×C711yescharacteristic
105#395C22×C711yescharacteristic
106#393C4×C711yescharacteristic
107#388—21no—
108#387—21no—
109#387—21no—
110#385—21no—
111#383C4×C711yescharacteristic
112#343—21no—
113#343C22×C712yes—
114#324C22×C712yes—
115#323—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—41no—
120C23×C7—41no—
121C23×C7C2×C22⋊C411yescharacteristic
122C7×Q8—42no—
123C7×Q8—42no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—22no—
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—41no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
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—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7#9911yescharacteristic
151C2×C4×C7#9911yescharacteristic
152C2×C4×C7#13512yes—
153C7×C8—82no—
154C7×C8—82no—
155C22×Q8C2×C4×C711yescharacteristic
156C23×C4C7×D411yescharacteristic
157C2×M16—22no—
158C4×Q8—41no—
159C4×Q8—41no—
160#117—41no—
161#117—41no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4C2×C4×C711yescharacteristic
166C2×C4⋊C4C7×D411yescharacteristic
167C2×C4⋊C4C23×C711yescharacteristic
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4C2×C4×C711yescharacteristic
171C2×C22⋊C4C7×D411yescharacteristic
172C2×C42—21no—
173C2×C42C2×C4×C711yescharacteristic
174C2×C42C7×D411yescharacteristic
175C4⋊C8—41no—
176C22⋊C8—22no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C4×C8—41no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—21no—
185#95—21no—
186#95C2×C4×C711yescharacteristic
187#95C2×C4×C711yescharacteristic
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—42no—
191C22×C7—41no—
192C22×C7—41no—
193C22×C7—41no—
194C22×C7—81no—
195C22×C7C2×#9911yescharacteristic
196C22×C7#41611yescharacteristic
197C22×C7#42111yescharacteristic
198C4×C7—22no—
199C4×C7—21no—
200C4×C7—21no—
201C4×C7—41no—
202C4×C7—42no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7—81no—
207C4×C7—81no—
208C4×C7—81no—
209C24C7×C22⋊C411yescharacteristic
210C2×Q8—42no—
211C2×Q8—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—41no—
220C22×C4C7×C22⋊C411yescharacteristic
221C22×C4C7×C22⋊C411yescharacteristic
222C22×C4C7×C22⋊C411yescharacteristic
223C22×C4C22×C4×C711yescharacteristic
224C22×C4C2×C7×D411yescharacteristic
225C22×C4C2×C7×D411yescharacteristic
226M16—42no—
227M16—42no—
228C2×C8—42no—
229C2×C8—42no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—42no—
246C2×C7—21no—
247C2×C7—21no—
248C2×C7—81no—
249C2×C7#151611yescharacteristic
250C2×C7#212011yescharacteristic
251C2×C7#213511yescharacteristic
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23C2×C7×C22⋊C411yescharacteristic
258Q8—42no—
259Q8—42no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
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×C4C7×#9911yescharacteristic
287C2×C4C7×#9911yescharacteristic
288C2×C4C7×#13512yes—
289C8—82no—
290C8—82no—
291C7#1184911yescharacteristic, direct product factor, Sylow
292C22—21no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22C2×C7×#9911yescharacteristic
300C22C7×#41611yescharacteristic
301C22C7×#42111yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—41no—
306C4—42no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4—81no—
312C4—81no—
313C2—21no—
314C2—21no—
315C2—81no—
316C2C7×#151611yescharacteristic
317C2C7×#212011yescharacteristic
318C2C7×#213511yescharacteristic
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 ◁ C22×C4×C7 ◁ C2×C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1354C2semidirect
5. C7×#1346C2semidirect
23. #11849C7direct (N × H)
60. #1354C2×C7semidirect
61. #1346C2×C7semidirect
291. C7#11849direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25992
C7×#18222
C7×#14582
C7×#13542
C7×#13462
#118497