Finite Group Databasebeta

← C7×#9486 #13186702

Subgroups — C7×#9486 #13186702

Order 1792.

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

normal subgroups only

This group has 542 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
1C7×#2982C211yesmaximal, characteristic, complemented
2C7×#2536C211yesmaximal, characteristic
3C7×#1801C211yesmaximal, characteristic, complemented
4C7×#1486C212yesmaximal
5C7×#1286C212yesmaximal, complemented
6C2×C7×C22⋊Q8—21no—
7C2×C4×C7×Q8C2211yescharacteristic
8C2×C4×C7×D4C2211yescharacteristic
9C7×C8×Q8—24no—
10C7×#431—22no—
11C7×#431C2212yes—
12C7×#423C2212yes—
13C2×C7×#101—21no—
14C7×#398C412yes—
15C7×#394C412yes—
16C7×#388—22no—
17C7×#382C2211yescharacteristic
18C7×#380—21no—
19C7×#343—21no—
20C7×D4⋊C8—24no—
21#9486C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×Q8D411yescharacteristic
23C22×C7×D4C2×C411yescharacteristic
24C23×C4×C7—21no—
25C2×C7×M16—22no—
26C7×C22⋊Q8—42no—
27C7×C22⋊Q8—42no—
28C4×C7×Q8—22no—
29C4×C7×Q8—22no—
30C4×C7×D4—22no—
31C4×C7×D4—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4C2×C412yes—
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—21no—
40C2×C42×C7D412yes—
41C2×C42×C7C2311yescharacteristic
42C7×C4⋊C8—24no—
43C7×C4⋊C8—22no—
44C7×C4⋊C8—22no—
45C7×C4⋊C8—22no—
46C7×#101—42no—
47C7×C22⋊C8—22no—
48C4×C7×C8—24no—
49C4×C7×C8D414yes—
50C7×#95—22no—
51C7×#95—21no—
52C7×#95C2×C412yes—
53#2982C2×C711yescharacteristic, complemented
54#2536C2×C711yescharacteristic
55#1801C2×C711yescharacteristic, complemented
56#1486C2×C712yes—
57#1286C2×C712yescomplemented
58C24×C7—21no—
59C2×C7×Q8—22no—
60C2×C7×Q8—22no—
61C2×C7×Q8D812yes—
62C2×C7×Q8SD1612yes—
63C2×C7×D4—22no—
64C2×C7×D4—42no—
65C22×C4×C7—22no—
66C22×C4×C7—21no—
67C22×C4×C7—42no—
68C22×C4×C7—41no—
69C22×C4×C7—41no—
70C22×C4×C7—42no—
71C22×C4×C7C22⋊C412yes—
72C22×C4×C7C22⋊C412yes—
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C7×M16—42no—
77C7×M16—42no—
78C2×C7×C8—24no—
79C2×C7×C8—42no—
80C7×C4⋊C4—24no—
81C7×C4⋊C4—22no—
82C7×C4⋊C4—22no—
83C7×C4⋊C4—44no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—41no—
86C7×C22⋊C4—44no—
87C7×C22⋊C4—44no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C42×C7—22no—
91C42×C7—22no—
92C42×C7C2×D412yes—
93C42×C7#4112yes—
94C2×C22⋊Q8—21no—
95C2×C4×Q8C22×C711yescharacteristic
96C2×C4×D4C22×C711yescharacteristic
97C8×Q8—24no—
98#431—22no—
99#431C22×C712yes—
100#423C22×C712yes—
101C2×#101—21no—
102#398C4×C712yes—
103#394C4×C712yes—
104#388—22no—
105#382C22×C711yescharacteristic
106#380—21no—
107#343—21no—
108D4⋊C8—24no—
109C23×C7—22no—
110C23×C7—41no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7—41no—
115C23×C7—41no—
116C23×C7C2×C22⋊C411yescharacteristic
117C7×Q8—24no—
118C7×Q8—24no—
119C7×D4—24no—
120C7×D4—42no—
121C2×C4×C7—22no—
122C2×C4×C7—22no—
123C2×C4×C7—22no—
124C2×C4×C7—41no—
125C2×C4×C7—41no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—41no—
130C2×C4×C7—42no—
131C2×C4×C7—41no—
132C2×C4×C7—41no—
133C2×C4×C7—82no—
134C2×C4×C7—82no—
135C2×C4×C7#10214yes—
136C2×C4×C7C4×D412yes—
137C2×C4×C7#12011yescharacteristic
138C2×C4×C7#12112yes—
139C2×C4×C7#12311yescharacteristic
140C2×C4×C7C2×D811yescharacteristic
141C2×C4×C7C2×SD1611yescharacteristic
142C2×C4×C7#13512yes—
143C7×C8—44no—
144C7×C8—82no—
145C22×Q8C7×D411yescharacteristic
146C22×D4C2×C4×C711yescharacteristic
147C23×C4—21no—
148C2×M16—22no—
149C22⋊Q8—42no—
150C22⋊Q8—42no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×D4—22no—
154C4×D4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×C4×C711yescharacteristic
158C2×C4⋊C4C2×C4×C712yes—
159C2×C4⋊C4C7×D411yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C42C7×D412yes—
164C2×C42C23×C711yescharacteristic
165C4⋊C8—24no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169#101—42no—
170C22⋊C8—22no—
171C4×C8—24no—
172C4×C8C7×D414yes—
173#95—22no—
174#95—21no—
175#95C2×C4×C712yes—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—42no—
179C22×C7—42no—
180C22×C7—42no—
181C22×C7—81no—
182C22×C7—81no—
183C22×C7—81no—
184C22×C7—81no—
185C22×C7#38611yescharacteristic
186C22×C7C2×#10211yescharacteristic
187C22×C7#41611yescharacteristic
188C4×C7—22no—
189C4×C7—24no—
190C4×C7—21no—
191C4×C7—21no—
192C4×C7—42no—
193C4×C7—82no—
194C4×C7—81no—
195C4×C7#46512yes—
196C4×C7#46612yes—
197C24—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8C7×D812yes—
201C2×Q8C7×SD1612yes—
202C2×D4—22no—
203C2×D4—42no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4C7×C22⋊C412yes—
211C22×C4C7×C22⋊C412yes—
212C22×C4C22×C4×C711yescharacteristic
213C22×C4C2×C7×D411yescharacteristic
214C22×C4C2×C7×D411yescharacteristic
215M16—42no—
216M16—42no—
217C2×C8—24no—
218C2×C8—42no—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—44no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C42—22no—
230C42—22no—
231C42C2×C7×D412yes—
232C42C7×#4112yes—
233C2×C7—21no—
234C2×C7—21no—
235C2×C7—82no—
236C2×C7#189111yescharacteristic
237C2×C7#189311yescharacteristic
238C2×C7#191211yescharacteristic
239C23—22no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23C2×C7×C22⋊C411yescharacteristic
247Q8—24no—
248Q8—24no—
249D4—24no—
250D4—42no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4C7×#10214yes—
266C2×C4C4×C7×D412yes—
267C2×C4C7×#12011yescharacteristic
268C2×C4C7×#12112yes—
269C2×C4C7×#12311yescharacteristic
270C2×C4C2×C7×D811yescharacteristic
271C2×C4C2×C7×SD1611yescharacteristic
272C2×C4C7×#13512yes—
273C8—44no—
274C8—82no—
275C7#948611yescharacteristic, direct product factor, Sylow
276C22—21no—
277C22—21no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22C7×#38611yescharacteristic
286C22C2×C7×#10211yescharacteristic
287C22C7×#41611yescharacteristic
288C4—22no—
289C4—24no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—82no—
294C4—81no—
295C4C7×#46512yes—
296C4C7×#46612yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2C7×#189111yescharacteristic
301C2C7×#189311yescharacteristic
302C2C7×#191211yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2982C2semidirect
3. C7×#1801C2semidirect
5. C7×#1286C2semidirect
21. #9486C7direct (N × H)
53. #2982C2×C7semidirect
55. #1801C2×C7semidirect
57. #1286C2×C7semidirect
275. C7#9486direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#29822
C7×#25362
C7×#18012
C7×#14862
C7×#12862
#94867