Finite Group Databasebeta

← C7×#9202 #13186418

Subgroups — C7×#9202 #13186418

Order 1792.

516 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 134 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×#2627C211yesmaximal, characteristic
2C7×#2313C211yesmaximal, characteristic
3C7×#1520C212yesmaximal
4C2×C7×#324C211yesmaximal, characteristic
5C7×#1289C212yesmaximal
6C2×C7×C22⋊Q8C2211yescharacteristic
7C22×C7×C4⋊C4C2211yescharacteristic
8C2×C7×C22⋊C8C2212yes—
9C7×#400C412yescomplemented
10C7×#398—22no—
11C7×#395C412yescomplemented
12C7×#387—24no—
13C7×#385—21no—
14C7×#381—21no—
15C7×#381—21no—
16C7×#380C2211yescharacteristic
17C7×#380C2212yes—
18C4×C7×C22⋊C4—21no—
19C7×#328—24no—
20C7×#324—22no—
21#9202C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×Q8C2×C411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C22×C7×C8—22no—
27C7×C22⋊Q8—22no—
28C7×C22⋊Q8—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4C2×C412yes—
37C2×C7×C4⋊C4D412yes—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C42×C7—22no—
42C7×C22⋊C8—22no—
43C7×C22⋊C8—22no—
44C7×#95—22no—
45C7×#95—24no—
46C7×#95—22no—
47C7×#95—22no—
48C7×#95—22no—
49C7×#95C2×C412yes—
50C7×#95D414yes—
51#2627C2×C711yescharacteristic
52#2313C2×C711yescharacteristic
53#1520C2×C712yes—
54C2×#324C2×C711yescharacteristic
55#1289C2×C712yes—
56C24×C7C2×D411yescharacteristic
57C2×C7×Q8—22no—
58C2×C7×Q8—81no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—22no—
67C22×C4×C7—21no—
68C22×C4×C7—24no—
69C22×C4×C7—81no—
70C22×C4×C7—81no—
71C22×C4×C7C22⋊C412yes—
72C22×C4×C7C22⋊C412yes—
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D412yes—
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7#4112yes—
77C2×C7×C8—42no—
78C2×C7×C8—82no—
79C7×C4⋊C4—24no—
80C7×C4⋊C4—24no—
81C7×C4⋊C4—22no—
82C7×C4⋊C4—42no—
83C7×C4⋊C4—82no—
84C7×C4⋊C4—81no—
85C7×C4⋊C4—82no—
86C7×C22⋊C4—41no—
87C7×C22⋊C4—41no—
88C7×C22⋊C4—42no—
89C7×C22⋊C4—42no—
90C7×C22⋊C4—42no—
91C42×C7—82no—
92C2×C22⋊Q8C22×C711yescharacteristic
93C22×C4⋊C4C22×C711yescharacteristic
94C2×C22⋊C8C22×C712yes—
95#400C4×C712yescomplemented
96#398—22no—
97#395C4×C712yescomplemented
98#387—24no—
99#385—21no—
100#381—21no—
101#381—21no—
102#380C22×C711yescharacteristic
103#380C22×C712yes—
104C4×C22⋊C4—21no—
105#328—24no—
106#324—22no—
107C23×C7—41no—
108C23×C7—41no—
109C23×C7C2×C22⋊C411yescharacteristic
110C23×C7#12011yescharacteristic
111C23×C7#12311yescharacteristic
112C23×C7#13512yes—
113C23×C7#13712yes—
114C7×Q8—82no—
115C2×C4×C7—22no—
116C2×C4×C7—22no—
117C2×C4×C7—22no—
118C2×C4×C7—22no—
119C2×C4×C7—41no—
120C2×C4×C7—42no—
121C2×C4×C7—42no—
122C2×C4×C7—41no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—44no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—44no—
132C2×C4×C7—42no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—42no—
136C2×C4×C7—44no—
137C2×C4×C7—81no—
138C2×C4×C7C4×D412yes—
139C2×C4×C7#12112yes—
140C7×C8—82no—
141C22×Q8C2×C4×C711yescharacteristic
142C23×C4C7×D411yescharacteristic
143C23×C4C7×D411yescharacteristic
144C23×C4C23×C711yescharacteristic
145C22×C8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4C2×C4×C711yescharacteristic
155C2×C4⋊C4C2×C4×C712yes—
156C2×C4⋊C4C7×D412yes—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—21no—
160C2×C42—22no—
161C22⋊C8—22no—
162C22⋊C8—22no—
163#95—22no—
164#95—24no—
165#95—22no—
166#95—22no—
167#95—22no—
168#95C2×C4×C712yes—
169#95C7×D414yes—
170C22×C7—21no—
171C22×C7—21no—
172C22×C7—21no—
173C22×C7—21no—
174C22×C7—21no—
175C22×C7—21no—
176C22×C7—42no—
177C22×C7—42no—
178C22×C7#38611yescharacteristic
179C22×C7#41611yescharacteristic
180C22×C7#41911yescharacteristic
181C22×C7#45212yes—
182C22×C7#48512yes—
183C4×C7—22no—
184C4×C7—22no—
185C4×C7—42no—
186C4×C7—42no—
187C4×C7—82no—
188C4×C7—84no—
189C4×C7—81no—
190C4×C7—81no—
191C24C2×C7×D411yescharacteristic
192C2×Q8—22no—
193C2×Q8—81no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—24no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4C7×C22⋊C412yes—
207C22×C4C7×C22⋊C412yes—
208C22×C4C22×C4×C711yescharacteristic
209C22×C4C2×C7×D412yes—
210C22×C4C2×C7×D411yescharacteristic
211C22×C4C7×#4112yes—
212C2×C8—42no—
213C2×C8—82no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—82no—
219C4⋊C4—81no—
220C4⋊C4—82no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C42—82no—
227C2×C7—21no—
228C2×C7—22no—
229C2×C7—21no—
230C2×C7—22no—
231C2×C7#187311yescharacteristic
232C2×C7#191211yescharacteristic
233C2×C7#191311yescharacteristic
234C23—41no—
235C23—41no—
236C23C2×C7×C22⋊C411yescharacteristic
237C23C7×#12011yescharacteristic
238C23C7×#12311yescharacteristic
239C23C7×#13512yes—
240C23C7×#13712yes—
241Q8—82no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—81no—
265C2×C4C4×C7×D412yes—
266C2×C4C7×#12112yes—
267C8—82no—
268C7#920211yescharacteristic, direct product factor, Sylow
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—42no—
276C22—42no—
277C22C7×#38611yescharacteristic
278C22C7×#41611yescharacteristic
279C22C7×#41911yescharacteristic
280C22C7×#45212yes—
281C22C7×#48512yes—
282C4—22no—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—82no—
287C4—84no—
288C4—81no—
289C4—81no—
290C2—21no—
291C2—22no—
292C2—22no—
293C2—21no—
294C2C7×#187311yescharacteristic
295C2C7×#191211yescharacteristic
296C2C7×#191311yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC24×C7C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
9. C7×#400C4semidirect
11. C7×#395C4semidirect
21. #9202C7direct (N × H)
95. #400C4×C7semidirect
97. #395C4×C7semidirect
268. C7#9202direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26272
C7×#23132
C7×#15202
C2×C7×#3242
C7×#12892
#92027