Finite Group Databasebeta

← C7×#11787 #13189003

Subgroups — C7×#11787 #13189003

Order 1792.

460 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2563C211yesmaximal, characteristic
2C4×C7×#99C211yesmaximal, characteristic
3C7×#1465C211yesmaximal, characteristic
4C7×#1351C214yesmaximal, complemented
5C2×C7×C4⋊Q8C411yescharacteristic, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×C7×#99—21no—
8C7×#407—21no—
9C7×#393C412yescomplemented
10C7×#393C2212yes—
11C7×#390—21no—
12C7×#382—21no—
13C7×#380—21no—
14C7×#380—21no—
15C4×C7×C4⋊C4C411yescharacteristic, complemented
16C4×C7×C22⋊C4—21no—
17C7×#342C2212yes—
18C7×#324—21no—
19C7×#323C2212yes—
20#11787C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8C2×C412yes—
22C22×C7×D4D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C2×C7×M16—41no—
26C7×C4⋊Q8—41no—
27C7×C4⋊Q8—41no—
28C4×C7×D4—41no—
29C4×C7×D4—41no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4C2×C411yescharacteristic
38C2×C7×C22⋊C4C2311yescharacteristic
39C2×C42×C7—21no—
40C2×C42×C7—41no—
41C2×C42×C7C2×C411yescharacteristic
42C7×#99—24no—
43C7×C22⋊C8—22no—
44C7×C22⋊C8—41no—
45C7×#95—22no—
46C7×#95—22no—
47C7×#95—41no—
48C7×#95C2×C412yes—
49#2563C2×C711yescharacteristic
50C4×#99C2×C711yescharacteristic
51#1465C2×C711yescharacteristic
52#1351C2×C714yescomplemented
53C24×C7C22⋊C411yescharacteristic
54C24×C7C2×D411yescharacteristic
55C2×C7×Q8—42no—
56C2×C7×Q8—82no—
57C2×C7×D4—22no—
58C2×C7×D4—22no—
59C2×C7×D4—41no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—42no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C7×M16—82no—
75C2×C7×C8—42no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4—41no—
81C7×C22⋊C4—24no—
82C7×C22⋊C4—41no—
83C7×C22⋊C4—41no—
84C7×C22⋊C4—42no—
85C7×C22⋊C4—82no—
86C7×C22⋊C4—82no—
87C42×C7—21no—
88C42×C7—21no—
89C42×C7—42no—
90C42×C7—82no—
91C2×C4⋊Q8C4×C711yescharacteristic, complemented
92C2×C4×D4C22×C711yescharacteristic
93C2×#99—21no—
94#407—21no—
95#393C4×C712yescomplemented
96#393C22×C712yes—
97#390—21no—
98#382—21no—
99#380—21no—
100#380—21no—
101C4×C4⋊C4C4×C711yescharacteristic, complemented
102C4×C22⋊C4—21no—
103#342C22×C712yes—
104#324—21no—
105#323C22×C712yes—
106C23×C7—21no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7—41no—
114C23×C7—41no—
115C23×C7C2×C22⋊C411yescharacteristic
116C23×C7#13712yes—
117C7×Q8—84no—
118C7×D4—44no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—22no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7—41no—
130C2×C4×C7—44no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—81no—
138C2×C4×C7—81no—
139C2×C4×C7—82no—
140C2×C4×C7—82no—
141C2×C4×C7#9911yescharacteristic
142C2×C4×C7#9911yescharacteristic
143C2×C4×C7#10412yes—
144C7×C8—82no—
145C22×Q8C2×C4×C712yes—
146C22×D4C7×D411yescharacteristic
147C23×C4C7×D411yescharacteristic
148C23×C4C7×D411yescharacteristic
149C2×M16—41no—
150C4⋊Q8—41no—
151C4⋊Q8—41no—
152C4×D4—41no—
153C4×D4—41no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C7×D411yescharacteristic
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4C2×C4×C711yescharacteristic
162C2×C22⋊C4C23×C711yescharacteristic
163C2×C42—21no—
164C2×C42—41no—
165C2×C42C2×C4×C711yescharacteristic
166#99—24no—
167C22⋊C8—22no—
168C22⋊C8—41no—
169#95—22no—
170#95—22no—
171#95—41no—
172#95C2×C4×C712yes—
173C22×C7—21no—
174C22×C7—21no—
175C22×C7—22no—
176C22×C7—42no—
177C22×C7—42no—
178C22×C7—41no—
179C22×C7—41no—
180C22×C7—41no—
181C22×C7—41no—
182C22×C7—41no—
183C22×C7—81no—
184C22×C7—81no—
185C22×C7C2×#9911yescharacteristic
186C22×C7#41911yescharacteristic
187C22×C7C2×#10411yescharacteristic
188C4×C7—21no—
189C4×C7—22no—
190C4×C7—41no—
191C4×C7—41no—
192C4×C7—41no—
193C4×C7—82no—
194C4×C7—84no—
195C4×C7—82no—
196C4×C7#35412yes—
197C24C7×C22⋊C411yescharacteristic
198C24C2×C7×D411yescharacteristic
199C2×Q8—42no—
200C2×Q8—82no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—41no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4C7×C22⋊C411yescharacteristic
214C22×C4C7×C22⋊C411yescharacteristic
215C22×C4C7×C22⋊C411yescharacteristic
216C22×C4C22×C4×C711yescharacteristic
217C22×C4C2×C7×D411yescharacteristic
218M16—82no—
219C2×C8—42no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C22⋊C4—24no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—42no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C42—21no—
232C42—21no—
233C42—42no—
234C42—82no—
235C2×C7—21no—
236C2×C7—21no—
237C2×C7—42no—
238C2×C7—81no—
239C2×C7#150811yescharacteristic
240C2×C7#212211yescharacteristic
241C2×C7C2×#35411yescharacteristic
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23C2×C7×C22⋊C411yescharacteristic
252C23C7×#13712yes—
253Q8—84no—
254D4—44no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4C7×#9911yescharacteristic
278C2×C4C7×#9911yescharacteristic
279C2×C4C7×#10412yes—
280C8—82no—
281C7#1178711yescharacteristic, direct product factor, Sylow
282C22—21no—
283C22—21no—
284C22—22no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22C2×C7×#9911yescharacteristic
295C22C7×#41911yescharacteristic
296C22C2×C7×#10411yescharacteristic
297C4—22no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—82no—
303C4—84no—
304C4—82no—
305C4C7×#35412yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2C7×#150811yescharacteristic
311C2C7×#212211yescharacteristic
312C2C2×C7×#35411yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1351C2semidirect
5. C2×C7×C4⋊Q8C4semidirect
9. C7×#393C4semidirect
15. C4×C7×C4⋊C4C4semidirect
20. #11787C7direct (N × H)
52. #1351C2×C7semidirect
91. C2×C4⋊Q8C4×C7semidirect
95. #393C4×C7semidirect
101. C4×C4⋊C4C4×C7semidirect
281. C7#11787direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25632
C4×C7×#992
C7×#14652
C7×#13512
#117877