Finite Group Databasebeta

← C7×#11786 #13189002

Subgroups — C7×#11786 #13189002

Order 1792.

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

normal subgroups only

This group has 462 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×#2487C211yesmaximal, characteristic
2C4×C7×#99C211yesmaximal, characteristic
3C7×#1466C211yesmaximal, characteristic
4C7×#1352C214yesmaximal, complemented
5C2×C4×C7×D4C2211yescharacteristic
6C22×C7×C4⋊C4C411yescharacteristic, complemented
7C7×#432—21no—
8C2×C7×#99—21no—
9C7×#400C412yescomplemented
10C7×#399C2212yes—
11C7×#389—21no—
12C7×#386—21no—
13C7×#385—21no—
14C7×#380—21no—
15C7×#380C411yescharacteristic, complemented
16C4×C7×C22⋊C4—21no—
17C7×#342C2212yes—
18C7×#323—21no—
19C7×#323C2212yes—
20#11786C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×D4D411yescharacteristic
22C23×C4×C7—21no—
23C23×C4×C7C2×C411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C4×C7×D4—41no—
26C4×C7×D4—41no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—41no—
31C2×C7×C4⋊C4—41no—
32C2×C7×C4⋊C4C2×C412yes—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—21no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4D411yescharacteristic
38C2×C7×C22⋊C4C2311yescharacteristic
39C2×C42×C7—41no—
40C2×C42×C7D411yescharacteristic
41C7×C4⋊C8—41no—
42C7×#99—24no—
43C7×C22⋊C8—22no—
44C7×#97—41no—
45C7×#95—22no—
46C7×#95—22no—
47C7×#95—41no—
48C7×#95C2×C412yes—
49#2487C2×C711yescharacteristic
50C4×#99C2×C711yescharacteristic
51#1466C2×C711yescharacteristic
52#1352C2×C714yescomplemented
53C24×C7C22⋊C411yescharacteristic
54C24×C7C2×D411yescharacteristic
55C2×C7×D4—22no—
56C2×C7×D4—22no—
57C2×C7×D4—41no—
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—42no—
65C22×C4×C7—42no—
66C22×C4×C7—42no—
67C22×C4×C7—42no—
68C22×C4×C7—82no—
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C2×C7×C8—42no—
75C7×C4⋊C4—41no—
76C7×C4⋊C4—42no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4—82no—
81C7×C22⋊C4—24no—
82C7×C22⋊C4—41no—
83C7×C22⋊C4—41no—
84C7×C22⋊C4—42no—
85C7×C22⋊C4—42no—
86C7×C22⋊C4—82no—
87C42×C7—21no—
88C42×C7—21no—
89C42×C7—82no—
90C2×C4×D4C22×C711yescharacteristic
91C22×C4⋊C4C4×C711yescharacteristic, complemented
92#432—21no—
93C2×#99—21no—
94#400C4×C712yescomplemented
95#399C22×C712yes—
96#389—21no—
97#386—21no—
98#385—21no—
99#380—21no—
100#380C4×C711yescharacteristic, complemented
101C4×C22⋊C4—21no—
102#342C22×C712yes—
103#323—21no—
104#323C22×C712yes—
105C23×C7—21no—
106C23×C7—21no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—41no—
110C23×C7—41no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7#10412yes—
114C23×C7C2×C22⋊C411yescharacteristic
115C23×C7#13712yes—
116C7×D4—44no—
117C2×C4×C7—22no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—42no—
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—82no—
139C2×C4×C7—82no—
140C2×C4×C7—84no—
141C2×C4×C7#9911yescharacteristic
142C2×C4×C7#9911yescharacteristic
143C7×C8—82no—
144C22×D4C7×D411yescharacteristic
145C23×C4—21no—
146C23×C4C2×C4×C711yescharacteristic
147C23×C4C7×D411yescharacteristic
148C4×D4—41no—
149C4×D4—41no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C2×C4×C712yes—
156C2×C4⋊C4C2×C4×C711yescharacteristic
157C2×C22⋊C4—22no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4C7×D411yescharacteristic
161C2×C22⋊C4C23×C711yescharacteristic
162C2×C42—41no—
163C2×C42C7×D411yescharacteristic
164C4⋊C8—41no—
165#99—24no—
166C22⋊C8—22no—
167#97—41no—
168#95—22no—
169#95—22no—
170#95—41no—
171#95C2×C4×C712yes—
172C22×C7—21no—
173C22×C7—21no—
174C22×C7—22no—
175C22×C7—22no—
176C22×C7—42no—
177C22×C7—42no—
178C22×C7—42no—
179C22×C7—41no—
180C22×C7—41no—
181C22×C7—41no—
182C22×C7—41no—
183C22×C7—81no—
184C22×C7C2×#9911yescharacteristic
185C22×C7#41911yescharacteristic
186C22×C7C2×#10411yescharacteristic
187C4×C7—21no—
188C4×C7—41no—
189C4×C7—41no—
190C4×C7—41no—
191C4×C7—41no—
192C4×C7—82no—
193C4×C7—84no—
194C4×C7—82no—
195C4×C7#35212yes—
196C24C7×C22⋊C411yescharacteristic
197C24C2×C7×D411yescharacteristic
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—41no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—82no—
212C22×C4C7×C22⋊C411yescharacteristic
213C22×C4C7×C22⋊C411yescharacteristic
214C22×C4C7×C22⋊C411yescharacteristic
215C22×C4C22×C4×C711yescharacteristic
216C22×C4C2×C7×D411yescharacteristic
217C2×C8—42no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—82no—
224C22⋊C4—24no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—82no—
230C42—21no—
231C42—21no—
232C42—82no—
233C2×C7—21no—
234C2×C7—21no—
235C2×C7—42no—
236C2×C7—42no—
237C2×C7#150811yescharacteristic
238C2×C7C2×#35211yescharacteristic
239C2×C7#212811yescharacteristic
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23C7×#10412yes—
249C23C2×C7×C22⋊C411yescharacteristic
250C23C7×#13712yes—
251D4—44no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—84no—
276C2×C4C7×#9911yescharacteristic
277C2×C4C7×#9911yescharacteristic
278C8—82no—
279C7#1178611yescharacteristic, direct product factor, Sylow
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22C2×C7×#9911yescharacteristic
293C22C7×#41911yescharacteristic
294C22C2×C7×#10411yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—82no—
301C4—84no—
302C4—82no—
303C4C7×#35212yes—
304C2—21no—
305C2—21no—
306C2—42no—
307C2—42no—
308C2C7×#150811yescharacteristic
309C2C2×C7×#35211yescharacteristic
310C2C7×#212811yescharacteristic
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×#1352C2semidirect
6. C22×C7×C4⋊C4C4semidirect
9. C7×#400C4semidirect
15. C7×#380C4semidirect
20. #11786C7direct (N × H)
52. #1352C2×C7semidirect
91. C22×C4⋊C4C4×C7semidirect
94. #400C4×C7semidirect
100. #380C4×C7semidirect
279. C7#11786direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24872
C4×C7×#992
C7×#14662
C7×#13522
#117867