Finite Group Databasebeta

← C7×#11788 #13189004

Subgroups — C7×#11788 #13189004

Order 1792.

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

normal subgroups only

This group has 516 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×#2542C211yesmaximal, characteristic
2C4×C7×#99C211yesmaximal, characteristic
3C7×#1465C211yesmaximal, characteristic
4C7×#1352C214yesmaximal, complemented
5C2×C7×C4⋊Q8C411yescharacteristic, complemented
6C2×C7×C22⋊Q8—21no—
7C2×C7×C22⋊Q8—21no—
8C2×C4×C7×D4C2211yescharacteristic
9C2×C7×#99—21no—
10C7×#407—21no—
11C7×#399C412yescomplemented
12C7×#399C2212yes—
13C7×#390—21no—
14C7×#384—21no—
15C4×C7×C4⋊C4C411yescharacteristic, complemented
16C4×C7×C22⋊C4—21no—
17C7×#342C2212yes—
18C7×#324—21no—
19C7×#323C2212yes—
20#11788C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×D4D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C2×C7×M16—41no—
26C7×C4⋊Q8—41no—
27C7×C4⋊Q8—41no—
28C7×C22⋊Q8—44no—
29C7×C22⋊Q8—44no—
30C4×C7×D4—41no—
31C4×C7×D4—41no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4C2×C412yes—
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4C2×C411yescharacteristic
42C2×C7×C22⋊C4C2311yescharacteristic
43C2×C42×C7—21no—
44C2×C42×C7—41no—
45C2×C42×C7C2×C411yescharacteristic
46C7×#99—24no—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—41no—
49C7×#95—41no—
50C7×#95C2×C412yes—
51#2542C2×C711yescharacteristic
52C4×#99C2×C711yescharacteristic
53#1465C2×C711yescharacteristic
54#1352C2×C714yescomplemented
55C24×C7C22⋊C411yescharacteristic
56C24×C7C2×D411yescharacteristic
57C2×C7×Q8—42no—
58C2×C7×Q8—42no—
59C2×C7×Q8—41no—
60C2×C7×Q8—41no—
61C2×C7×D4—22no—
62C2×C7×D4—22no—
63C2×C7×D4—41no—
64C22×C4×C7—22no—
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—22no—
70C22×C4×C7—42no—
71C22×C4×C7—42no—
72C22×C4×C7—42no—
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22×C411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C7×M16—82no—
79C2×C7×C8—42no—
80C7×C4⋊C4—24no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—42no—
83C7×C4⋊C4—44no—
84C7×C4⋊C4—44no—
85C7×C4⋊C4—42no—
86C7×C22⋊C4—24no—
87C7×C22⋊C4—41no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—44no—
90C7×C22⋊C4—44no—
91C7×C22⋊C4—42no—
92C42×C7—21no—
93C42×C7—21no—
94C42×C7—42no—
95C42×C7—82no—
96C2×C4⋊Q8C4×C711yescharacteristic, complemented
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×C4×D4C22×C711yescharacteristic
100C2×#99—21no—
101#407—21no—
102#399C4×C712yescomplemented
103#399C22×C712yes—
104#390—21no—
105#384—21no—
106C4×C4⋊C4C4×C711yescharacteristic, complemented
107C4×C22⋊C4—21no—
108#342C22×C712yes—
109#324—21no—
110#323C22×C712yes—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—41no—
117C23×C7—41no—
118C23×C7—41no—
119C23×C7—41no—
120C23×C7C2×C22⋊C411yescharacteristic
121C23×C7#13712yes—
122C7×Q8—82no—
123C7×Q8—82no—
124C7×D4—44no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—22no—
128C2×C4×C7—24no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—41no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—44no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—81no—
144C2×C4×C7—81no—
145C2×C4×C7—82no—
146C2×C4×C7—82no—
147C2×C4×C7#9911yescharacteristic
148C2×C4×C7#9911yescharacteristic
149C2×C4×C7#10412yes—
150C7×C8—82no—
151C22×Q8—21no—
152C22×D4C7×D411yescharacteristic
153C23×C4C7×D411yescharacteristic
154C23×C4C7×D411yescharacteristic
155C2×M16—41no—
156C4⋊Q8—41no—
157C4⋊Q8—41no—
158C22⋊Q8—44no—
159C22⋊Q8—44no—
160C4×D4—41no—
161C4×D4—41no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C2×C4×C712yes—
167C2×C4⋊C4C7×D411yescharacteristic
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4C2×C4×C711yescharacteristic
172C2×C22⋊C4C23×C711yescharacteristic
173C2×C42—21no—
174C2×C42—41no—
175C2×C42C2×C4×C711yescharacteristic
176#99—24no—
177C22⋊C8—22no—
178C22⋊C8—41no—
179#95—41no—
180#95C2×C4×C712yes—
181C22×C7—21no—
182C22×C7—21no—
183C22×C7—22no—
184C22×C7—42no—
185C22×C7—42no—
186C22×C7—41no—
187C22×C7—41no—
188C22×C7—41no—
189C22×C7—41no—
190C22×C7—41no—
191C22×C7—81no—
192C22×C7—81no—
193C22×C7C2×#9911yescharacteristic
194C22×C7#41911yescharacteristic
195C22×C7C2×#10411yescharacteristic
196C4×C7—21no—
197C4×C7—22no—
198C4×C7—44no—
199C4×C7—41no—
200C4×C7—41no—
201C4×C7—41no—
202C4×C7—84no—
203C4×C7—82no—
204C4×C7#35312yes—
205C24C7×C22⋊C411yescharacteristic
206C24C2×C7×D411yescharacteristic
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—41no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C7×C22⋊C411yescharacteristic
224C22×C4C7×C22⋊C411yescharacteristic
225C22×C4C7×C22⋊C411yescharacteristic
226C22×C4C22×C4×C711yescharacteristic
227C22×C4C2×C7×D411yescharacteristic
228M16—82no—
229C2×C8—42no—
230C4⋊C4—24no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—42no—
236C22⋊C4—24no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C22⋊C4—42no—
242C42—21no—
243C42—21no—
244C42—42no—
245C42—82no—
246C2×C7—21no—
247C2×C7—21no—
248C2×C7—42no—
249C2×C7—81no—
250C2×C7#150811yescharacteristic
251C2×C7#212311yescharacteristic
252C2×C7C2×#35311yescharacteristic
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23C2×C7×C22⋊C411yescharacteristic
263C23C7×#13712yes—
264Q8—82no—
265Q8—82no—
266D4—44no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4C7×#9911yescharacteristic
290C2×C4C7×#9911yescharacteristic
291C2×C4C7×#10412yes—
292C8—82no—
293C7#1178811yescharacteristic, direct product factor, Sylow
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22—81no—
306C22C2×C7×#9911yescharacteristic
307C22C7×#41911yescharacteristic
308C22C2×C7×#10411yescharacteristic
309C4—22no—
310C4—21no—
311C4—44no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—84no—
316C4—82no—
317C4C7×#35312yes—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—81no—
322C2C7×#150811yescharacteristic
323C2C7×#212311yescharacteristic
324C2C2×C7×#35311yescharacteristic
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
5. C2×C7×C4⋊Q8C4semidirect
11. C7×#399C4semidirect
15. C4×C7×C4⋊C4C4semidirect
20. #11788C7direct (N × H)
54. #1352C2×C7semidirect
96. C2×C4⋊Q8C4×C7semidirect
102. #399C4×C7semidirect
106. C4×C4⋊C4C4×C7semidirect
293. C7#11788direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25422
C4×C7×#992
C7×#14652
C7×#13522
#117887