Finite Group Databasebeta

← C7×#12856 #13190072

Subgroups — C7×#12856 #13190072

Order 1792.

914 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C2×C7×#390C211yesmaximal, characteristic
2C7×#1781C212yesmaximal
3C7×#1393C214yesmaximal, complemented
4C23×C7×D4C2211yescharacteristic
5C22×C7×C4⋊C4C411yescharacteristic, complemented
6C22×C7×C22⋊C4—22no—
7C22×C42×C7C411yescharacteristic, complemented
8C2×C7×#99—22no—
9C2×C7×#99C2214yes—
10C7×#390—22no—
11C7×#390C412yescomplemented
12C7×#390C2212yes—
13C7×#379—22no—
14#12856C711yesmaximal, characteristic, direct product factor, Sylow
15C25×C7D412yes—
16C22×C7×D4—24no—
17C22×C7×D4—42no—
18C22×C7×D4—41no—
19C22×C7×D4—41no—
20C22×C7×D4C2×C411yescharacteristic
21C22×C7×D4D412yes—
22C22×C7×D4C2311yescharacteristic
23C23×C4×C7—21no—
24C23×C4×C7—21no—
25C23×C4×C7C2×C411yescharacteristic
26C2×C7×C4⋊C4—21no—
27C2×C7×C4⋊C4—41no—
28C2×C7×C4⋊C4—41no—
29C2×C7×C4⋊C4C2×C412yes—
30C2×C7×C22⋊C4—24no—
31C2×C7×C22⋊C4—24no—
32C2×C7×C22⋊C4—24no—
33C2×C7×C22⋊C4—48no—
34C2×C7×C22⋊C4—42no—
35C2×C7×C22⋊C4—42no—
36C2×C42×C7—21no—
37C2×C42×C7—42no—
38C2×C42×C7C2×C412yes—
39C7×#99—28no—
40C7×#99—44no—
41C2×#390C2×C711yescharacteristic
42#1781C2×C712yes—
43#1393C2×C714yescomplemented
44C24×C7—24no—
45C24×C7—22no—
46C24×C7—22no—
47C24×C7—42no—
48C24×C7—44no—
49C24×C7—44no—
50C24×C7C22⋊C412yes—
51C24×C7C22⋊C411yescharacteristic
52C24×C7C2×D412yes—
53C2×C7×D4—24no—
54C2×C7×D4—24no—
55C2×C7×D4—22no—
56C2×C7×D4—41no—
57C2×C7×D4—41no—
58C2×C7×D4—42no—
59C2×C7×D4—41no—
60C2×C7×D4—41no—
61C2×C7×D4—48no—
62C2×C7×D4—48no—
63C2×C7×D4#4114yes—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—22no—
67C22×C4×C7—22no—
68C22×C4×C7—24no—
69C22×C4×C7—22no—
70C22×C4×C7—22no—
71C22×C4×C7—42no—
72C22×C4×C7—42no—
73C22×C4×C7—44no—
74C22×C4×C7—41no—
75C22×C4×C7—41no—
76C22×C4×C7—44no—
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22×C411yescharacteristic
79C7×C4⋊C4—42no—
80C7×C4⋊C4—42no—
81C7×C22⋊C4—48no—
82C7×C22⋊C4—48no—
83C7×C22⋊C4—44no—
84C7×C22⋊C4—44no—
85C7×C22⋊C4—84no—
86C7×C22⋊C4—84no—
87C42×C7—42no—
88C42×C7—42no—
89C23×D4C22×C711yescharacteristic
90C22×C4⋊C4C4×C711yescharacteristic, complemented
91C22×C22⋊C4—22no—
92C22×C42C4×C711yescharacteristic, complemented
93C2×#99—22no—
94C2×#99C22×C714yes—
95#390—22no—
96#390C4×C712yescomplemented
97#390C22×C712yes—
98#379—22no—
99C23×C7—22no—
100C23×C7—22no—
101C23×C7—21no—
102C23×C7—21no—
103C23×C7—22no—
104C23×C7—22no—
105C23×C7—22no—
106C23×C7—22no—
107C23×C7—22no—
108C23×C7—44no—
109C23×C7—44no—
110C23×C7—42no—
111C23×C7—42no—
112C23×C7—42no—
113C23×C7—42no—
114C23×C7—416no—
115C23×C7—44no—
116C23×C7—44no—
117C23×C7—82no—
118C23×C7—82no—
119C23×C7—88no—
120C23×C7#9911yescharacteristic
121C23×C7#9911yescharacteristic
122C23×C7C2×C22⋊C411yescharacteristic
123C23×C7#12314yes—
124C7×D4—44no—
125C7×D4—48no—
126C7×D4—44no—
127C2×C4×C7—24no—
128C2×C4×C7—24no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—24no—
134C2×C4×C7—48no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—44no—
138C2×C4×C7—42no—
139C2×C4×C7—82no—
140C2×C4×C7—82no—
141C2×C4×C7—84no—
142C2×C4×C7—84no—
143C2×C4×C7—84no—
144C2×C4×C7#9912yes—
145C2×C4×C7#11712yes—
146C25C7×D412yes—
147C22×D4—24no—
148C22×D4—42no—
149C22×D4—41no—
150C22×D4—41no—
151C22×D4C2×C4×C711yescharacteristic
152C22×D4C7×D412yes—
153C22×D4C23×C711yescharacteristic
154C23×C4—21no—
155C23×C4—21no—
156C23×C4C2×C4×C711yescharacteristic
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4C2×C4×C712yes—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—48no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C42—21no—
168C2×C42—42no—
169C2×C42C2×C4×C712yes—
170#99—28no—
171#99—44no—
172C22×C7—24no—
173C22×C7—24no—
174C22×C7—21no—
175C22×C7—21no—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—21no—
179C22×C7—21no—
180C22×C7—24no—
181C22×C7—44no—
182C22×C7—44no—
183C22×C7—42no—
184C22×C7—42no—
185C22×C7—42no—
186C22×C7—42no—
187C22×C7—42no—
188C22×C7—82no—
189C22×C7—88no—
190C22×C7—88no—
191C22×C7—82no—
192C22×C7#35312yes—
193C22×C7#35412yes—
194C22×C7#38111yescharacteristic
195C22×C7C2×#9911yescharacteristic
196C22×C7C2×#9911yescharacteristic
197C4×C7—22no—
198C4×C7—22no—
199C4×C7—44no—
200C4×C7—82no—
201C4×C7—164no—
202C24—24no—
203C24—22no—
204C24—22no—
205C24—42no—
206C24—44no—
207C24—44no—
208C24C7×C22⋊C411yescharacteristic
209C24C7×C22⋊C412yes—
210C24C2×C7×D412yes—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—22no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—42no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—48no—
220C2×D4—48no—
221C2×D4C7×#4114yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—44no—
235C22×C4C7×C22⋊C411yescharacteristic
236C22×C4C22×C4×C711yescharacteristic
237C4⋊C4—42no—
238C4⋊C4—42no—
239C22⋊C4—48no—
240C22⋊C4—48no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4—84no—
244C22⋊C4—84no—
245C42—42no—
246C42—42no—
247C2×C7—21no—
248C2×C7—21no—
249C2×C7—22no—
250C2×C7—22no—
251C2×C7—44no—
252C2×C7—82no—
253C2×C7#181911yescharacteristic
254C2×C7C2×#35311yescharacteristic
255C2×C7C2×#35411yescharacteristic
256C23—22no—
257C23—22no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—44no—
266C23—44no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—416no—
272C23—44no—
273C23—44no—
274C23—82no—
275C23—82no—
276C23—88no—
277C23C7×#9911yescharacteristic
278C23C7×#9911yescharacteristic
279C23C2×C7×C22⋊C411yescharacteristic
280C23C7×#12314yes—
281D4—44no—
282D4—48no—
283D4—44no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—48no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4C7×#9912yes—
302C2×C4C7×#11712yes—
303C7#1285611yescharacteristic, direct product factor, Sylow
304C22—24no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—24no—
313C22—44no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—82no—
321C22—88no—
322C22—88no—
323C22—82no—
324C22C7×#35312yes—
325C22C7×#35412yes—
326C22C7×#38111yescharacteristic
327C22C2×C7×#9911yescharacteristic
328C22C2×C7×#9911yescharacteristic
329C4—22no—
330C4—22no—
331C4—44no—
332C4—82no—
333C4—164no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—22no—
338C2—44no—
339C2—82no—
340C2C7×#181911yescharacteristic
341C2C2×C7×#35311yescharacteristic
342C2C2×C7×#35411yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC22×D4C23×C7
prime radicalC23×C7×D4C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C24×C7 ◁ C23×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1393C2semidirect
5. C22×C7×C4⋊C4C4semidirect
7. C22×C42×C7C4semidirect
11. C7×#390C4semidirect
14. #12856C7direct (N × H)
43. #1393C2×C7semidirect
90. C22×C4⋊C4C4×C7semidirect
92. C22×C42C4×C7semidirect
96. #390C4×C7semidirect
303. C7#12856direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3902
C7×#17812
C7×#13932
#128567