Finite Group Databasebeta

← C7×#36600 #13213816

Subgroups — C7×#36600 #13213816

Order 1792.

1574 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2859C211yesmaximal, characteristic
2C7×#2859C212yesmaximal
3C7×#2585C218yesmaximal
4C7×#2570C214yesmaximal
5C7×#2552C212yesmaximal
6C7×#2453C214yesmaximal
7C7×#2370C214yesmaximal
8C7×#2355C212yesmaximal
9C4×C7×#125C211yesmaximal, characteristic
10C4×C7×#125C212yesmaximal
11C42×C7×Q8C211yesmaximal, characteristic
12C2×C7×#125C2214yes—
13C2×C7×#125C2211yescharacteristic
14C2×C7×#125C2212yes—
15C2×C4×C7×Q8C2212yes—
16C2×C4×C7×Q8C2211yescharacteristic
17C7×#400C2218yes—
18C7×#398C2218yes—
19C7×#398C2218yes—
20C7×#391C2214yes—
21C7×#389C2214yes—
22C7×#389C2218yes—
23C7×#389C2214yes—
24C7×#389C2214yes—
25C7×#389C2212yes—
26C7×#389C2214yes—
27C7×#389C2218yes—
28C7×#387C2218yes—
29C7×#387C2214yes—
30C7×#387C2218yes—
31C7×#387C2218yes—
32C7×#387C2218yes—
33C7×#384C2214yes—
34C7×#384C2212yes—
35C7×#382C2211yescharacteristic
36C7×#382C2212yes—
37C7×#382C2214yes—
38C4×C7×C4⋊C4C2214yes—
39C4×C7×C4⋊C4C2211yescharacteristic
40C4×C7×C4⋊C4C2212yes—
41C4×C7×C4⋊C4C2214yes—
42C4×C7×C4⋊C4C2212yes—
43C4×C7×C4⋊C4C2212yes—
44C4×C7×C4⋊C4C2214yes—
45C4×C7×C4⋊C4C2218yes—
46C4×C7×C4⋊C4C2214yes—
47C43×C7C2211yescharacteristic
48C43×C7C2212yes—
49#36600C711yesmaximal, characteristic, direct product factor, Sylow
50C22×C7×Q8C2311yescharacteristic
51C7×#125—216no—
52C7×#125—24no—
53C7×#125—28no—
54C4×C7×Q8—28no—
55C4×C7×Q8Q818yes—
56C2×C7×C4⋊C4C2312yes—
57C2×C7×C4⋊C4C2314yes—
58C2×C7×C4⋊C4C2312yes—
59C2×C7×C4⋊C4C2314yes—
60C2×C7×C4⋊C4C2314yes—
61C2×C7×C4⋊C4C2318yes—
62C2×C7×C4⋊C4C2314yes—
63C2×C7×C4⋊C4C2318yes—
64C2×C7×C4⋊C4C2318yes—
65C2×C7×C4⋊C4C2314yes—
66C2×C7×C4⋊C4C2314yes—
67C2×C7×C4⋊C4C2318yes—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2312yes—
70C2×C7×C4⋊C4C2312yes—
71C2×C42×C7C2312yes—
72C2×C42×C7C2311yescharacteristic
73C2×C42×C7C2312yes—
74C2×C42×C7C2314yes—
75C2×C42×C7C2314yes—
76C2×C42×C7C2314yes—
77C2×C42×C7C2312yes—
78C2×C42×C7C2318yes—
79C2×C42×C7C2312yes—
80C2×C42×C7C2311yescharacteristic
81C2×C42×C7C2311yescharacteristic
82C7×#95C2314yes—
83C7×#95C2314yes—
84C7×#95C2312yes—
85C7×#95C2318yes—
86C7×#95C2314yes—
87C7×#95C2314yes—
88C7×#95C2318yes—
89C7×#95C2318yes—
90C7×#95C2318yes—
91C7×#95C2318yes—
92#2859C2×C711yescharacteristic
93#2859C2×C712yes—
94#2585C2×C718yes—
95#2570C2×C714yes—
96#2552C2×C712yes—
97#2453C2×C714yes—
98#2370C2×C714yes—
99#2355C2×C712yes—
100C4×#125C2×C712yes—
101C4×#125C2×C711yescharacteristic
102C42×Q8C2×C711yescharacteristic
103C2×C7×Q8—24no—
104C2×C7×Q8C2×Q814yes—
105C22×C4×C7C2412yes—
106C22×C4×C7C2414yes—
107C22×C4×C7C2414yes—
108C22×C4×C7C2418yes—
109C22×C4×C7C2412yes—
110C22×C4×C7C2412yes—
111C22×C4×C7C2411yescharacteristic
112C22×C4×C7C2414yes—
113C22×C4×C7C2412yes—
114C22×C4×C7C2411yescharacteristic
115C22×C4×C7C2411yescharacteristic
116C7×C4⋊C4—28no—
117C7×C4⋊C4—216no—
118C7×C4⋊C4—216no—
119C7×C4⋊C4—28no—
120C7×C4⋊C4—216no—
121C7×C4⋊C4—28no—
122C7×C4⋊C4—24no—
123C7×C4⋊C4—28no—
124C7×C4⋊C4—24no—
125C7×C4⋊C4—44no—
126C7×C4⋊C4—48no—
127C7×C4⋊C4—44no—
128C7×C4⋊C4C2×Q818yes—
129C7×C4⋊C4C2×Q814yes—
130C7×C4⋊C4#4118yes—
131C42×C7—28no—
132C42×C7—24no—
133C42×C7—28no—
134C42×C7—24no—
135C42×C7—22no—
136C42×C7—44no—
137C42×C7—42no—
138C42×C7—48no—
139C42×C7C2×Q818yes—
140C42×C7C2×Q814yes—
141C42×C7#4114yes—
142C2×#125C22×C712yes—
143C2×#125C22×C711yescharacteristic
144C2×#125C22×C714yes—
145C2×C4×Q8C22×C712yes—
146C2×C4×Q8C22×C711yescharacteristic
147#400C22×C718yes—
148#398C22×C718yes—
149#398C22×C718yes—
150#391C22×C714yes—
151#389C22×C712yes—
152#389C22×C714yes—
153#389C22×C714yes—
154#389C22×C718yes—
155#389C22×C718yes—
156#389C22×C714yes—
157#389C22×C714yes—
158#387C22×C714yes—
159#387C22×C718yes—
160#387C22×C718yes—
161#387C22×C718yes—
162#387C22×C718yes—
163#384C22×C714yes—
164#384C22×C712yes—
165#382C22×C711yescharacteristic
166#382C22×C712yes—
167#382C22×C714yes—
168C4×C4⋊C4C22×C714yes—
169C4×C4⋊C4C22×C718yes—
170C4×C4⋊C4C22×C712yes—
171C4×C4⋊C4C22×C714yes—
172C4×C4⋊C4C22×C712yes—
173C4×C4⋊C4C22×C714yes—
174C4×C4⋊C4C22×C714yes—
175C4×C4⋊C4C22×C712yes—
176C4×C4⋊C4C22×C711yescharacteristic
177C43C22×C711yescharacteristic
178C43C22×C712yes—
179C23×C7C2511yescharacteristic
180C7×Q8—28no—
181C2×C4×C7—22no—
182C2×C4×C7—24no—
183C2×C4×C7—24no—
184C2×C4×C7—22no—
185C2×C4×C7—24no—
186C2×C4×C7—22no—
187C2×C4×C7—24no—
188C2×C4×C7—24no—
189C2×C4×C7—22no—
190C2×C4×C7—24no—
191C2×C4×C7—28no—
192C2×C4×C7—24no—
193C2×C4×C7—28no—
194C2×C4×C7—22no—
195C2×C4×C7—24no—
196C2×C4×C7—24no—
197C2×C4×C7—28no—
198C2×C4×C7—22no—
199C2×C4×C7—24no—
200C2×C4×C7C22×Q814yes—
201C2×C4×C7C22×Q812yes—
202C2×C4×C7C22×Q814yes—
203C2×C4×C7C22×Q812yes—
204C2×C4×C7C22×Q812yes—
205C2×C4×C7C2×#4118yes—
206C2×C4×C7C2×#4114yes—
207C2×C4×C7C2×#4114yes—
208C2×C4×C7C2×#4112yes—
209C2×C4×C72−1+412yes—
210C22×Q8C23×C711yescharacteristic
211#125—216no—
212#125—28no—
213#125—24no—
214C4×Q8—28no—
215C4×Q8C7×Q818yes—
216C2×C4⋊C4C23×C714yes—
217C2×C4⋊C4C23×C714yes—
218C2×C4⋊C4C23×C712yes—
219C2×C4⋊C4C23×C714yes—
220C2×C4⋊C4C23×C718yes—
221C2×C4⋊C4C23×C718yes—
222C2×C4⋊C4C23×C714yes—
223C2×C4⋊C4C23×C718yes—
224C2×C4⋊C4C23×C712yes—
225C2×C4⋊C4C23×C714yes—
226C2×C4⋊C4C23×C714yes—
227C2×C4⋊C4C23×C718yes—
228C2×C4⋊C4C23×C711yescharacteristic
229C2×C4⋊C4C23×C712yes—
230C2×C4⋊C4C23×C712yes—
231C2×C42C23×C712yes—
232C2×C42C23×C714yes—
233C2×C42C23×C714yes—
234C2×C42C23×C711yescharacteristic
235C2×C42C23×C712yes—
236C2×C42C23×C714yes—
237C2×C42C23×C712yes—
238C2×C42C23×C718yes—
239C2×C42C23×C712yes—
240C2×C42C23×C711yescharacteristic
241C2×C42C23×C711yescharacteristic
242#95C23×C714yes—
243#95C23×C714yes—
244#95C23×C712yes—
245#95C23×C718yes—
246#95C23×C714yes—
247#95C23×C714yes—
248#95C23×C718yes—
249#95C23×C718yes—
250#95C23×C718yes—
251#95C23×C718yes—
252C22×C7C23×Q811yescharacteristic
253C22×C7C22×#4112yes—
254C22×C7C22×#4111yescharacteristic
255C22×C7C2×2−1+411yescharacteristic
256C22×C7#58512yes—
257C4×C7—24no—
258C4×C7—24no—
259C4×C7—24no—
260C4×C7—22no—
261C4×C7—28no—
262C4×C7—44no—
263C4×C7—48no—
264C4×C7—42no—
265C4×C7—44no—
266C4×C7#53314yes—
267C4×C7#55714yes—
268C2×Q8—24no—
269C2×Q8C2×C7×Q814yes—
270C22×C4C24×C712yes—
271C22×C4C24×C714yes—
272C22×C4C24×C714yes—
273C22×C4C24×C718yes—
274C22×C4C24×C712yes—
275C22×C4C24×C712yes—
276C22×C4C24×C711yescharacteristic
277C22×C4C24×C714yes—
278C22×C4C24×C712yes—
279C22×C4C24×C711yescharacteristic
280C22×C4C24×C711yescharacteristic
281C4⋊C4—28no—
282C4⋊C4—216no—
283C4⋊C4—216no—
284C4⋊C4—216no—
285C4⋊C4—28no—
286C4⋊C4—28no—
287C4⋊C4—28no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—44no—
291C4⋊C4—48no—
292C4⋊C4—44no—
293C4⋊C4C2×C7×Q818yes—
294C4⋊C4C2×C7×Q814yes—
295C4⋊C4C7×#4118yes—
296C42—28no—
297C42—24no—
298C42—28no—
299C42—24no—
300C42—22no—
301C42—44no—
302C42—42no—
303C42—48no—
304C42C2×C7×Q818yes—
305C42C2×C7×Q814yes—
306C42C7×#4114yes—
307C2×C7C2×#53311yescharacteristic
308C2×C7C2×#55711yescharacteristic
309C2×C7#348012yes—
310C2×C7#350411yescharacteristic
311C2×C7#351512yes—
312C23C25×C711yescharacteristic
313Q8—28no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—28no—
325C2×C4—24no—
326C2×C4—28no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—28no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4C22×C7×Q814yes—
334C2×C4C22×C7×Q812yes—
335C2×C4C22×C7×Q814yes—
336C2×C4C22×C7×Q812yes—
337C2×C4C22×C7×Q812yes—
338C2×C4C2×C7×#4118yes—
339C2×C4C2×C7×#4114yes—
340C2×C4C2×C7×#4114yes—
341C2×C4C2×C7×#4112yes—
342C2×C4C7×2−1+412yes—
343C7#3660011yescharacteristic, direct product factor, Sylow
344C22C23×C7×Q811yescharacteristic
345C22C22×C7×#4112yes—
346C22C22×C7×#4111yescharacteristic
347C22C2×C7×2−1+411yescharacteristic
348C22C7×#58512yes—
349C4—24no—
350C4—24no—
351C4—24no—
352C4—22no—
353C4—28no—
354C4—44no—
355C4—48no—
356C4—42no—
357C4—44no—
358C4C7×#53314yes—
359C4C7×#55714yes—
360C2C2×C7×#53311yescharacteristic
361C2C2×C7×#55711yescharacteristic
362C2C7×#348012yes—
363C2C7×#350411yescharacteristic
364C2C7×#351512yes—
subgroupG/H
center, socle, prime radicalC23×C7C25
commutator, Frattini, cosocleC23C25×C7
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C7 ◁ G

Complements

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

NH ≅ G/Ntype
49. #36600C7direct (N × H)
343. C7#36600direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28592
C7×#25852
C7×#25702
C7×#25522
C7×#24532
C7×#23702
C7×#23552
C4×C7×#1252
C42×C7×Q82
#366007