Finite Group Databasebeta

← C7×#36691 #13213907

Subgroups — C7×#36691 #13213907

Order 1792.

1474 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2845C212yesmaximal, complemented
2C7×#2844C212yesmaximal, complemented
3C7×#2843C214yesmaximal, complemented
4C7×#2841C218yesmaximal
5C7×#2402C214yesmaximal, complemented
6C7×#2399C214yesmaximal, complemented
7C7×#2396C214yesmaximal
8C7×#2285C211yesmaximal, characteristic
9C7×#2283C212yesmaximal
10C2×C4×C7×Q8C2214yes—
11C2×C7×#117C2212yes—
12C22×C42×C7C2211yescharacteristic
13C7×#400C2214yes—
14C7×#400C2218yes—
15C7×#400C2218yes—
16C7×#399C2214yes—
17C7×#398C2218yes—
18C7×#398C2218yes—
19C7×#398C2218yes—
20C7×#398C2218yes—
21C7×#398C2218yes—
22C7×#396C2218yes—
23C7×#395C2214yes—
24C7×#395C2218yes—
25C7×#395C2218yes—
26C7×#393C2214yes—
27C7×#384C2212yes—
28C7×#383C2214yes—
29C7×#382C2214yes—
30C7×#382C2211yescharacteristic
31C7×#382C2211yescharacteristic
32C7×#381C2214yes—
33C7×#380C2214yes—
34C7×#380C2214yes—
35C4×C7×C4⋊C4C2218yes—
36C4×C7×C4⋊C4C2214yes—
37C4×C7×C4⋊C4C2212yes—
38C4×C7×C4⋊C4C2214yes—
39C4×C7×C4⋊C4C2212yes—
40C4×C7×C22⋊C4C2214yes—
41C7×#376C2214yes—
42#36691C711yesmaximal, characteristic, direct product factor, Sylow
43C22×C7×Q8C2314yes—
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2312yes—
46C4×C7×Q8—48no—
47C7×#117—44no—
48C2×C7×C4⋊C4C2312yes—
49C2×C7×C4⋊C4C2312yes—
50C2×C7×C4⋊C4C2314yes—
51C2×C7×C4⋊C4C2314yes—
52C2×C7×C4⋊C4C2318yes—
53C2×C7×C4⋊C4C2318yes—
54C2×C7×C4⋊C4C2318yes—
55C2×C7×C4⋊C4C2318yes—
56C2×C7×C4⋊C4C2314yes—
57C2×C7×C4⋊C4C2318yes—
58C2×C7×C22⋊C4C2318yes—
59C2×C7×C22⋊C4C2314yes—
60C2×C42×C7Q818yescomplemented
61C2×C42×C7C2314yes—
62C2×C42×C7C2314yes—
63C2×C42×C7C2314yes—
64C2×C42×C7C2311yescharacteristic
65C2×C42×C7C2311yescharacteristic
66C2×C42×C7C2312yes—
67C7×#95C2318yes—
68C7×#95C2318yes—
69C7×#95C2318yes—
70C7×#95C2314yes—
71C7×#95C2314yes—
72C7×#95C2314yes—
73C7×#95C2314yes—
74C7×#95C2318yes—
75C7×#95C2314yes—
76C7×#95C2318yes—
77C7×#95C2314yes—
78#2845C2×C712yescomplemented
79#2844C2×C712yescomplemented
80#2843C2×C714yescomplemented
81#2841C2×C718yes—
82#2402C2×C714yescomplemented
83#2399C2×C714yescomplemented
84#2396C2×C714yes—
85#2285C2×C711yescharacteristic
86#2283C2×C712yes—
87C24×C7C2411yescharacteristic
88C2×C7×Q8—44no—
89C2×C7×Q8—48no—
90C22×C4×C7—22no—
91C22×C4×C7—22no—
92C22×C4×C7—44no—
93C22×C4×C7C2×Q814yes—
94C22×C4×C7C2×Q818yes—
95C22×C4×C7C2418yes—
96C22×C4×C7C2414yes—
97C22×C4×C7C2418yes—
98C22×C4×C7C2414yes—
99C22×C4×C7C2411yescharacteristic
100C22×C4×C7C2412yes—
101C22×C4×C7C2412yes—
102C22×C4×C7C2411yescharacteristic
103C7×C4⋊C4—44no—
104C7×C4⋊C4—48no—
105C7×C4⋊C4—44no—
106C7×C4⋊C4—48no—
107C7×C4⋊C4—48no—
108C7×C4⋊C4—48no—
109C7×C4⋊C4—42no—
110C7×C4⋊C4—42no—
111C7×C4⋊C4—48no—
112C7×C4⋊C4—44no—
113C7×C22⋊C4—48no—
114C7×C22⋊C4—44no—
115C42×C7—28no—
116C42×C7—28no—
117C42×C7—28no—
118C42×C7C2×Q814yes—
119C42×C7C2×Q814yes—
120C42×C7C2×Q818yes—
121C2×C4×Q8C22×C714yes—
122C2×#117C22×C712yes—
123C22×C42C22×C711yescharacteristic
124#400C22×C718yes—
125#400C22×C718yes—
126#400C22×C714yes—
127#399C22×C714yes—
128#398C22×C718yes—
129#398C22×C718yes—
130#398C22×C718yes—
131#398C22×C718yes—
132#398C22×C718yes—
133#396C22×C718yes—
134#395C22×C718yes—
135#395C22×C718yes—
136#395C22×C714yes—
137#393C22×C714yes—
138#384C22×C712yes—
139#383C22×C714yes—
140#382C22×C711yescharacteristic
141#382C22×C714yes—
142#382C22×C711yescharacteristic
143#381C22×C714yes—
144#380C22×C714yes—
145#380C22×C714yes—
146C4×C4⋊C4C22×C718yes—
147C4×C4⋊C4C22×C714yes—
148C4×C4⋊C4C22×C712yes—
149C4×C4⋊C4C22×C714yes—
150C4×C4⋊C4C22×C712yes—
151C4×C22⋊C4C22×C714yes—
152#376C22×C714yes—
153C23×C7—21no—
154C23×C7—21no—
155C23×C7—24no—
156C23×C7C22×Q812yes—
157C23×C7C2511yescharacteristic
158C7×Q8—416no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—24no—
162C2×C4×C7—24no—
163C2×C4×C7—28no—
164C2×C4×C7—24no—
165C2×C4×C7—24no—
166C2×C4×C7—28no—
167C2×C4×C7—28no—
168C2×C4×C7—28no—
169C2×C4×C7—28no—
170C2×C4×C7—28no—
171C2×C4×C7—28no—
172C2×C4×C7—28no—
173C2×C4×C7—28no—
174C2×C4×C7—48no—
175C2×C4×C7C22×Q812yes—
176C2×C4×C7C22×Q814yes—
177C2×C4×C7C22×Q814yes—
178C2×C4×C7C22×Q812yes—
179C2×C4×C7H(5,𝔽2)12yes—
180C2×C4×C72−1+412yes—
181C22×Q8C23×C714yes—
182C23×C4C23×C711yescharacteristic
183C23×C4C23×C712yes—
184C4×Q8—48no—
185#117—44no—
186C2×C4⋊C4C23×C712yes—
187C2×C4⋊C4C23×C712yes—
188C2×C4⋊C4C23×C714yes—
189C2×C4⋊C4C23×C718yes—
190C2×C4⋊C4C23×C714yes—
191C2×C4⋊C4C23×C714yes—
192C2×C4⋊C4C23×C718yes—
193C2×C4⋊C4C23×C718yes—
194C2×C4⋊C4C23×C718yes—
195C2×C4⋊C4C23×C718yes—
196C2×C22⋊C4C23×C714yes—
197C2×C22⋊C4C23×C718yes—
198C2×C42C7×Q818yescomplemented
199C2×C42C23×C714yes—
200C2×C42C23×C714yes—
201C2×C42C23×C714yes—
202C2×C42C23×C712yes—
203C2×C42C23×C711yescharacteristic
204C2×C42C23×C711yescharacteristic
205#95C23×C718yes—
206#95C23×C718yes—
207#95C23×C718yes—
208#95C23×C714yes—
209#95C23×C714yes—
210#95C23×C714yes—
211#95C23×C714yes—
212#95C23×C718yes—
213#95C23×C714yes—
214#95C23×C718yes—
215#95C23×C714yes—
216C22×C7—22no—
217C22×C7—24no—
218C22×C7—42no—
219C22×C7—42no—
220C22×C7C23×Q811yescharacteristic
221C22×C7C2×H(5,𝔽2)11yescharacteristic
222C22×C7C2×2−1+411yescharacteristic
223C22×C7#58514yes—
224C4×C7—22no—
225C4×C7—24no—
226C4×C7—24no—
227C4×C7—44no—
228C4×C7—44no—
229C4×C7—48no—
230C4×C7—48no—
231C4×C7#54414yes—
232C24C24×C711yescharacteristic
233C2×Q8—44no—
234C2×Q8—48no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—44no—
238C22×C4C2×C7×Q814yes—
239C22×C4C2×C7×Q818yes—
240C22×C4C24×C718yes—
241C22×C4C24×C714yes—
242C22×C4C24×C718yes—
243C22×C4C24×C714yes—
244C22×C4C24×C711yescharacteristic
245C22×C4C24×C712yes—
246C22×C4C24×C712yes—
247C22×C4C24×C711yescharacteristic
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—48no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—48no—
254C4⋊C4—44no—
255C4⋊C4—48no—
256C4⋊C4—48no—
257C4⋊C4—48no—
258C22⋊C4—44no—
259C22⋊C4—48no—
260C42—28no—
261C42—28no—
262C42—28no—
263C42C2×C7×Q814yes—
264C42C2×C7×Q814yes—
265C42C2×C7×Q818yes—
266C2×C7—42no—
267C2×C7C2×#54411yescharacteristic
268C2×C7#345812yes—
269C2×C7#352912yes—
270C2×C7#353112yes—
271C23—21no—
272C23—21no—
273C23—24no—
274C23C22×C7×Q812yes—
275C23C25×C711yescharacteristic
276Q8—416no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—28no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—28no—
285C2×C4—28no—
286C2×C4—28no—
287C2×C4—28no—
288C2×C4—28no—
289C2×C4—28no—
290C2×C4—28no—
291C2×C4—28no—
292C2×C4—48no—
293C2×C4C22×C7×Q812yes—
294C2×C4C22×C7×Q814yes—
295C2×C4C22×C7×Q814yes—
296C2×C4C22×C7×Q812yes—
297C2×C4C7×H(5,𝔽2)12yes—
298C2×C4C7×2−1+412yes—
299C7#3669111yescharacteristic, direct product factor, Sylow
300C22—22no—
301C22—24no—
302C22—42no—
303C22—42no—
304C22C23×C7×Q811yescharacteristic
305C22C2×C7×H(5,𝔽2)11yescharacteristic
306C22C2×C7×2−1+411yescharacteristic
307C22C7×#58514yes—
308C4—22no—
309C4—24no—
310C4—24no—
311C4—44no—
312C4—48no—
313C4—44no—
314C4—48no—
315C4C7×#54414yes—
316C2—42no—
317C2C2×C7×#54411yescharacteristic
318C2C7×#345812yes—
319C2C7×#352912yes—
320C2C7×#353112yes—
subgroupG/H
centerC22×C4×C7C24
commutator, Frattini, cosocleC23C25×C7
socleC23×C7C25
prime radicalC24×C7C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2845C2semidirect
2. C7×#2844C2semidirect
3. C7×#2843C2semidirect
5. C7×#2402C2semidirect
6. C7×#2399C2semidirect
42. #36691C7direct (N × H)
60. C2×C42×C7Q8semidirect
78. #2845C2×C7semidirect
79. #2844C2×C7semidirect
80. #2843C2×C7semidirect
82. #2402C2×C7semidirect
83. #2399C2×C7semidirect
198. C2×C42C7×Q8semidirect
299. C7#36691direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28452
C7×#28442
C7×#28432
C7×#28412
C7×#24022
C7×#23992
C7×#23962
C7×#22852
C7×#22832
#366917