Finite Group Databasebeta

← C2×C7×#2537 #13212138

Subgroups — C2×C7×#2537 #13212138

Order 1792.

4782 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#121C214yesmaximal, complemented
2C22×C4×C7×D4C212yesmaximal, complemented
3C7×#2537C2116yesmaximal, direct product factor
4C2×C7×#394C214yesmaximal, complemented
5C2×C7×#384C211yesmaximal, characteristic, complemented
6C2×C7×#380C214yesmaximal, complemented
7C23×C7×D4C2214yes—
8C23×C7×D4C2212yes—
9C24×C4×C7C2214yes—
10C2×C7×#121—232no—
11C2×C7×#121C22132yescomplemented
12C2×C4×C7×D4—216no—
13C2×C4×C7×D4C22116yescomplemented
14C22×C7×C4⋊C4C2212yes—
15C22×C7×C4⋊C4C2214yes—
16C22×C7×C4⋊C4C2212yes—
17C22×C7×C22⋊C4C2214yes—
18C22×C7×C22⋊C4C2218yes—
19C22×C42×C7C2211yescharacteristic
20C7×#394C22132yescomplemented
21C7×#384C2218yescomplemented
22C7×#380C22132yescomplemented
23C2×C7×#95C2214yes—
24C2×#2537C711yesmaximal, characteristic, direct product factor, Sylow
25C25×C7C2314yes—
26C22×C7×D4—232no—
27C22×C7×D4—28no—
28C22×C7×D4—28no—
29C22×C7×D4—48no—
30C22×C7×D4D4116yescomplemented
31C22×C7×D4C23116yes—
32C22×C7×D4C2318yes—
33C23×C4×C7—216no—
34C23×C4×C7—216no—
35C23×C4×C7—216no—
36C23×C4×C7C23116yes—
37C23×C4×C7C2314yes—
38C23×C4×C7C2314yes—
39C23×C4×C7C2311yescharacteristic
40C23×C4×C7C2312yes—
41C7×#121—2128no—
42C4×C7×D4—264no—
43C2×C7×C4⋊C4—28no—
44C2×C7×C4⋊C4—216no—
45C2×C7×C4⋊C4—44no—
46C2×C7×C4⋊C4C2318yes—
47C2×C7×C4⋊C4C23116yes—
48C2×C7×C4⋊C4C2318yes—
49C2×C7×C22⋊C4—232no—
50C2×C7×C22⋊C4—216no—
51C2×C7×C22⋊C4C23116yes—
52C2×C7×C22⋊C4C23132yes—
53C2×C42×C7D418yescomplemented
54C2×C42×C7C2314yes—
55C7×#95C23116yes—
56C22×#121C2×C714yescomplemented
57C22×C4×D4C2×C712yescomplemented
58#2537C2×C7116yesdirect product factor
59C2×#394C2×C714yescomplemented
60C2×#384C2×C711yescharacteristic, complemented
61C2×#380C2×C714yescomplemented
62C24×C7—28no—
63C24×C7—28no—
64C24×C7—28no—
65C24×C7—28no—
66C24×C7—28no—
67C24×C7—28no—
68C24×C7C2×D4116yes—
69C24×C7C2418yes—
70C24×C7C2411yescharacteristic
71C2×C7×D4—264no—
72C2×C7×D4—264no—
73C2×C7×D4—216no—
74C2×C7×D4—216no—
75C2×C7×D4—416no—
76C2×C7×D4—464no—
77C2×C7×D4C2×D4132yescomplemented
78C22×C4×C7—22no—
79C22×C4×C7—22no—
80C22×C4×C7—28no—
81C22×C4×C7—28no—
82C22×C4×C7—216no—
83C22×C4×C7—28no—
84C22×C4×C7—28no—
85C22×C4×C7—232no—
86C22×C4×C7—232no—
87C22×C4×C7—232no—
88C22×C4×C7—464no—
89C22×C4×C7C2×D414yes—
90C22×C4×C7C2×D418yes—
91C22×C4×C7C2×D418yes—
92C22×C4×C7#4118yes—
93C22×C4×C7C2418yes—
94C22×C4×C7C2418yes—
95C22×C4×C7C2412yes—
96C22×C4×C7C2414yes—
97C7×C4⋊C4—216no—
98C7×C4⋊C4—232no—
99C7×C4⋊C4—48no—
100C7×C22⋊C4—264no—
101C7×C22⋊C4—232no—
102C42×C7C2×D4116yescomplemented
103C23×D4C22×C714yes—
104C23×D4C22×C712yes—
105C24×C4C22×C714yes—
106C2×#121—232no—
107C2×#121C22×C7132yescomplemented
108C2×C4×D4—216no—
109C2×C4×D4C22×C7116yescomplemented
110C22×C4⋊C4C22×C712yes—
111C22×C4⋊C4C22×C714yes—
112C22×C4⋊C4C22×C712yes—
113C22×C22⋊C4C22×C714yes—
114C22×C22⋊C4C22×C718yes—
115C22×C42C22×C711yescharacteristic
116#394C22×C7132yescomplemented
117#384C22×C718yescomplemented
118#380C22×C7132yescomplemented
119C2×#95C22×C714yes—
120C23×C7—232no—
121C23×C7—232no—
122C23×C7—232no—
123C23×C7—28no—
124C23×C7—28no—
125C23×C7—28no—
126C23×C7—28no—
127C23×C7—28no—
128C23×C7—28no—
129C23×C7—416no—
130C23×C7—416no—
131C23×C7—416no—
132C23×C7—416no—
133C23×C7C22×D4116yes—
134C23×C7C22×D412yes—
135C23×C7C22×D414yes—
136C23×C7C2×#4114yes—
137C23×C7H(5,𝔽2)12yes—
138C23×C72−1+412yes—
139C23×C7C2511yescharacteristic
140C7×D4—264no—
141C7×D4—464no—
142C2×C4×C7—28no—
143C2×C4×C7—22no—
144C2×C4×C7—22no—
145C2×C4×C7—28no—
146C2×C4×C7—28no—
147C2×C4×C7—216no—
148C2×C4×C7—28no—
149C2×C4×C7—28no—
150C2×C4×C7—416no—
151C2×C4×C7—416no—
152C2×C4×C7—464no—
153C2×C4×C7#121132yescomplemented
154C2×C4×C7C22×D414yes—
155C2×C4×C7C22×D418yes—
156C2×C4×C7C22×D418yes—
157C2×C4×C7C2×#4118yes—
158C25C23×C714yes—
159C22×D4—232no—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—48no—
163C22×D4C7×D4116yescomplemented
164C22×D4C23×C7116yes—
165C22×D4C23×C718yes—
166C23×C4—216no—
167C23×C4—216no—
168C23×C4—216no—
169C23×C4C23×C714yes—
170C23×C4C23×C714yes—
171C23×C4C23×C7116yes—
172C23×C4C23×C711yescharacteristic
173C23×C4C23×C712yes—
174#121—2128no—
175C4×D4—264no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—216no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4C23×C718yes—
180C2×C4⋊C4C23×C7116yes—
181C2×C4⋊C4C23×C718yes—
182C2×C22⋊C4—232no—
183C2×C22⋊C4—216no—
184C2×C22⋊C4C23×C7116yes—
185C2×C22⋊C4C23×C7132yes—
186C2×C42C7×D418yescomplemented
187C2×C42C23×C714yes—
188#95C23×C7116yes—
189C22×C7—216no—
190C22×C7—216no—
191C22×C7—216no—
192C22×C7—48no—
193C22×C7—48no—
194C22×C7—48no—
195C22×C7—48no—
196C22×C7—464no—
197C22×C7C2×#12118yes—
198C22×C7#53714yes—
199C22×C7D4214yes—
200C22×C7#54718yes—
201C22×C7#55514yes—
202C22×C7C23×D411yescharacteristic
203C22×C7C23×D412yes—
204C22×C7C22×#4112yes—
205C22×C7C2×H(5,𝔽2)11yescharacteristic
206C22×C7C2×2−1+411yescharacteristic
207C4×C7—24no—
208C4×C7—48no—
209C4×C7—48no—
210C4×C7C2×#121116yescomplemented
211C24—28no—
212C24—28no—
213C24—28no—
214C24—28no—
215C24—28no—
216C24—28no—
217C24C2×C7×D4116yes—
218C24C24×C718yes—
219C24C24×C711yescharacteristic
220C2×D4—264no—
221C2×D4—264no—
222C2×D4—216no—
223C2×D4—216no—
224C2×D4—416no—
225C2×D4—464no—
226C2×D4C2×C7×D4132yescomplemented
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—28no—
230C22×C4—28no—
231C22×C4—216no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—232no—
235C22×C4—232no—
236C22×C4—232no—
237C22×C4—464no—
238C22×C4C2×C7×D414yes—
239C22×C4C2×C7×D418yes—
240C22×C4C2×C7×D418yes—
241C22×C4C7×#4118yes—
242C22×C4C24×C718yes—
243C22×C4C24×C718yes—
244C22×C4C24×C712yes—
245C22×C4C24×C714yes—
246C4⋊C4—216no—
247C4⋊C4—232no—
248C4⋊C4—48no—
249C22⋊C4—264no—
250C22⋊C4—232no—
251C42C2×C7×D4116yescomplemented
252C2×C7—416no—
253C2×C7#253718yesdirect product factor
254C2×C7C22×#12112yes—
255C2×C7C2×#53711yescharacteristic
256C2×C7C2×D4211yescharacteristic
257C2×C7C2×#54712yes—
258C2×C7C2×#55511yescharacteristic
259C23—28no—
260C23—28no—
261C23—28no—
262C23—28no—
263C23—28no—
264C23—28no—
265C23—232no—
266C23—232no—
267C23—232no—
268C23—416no—
269C23—416no—
270C23—416no—
271C23—416no—
272C23C22×C7×D4116yes—
273C23C22×C7×D412yes—
274C23C22×C7×D414yes—
275C23C2×C7×#4114yes—
276C23C7×H(5,𝔽2)12yes—
277C23C7×2−1+412yes—
278C23C25×C711yescharacteristic
279D4—264no—
280D4—464no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—28no—
284C2×C4—28no—
285C2×C4—28no—
286C2×C4—216no—
287C2×C4—28no—
288C2×C4—28no—
289C2×C4—416no—
290C2×C4—416no—
291C2×C4—464no—
292C2×C4C7×#121132yescomplemented
293C2×C4C22×C7×D414yes—
294C2×C4C22×C7×D418yes—
295C2×C4C22×C7×D418yes—
296C2×C4C2×C7×#4118yes—
297C7C2×#253711yescharacteristic, direct product factor, Sylow
298C22—216no—
299C22—216no—
300C22—216no—
301C22—48no—
302C22—48no—
303C22—464no—
304C22—48no—
305C22—48no—
306C22C2×C7×#12118yes—
307C22C7×#53714yes—
308C22C7×D4214yes—
309C22C7×#54718yes—
310C22C7×#55514yes—
311C22C23×C7×D411yescharacteristic
312C22C23×C7×D412yes—
313C22C22×C7×#4112yes—
314C22C2×C7×H(5,𝔽2)11yescharacteristic
315C22C2×C7×2−1+411yescharacteristic
316C4—24no—
317C4—48no—
318C4—48no—
319C4C2×C7×#121116yescomplemented
320C2—416no—
321C2C7×#253718yesdirect product factor
322C2C22×C7×#12112yes—
323C2C2×C7×#53711yescharacteristic
324C2C2×C7×D4211yescharacteristic
325C2C2×C7×#54712yes—
326C2C2×C7×#55511yescharacteristic
subgroupG/H
center, socleC24×C7C24
commutator, Frattini, cosocleC23C25×C7
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C7×#121C2semidirect
2. C22×C4×C7×D4C2semidirect
3. C7×#2537C2direct (N × H)
4. C2×C7×#394C2semidirect
5. C2×C7×#384C2semidirect
6. C2×C7×#380C2semidirect
11. C2×C7×#121C22semidirect
13. C2×C4×C7×D4C22semidirect
20. C7×#394C22semidirect
21. C7×#384C22semidirect
22. C7×#380C22semidirect
24. C2×#2537C7direct (N × H)
30. C22×C7×D4D4semidirect
53. C2×C42×C7D4semidirect
56. C22×#121C2×C7semidirect
57. C22×C4×D4C2×C7semidirect
58. #2537C2×C7direct (N × H)
59. C2×#394C2×C7semidirect
60. C2×#384C2×C7semidirect
61. C2×#380C2×C7semidirect
77. C2×C7×D4C2×D4semidirect
102. C42×C7C2×D4semidirect
107. C2×#121C22×C7semidirect
109. C2×C4×D4C22×C7semidirect
116. #394C22×C7semidirect
117. #384C22×C7semidirect
118. #380C22×C7semidirect
153. C2×C4×C7#121semidirect
163. C22×D4C7×D4semidirect
186. C2×C42C7×D4semidirect
210. C4×C7C2×#121semidirect
226. C2×D4C2×C7×D4semidirect
251. C42C2×C7×D4semidirect
253. C2×C7#2537direct (N × H)
292. C2×C4C7×#121semidirect
297. C7C2×#2537direct (N × H)
319. C4C2×C7×#121semidirect
321. C2C7×#2537direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×#1212
C22×C4×C7×D42
C7×#25372
C2×C7×#3942
C2×C7×#3842
C2×C7×#3802
C2×#25377