Finite Group Databasebeta

← C7×#60065 #13237281

Subgroups — C7×#60065 #13237281

Order 1792.

4874 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#585C211yesmaximal, characteristic
2C2×C7×#568C211yesmaximal, characteristic, complemented
3C22×C7×#131C211yesmaximal, characteristic, complemented
4C7×#2880C2116yesmaximal, complemented
5C7×#2879C216yesmaximal, complemented
6C2×C7×#408C216yesmaximal, complemented
7C7×#585—28no—
8C7×#585—28no—
9C2×C7×2−1+4C414yes—
10C2×C7×2−1+4C412yes—
11C2×C7×H(5,𝔽2)C414yes—
12C2×C7×H(5,𝔽2)C416yes—
13C22×C7×#41C2216yes—
14C22×C7×#41C2218yes—
15C22×C7×#41C2211yescharacteristic
16C7×#568—28no—
17C2×C7×#131—28no—
18C2×C7×#131C2218yescomplemented
19C2×C7×#131C2218yescomplemented
20C22×C7×M16C2213yescomplemented
21C22×C7×M16C2216yescomplemented
22C23×C7×C8C2213yescomplemented
23C7×#408C2214yescomplemented
24C7×#408C22112yescomplemented
25C7×#408C22124yescomplemented
26C7×#408C22124yescomplemented
27C7×#407C22124yescomplemented
28C2×C7×C22⋊C8C22124yescomplemented
29#60065C711yesmaximal, characteristic, direct product factor, Sylow
30C7×2−1+4—232no—
31C7×2−1+4—216no—
32C7×H(5,𝔽2)—232no—
33C7×H(5,𝔽2)—248no—
34C2×C7×#41—232no—
35C2×C7×#41—232no—
36C2×C7×#41—224no—
37C2×C7×#41—224no—
38C2×C7×#41C2×C418yes—
39C2×C7×#41C2×C4124yes—
40C2×C7×#41C2×C4112yes—
41C2×C7×#41C2×C4112yes—
42C2×C7×#41C2×C414yes—
43C2×C7×#41D418yes—
44C2×C7×#41D418yes—
45C2×C7×#41C2314yes—
46C2×C7×#41C23112yes—
47C2×C7×#41C2314yes—
48C22×C7×Q8C2×C418yes—
49C22×C7×Q8C2×C416yes—
50C22×C7×Q8C2×C411yescharacteristic
51C22×C7×D4C2×C4112yes—
52C22×C7×D4C2×C4124yes—
53C22×C7×D4C2×C416yes—
54C22×C7×D4C2×C413yes—
55C23×C4×C7C23112yes—
56C23×C4×C7C2313yes—
57C7×#131—232no—
58C7×#131—232no—
59C2×C7×M16—212no—
60C2×C7×M16—224no—
61C2×C7×M16C23112yes—
62C2×C7×M16C23112yes—
63C2×C7×M16C2314yes—
64C22×C7×C8—212no—
65C22×C7×C8C23112yes—
66C22×C7×C8C23112yes—
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2313yes—
69C7×C22⋊C8C23116yes—
70C7×C22⋊C8C23148yes—
71C2×#585C2×C711yescharacteristic
72C2×#568C2×C711yescharacteristic, complemented
73C22×#131C2×C711yescharacteristic, complemented
74#2880C2×C7116yescomplemented
75#2879C2×C716yescomplemented
76C2×#408C2×C716yescomplemented
77C24×C7C22×C4112yes—
78C24×C7C22×C413yes—
79C7×#41—232no—
80C7×#41—296no—
81C7×#41—248no—
82C7×#41—248no—
83C7×#41—28no—
84C7×#41—224no—
85C7×#41—224no—
86C7×#41—28no—
87C7×#41C22⋊C4132yes—
88C7×#41C2×D4116yes—
89C7×#41C2×D4116yes—
90C2×C7×Q8—232no—
91C2×C7×Q8—224no—
92C2×C7×Q8C22⋊C418yes—
93C2×C7×Q8C22×C4112yes—
94C2×C7×Q8C22×C414yes—
95C2×C7×Q8C22×C414yes—
96C2×C7×D4—248no—
97C2×C7×D4—296no—
98C2×C7×D4—224no—
99C2×C7×D4C22⋊C4124yes—
100C2×C7×D4C22×C4124yes—
101C2×C7×D4C22×C418yes—
102C2×C7×D4C22×C4112yes—
103C2×C7×D4C22×C414yes—
104C2×C7×D4C22×C4112yes—
105C22×C4×C7—224no—
106C22×C4×C7—224no—
107C22×C4×C7C22×C4112yes—
108C22×C4×C7C22×C4124yes—
109C22×C4×C7C22×C416yes—
110C22×C4×C7C22×C413yes—
111C22×C4×C7C2×D4112yes—
112C22×C4×C7C2×D4112yes—
113C22×C4×C7C2418yes—
114C22×C4×C7C2411yescharacteristic
115C22×C4×C7C2416yes—
116C7×M16—28no—
117C7×M16—224no—
118C7×M16—224no—
119C2×C7×C8—22no—
120C2×C7×C8—26no—
121C2×C7×C8—224no—
122C2×C7×C8—224no—
123C2×C7×C8C2412yes—
124C2×C7×C8C2418yes—
125C2×C7×C8C2416yes—
126#585—28no—
127#585—28no—
128C2×2−1+4C4×C714yes—
129C2×2−1+4C4×C712yes—
130C2×H(5,𝔽2)C4×C714yes—
131C2×H(5,𝔽2)C4×C716yes—
132C22×#41C22×C716yes—
133C22×#41C22×C718yes—
134C22×#41C22×C711yescharacteristic
135#568—28no—
136C2×#131—28no—
137C2×#131C22×C718yescomplemented
138C2×#131C22×C718yescomplemented
139C22×M16C22×C716yescomplemented
140C22×M16C22×C713yescomplemented
141C23×C8C22×C713yescomplemented
142#408C22×C714yescomplemented
143#408C22×C7112yescomplemented
144#408C22×C7124yescomplemented
145#408C22×C7124yescomplemented
146#407C22×C7124yescomplemented
147C2×C22⋊C8C22×C7124yescomplemented
148C23×C7—224no—
149C23×C7—46no—
150C23×C7—424no—
151C23×C7C2×C22⋊C4112yes—
152C23×C7C23×C418yes—
153C23×C7C23×C416yes—
154C23×C7C23×C411yescharacteristic
155C7×Q8—224no—
156C7×Q8—28no—
157C7×Q8C2×C22⋊C4116yes—
158C7×D4—248no—
159C7×D4—216no—
160C7×D4—224no—
161C7×D4—28no—
162C7×D4C2×C22⋊C4148yes—
163C2×C4×C7—224no—
164C2×C4×C7—248no—
165C2×C4×C7—28no—
166C2×C4×C7—28no—
167C2×C4×C7C2×C22⋊C4124yes—
168C2×C4×C7C2×C22⋊C4112yes—
169C2×C4×C7C23×C418yes—
170C2×C4×C7C23×C416yes—
171C2×C4×C7C23×C411yescharacteristic
172C2×C4×C7C22×D412yes—
173C2×C4×C7C22×D4112yes—
174C2×C4×C7C22×D4112yes—
175C2×C4×C7C22×D412yes—
176C2×C4×C7C2511yescharacteristic
177C7×C8—22no—
178C7×C8—26no—
179C7×C8—28no—
1802−1+4—232no—
1812−1+4—216no—
182H(5,𝔽2)—232no—
183H(5,𝔽2)—248no—
184C2×#41—232no—
185C2×#41—232no—
186C2×#41—224no—
187C2×#41—224no—
188C2×#41C2×C4×C718yes—
189C2×#41C2×C4×C7124yes—
190C2×#41C2×C4×C7112yes—
191C2×#41C2×C4×C7112yes—
192C2×#41C2×C4×C714yes—
193C2×#41C7×D418yes—
194C2×#41C7×D418yes—
195C2×#41C23×C714yes—
196C2×#41C23×C7112yes—
197C2×#41C23×C714yes—
198C22×Q8C2×C4×C718yes—
199C22×Q8C2×C4×C716yes—
200C22×Q8C2×C4×C711yescharacteristic
201C22×D4C2×C4×C7112yes—
202C22×D4C2×C4×C7124yes—
203C22×D4C2×C4×C716yes—
204C22×D4C2×C4×C713yes—
205C23×C4C23×C7112yes—
206C23×C4C23×C713yes—
207#131—232no—
208#131—232no—
209C2×M16—212no—
210C2×M16—224no—
211C2×M16C23×C714yes—
212C2×M16C23×C7112yes—
213C2×M16C23×C7112yes—
214C22×C8—212no—
215C22×C8C23×C711yescharacteristic
216C22×C8C23×C713yes—
217C22×C8C23×C7112yes—
218C22×C8C23×C7112yes—
219C22⋊C8C23×C7116yes—
220C22⋊C8C23×C7148yes—
221C22×C7—22no—
222C22×C7—212no—
223C22×C7—28no—
224C22×C7—216no—
225C22×C7—412no—
226C22×C7—824no—
227C22×C7C22×C22⋊C4112yes—
228C22×C7C22×C22⋊C412yes—
229C22×C7C24×C411yescharacteristic
230C4×C7—28no—
231C4×C7C22×C22⋊C4112yes—
232C4×C7C22×C22⋊C412yes—
233C4×C7C23×D411yescharacteristic
234C4×C7C23×D411yescharacteristic
235C24C22×C4×C7112yes—
236C24C22×C4×C713yes—
237#41—232no—
238#41—296no—
239#41—248no—
240#41—248no—
241#41—28no—
242#41—224no—
243#41—224no—
244#41—28no—
245#41C7×C22⋊C4132yes—
246#41C2×C7×D4116yes—
247#41C2×C7×D4116yes—
248C2×Q8—232no—
249C2×Q8—224no—
250C2×Q8C7×C22⋊C418yes—
251C2×Q8C22×C4×C7112yes—
252C2×Q8C22×C4×C714yes—
253C2×Q8C22×C4×C714yes—
254C2×D4—248no—
255C2×D4—296no—
256C2×D4—224no—
257C2×D4C7×C22⋊C4124yes—
258C2×D4C22×C4×C7124yes—
259C2×D4C22×C4×C718yes—
260C2×D4C22×C4×C7112yes—
261C2×D4C22×C4×C714yes—
262C2×D4C22×C4×C7112yes—
263C22×C4—224no—
264C22×C4—224no—
265C22×C4C22×C4×C7112yes—
266C22×C4C22×C4×C7124yes—
267C22×C4C22×C4×C716yes—
268C22×C4C22×C4×C713yes—
269C22×C4C2×C7×D4112yes—
270C22×C4C2×C7×D4112yes—
271C22×C4C24×C718yes—
272C22×C4C24×C711yescharacteristic
273C22×C4C24×C716yes—
274M16—28no—
275M16—224no—
276M16—224no—
277C2×C8—22no—
278C2×C8—26no—
279C2×C8—224no—
280C2×C8—224no—
281C2×C8C24×C712yes—
282C2×C8C24×C718yes—
283C2×C8C24×C716yes—
284C2×C7—22no—
285C2×C7—212no—
286C2×C7—48no—
287C2×C7C23×C22⋊C411yescharacteristic
288C2×C7#357312yes—
289C23—224no—
290C23—46no—
291C23—424no—
292C23C2×C7×C22⋊C4112yes—
293C23C23×C4×C718yes—
294C23C23×C4×C716yes—
295C23C23×C4×C711yescharacteristic
296Q8—224no—
297Q8—28no—
298Q8C2×C7×C22⋊C4116yes—
299D4—248no—
300D4—216no—
301D4—224no—
302D4—28no—
303D4C2×C7×C22⋊C4148yes—
304C2×C4—224no—
305C2×C4—28no—
306C2×C4—28no—
307C2×C4—248no—
308C2×C4C2×C7×C22⋊C4124yes—
309C2×C4C2×C7×C22⋊C4112yes—
310C2×C4C23×C4×C718yes—
311C2×C4C23×C4×C716yes—
312C2×C4C23×C4×C711yescharacteristic
313C2×C4C22×C7×D412yes—
314C2×C4C22×C7×D4112yes—
315C2×C4C22×C7×D4112yes—
316C2×C4C22×C7×D412yes—
317C2×C4C25×C711yescharacteristic
318C8—22no—
319C8—26no—
320C8—28no—
321C7#6006511yescharacteristic, direct product factor, Sylow
322C22—22no—
323C22—212no—
324C22—216no—
325C22—28no—
326C22—412no—
327C22—824no—
328C22C22×C7×C22⋊C4112yes—
329C22C22×C7×C22⋊C412yes—
330C22C24×C4×C711yescharacteristic
331C4—28no—
332C4C22×C7×C22⋊C4112yes—
333C4C22×C7×C22⋊C412yes—
334C4C23×C7×D411yescharacteristic
335C4C23×C7×D411yescharacteristic
336C2—22no—
337C2—212no—
338C2—48no—
339C2C23×C7×C22⋊C411yescharacteristic
340C2C7×#357312yes—
subgroupG/H
centerC2×C4×C7C25
commutatorC22C24×C4×C7
Frattini, cosocleC2×C4C25×C7
socleC22×C7C24×C4
prime radicalC2×C7×#585C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#568C2semidirect
3. C22×C7×#131C2semidirect
4. C7×#2880C2semidirect
5. C7×#2879C2semidirect
6. C2×C7×#408C2semidirect
18. C2×C7×#131C22semidirect
19. C2×C7×#131C22semidirect
20. C22×C7×M16C22semidirect
21. C22×C7×M16C22semidirect
22. C23×C7×C8C22semidirect
23. C7×#408C22semidirect
24. C7×#408C22semidirect
25. C7×#408C22semidirect
26. C7×#408C22semidirect
27. C7×#407C22semidirect
28. C2×C7×C22⋊C8C22semidirect
29. #60065C7direct (N × H)
72. C2×#568C2×C7semidirect
73. C22×#131C2×C7semidirect
74. #2880C2×C7semidirect
75. #2879C2×C7semidirect
76. C2×#408C2×C7semidirect
137. C2×#131C22×C7semidirect
138. C2×#131C22×C7semidirect
139. C22×M16C22×C7semidirect
140. C22×M16C22×C7semidirect
141. C23×C8C22×C7semidirect
142. #408C22×C7semidirect
143. #408C22×C7semidirect
144. #408C22×C7semidirect
145. #408C22×C7semidirect
146. #407C22×C7semidirect
147. C2×C22⋊C8C22×C7semidirect
321. C7#60065direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5852
C2×C7×#5682
C22×C7×#1312
C7×#28802
C7×#28792
C2×C7×#4082
#600657