Finite Group Databasebeta

← C7×#60333 #13237549

Subgroups — C7×#60333 #13237549

Order 1792.

1926 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C7×2−1+4C211yesmaximal, characteristic
2C7×#2991C214yesmaximal, complemented
3C7×#2989C216yesmaximal, complemented
4C7×#2984C214yesmaximal, complemented
5C7×#2982C211yesmaximal, characteristic, complemented
6C7×#2971C214yesmaximal, complemented
7C7×#2965C216yesmaximal, complemented
8C7×Q8×M16C214yesmaximal, complemented
9C2×C7×C8×Q8C211yesmaximal, characteristic, complemented
10C2×C7×2−1+4C411yescharacteristic
11C7×#520C414yes—
12C7×#520C416yes—
13C7×#519C414yes—
14C7×#519C411yescharacteristic
15C4×C7×#41C2216yes—
16C4×C7×#41C2214yes—
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2214yes—
19C7×#446C2216yes—
20C7×#446C22112yes—
21C7×#446C2216yes—
22C7×C8×Q8C2214yes—
23C7×C8×Q8C2214yes—
24C7×C8×Q8C2214yes—
25C7×C8×Q8C2214yes—
26C7×#436C2216yes—
27C7×#436C2216yes—
28C7×#436C22112yes—
29C7×#435C2214yes—
30C7×#435C22112yes—
31C7×#432C22112yes—
32C7×#431C2216yes—
33C7×#431C22112yes—
34C7×#423C22112yes—
35C2×C7×C4⋊C8C2213yes—
36C4×C7×M16C22112yes—
37C2×C4×C7×C8C2213yes—
38#60333C711yesmaximal, characteristic, direct product factor, Sylow
39C7×2−1+4—28no—
40C2×C7×#41C2×C414yes—
41C2×C7×#41C2×C416yes—
42C22×C7×Q8C2×C414yes—
43C22×C7×Q8C2×C411yescharacteristic
44C2×C7×M16C2314yes—
45C2×C7×M16C23112yes—
46C22×C7×C8C2311yescharacteristic
47C22×C7×C8C2313yes—
48C4×C7×Q8C2×C4112yes—
49C4×C7×Q8C2×C414yes—
50C4×C7×Q8C2×C414yes—
51C4×C7×Q8C2×C416yes—
52C4×C7×Q8C2×C414yes—
53C4×C7×Q8C2314yes—
54C4×C7×Q8C2316yes—
55C4×C7×D4C2×C4112yes—
56C4×C7×D4C2×C416yes—
57C4×C7×D4C2×C4112yes—
58C4×C7×D4C2314yes—
59C4×C7×D4C2316yes—
60C7×#117C2×C4112yes—
61C7×#117C2×C4112yes—
62C7×#117C2×C416yes—
63C2×C7×C4⋊C4C2×C4112yes—
64C2×C7×C4⋊C4C2×C413yes—
65C2×C42×C7C2313yes—
66C2×C42×C7C23112yes—
67C7×C4⋊C8C23112yes—
68C7×C4⋊C8C23112yes—
69C7×C4⋊C8C2316yes—
70C7×C4⋊C8C2314yes—
71C7×C4⋊C8C2316yes—
72C7×C22⋊C8C2314yes—
73C7×C22⋊C8C2314yes—
74C7×C22⋊C8C23112yes—
75C7×#97C2314yes—
76C7×#97C23112yes—
77C4×C7×C8C2316yes—
78C4×C7×C8C2316yes—
79C4×C7×C8C23112yes—
80C4×2−1+4C2×C711yescharacteristic
81#2991C2×C714yescomplemented
82#2989C2×C716yescomplemented
83#2984C2×C714yescomplemented
84#2982C2×C711yescharacteristic, complemented
85#2971C2×C714yescomplemented
86#2965C2×C716yescomplemented
87Q8×M16C2×C714yescomplemented
88C2×C8×Q8C2×C711yescharacteristic, complemented
89C7×#41—216no—
90C7×#41—224no—
91C2×C7×Q8—216no—
92C2×C7×Q8M1618yescomplemented
93C2×C7×Q8C22×C416yes—
94C2×C7×Q8C22×C414yes—
95C2×C7×D4C22×C414yes—
96C2×C7×D4C22×C416yes—
97C22×C4×C7C22×C4112yes—
98C22×C4×C7C22×C4112yes—
99C22×C4×C7C22×C413yes—
100C22×C4×C7C22×C413yes—
101C22×C4×C7C2411yescharacteristic
102C22×C4×C7C2414yes—
103C7×M16—28no—
104C7×M16—412no—
105C2×C7×C8—22no—
106C2×C7×C8—43no—
107C2×C7×C8C2411yescharacteristic
108C2×C7×C8C2414yes—
109C2×C7×C8C2414yes—
110C2×C7×C8C2416yes—
111C2×C7×C8C2411yescharacteristic
112C7×C4⋊C4C22×C4112yes—
113C7×C4⋊C4C22×C4112yes—
114C7×C4⋊C4C22×C416yes—
115C7×C4⋊C4C22×C414yes—
116C7×C4⋊C4C22×C416yes—
117C7×C22⋊C4C22×C414yes—
118C7×C22⋊C4C22×C4112yes—
119C7×C22⋊C4C22×C414yes—
120C42×C7C22×C4112yes—
121C42×C7C22×C416yes—
122C42×C7C22×C4112yes—
123C42×C7C2416yes—
124C42×C7C2414yes—
125C2×2−1+4C4×C711yescharacteristic
126#520C4×C714yes—
127#520C4×C716yes—
128#519C4×C714yes—
129#519C4×C711yescharacteristic
130C4×#41C22×C716yes—
131C4×#41C22×C714yes—
132C2×C4×Q8C22×C711yescharacteristic
133C2×C4×Q8C22×C714yes—
134#446C22×C716yes—
135#446C22×C7112yes—
136#446C22×C716yes—
137C8×Q8C22×C714yes—
138C8×Q8C22×C714yes—
139C8×Q8C22×C714yes—
140C8×Q8C22×C714yes—
141#436C22×C716yes—
142#436C22×C716yes—
143#436C22×C7112yes—
144#435C22×C714yes—
145#435C22×C7112yes—
146#432C22×C7112yes—
147#431C22×C716yes—
148#431C22×C7112yes—
149#423C22×C7112yes—
150C2×C4⋊C8C22×C713yes—
151C4×M16C22×C7112yes—
152C2×C4×C8C22×C713yes—
153C23×C7C23×C414yes—
154C23×C7C23×C411yescharacteristic
155C7×Q8—212no—
156C7×Q8C2×M16116yes—
157C7×D4—28no—
158C7×D4—212no—
159C2×C4×C7—224no—
160C2×C4×C7—21no—
161C2×C4×C7—24no—
162C2×C4×C7—21no—
163C2×C4×C7—24no—
164C2×C4×C7—43no—
165C2×C4×C7—412no—
166C2×C4×C7C2×M16112yes—
167C2×C4×C7C23×C411yescharacteristic
168C2×C4×C7C23×C414yes—
169C2×C4×C7C23×C416yes—
170C2×C4×C7C23×C414yes—
171C2×C4×C7C23×C414yes—
172C2×C4×C7C23×C416yes—
173C2×C4×C7C2511yescharacteristic
174C7×C8—21no—
175C7×C8—24no—
176C7×C8—26no—
177C7×C8—24no—
178C7×C82−1+412yescomplemented
1792−1+4—28no—
180C2×#41C2×C4×C714yes—
181C2×#41C2×C4×C716yes—
182C22×Q8C2×C4×C714yes—
183C22×Q8C2×C4×C711yescharacteristic
184C2×M16C23×C714yes—
185C2×M16C23×C7112yes—
186C22×C8C23×C711yescharacteristic
187C22×C8C23×C713yes—
188C4×Q8C2×C4×C7112yes—
189C4×Q8C2×C4×C714yes—
190C4×Q8C2×C4×C714yes—
191C4×Q8C2×C4×C716yes—
192C4×Q8C2×C4×C714yes—
193C4×Q8C23×C714yes—
194C4×Q8C23×C716yes—
195C4×D4C2×C4×C7112yes—
196C4×D4C2×C4×C716yes—
197C4×D4C2×C4×C7112yes—
198C4×D4C23×C714yes—
199C4×D4C23×C716yes—
200#117C2×C4×C7112yes—
201#117C2×C4×C7112yes—
202#117C2×C4×C716yes—
203C2×C4⋊C4C2×C4×C7112yes—
204C2×C4⋊C4C2×C4×C713yes—
205C2×C42C23×C713yes—
206C2×C42C23×C7112yes—
207C4⋊C8C23×C7112yes—
208C4⋊C8C23×C7112yes—
209C4⋊C8C23×C716yes—
210C4⋊C8C23×C714yes—
211C4⋊C8C23×C716yes—
212C22⋊C8C23×C714yes—
213C22⋊C8C23×C714yes—
214C22⋊C8C23×C7112yes—
215#97C23×C714yes—
216#97C23×C7112yes—
217C4×C8C23×C716yes—
218C4×C8C23×C716yes—
219C4×C8C23×C7112yes—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7—24no—
223C22×C7—24no—
224C22×C7—24no—
225C22×C7C22×M1612yes—
226C22×C7C24×C411yescharacteristic
227C4×C7—21no—
228C4×C7—24no—
229C4×C7—24no—
230C4×C7—26no—
231C4×C7—24no—
232C4×C7C22×M16112yes—
233C4×C7C2×2−1+411yescharacteristic
234C4×C7#58511yescharacteristic
235#41—216no—
236#41—224no—
237C2×Q8—216no—
238C2×Q8C7×M1618yescomplemented
239C2×Q8C22×C4×C716yes—
240C2×Q8C22×C4×C714yes—
241C2×D4C22×C4×C714yes—
242C2×D4C22×C4×C716yes—
243C22×C4C22×C4×C7112yes—
244C22×C4C22×C4×C7112yes—
245C22×C4C22×C4×C713yes—
246C22×C4C22×C4×C713yes—
247C22×C4C24×C711yescharacteristic
248C22×C4C24×C714yes—
249M16—28no—
250M16—412no—
251C2×C8—22no—
252C2×C8—43no—
253C2×C8C24×C711yescharacteristic
254C2×C8C24×C714yes—
255C2×C8C24×C714yes—
256C2×C8C24×C716yes—
257C2×C8C24×C711yescharacteristic
258C4⋊C4C22×C4×C7112yes—
259C4⋊C4C22×C4×C7112yes—
260C4⋊C4C22×C4×C716yes—
261C4⋊C4C22×C4×C714yes—
262C4⋊C4C22×C4×C716yes—
263C22⋊C4C22×C4×C714yes—
264C22⋊C4C22×C4×C7112yes—
265C22⋊C4C22×C4×C714yes—
266C42C22×C4×C7112yes—
267C42C22×C4×C716yes—
268C42C22×C4×C7112yes—
269C42C24×C716yes—
270C42C24×C714yes—
271C2×C7—22no—
272C2×C7—44no—
273C2×C7C4×2−1+411yescharacteristic
274C2×C7C23×M1611yescharacteristic
275C2×C7#357311yescharacteristic
276C23C23×C4×C714yes—
277C23C23×C4×C711yescharacteristic
278Q8—212no—
279Q8C2×C7×M16116yes—
280D4—28no—
281D4—212no—
282C2×C4—21no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—224no—
287C2×C4—43no—
288C2×C4—412no—
289C2×C4C2×C7×M16112yes—
290C2×C4C23×C4×C711yescharacteristic
291C2×C4C23×C4×C714yes—
292C2×C4C23×C4×C716yes—
293C2×C4C23×C4×C714yes—
294C2×C4C23×C4×C714yes—
295C2×C4C23×C4×C716yes—
296C2×C4C25×C711yescharacteristic
297C8—21no—
298C8—24no—
299C8—26no—
300C8—24no—
301C8C7×2−1+412yescomplemented
302C7#6033311yescharacteristic, direct product factor, Sylow
303C22—21no—
304C22—21no—
305C22—24no—
306C22—24no—
307C22—24no—
308C22C22×C7×M1612yes—
309C22C24×C4×C711yescharacteristic
310C4—24no—
311C4—26no—
312C4—24no—
313C4—21no—
314C4—24no—
315C4C22×C7×M16112yes—
316C4C2×C7×2−1+411yescharacteristic
317C4C7×#58511yescharacteristic
318C2—22no—
319C2—44no—
320C2C4×C7×2−1+411yescharacteristic
321C2C23×C7×M1611yescharacteristic
322C2C7×#357311yescharacteristic
subgroupG/H
centerC2×C4×C7C25
commutatorC22C24×C4×C7
Frattini, cosocleC2×C4C25×C7
socleC22×C7C24×C4
prime radicalC2×C7×2−1+4C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2991C2semidirect
3. C7×#2989C2semidirect
4. C7×#2984C2semidirect
5. C7×#2982C2semidirect
6. C7×#2971C2semidirect
7. C7×#2965C2semidirect
8. C7×Q8×M16C2semidirect
9. C2×C7×C8×Q8C2semidirect
38. #60333C7direct (N × H)
81. #2991C2×C7semidirect
82. #2989C2×C7semidirect
83. #2984C2×C7semidirect
84. #2982C2×C7semidirect
85. #2971C2×C7semidirect
86. #2965C2×C7semidirect
87. Q8×M16C2×C7semidirect
88. C2×C8×Q8C2×C7semidirect
92. C2×C7×Q8M16semidirect
178. C7×C82−1+4semidirect
238. C2×Q8C7×M16semidirect
301. C8C7×2−1+4semidirect
302. C7#60333direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C7×2−1+42
C7×#29912
C7×#29892
C7×#29842
C7×#29822
C7×#29712
C7×#29652
C7×Q8×M162
C2×C7×C8×Q82
#603337