Finite Group Databasebeta

← C7×#10621056 #72269382

Subgroups — C7×#10621056 #72269382

Order 4704.

434 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#66560C211yesmaximal, characteristic, complemented
2C3×C7×#1141C211yesmaximal, characteristic, complemented
3C7×#66633C211yesmaximal, characteristic, complemented
4C7×#66567C211yesmaximal, characteristic, complemented
5C7×#66669C211yesmaximal, characteristic, complemented
6C7×#66558C211yesmaximal, characteristic, complemented
7D3×C72×D4C211yesmaximal, characteristic, complemented
8C7×#6495—31nomaximal, Hall
9C22×D3×C72—21no—
10C3×C72×D4C2211yescharacteristic
11C7×Dic42C2211yescharacteristic, complemented
12C72×#67—21no—
13C72×D12C2211yescharacteristic
14C4×D3×C72C2211yescharacteristic
15C3×C7×D28C2211yescharacteristic, complemented
16C7×#4245—21no—
17C7×Dic3×D7—21no—
18C7×C3⋊(C7⋊C8)C2211yescharacteristic, complemented
19C3×C7×C7⋊C8C2211yescharacteristic, complemented
20C2×C72×D4—31no—
21C7×#1157—31no—
22C7×#1142—31no—
23C7×#1142—31no—
24C7×#1141—31no—
25C7×#1141D311yescharacteristic, complemented
26C7×#1136—31no—
27C7×#911—71nomaximal
28#10621056C711yesmaximal, characteristic, direct product factor
29C22×C3×C72—21no—
30C2×D3×C72—21no—
31C2×D3×C72—42no—
32C2×D3×C72D411yescharacteristic
33C2×C3×C7×D7—21no—
34C3×C4×C72C2311yescharacteristic
35C7×Dic21—21no—
36C72×Dic3D411yescharacteristic
37C23×C72—61no—
38C72×D4—31no—
39C72×D4—61no—
40C72×D4C2×D311yescharacteristic
41C2×C4×C72—31no—
42C7×C22⋊D7—61no—
43C7×D28C2×D311yescharacteristic, complemented
44C4×C7×D7—61no—
45C7×Dic14—31no—
46C7×C7⋊C8—31no—
47C7×C7⋊C8C2×D311yescharacteristic, complemented
48C7×#237—71no—
49D3×C7×D4—23no—
50D3×C7×D4D711yescharacteristic, complemented
51D3×C7×D4C2×C711yescharacteristic, complemented
52#66669C2×C711yescharacteristic, complemented
53C3×C7×D8—71no—
54#66633C2×C711yescharacteristic, complemented
55C7×#214—71no—
56C7×#213—71no—
57C7×#204—71no—
58C7×ΓL(1,𝔽25)—71no—
59C3×#1141C2×C711yescharacteristic, complemented
60#66567C2×C711yescharacteristic, complemented
61#66560C2×C711yescharacteristic, complemented
62#66558C2×C711yescharacteristic, complemented
63C2×C3×C72—41no—
64C2×C3×C72C2×D411yescharacteristic
65D3×C72—21no—
66D3×C72—41no—
67C3×C7×D7—41no—
68C7×AGL(1,ℤ/8)—211no—
69#6495—31no—
70C22×C72—21no—
71C22×C72—31no—
72C22×C72—61no—
73C22×C72—122no—
74C2×C7×D7—21no—
75C4×C72—31no—
76C4×C72C22×D311yescharacteristic
77C7×Dic7—61no—
78C22×D3×C7—26no—
79C22×D3×C7—21no—
80C22×D3×C7—21no—
81C3×C7×D4—23no—
82C3×C7×D4—71no—
83C3×C7×D4C2×D711yescharacteristic
84C3×C7×D4C22×C711yescharacteristic
85Dic42C22×C711yescharacteristic, complemented
86C7×#67—26no—
87C7×#67—21no—
88C7×#67—21no—
89C7×#67—141no—
90C2×C7×Dic3—141no—
91C7×D12—23no—
92C7×D12C2×D711yescharacteristic
93C7×D12C22×C711yescharacteristic
94C4×D3×C7—23no—
95C4×D3×C7C2×D711yescharacteristic
96C4×D3×C7C22×C711yescharacteristic
97C7×Dic6—71no—
98C3×D28C22×C711yescharacteristic, complemented
99#4245—21no—
100Dic3×D7—21no—
101C3×C7×C8—71no—
102C3⋊(C7⋊C8)C22×C711yescharacteristic, complemented
103C3×C7⋊C8C22×C711yescharacteristic, complemented
104C7×C3⋊C8—71no—
105C3×C72AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
106C7×#41—211no—
107C2×C7×D4—31no—
108C2×C7×D4—31no—
109C2×C7×D4—63no—
110#1157—31no—
111C7×SD16—211no—
112C7×SD16—211no—
113C7×D8—71no—
114C7×D8—211no—
115C7×M16—211no—
116#1142—31no—
117#1142—31no—
118#1141—31no—
119#1141D3×C711yescharacteristic, complemented
120#1136—31no—
121C2×C72—41no—
122C2×C72—61no—
123C2×C72—121no—
124C2×C72D3×D411yescharacteristic
125C7×D7—41no—
126#911—71noHall
127C22×C3×C7—26no—
128C22×C3×C7—21no—
129C22×C3×C7—21no—
130C22×C3×C7—141no—
131C2×D3×C7—26no—
132C2×D3×C7—21no—
133C2×D3×C7—21no—
134C2×D3×C7—23no—
135C2×D3×C7—412no—
136C2×D3×C7—42no—
137C2×D3×C7—42no—
138C2×D3×C7C22⋊D711yescharacteristic
139C2×D3×C7C7×D411yescharacteristic
140C2×C3×D7—21no—
141C3×C4×C7—23no—
142C3×C4×C7C22×D711yescharacteristic
143C3×C4×C7C23×C711yescharacteristic
144Dic21—21no—
145C7×Dic3—23no—
146C7×Dic3—141no—
147C7×Dic3C22⋊D711yescharacteristic
148C7×Dic3C7×D411yescharacteristic
149C23×C7—61no—
150C23×C7—61no—
151C23×C7—66no—
152C7×Q8—211no—
153C7×D4—23no—
154C7×D4—31no—
155C7×D4—31no—
156C7×D4—63no—
157C7×D4—61no—
158C7×D4—61no—
159C7×D4—66no—
160C7×D4—71no—
161C7×D4—421no—
162C7×D4D3×D711yescharacteristic
163C7×D4C2×D3×C711yescharacteristic
164C2×C4×C7—31no—
165C2×C4×C7—31no—
166C2×C4×C7—63no—
167C2×C4×C7—421no—
168C22⋊D7—61no—
169D28C2×D3×C711yescharacteristic, complemented
170C4×D7—61no—
171Dic14—31no—
172C7×C8—71no—
173C7×C8—211no—
174C7⋊C8—31no—
175C7⋊C8C2×D3×C711yescharacteristic, complemented
176C72#91111yescharacteristic, complemented, Sylow
177#237—71no—
178D3×D4C7×D711yescharacteristic, complemented
179C3×D8—71no—
180#214—71no—
181#213—71no—
182#204—71no—
183ΓL(1,𝔽25)—71no—
184C2×C3×C7—23no—
185C2×C3×C7—46no—
186C2×C3×C7—41no—
187C2×C3×C7—41no—
188C2×C3×C7—281no—
189C2×C3×C7C2×C22⋊D711yescharacteristic
190C2×C3×C7C2×C7×D411yescharacteristic
191C3×D7—41no—
192D3×C7—26no—
193D3×C7—21no—
194D3×C7—21no—
195D3×C7—46no—
196D3×C7—41no—
197D3×C7—41no—
198AGL(1,ℤ/8)—211noSylow
199C22×C7—26no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—31no—
203C22×C7—31no—
204C22×C7—63no—
205C22×C7—66no—
206C22×C7—61no—
207C22×C7—61no—
208C22×C7—1212no—
209C22×C7—122no—
210C22×C7—122no—
211C22×C7—141no—
212C2×D7—21no—
213C4×C7—23no—
214C4×C7—31no—
215C4×C7—31no—
216C4×C7—63no—
217C4×C7—421no—
218C4×C7C2×D3×D711yescharacteristic
219C4×C7C22×D3×C711yescharacteristic
220Dic7—61no—
221C22×D3—21no—
222C3×D4—71no—
223C3×D4C2×C7×D711yescharacteristic
224#67—21no—
225#67—141no—
226C2×Dic3—141no—
227D12C2×C7×D711yescharacteristic
228C4×D3C2×C7×D711yescharacteristic
229Dic6—71no—
230C3×C8—71no—
231C3⋊C8—71no—
232C3×C7—23no—
233C3×C7#649511yescharacteristic, complemented
234C3×C7C7×AGL(1,ℤ/8)11yescharacteristic, complemented
235#41—211no—
236C2×D4—31no—
237SD16—211no—
238SD16—211no—
239D8—71no—
240D8—211no—
241M16—211no—
242C2×C7—23no—
243C2×C7—46no—
244C2×C7—41no—
245C2×C7—41no—
246C2×C7—66no—
247C2×C7—61no—
248C2×C7—61no—
249C2×C7—126no—
250C2×C7—121no—
251C2×C7—121no—
252C2×C7—281no—
253C2×C7D3×C22⋊D711yescharacteristic
254C2×C7D3×C7×D411yescharacteristic
255D7—41no—
256C22×C3—21no—
257C22×C3—141no—
258C2×D3—21no—
259C2×D3—42no—
260C2×D3C7×C22⋊D711yescharacteristic
261C3×C4C22×C7×D711yescharacteristic
262Dic3—141no—
263Dic3C7×C22⋊D711yescharacteristic
264C23—61no—
265Q8—211no—
266D4—31no—
267D4—61no—
268D4—71no—
269D4—421no—
270D4D3×C7×D711yescharacteristic
271C2×C4—31no—
272C2×C4—421no—
273C8—71no—
274C8—211no—
275C7—23no—
276C7#1062105611yescharacteristic, direct product factor
277C7C7×#91111yescharacteristic, complemented
278C2×C3—41no—
279C2×C3—281no—
280C2×C3C2×C7×C22⋊D711yescharacteristic
281D3—21no—
282D3—41no—
283C22—21no—
284C22—31no—
285C22—61no—
286C22—122no—
287C22—141no—
288C4—31no—
289C4—421no—
290C4C2×D3×C7×D711yescharacteristic
291C3C7×#649511yescharacteristic, complemented, Sylow
292C2—41no—
293C2—61no—
294C2—121no—
295C2—281no—
296C2D3×C7×C22⋊D711yescharacteristic
subgroupG/H
centerC2×C7D3×C22⋊D7
commutator, cosocleC3×C4×C7C23×C7
FrattiniC4C2×D3×C7×D7
FittingC3×C72×D4C22
socleC2×C3×C72C2×D4
hypercenterC7×D4D3×D7
hypocenterC3×C7C7×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C7 ◁ C4×C7 ◁ C7×D4

Complements

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

NH ≅ G/Ntype
1. C7×#66560C2semidirect
2. C3×C7×#1141C2semidirect
3. C7×#66633C2semidirect
4. C7×#66567C2semidirect
5. C7×#66669C2semidirect
6. C7×#66558C2semidirect
7. D3×C72×D4C2semidirect
11. C7×Dic42C22semidirect
15. C3×C7×D28C22semidirect
18. C7×C3⋊(C7⋊C8)C22semidirect
19. C3×C7×C7⋊C8C22semidirect
25. C7×#1141D3semidirect
28. #10621056C7direct (N × H)
43. C7×D28C2×D3semidirect
47. C7×C7⋊C8C2×D3semidirect
50. D3×C7×D4D7semidirect
51. D3×C7×D4C2×C7semidirect
52. #66669C2×C7semidirect
54. #66633C2×C7semidirect
59. C3×#1141C2×C7semidirect
60. #66567C2×C7semidirect
61. #66560C2×C7semidirect
62. #66558C2×C7semidirect
85. Dic42C22×C7semidirect
98. C3×D28C22×C7semidirect
102. C3⋊(C7⋊C8)C22×C7semidirect
103. C3×C7⋊C8C22×C7semidirect
105. C3×C72AGL(1,ℤ/8)semidirect
119. #1141D3×C7semidirect
169. D28C2×D3×C7semidirect
175. C7⋊C8C2×D3×C7semidirect
176. C72#911semidirect
178. D3×D4C7×D7semidirect
233. C3×C7#6495semidirect
234. C3×C7C7×AGL(1,ℤ/8)semidirect
276. C7#10621056direct (N × H)
277. C7C7×#911semidirect
291. C3C7×#6495semidirect

Maximal subgroups

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

subgroupindex
C7×#665602
C3×C7×#11412
C7×#666332
C7×#665672
C7×#666692
C7×#665582
D3×C72×D42
C7×#64953
C7×#9117
#106210567