Finite Group Databasebeta

← C7×#4825 #13055860

Subgroups — C7×#4825 #13055860

Order 1344.

304 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 78 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×#1009C211yesmaximal, characteristic
2C7×#951C211yesmaximal, characteristic, complemented
3C7×#909C211yesmaximal, characteristic, complemented
4C7×#875C211yesmaximal, characteristic, complemented
5C3×C7×#104C211yesmaximal, characteristic, complemented
6C7×#823C211yesmaximal, characteristic, complemented
7C7×#805C211yesmaximal, characteristic, complemented
8C7×#455—31nomaximal, Hall
9C3×C7×#41C2211yescharacteristic
10C7×#239—21no—
11D3×C7×D4—22no—
12D3×C7×D4—21no—
13C7×#235—21no—
14C7×#235C2211yescharacteristic
15C2×C7×D12—21no—
16C2×C7×D12C2211yescharacteristic
17C2×C7×Dic6C2211yescharacteristic, complemented
18C3×C7×M16C2211yescharacteristic, complemented
19C3×C42×C7C2211yescharacteristic, complemented
20C7×#216—21no—
21C7×#214—21no—
22C7×(C2×D3)⋊C4—22no—
23C7×#208C2211yescharacteristic, complemented
24C7×Dic12—21no—
25C7×#204—21no—
26C7×H(5,𝔽2)—31no—
27C7×#137—31no—
28C7×#137—31no—
29C7×#124—31no—
30C7×#104—31no—
31C7×#104D311yescharacteristic, complemented
32C7×#100—31no—
33#4825C711yesmaximal, characteristic, direct product factor, Hall
34C22×D3×C7—22no—
35C22×D3×C7D412yes—
36C3×C7×Q8D411yescharacteristic
37C3×C7×D4—21no—
38C3×C7×D4D411yescharacteristic
39C2×C3×C4×C7—21no—
40C2×C3×C4×C7—21no—
41C2×C3×C4×C7C2311yescharacteristic
42C7×#67—22no—
43C7×#67—21no—
44C2×C7×Dic3—21no—
45C7×D12—22no—
46C7×D12—21no—
47C7×D12—21no—
48C7×D12D411yescharacteristic
49C4×D3×C7—22no—
50C4×D3×C7—21no—
51C7×Dic6—21no—
52C7×Dic6—21no—
53C7×Dic6D411yescharacteristic
54C3×C7×C8—21no—
55C7×C3⋊C8—21no—
56C7×#41—31no—
57C7×#41—61no—
58C7×#41—61no—
59C7×#41C2×D311yescharacteristic
60C2×C7×Q8—31no—
61C2×C7×D4—31no—
62C2×C7×D4—61no—
63C2×C7×D4—62no—
64C2×C7×D4—61no—
65C7×Q16—61no—
66C7×Q16—61no—
67C7×SD16—61no—
68C7×SD16—61no—
69C7×M16—31no—
70C7×M16C2×D311yescharacteristic, complemented
71C7×C22⋊C4—62no—
72C42×C7C2×D311yescharacteristic, complemented
73#1009C2×C711yescharacteristic
74#951C2×C711yescharacteristic, complemented
75#909C2×C711yescharacteristic, complemented
76#875C2×C711yescharacteristic, complemented
77C3×#104C2×C711yescharacteristic, complemented
78#823C2×C711yescharacteristic, complemented
79#805C2×C711yescharacteristic, complemented
80C22×C3×C7—21no—
81C22×C3×C7C2×D411yescharacteristic
82C2×D3×C7—22no—
83C2×D3×C7—21no—
84C2×D3×C7—42no—
85C2×D3×C7—82no—
86C3×C4×C7—21no—
87C3×C4×C7—41no—
88C3×C4×C7C2×D411yescharacteristic
89C3×C4×C7C2×D411yescharacteristic
90C7×Dic3—21no—
91C7×Dic3—41no—
92#455—31noSylow
93C23×C7—32no—
94C23×C7—62no—
95C7×Q8—31no—
96C7×Q8—61no—
97C7×Q8—61no—
98C7×Q8D1211yescharacteristic
99C7×D4—21no—
100C7×D4—31no—
101C7×D4—62no—
102C7×D4—62no—
103C7×D4—61no—
104C7×D4—61no—
105C7×D4—61no—
106C7×D4D1211yescharacteristic
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—62no—
110C2×C4×C7—61no—
111C2×C4×C7—61no—
112C2×C4×C7C22×D311yescharacteristic
113C7×C8—21no—
114C7×C8—61no—
115C3×#41C22×C711yescharacteristic
116#239—21no—
117D3×D4—22no—
118D3×D4—21no—
119#235—21no—
120#235C22×C711yescharacteristic
121C2×D12—21no—
122C2×D12C22×C711yescharacteristic
123C2×Dic6C22×C711yescharacteristic, complemented
124C3×M16C22×C711yescharacteristic, complemented
125C3×C42C22×C711yescharacteristic, complemented
126#216—21no—
127#214—21no—
128(C2×D3)⋊C4—22no—
129#208C22×C711yescharacteristic, complemented
130Dic12—21no—
131#204—21no—
132C2×C3×C7—21no—
133C2×C3×C7—41no—
134C2×C3×C7#12011yescharacteristic
135D3×C7—42no—
136D3×C7—41no—
137H(5,𝔽2)—31no—
138#137—31no—
139#137—31no—
140#124—31no—
141#104—31no—
142#104D3×C711yescharacteristic, complemented
143#100—31no—
144C22×C7—21no—
145C22×C7—61no—
146C22×C7—62no—
147C22×C7—122no—
148C22×C7—242no—
149C22×C7D3×D411yescharacteristic
150C4×C7—21no—
151C4×C7—41no—
152C4×C7—61no—
153C4×C7—121no—
154C4×C7C2×D1211yescharacteristic
155C4×C7D3×D411yescharacteristic
156C22×D3—22no—
157C22×D3C7×D412yes—
158C3×Q8C7×D411yescharacteristic
159C3×D4—21no—
160C3×D4C7×D411yescharacteristic
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4C23×C711yescharacteristic
164#67—22no—
165#67—21no—
166C2×Dic3—21no—
167D12—22no—
168D12—21no—
169D12—21no—
170D12C7×D411yescharacteristic
171C4×D3—22no—
172C4×D3—21no—
173Dic6—21no—
174Dic6—21no—
175Dic6C7×D411yescharacteristic
176C3×C8—21no—
177C3⋊C8—21no—
178C3×C7#45511yescharacteristic, complemented, Hall
179#41—31no—
180#41—61no—
181#41—61no—
182#41C2×D3×C711yescharacteristic
183C2×Q8—31no—
184C2×D4—31no—
185C2×D4—61no—
186C2×D4—62no—
187C2×D4—61no—
188Q16—61no—
189Q16—61no—
190SD16—61no—
191SD16—61no—
192M16—31no—
193M16C2×D3×C711yescharacteristic, complemented
194C22⋊C4—62no—
195C42C2×D3×C711yescharacteristic, complemented
196C2×C7—21no—
197C2×C7—41no—
198C2×C7—122no—
199C2×C7—121no—
200C2×C7#88211yescharacteristic
201C22×C3—21no—
202C22×C3C2×C7×D411yescharacteristic
203C2×D3—22no—
204C2×D3—21no—
205C2×D3—42no—
206C2×D3—82no—
207C3×C4—21no—
208C3×C4—41no—
209C3×C4C2×C7×D411yescharacteristic
210C3×C4C2×C7×D411yescharacteristic
211Dic3—21no—
212Dic3—41no—
213C23—32no—
214C23—62no—
215Q8—31no—
216Q8—61no—
217Q8—61no—
218Q8C7×D1211yescharacteristic
219D4—21no—
220D4—31no—
221D4—62no—
222D4—62no—
223D4—61no—
224D4—61no—
225D4—61no—
226D4C7×D1211yescharacteristic
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—62no—
230C2×C4—61no—
231C2×C4—61no—
232C2×C4C22×D3×C711yescharacteristic
233C8—21no—
234C8—61no—
235C7#482511yescharacteristic, direct product factor, Sylow
236C2×C3—21no—
237C2×C3—41no—
238C2×C3C7×#12011yescharacteristic
239D3—42no—
240D3—41no—
241C22—21no—
242C22—61no—
243C22—62no—
244C22—122no—
245C22—242no—
246C22D3×C7×D411yescharacteristic
247C4—21no—
248C4—41no—
249C4—61no—
250C4—121no—
251C4C2×C7×D1211yescharacteristic
252C4D3×C7×D411yescharacteristic
253C3C7×#45511yescharacteristic, complemented, Sylow
254C2—21no—
255C2—41no—
256C2—122no—
257C2—121no—
258C2C7×#88211yescharacteristic
subgroupG/H
centerC2×C7#882
commutator, cosocleC2×C3×C4C23×C7
FrattiniC2×C4C22×D3×C7
FittingC3×C7×#104C2
socleC2×C3×C7#120
prime radicalC7×#1009C2
hypercenterC7×#104D3
hypocenterC3C7×#455
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C7×#104

Complements

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

NH ≅ G/Ntype
2. C7×#951C2semidirect
3. C7×#909C2semidirect
4. C7×#875C2semidirect
5. C3×C7×#104C2semidirect
6. C7×#823C2semidirect
7. C7×#805C2semidirect
17. C2×C7×Dic6C22semidirect
18. C3×C7×M16C22semidirect
19. C3×C42×C7C22semidirect
23. C7×#208C22semidirect
31. C7×#104D3semidirect
33. #4825C7direct (N × H)
70. C7×M16C2×D3semidirect
72. C42×C7C2×D3semidirect
74. #951C2×C7semidirect
75. #909C2×C7semidirect
76. #875C2×C7semidirect
77. C3×#104C2×C7semidirect
78. #823C2×C7semidirect
79. #805C2×C7semidirect
123. C2×Dic6C22×C7semidirect
124. C3×M16C22×C7semidirect
125. C3×C42C22×C7semidirect
129. #208C22×C7semidirect
142. #104D3×C7semidirect
178. C3×C7#455semidirect
193. M16C2×D3×C7semidirect
195. C42C2×D3×C7semidirect
235. C7#4825direct (N × H)
253. C3C7×#455semidirect

Maximal subgroups

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

subgroupindex
C7×#10092
C7×#9512
C7×#9092
C7×#8752
C3×C7×#1042
C7×#8232
C7×#8052
C7×#4553
#48257