Finite Group Databasebeta

← C7×#5032 #13056067

Subgroups — C7×#5032 #13056067

Order 1344.

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

normal subgroups only

This group has 396 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×#1004C211yesmaximal, characteristic, complemented
2C3×C7×#120C211yesmaximal, characteristic, complemented
3C7×#953C212yesmaximal, complemented
4C7×#834C211yesmaximal, characteristic, complemented
5C7×#806C212yesmaximal, complemented
6C7×B(4,𝔽2)—31nomaximal, Hall
7C24×C3×C7C2211yescharacteristic, complemented
8C2×C3×C7×D4—21no—
9C2×C3×C7×D4C2211yescharacteristic
10C2×C7×#67—21no—
11C2×C7×#67—22no—
12C2×C7×#67C2212yes—
13C7×#237—22no—
14D3×C7×D4—22no—
15C7×#235—21no—
16C3×C7×C22⋊C4—21no—
17C3×C7×C22⋊C4C2211yescharacteristic, complemented
18C7×#217—22no—
19C7×#217C2212yescomplemented
20C7×H(5,𝔽2)—31no—
21C7×#120—32no—
22C7×#120D311yescharacteristic, complemented
23C7×#99—31no—
24C7×#99—32no—
25#5032C711yesmaximal, characteristic, direct product factor, Hall
26C23×C3×C7—22no—
27C23×C3×C7—21no—
28C23×C3×C7—42no—
29C23×C3×C7D411yescharacteristic, complemented
30C23×C3×C7C2311yescharacteristic
31C22×D3×C7—21no—
32C22×D3×C7D412yescomplemented
33C3×C7×D4—22no—
34C3×C7×D4—42no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7D411yescharacteristic, complemented
37C7×#67—22no—
38C7×#67—22no—
39C7×#67—22no—
40C7×#67—44no—
41C2×C7×Dic3—21no—
42C2×C7×Dic3—22no—
43C2×C7×Dic3D412yescomplemented
44C7×D12—21no—
45C4×D3×C7—22no—
46C7×Dic6—21no—
47C24×C7C2×D311yescharacteristic, complemented
48C7×#41—62no—
49C7×#41—61no—
50C2×C7×D4—21no—
51C2×C7×D4—32no—
52C2×C7×D4—62no—
53C2×C7×D4—61no—
54C2×C7×D4—62no—
55C2×C7×D4C2×D311yescharacteristic
56C7×C22⋊C4—21no—
57C7×C22⋊C4—32no—
58C7×C22⋊C4—62no—
59C7×C22⋊C4C2×D311yescharacteristic, complemented
60#1004C2×C711yescharacteristic, complemented
61C3×#120C2×C711yescharacteristic, complemented
62#953C2×C712yescomplemented
63#834C2×C711yescharacteristic, complemented
64#806C2×C712yescomplemented
65C22×C3×C7—24no—
66C22×C3×C7—22no—
67C22×C3×C7—21no—
68C22×C3×C7—41no—
69C22×C3×C7—41no—
70C22×C3×C7—42no—
71C22×C3×C7—41no—
72C22×C3×C7—41no—
73C22×C3×C7C2×D412yes—
74C22×C3×C7C2×D411yescharacteristic
75C2×D3×C7—22no—
76C2×D3×C7—42no—
77C2×D3×C7—81no—
78C3×C4×C7—21no—
79C3×C4×C7—41no—
80C7×Dic3—22no—
81C7×Dic3—42no—
82B(4,𝔽2)—31noSylow
83C23×C7—22no—
84C23×C7—21no—
85C23×C7—32no—
86C23×C7—42no—
87C23×C7—61no—
88C23×C7#6711yescharacteristic, complemented
89C23×C7C22×D311yescharacteristic
90C7×Q8—61no—
91C7×D4—22no—
92C7×D4—42no—
93C7×D4—61no—
94C7×D4—62no—
95C7×D4—62no—
96C7×D4—62no—
97C7×D4—124no—
98C2×C4×C7—21no—
99C2×C4×C7—32no—
100C2×C4×C7—62no—
101C2×C4×C7—61no—
102C2×C4×C7—62no—
103C2×C4×C7#6711yescharacteristic, complemented
104C24×C3C22×C711yescharacteristic, complemented
105C2×C3×D4—21no—
106C2×C3×D4C22×C711yescharacteristic
107C2×#67—21no—
108C2×#67—22no—
109C2×#67C22×C712yes—
110#237—22no—
111D3×D4—22no—
112#235—21no—
113C3×C22⋊C4—21no—
114C3×C22⋊C4C22×C711yescharacteristic, complemented
115#217—22no—
116#217C22×C712yescomplemented
117C2×C3×C7—22no—
118C2×C3×C7—21no—
119C2×C3×C7—41no—
120C2×C3×C7—42no—
121C2×C3×C7#12011yescharacteristic
122D3×C7—42no—
123H(5,𝔽2)—31no—
124#120—32no—
125#120D3×C711yescharacteristic, complemented
126#99—31no—
127#99—32no—
128C22×C7—22no—
129C22×C7—24no—
130C22×C7—21no—
131C22×C7—41no—
132C22×C7—41no—
133C22×C7—41no—
134C22×C7—42no—
135C22×C7—41no—
136C22×C7—62no—
137C22×C7—122no—
138C22×C7—241no—
139C22×C7D3×D412yes—
140C22×C7C2×#6711yescharacteristic
141C4×C7—21no—
142C4×C7—41no—
143C4×C7—62no—
144C4×C7—122no—
145C23×C3—22no—
146C23×C3—21no—
147C23×C3—42no—
148C23×C3C7×D411yescharacteristic, complemented
149C23×C3C23×C711yescharacteristic
150C22×D3—21no—
151C22×D3C7×D412yescomplemented
152C3×D4—22no—
153C3×D4—42no—
154C2×C3×C4—21no—
155C2×C3×C4C7×D411yescharacteristic, complemented
156#67—22no—
157#67—22no—
158#67—22no—
159#67—44no—
160C2×Dic3—21no—
161C2×Dic3—22no—
162C2×Dic3C7×D412yescomplemented
163D12—21no—
164C4×D3—22no—
165Dic6—21no—
166C3×C7B(4,𝔽2)11yescharacteristic, complemented, Hall
167C24C2×D3×C711yescharacteristic, complemented
168#41—62no—
169#41—61no—
170C2×D4—21no—
171C2×D4—32no—
172C2×D4—62no—
173C2×D4—61no—
174C2×D4—62no—
175C2×D4C2×D3×C711yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—32no—
178C22⋊C4—62no—
179C22⋊C4C2×D3×C711yescharacteristic, complemented
180C2×C7—21no—
181C2×C7—22no—
182C2×C7—42no—
183C2×C7—41no—
184C2×C7—122no—
185C2×C7#93711yescharacteristic
186C22×C3—24no—
187C22×C3—22no—
188C22×C3—21no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—42no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3C2×C7×D412yes—
195C22×C3C2×C7×D411yescharacteristic
196C2×D3—22no—
197C2×D3—42no—
198C2×D3—81no—
199C3×C4—21no—
200C3×C4—41no—
201Dic3—22no—
202Dic3—42no—
203C23—22no—
204C23—21no—
205C23—32no—
206C23—42no—
207C23—61no—
208C23C7×#6711yescharacteristic, complemented
209C23C22×D3×C711yescharacteristic
210Q8—61no—
211D4—22no—
212D4—42no—
213D4—61no—
214D4—62no—
215D4—62no—
216D4—62no—
217D4—124no—
218C2×C4—21no—
219C2×C4—32no—
220C2×C4—62no—
221C2×C4—61no—
222C2×C4—62no—
223C2×C4C7×#6711yescharacteristic, complemented
224C7#503211yescharacteristic, direct product factor, Sylow
225C2×C3—22no—
226C2×C3—21no—
227C2×C3—41no—
228C2×C3—42no—
229C2×C3C7×#12011yescharacteristic
230D3—42no—
231C22—22no—
232C22—24no—
233C22—21no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—42no—
238C22—41no—
239C22—62no—
240C22—122no—
241C22—241no—
242C22D3×C7×D412yes—
243C22C2×C7×#6711yescharacteristic
244C4—21no—
245C4—41no—
246C4—62no—
247C4—122no—
248C3C7×B(4,𝔽2)11yescharacteristic, complemented, Sylow
249C2—21no—
250C2—22no—
251C2—42no—
252C2—41no—
253C2—122no—
254C2C7×#93711yescharacteristic
subgroupG/H
centerC2×C7#937
commutator, cosocleC23×C3C23×C7
FrattiniC23C22×D3×C7
FittingC3×C7×#120C2
socleC2×C3×C7#120
hypercenterC7×#120D3
hypocenterC3C7×B(4,𝔽2)
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C7 ◁ C23×C7 ◁ C7×#120

Complements

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

NH ≅ G/Ntype
1. C7×#1004C2semidirect
2. C3×C7×#120C2semidirect
3. C7×#953C2semidirect
4. C7×#834C2semidirect
5. C7×#806C2semidirect
7. C24×C3×C7C22semidirect
17. C3×C7×C22⋊C4C22semidirect
19. C7×#217C22semidirect
22. C7×#120D3semidirect
25. #5032C7direct (N × H)
29. C23×C3×C7D4semidirect
32. C22×D3×C7D4semidirect
36. C2×C3×C4×C7D4semidirect
43. C2×C7×Dic3D4semidirect
47. C24×C7C2×D3semidirect
59. C7×C22⋊C4C2×D3semidirect
60. #1004C2×C7semidirect
61. C3×#120C2×C7semidirect
62. #953C2×C7semidirect
63. #834C2×C7semidirect
64. #806C2×C7semidirect
88. C23×C7#67semidirect
103. C2×C4×C7#67semidirect
104. C24×C3C22×C7semidirect
114. C3×C22⋊C4C22×C7semidirect
116. #217C22×C7semidirect
125. #120D3×C7semidirect
148. C23×C3C7×D4semidirect
151. C22×D3C7×D4semidirect
155. C2×C3×C4C7×D4semidirect
162. C2×Dic3C7×D4semidirect
166. C3×C7B(4,𝔽2)semidirect
167. C24C2×D3×C7semidirect
179. C22⋊C4C2×D3×C7semidirect
208. C23C7×#67semidirect
223. C2×C4C7×#67semidirect
224. C7#5032direct (N × H)
248. C3C7×B(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C7×#10042
C3×C7×#1202
C7×#9532
C7×#8342
C7×#8062
C7×B(4,𝔽2)3
#50327