Finite Group Databasebeta

← C2×D7×GO+(4,𝔽2) #72217618

Subgroups — C2×D7×GO+(4,𝔽2) #72217618

Order 2016.

740 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C2×D32×D7C212yesmaximal, complemented
2C2×#157⋊D7C211yesmaximal, characteristic, complemented
3C2×D32⋊D7C212yesmaximal, complemented
4C2×C7×GO+(4,𝔽2)C211yesmaximal, characteristic, complemented
5C2×D7×#157C211yesmaximal, characteristic, complemented
6D7×GO+(4,𝔽2)C218yesmaximal, direct product factor
7C2×(C3⋊D3)⋊D7C2212yescomplemented
8C2×D3×D21—22no—
9C2×D32×C7C2212yescomplemented
10C2×D7×C3⋊D3C2211yescharacteristic
11C2×C3×D3×D7—22no—
12C2×C7⋊#157C2211yescharacteristic, complemented
13C2×C7×#157C2211yescharacteristic, complemented
14D32×D7—24no—
15D32×D7C2218yescomplemented
16#157⋊D7C2214yescomplemented
17D32⋊D7C2218yescomplemented
18C7×GO+(4,𝔽2)C2214yescomplemented
19D7×#157C2214yescomplemented
20C22×D3×D7—62no—
21C22×GO+(4,𝔽2)—71nomaximal, Hall
22C2×C3⋊D21D411yescharacteristic, complemented
23C2×C7×C3⋊D3C2311yescharacteristic
24C2×C3×D21—22no—
25C2×C3×D3×C7—22no—
26C2×C32×D7D411yescharacteristic, complemented
27(C3⋊D3)⋊D7—22no—
28(C3⋊D3)⋊D7C2314yescomplemented
29D3×D21—28no—
30D32×C7—22no—
31D32×C7C2314yescomplemented
32D7×C3⋊D3D412yescomplemented
33D7×C3⋊D3C2312yes—
34C3×D3×D7—28no—
35C7⋊#157C2312yescomplemented
36C7×#157C2312yescomplemented
37C2×D4×D7—91nomaximal, Hall
38C22×D21—62no—
39C22×D3×C7—62no—
40C22×C3×D7—62no—
41C2×D3×D7—22no—
42C2×D3×D7—68no—
43C2×D3×D7—62no—
44C2×D3×D7—62no—
45C2×D3×D7—68no—
46C2×D3×D7—62no—
47C22×D32—72no—
48C22×#157—71no—
49C2×GO+(4,𝔽2)—72no—
50C2×GO+(4,𝔽2)—71no—
51C2×GO+(4,𝔽2)—78no—
52C2×GO+(4,𝔽2)D711yescharacteristic, direct product factor
53C2×C32×C7C2×D411yescharacteristic, complemented
54C3⋊D21C2×D412yescomplemented
55C7×C3⋊D3C2×D411yescharacteristic, complemented
56C7×C3⋊D3C2411yescharacteristic
57C3×D21—24no—
58C3×D3×C7—24no—
59C32×D7C2×D412yescomplemented
60C23×D7—92no—
61C2×C7×D4—91no—
62C2×C22⋊D7—92no—
63D4×D7—98no—
64C2×D28—91no—
65C2×C4×D7—91no—
66C22×C3×C7—62no—
67C2×D21—22no—
68C2×D21—62no—
69C2×D21—68no—
70C2×D3×C7—22no—
71C2×D3×C7—64no—
72C2×D3×C7—62no—
73C2×D3×C7—64no—
74C2×C3×D7—22no—
75C2×C3×D7—68no—
76C2×C3×D7—62no—
77D3×D7—28no—
78D3×D7—64no—
79D3×D7—64no—
80D3×D7—64no—
81D3×D7—68no—
82D3×D7—64no—
83C22×C3⋊D3—71no—
84C22×C3×D3—142no—
85C2×D32—72no—
86C2×D32—78no—
87C2×D32—144no—
88C2×D32—142no—
89C2×D32C2×D712yescomplemented
90C2×#157—71no—
91C2×#157—74no—
92C2×#157C2×D711yescharacteristic, complemented
93GO+(4,𝔽2)—78no—
94GO+(4,𝔽2)—74no—
95GO+(4,𝔽2)C2×D714yesdirect product factor
96C32×C7C22×D411yescharacteristic, complemented, Hall
97C23×C7—92no—
98C22×D7—62no—
99C22×D7—92no—
100C22×D7—98no—
101C22×D7—91no—
102C22×D7—184no—
103C22×D7—182no—
104C7×D4—94no—
105C2×C4×C7—91no—
106C22⋊D7—98no—
107C2×Dic7—91no—
108D28—94no—
109C4×D7—94no—
110C23×D3—422no—
111C2×C3×C7—22no—
112C2×C3×C7—64no—
113D21—24no—
114D21—64no—
115C3×D7—24no—
116C3×D7—64no—
117D3×C7—24no—
118D3×C7—62no—
119D3×C7—62no—
120C22×C32—71no—
121C2×C3⋊D3—71no—
122C2×C3⋊D3—72no—
123C2×C3⋊D3—72no—
124C2×C3⋊D3C22×D711yescharacteristic
125C2×C3×D3—22no—
126C2×C3×D3—148no—
127C2×C3×D3—142no—
128D32—22no—
129D32—74no—
130D32—148no—
131D32—142no—
132D32C22×D714yescomplemented
133#157—72no—
134#157C22×D712yescomplemented
135C22×D4—631noSylow
136C22×C7—62no—
137C22×C7—94no—
138C22×C7—91no—
139C22×C7—182no—
140C2×D7—62no—
141C2×D7—68no—
142C2×D7—94no—
143C2×D7—91no—
144C2×D7—92no—
145C2×D7—92no—
146C2×D7—182no—
147C2×D7—188no—
148C2×D7GO+(4,𝔽2)11yescharacteristic, direct product factor
149C4×C7—92no—
150Dic7—92no—
151C23×C3—422no—
152C22×D3—62no—
153C22×D3—142no—
154C22×D3—428no—
155C22×D3—422no—
156C22×D3—422no—
157C22×D3—422no—
158C22×D3—428no—
159C22×D3—422no—
160C3×C7—22no—
161C2×C32—72no—
162C2×C32D4×D711yescharacteristic, complemented
163C3⋊D3—72no—
164C3⋊D3D4×D711yescharacteristic, complemented
165C3⋊D3C23×D711yescharacteristic
166C3×D3—24no—
167C3×D3—144no—
168C24—632no—
169C2×D4—91no—
170C2×D4—631no—
171C2×D4—638no—
172C2×D4—632no—
173C22×C4—631no—
174C2×C7—64no—
175C2×C7—91no—
176C2×C7—91no—
177C2×C7C2×GO+(4,𝔽2)11yescharacteristic, complemented
178D7—64no—
179D7—92no—
180D7C2×GO+(4,𝔽2)12yesdirect product factor
181C22×C3—62no—
182C22×C3—142no—
183C22×C3—422no—
184C22×C3—428no—
185C2×D3—22no—
186C2×D3—64no—
187C2×D3—62no—
188C2×D3—64no—
189C2×D3—148no—
190C2×D3—142no—
191C2×D3—424no—
192C2×D3—424no—
193C2×D3—424no—
194C2×D3—422no—
195C2×D3—428no—
196C2×D3—424no—
197C2×D3—428no—
198C32C2×D4×D711yescharacteristic, complemented, Sylow
199C23—92no—
200C23—422no—
201C23—631no—
202C23—638no—
203C23—632no—
204C23—1262no—
205C23—1264no—
206D4—94no—
207D4—634no—
208D4—638no—
209C2×C4—91no—
210C2×C4—634no—
211C2×C4—631no—
212C7C22×GO+(4,𝔽2)11yescharacteristic, complemented, Sylow
213C2×C3—22no—
214C2×C3—64no—
215C2×C3—144no—
216C2×C3—424no—
217D3—24no—
218D3—62no—
219D3—62no—
220D3—144no—
221D3—424no—
222C22—62no—
223C22—71no—
224C22—94no—
225C22—91no—
226C22—182no—
227C22—428no—
228C22—422no—
229C22—632no—
230C22—632no—
231C22—634no—
232C22—631no—
233C22—1268no—
234C22—1262no—
235C4—92no—
236C4—632no—
237C3—22no—
238C2—64no—
239C2—72no—
240C2—91no—
241C2—91no—
242C2—424no—
243C2—632no—
244C2D7×GO+(4,𝔽2)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2D7×GO+(4,𝔽2)
commutator, cosocleC7×C3⋊D3C24
FrattiniC1G
Fitting, socleC2×C32×C7C2×D4
hypocenterC32×C7C22×D4
LCSG ▷ C7×C3⋊D3 ▷ C32×C7
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×D32×D7C2semidirect
2. C2×#157⋊D7C2semidirect
3. C2×D32⋊D7C2semidirect
4. C2×C7×GO+(4,𝔽2)C2semidirect
5. C2×D7×#157C2semidirect
6. D7×GO+(4,𝔽2)C2direct (N × H)
7. C2×(C3⋊D3)⋊D7C22semidirect
9. C2×D32×C7C22semidirect
12. C2×C7⋊#157C22semidirect
13. C2×C7×#157C22semidirect
15. D32×D7C22semidirect
16. #157⋊D7C22semidirect
17. D32⋊D7C22semidirect
18. C7×GO+(4,𝔽2)C22semidirect
19. D7×#157C22semidirect
22. C2×C3⋊D21D4semidirect
26. C2×C32×D7D4semidirect
28. (C3⋊D3)⋊D7C23semidirect
31. D32×C7C23semidirect
32. D7×C3⋊D3D4semidirect
35. C7⋊#157C23semidirect
36. C7×#157C23semidirect
52. C2×GO+(4,𝔽2)D7direct (N × H)
53. C2×C32×C7C2×D4semidirect
54. C3⋊D21C2×D4semidirect
55. C7×C3⋊D3C2×D4semidirect
59. C32×D7C2×D4semidirect
89. C2×D32C2×D7semidirect
92. C2×#157C2×D7semidirect
95. GO+(4,𝔽2)C2×D7direct (N × H)
96. C32×C7C22×D4semidirect
132. D32C22×D7semidirect
134. #157C22×D7semidirect
148. C2×D7GO+(4,𝔽2)direct (N × H)
162. C2×C32D4×D7semidirect
164. C3⋊D3D4×D7semidirect
177. C2×C7C2×GO+(4,𝔽2)semidirect
180. D7C2×GO+(4,𝔽2)direct (N × H)
198. C32C2×D4×D7semidirect
212. C7C22×GO+(4,𝔽2)semidirect
244. C2D7×GO+(4,𝔽2)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D32×D72
C2×#157⋊D72
C2×D32⋊D72
C2×C7×GO+(4,𝔽2)2
C2×D7×#1572
D7×GO+(4,𝔽2)2
C22×GO+(4,𝔽2)7
C2×D4×D79