Finite Group Databasebeta

← #90238

Subgroups — #90238

Order 448.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×#1157C211yesmaximal, characteristic, complemented
2C22×C4×D7C211yesmaximal, characteristic, complemented
3C4×C7×D4C211yesmaximal, characteristic, complemented
4#6502C211yesmaximal, characteristic, complemented
5#6500C211yesmaximal, characteristic, complemented
6C4×C22⋊D7C211yesmaximal, characteristic, complemented
7C4×C22⋊D7C211yesmaximal, characteristic, complemented
8#6459C211yesmaximal, characteristic, complemented
9#6457C211yesmaximal, characteristic, complemented
10#6447C211yesmaximal, characteristic, complemented
11#6446C211yesmaximal, characteristic, complemented
12#6445C211yesmaximal, characteristic, complemented
13#6441C211yesmaximal, characteristic, complemented
14C4×D28C211yesmaximal, characteristic, complemented
15#6435C211yesmaximal, characteristic, complemented
16C23×D7C2211yescharacteristic
17C2×C7×D4C2211yescharacteristic
18C22×C4×C7C2211yescharacteristic
19C22×C4×C7C2211yescharacteristic, complemented
20C2×C22⋊D7C2211yescharacteristic
21C2×C22⋊D7C2211yescharacteristic
22C2×C22⋊D7C2211yescharacteristic
23C2×C22⋊D7C2211yescharacteristic, complemented
24C22×Dic7C2211yescharacteristic
25#1157—24no—
26C2×D28C2211yescharacteristic
27C2×D28C2211yescharacteristic
28C2×C4×D7—24no—
29C2×C4×D7C2211yescharacteristic
30C2×C4×D7C2211yescharacteristic
31C2×C4×D7C2211yescharacteristic
32C2×C4×D7C2211yescharacteristic
33C2×C4×D7C2211yescharacteristic, complemented
34C2×Dic14C2211yescharacteristic, complemented
35C7×C4⋊C4C2211yescharacteristic, complemented
36C7×C22⋊C4C2211yescharacteristic
37C7×C22⋊C4C2211yescharacteristic, complemented
38C42×C7C2211yescharacteristic, complemented
39C22⋊Dic7C2211yescharacteristic
40C22⋊Dic7C2211yescharacteristic, complemented
41(C2×D7)⋊C4C2211yescharacteristic
42(C2×D7)⋊C4C2211yescharacteristic
43(C2×D7)⋊C4C2211yescharacteristic
44(C2×D7)⋊C4C2211yescharacteristic
45(C2×D7)⋊C4C2211yescharacteristic, complemented
46(C2×D7)⋊C4C2211yescharacteristic, complemented
47C4⋊Dic7C2211yescharacteristic, complemented
48Dic7⋊C4C2211yescharacteristic, complemented
49Dic7⋊C4C2211yescharacteristic, complemented
50Dic7⋊C4C2211yescharacteristic, complemented
51Dic7⋊C4C2211yescharacteristic, complemented
52C4×Dic7C2211yescharacteristic, complemented
53#525—71nomaximal, Sylow
54C23×C7C2311yescharacteristic
55C23×C7C2311yescharacteristic
56C22×D7—22no—
57C22×D7—22no—
58C22×D7—22no—
59C22×D7C2311yescharacteristic
60C22×D7C2311yescharacteristic
61C22×D7C2311yescharacteristic
62C7×D4—22no—
63C2×C4×C7—22no—
64C2×C4×C7—22no—
65C2×C4×C7C2311yescharacteristic
66C2×C4×C7C2311yescharacteristic
67C2×C4×C7C2311yescharacteristic
68C2×C4×C7C2311yescharacteristic
69C2×C4×C7C2311yescharacteristic
70C22⋊D7—22no—
71C22⋊D7—22no—
72C22⋊D7—22no—
73C22⋊D7—22no—
74C2×Dic7—22no—
75C2×Dic7C2311yescharacteristic
76C2×Dic7C2311yescharacteristic
77C2×Dic7C2311yescharacteristic
78C2×Dic7C2311yescharacteristic
79C2×Dic7C2311yescharacteristic
80D28—22no—
81D28—22no—
82C4×D7—22no—
83C4×D7—22no—
84C4×D7—22no—
85C4×D7—22no—
86C4×D7D414yescomplemented
87Dic14—22no—
88C2×#41—71no—
89C23×C4—71no—
90#123—71no—
91#123—71no—
92C22⋊Q8—71no—
93C22⋊Q8—71no—
94#121—71no—
95#121—71no—
96#120—71no—
97#120—71no—
98C4×D4—71no—
99C4×D4—71no—
100C4×D4—71no—
101C4×D4D711yescharacteristic, complemented
102#117—71no—
103C22×C7—21no—
104C22×C7—21no—
105C22×C7—21no—
106C22×C7—21no—
107C22×C7—21no—
108C22×C7#4112yescomplemented
109C22×C7C2411yescharacteristic
110C2×D7—28no—
111C2×D7—21no—
112C2×D7—21no—
113C2×D7—21no—
114C2×D7—21no—
115C2×D7—21no—
116C2×D7—21no—
117C2×D7—21no—
118C2×D7C2×D412yes—
119C2×D7#4112yescomplemented
120C4×C7—21no—
121C4×C7—21no—
122C4×C7—21no—
123C4×C7—21no—
124C4×C7C2×D412yes—
125Dic7—21no—
126Dic7—21no—
127Dic7—21no—
128Dic7—21no—
129Dic7C2×D412yes—
130C24—71no—
131#41—144no—
132C2×Q8—71no—
133C2×D4—71no—
134C2×D4—71no—
135C2×D4—71no—
136C2×D4—71no—
137C2×D4—71no—
138C2×D4—71no—
139C2×D4C2×D711yescharacteristic
140C22×C4—71no—
141C22×C4—71no—
142C22×C4—71no—
143C22×C4—71no—
144C22×C4—71no—
145C22×C4—71no—
146C22×C4—144no—
147C22×C4C2×D711yescharacteristic
148C22×C4C2×D711yescharacteristic, complemented
149C4⋊C4—71no—
150C4⋊C4—71no—
151C4⋊C4—71no—
152C4⋊C4—71no—
153C4⋊C4—71no—
154C4⋊C4C2×D711yescharacteristic, complemented
155C22⋊C4—71no—
156C22⋊C4—71no—
157C22⋊C4—71no—
158C22⋊C4—71no—
159C22⋊C4—71no—
160C22⋊C4—71no—
161C22⋊C4—71no—
162C22⋊C4—71no—
163C22⋊C4C2×D711yescharacteristic
164C22⋊C4C2×D711yescharacteristic, complemented
165C42—71no—
166C42C2×D711yescharacteristic, complemented
167C2×C7—22no—
168C2×C7—41no—
169C2×C7C22×D411yescharacteristic
170C2×C7C2×#4111yescharacteristic
171C2×C7C2×#4111yescharacteristic
172D7—22no—
173D7—22no—
174D7—41no—
175C23—71no—
176C23—71no—
177C23—71no—
178C23—142no—
179C23—142no—
180C23—142no—
181C23C22×D711yescharacteristic
182C23C22×D711yescharacteristic
183Q8—142no—
184D4—22no—
185D4—142no—
186D4—142no—
187D4—142no—
188D4—142no—
189D4—142no—
190D4—142no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—71no—
194C2×C4—74no—
195C2×C4—71no—
196C2×C4—71no—
197C2×C4—71no—
198C2×C4—71no—
199C2×C4—142no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4—142no—
203C2×C4—142no—
204C2×C4C22×D711yescharacteristic
205C2×C4C22×D711yescharacteristic
206C2×C4C22×D711yescharacteristic
207C2×C4C22×D711yescharacteristic
208C2×C4C22×D711yescharacteristic
209C7#52511yescharacteristic, complemented, Sylow
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—72no—
216C22—72no—
217C22—148no—
218C22—141no—
219C22—141no—
220C22—141no—
221C22—141no—
222C22—141no—
223C22—141no—
224C22—141no—
225C22#115712yescomplemented
226C22C23×D711yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—72no—
232C4—141no—
233C4—141no—
234C4—141no—
235C4—141no—
236C4D4×D712yes—
237C2—22no—
238C2—41no—
239C2—142no—
240C2—142no—
241C2—281no—
242C2C2×#115711yescharacteristic
243C2C2×D4×D711yescharacteristic
244C2D7×#4111yescharacteristic
subgroupG/H
centerC2×C4C22×D7
commutator, socle, cosocleC22×C7C24
FrattiniC22C23×D7
FittingC4×C7×D4C2
hypercenterC4×D4D7
hypocenterC7#525
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#1157C2semidirect
2. C22×C4×D7C2semidirect
3. C4×C7×D4C2semidirect
4. #6502C2semidirect
5. #6500C2semidirect
6. C4×C22⋊D7C2semidirect
7. C4×C22⋊D7C2semidirect
8. #6459C2semidirect
9. #6457C2semidirect
10. #6447C2semidirect
11. #6446C2semidirect
12. #6445C2semidirect
13. #6441C2semidirect
14. C4×D28C2semidirect
15. #6435C2semidirect
19. C22×C4×C7C22semidirect
23. C2×C22⋊D7C22semidirect
33. C2×C4×D7C22semidirect
34. C2×Dic14C22semidirect
35. C7×C4⋊C4C22semidirect
37. C7×C22⋊C4C22semidirect
38. C42×C7C22semidirect
40. C22⋊Dic7C22semidirect
45. (C2×D7)⋊C4C22semidirect
46. (C2×D7)⋊C4C22semidirect
47. C4⋊Dic7C22semidirect
48. Dic7⋊C4C22semidirect
49. Dic7⋊C4C22semidirect
50. Dic7⋊C4C22semidirect
51. Dic7⋊C4C22semidirect
52. C4×Dic7C22semidirect
86. C4×D7D4semidirect
101. C4×D4D7semidirect
108. C22×C7#41semidirect
119. C2×D7#41semidirect
148. C22×C4C2×D7semidirect
154. C4⋊C4C2×D7semidirect
164. C22⋊C4C2×D7semidirect
166. C42C2×D7semidirect
209. C7#525semidirect
225. C22#1157semidirect

Maximal subgroups

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

subgroupindex
C2×#11572
C22×C4×D72
C4×C7×D42
#65022
#65002
C4×C22⋊D72
#64592
#64572
#64472
#64462
#64452
#64412
C4×D282
#64352
#5257