Finite Group Databasebeta

← #90328

Subgroups — #90328

Order 448.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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×C22⋊Q8C211yesmaximal, characteristic, complemented
2#6510C211yesmaximal, characteristic, complemented
3#6509C211yesmaximal, characteristic, complemented
4C4×C22⋊D7C211yesmaximal, characteristic, complemented
5C2×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
6#6461C211yesmaximal, characteristic, complemented
7#6461C211yesmaximal, characteristic, complemented
8#6457C211yesmaximal, characteristic, complemented
9#6457C211yesmaximal, characteristic, complemented
10#6456C211yesmaximal, characteristic, complemented
11#6455C211yesmaximal, characteristic, complemented
12#6449C211yesmaximal, characteristic, complemented
13#6445C211yesmaximal, characteristic
14D7×C22⋊C4C211yesmaximal, characteristic, complemented
15#6440C211yesmaximal, characteristic, complemented
16C23×D7C2211yescharacteristic
17C2×C7×Q8C2211yescharacteristic, complemented
18C22×C4×C7C2211yescharacteristic
19C2×C22⋊D7C2211yescharacteristic
20C22×Dic7C2211yescharacteristic
21C2×D28C2211yescharacteristic
22C2×D28C2211yescharacteristic
23C2×C4×D7C2211yescharacteristic
24C2×C4×D7C2211yescharacteristic
25C2×C4×D7C2211yescharacteristic, complemented
26C7×C4⋊C4C2211yescharacteristic, complemented
27C7×C4⋊C4C2211yescharacteristic, complemented
28C7×C4⋊C4C2211yescharacteristic, complemented
29C7×C22⋊C4C2211yescharacteristic
30C7×C22⋊C4C2211yescharacteristic
31C22⋊Dic7C2211yescharacteristic
32(C2×D7)⋊C4C2211yescharacteristic
33(C2×D7)⋊C4C2211yescharacteristic
34(C2×D7)⋊C4C2211yescharacteristic
35(C2×D7)⋊C4C2211yescharacteristic
36(C2×D7)⋊C4C2211yescharacteristic
37(C2×D7)⋊C4C2211yescharacteristic
38(C2×D7)⋊C4C2211yescharacteristic
39(C2×D7)⋊C4C2211yescharacteristic
40(C2×D7)⋊C4C2211yescharacteristic, complemented
41(C2×D7)⋊C4C2211yescharacteristic, complemented
42(C2×D7)⋊C4C2211yescharacteristic, complemented
43C4⋊Dic7C2211yescharacteristic, complemented
44C4⋊Dic7C2211yescharacteristic, complemented
45Dic7⋊C4C2211yescharacteristic, complemented
46Dic7⋊C4C2211yescharacteristic, complemented
47Dic7⋊C4C2211yescharacteristic, complemented
48C4×Dic7C2211yescharacteristic, complemented
49C4×Dic7C2211yescharacteristic, complemented
50C4×Dic7C2211yescharacteristic, complemented
51#551—71nomaximal, Sylow
52C23×C7C2311yescharacteristic
53C22×D7—22no—
54C22×D7—22no—
55C22×D7—41no—
56C22×D7C2311yescharacteristic
57C22×D7C2311yescharacteristic
58C22×D7C2311yescharacteristic
59C7×Q8—41no—
60C2×C4×C7—41no—
61C2×C4×C7C2311yescharacteristic
62C2×C4×C7C2311yescharacteristic
63C2×C4×C7C2311yescharacteristic
64C2×C4×C7C2311yescharacteristic
65C2×C4×C7C2311yescharacteristic
66C2×C4×C7C2311yescharacteristic
67C22⋊D7—22no—
68C2×Dic7—22no—
69C2×Dic7C2311yescharacteristic
70C2×Dic7C2311yescharacteristic
71C2×Dic7C2311yescharacteristic
72C2×Dic7C2311yescharacteristic
73C2×Dic7C2311yescharacteristic
74D28—22no—
75D28—41no—
76C4×D7—22no—
77C4×D7—41no—
78C4×D7—41no—
79#126—71no—
80#126—71no—
81#124—71no—
82#123—71no—
83#123—71no—
84#123—71no—
85C22⋊Q8—71no—
86C22⋊Q8D711yescharacteristic, complemented
87#120—71no—
88C4×D4—71no—
89C4×D4—71no—
90#117—71no—
91#117—71no—
92C2×C22⋊C4—71no—
93C2×C22⋊C4—71no—
94C22×C7—21no—
95C22×C7—21no—
96C22×C7#4112yes—
97C22×C7C2411yescharacteristic
98C2×D7—21no—
99C2×D7—21no—
100C2×D7—21no—
101C2×D7—21no—
102C2×D7—21no—
103C2×D7—21no—
104C2×D7—21no—
105C2×D7—21no—
106C2×D7—44no—
107C2×D7#4112yes—
108C4×C7—21no—
109C4×C7—21no—
110C4×C7—21no—
111C4×C7—21no—
112C4×C7—21no—
113C4×C7—21no—
114Dic7—21no—
115Dic7—21no—
116Dic7—21no—
117Dic7—21no—
118Dic7—21no—
119C24—71no—
120C2×Q8C2×D711yescharacteristic, complemented
121C2×D4—71no—
122C2×D4—71no—
123C2×D4—71no—
124C22×C4—71no—
125C22×C4—71no—
126C22×C4—71no—
127C22×C4—71no—
128C22×C4C2×D711yescharacteristic
129C4⋊C4—71no—
130C4⋊C4—71no—
131C4⋊C4—71no—
132C4⋊C4—71no—
133C4⋊C4—71no—
134C4⋊C4C2×D711yescharacteristic, complemented
135C4⋊C4C2×D711yescharacteristic, complemented
136C4⋊C4C2×D711yescharacteristic, complemented
137C22⋊C4—71no—
138C22⋊C4—71no—
139C22⋊C4—71no—
140C22⋊C4—71no—
141C22⋊C4—71no—
142C22⋊C4—71no—
143C22⋊C4—71no—
144C22⋊C4—71no—
145C22⋊C4—71no—
146C22⋊C4—71no—
147C22⋊C4—71no—
148C22⋊C4—71no—
149C22⋊C4C2×D711yescharacteristic
150C22⋊C4C2×D711yescharacteristic
151C42—71no—
152C42—71no—
153C42—71no—
154C2×C7—22no—
155C2×C7C2×#4111yescharacteristic
156C2×C7C2×#4111yescharacteristic
157C2×C7H(5,𝔽2)11yescharacteristic
158D7—22no—
159D7—41no—
160D7—41no—
161C23—71no—
162C23—71no—
163C23—71no—
164C23—142no—
165C23—142no—
166C23—281no—
167C23C22×D711yescharacteristic
168Q8—41no—
169D4—142no—
170D4—142no—
171D4—281no—
172C2×C4—41no—
173C2×C4—71no—
174C2×C4—71no—
175C2×C4—71no—
176C2×C4—71no—
177C2×C4—71no—
178C2×C4—142no—
179C2×C4—142no—
180C2×C4—281no—
181C2×C4—281no—
182C2×C4C22×D711yescharacteristic
183C2×C4C22×D711yescharacteristic
184C2×C4C22×D711yescharacteristic
185C2×C4C22×D711yescharacteristic
186C2×C4C22×D711yescharacteristic
187C2×C4C22×D711yescharacteristic
188C7#55111yescharacteristic, complemented, Sylow
189C22—21no—
190C22—21no—
191C22—72no—
192C22—141no—
193C22—141no—
194C22—141no—
195C22—141no—
196C22—141no—
197C22—141no—
198C22—141no—
199C22—141no—
200C22—284no—
201C22#116112yes—
202C22C23×D711yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—141no—
210C4—141no—
211C4—141no—
212C4—141no—
213C4—141no—
214C2—22no—
215C2—142no—
216C2—281no—
217C2—281no—
218C2#654811yescharacteristic
219C2C2×#116111yescharacteristic
220C2D7×#4111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×C22⋊Q8C2
prime radical#6445C2
hypercenterC22⋊Q8D7
hypocenterC7#551
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C7×C22⋊Q8C2semidirect
2. #6510C2semidirect
3. #6509C2semidirect
4. C4×C22⋊D7C2semidirect
5. C2×(C2×D7)⋊C4C2semidirect
6. #6461C2semidirect
7. #6461C2semidirect
8. #6457C2semidirect
9. #6457C2semidirect
10. #6456C2semidirect
11. #6455C2semidirect
12. #6449C2semidirect
14. D7×C22⋊C4C2semidirect
15. #6440C2semidirect
17. C2×C7×Q8C22semidirect
25. C2×C4×D7C22semidirect
26. C7×C4⋊C4C22semidirect
27. C7×C4⋊C4C22semidirect
28. C7×C4⋊C4C22semidirect
40. (C2×D7)⋊C4C22semidirect
41. (C2×D7)⋊C4C22semidirect
42. (C2×D7)⋊C4C22semidirect
43. C4⋊Dic7C22semidirect
44. C4⋊Dic7C22semidirect
45. Dic7⋊C4C22semidirect
46. Dic7⋊C4C22semidirect
47. Dic7⋊C4C22semidirect
48. C4×Dic7C22semidirect
49. C4×Dic7C22semidirect
50. C4×Dic7C22semidirect
86. C22⋊Q8D7semidirect
120. C2×Q8C2×D7semidirect
134. C4⋊C4C2×D7semidirect
135. C4⋊C4C2×D7semidirect
136. C4⋊C4C2×D7semidirect
188. C7#551semidirect

Maximal subgroups

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

subgroupindex
C7×C22⋊Q82
#65102
#65092
C4×C22⋊D72
C2×(C2×D7)⋊C42
#64612
#64572
#64562
#64552
#64492
#64452
D7×C22⋊C42
#64402
#5517