Finite Group Databasebeta

← #90338

Subgroups — #90338

Order 448.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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
4#6493C211yesmaximal, characteristic, complemented
5#6492C211yesmaximal, characteristic, complemented
6#6460C211yesmaximal, characteristic, complemented
7#6459C211yesmaximal, characteristic, complemented
8#6458C211yesmaximal, characteristic, complemented
9#6457C211yesmaximal, characteristic, complemented
10#6457C211yesmaximal, characteristic, complemented
11#6452C211yesmaximal, characteristic, complemented
12#6448C211yesmaximal, characteristic, complemented
13#6447C211yesmaximal, characteristic, complemented
14#6447C211yesmaximal, characteristic, complemented
15#6446C211yesmaximal, characteristic, complemented
16C2×C7×Q8C2211yescharacteristic
17C22×C4×C7C2211yescharacteristic
18C2×C22⋊D7C2211yescharacteristic
19C2×C22⋊D7C2211yescharacteristic
20C2×C22⋊D7C2211yescharacteristic
21C2×D28C2211yescharacteristic
22C2×D28C2211yescharacteristic
23C2×D28C2211yescharacteristic
24C2×C4×D7C2211yescharacteristic
25C2×C4×D7C2211yescharacteristic
26C2×C4×D7C2211yescharacteristic
27C2×Dic14C2211yescharacteristic
28C7×C4⋊C4C2211yescharacteristic
29C7×C4⋊C4C2211yescharacteristic
30C7×C4⋊C4C2211yescharacteristic
31C7×C22⋊C4C2211yescharacteristic
32C7×C22⋊C4C2211yescharacteristic
33C22⋊Dic7C2211yescharacteristic
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
41(C2×D7)⋊C4C2211yescharacteristic
42(C2×D7)⋊C4C2211yescharacteristic
43C4⋊Dic7C2211yescharacteristic
44C4⋊Dic7C2211yescharacteristic
45Dic7⋊C4C2211yescharacteristic
46Dic7⋊C4C2211yescharacteristic
47Dic7⋊C4C2211yescharacteristic
48Dic7⋊C4C2211yescharacteristic
49Dic7⋊C4C2211yescharacteristic
50C4×Dic7C2211yescharacteristic
51#562—71nomaximal, Sylow
52C23×C7C2311yescharacteristic
53C22×D7C2311yescharacteristic
54C22×D7C2311yescharacteristic
55C22×D7C2311yescharacteristic
56C7×Q8—41no—
57C2×C4×C7—41no—
58C2×C4×C7C2311yescharacteristic
59C2×C4×C7C2311yescharacteristic
60C2×C4×C7C2311yescharacteristic
61C2×C4×C7C2311yescharacteristic
62C2×C4×C7C2311yescharacteristic
63C2×C4×C7C2311yescharacteristic
64C22⋊D7—41no—
65C22⋊D7—41no—
66C22⋊D7—41no—
67C2×Dic7C2311yescharacteristic
68C2×Dic7C2311yescharacteristic
69C2×Dic7C2311yescharacteristic
70C2×Dic7C2311yescharacteristic
71C2×Dic7C2311yescharacteristic
72D28—41no—
73D28—41no—
74D28—41no—
75C4×D7—41no—
76C4×D7—41no—
77C4×D7—41no—
78Dic14—41no—
79#125—71no—
80#124—71no—
81#124—71no—
82#123—71no—
83#123—71no—
84#123—71no—
85#123—71no—
86C22⋊Q8—71no—
87C22⋊Q8—71no—
88C22⋊Q8—71no—
89C22⋊Q8D711yescharacteristic, complemented
90#121—71no—
91#121—71no—
92#121—71no—
93#121—71no—
94C22×C7—21no—
95C22×C7—21no—
96C22×C7—21no—
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—21no—
107C4×C7—21no—
108C4×C7—21no—
109C4×C7—21no—
110C4×C7—21no—
111C4×C7—21no—
112C4×C7—21no—
113Dic7—21no—
114Dic7—21no—
115Dic7—21no—
116Dic7—21no—
117Dic7—21no—
118C2×Q8—71no—
119C2×Q8C2×D711yescharacteristic
120C2×D4—71no—
121C2×D4—71no—
122C2×D4—71no—
123C2×D4—71no—
124C2×D4—71no—
125C2×D4—71no—
126C22×C4—71no—
127C22×C4—71no—
128C22×C4—71no—
129C22×C4C2×D711yescharacteristic
130C4⋊C4—71no—
131C4⋊C4—71no—
132C4⋊C4—71no—
133C4⋊C4—71no—
134C4⋊C4—71no—
135C4⋊C4—71no—
136C4⋊C4—71no—
137C4⋊C4C2×D711yescharacteristic
138C4⋊C4C2×D711yescharacteristic
139C4⋊C4C2×D711yescharacteristic
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⋊C4—71no—
150C22⋊C4C2×D711yescharacteristic
151C22⋊C4C2×D711yescharacteristic
152C42—71no—
153C2×C7—41no—
154C2×C7H(5,𝔽2)11yescharacteristic
155C2×C7H(5,𝔽2)11yescharacteristic
156C2×C72−1+411yescharacteristic
157D7—41no—
158D7—41no—
159D7—41no—
160C23—71no—
161C23—71no—
162C23—71no—
163C23C22×D711yescharacteristic
164Q8—41no—
165Q8—281no—
166D4—281no—
167D4—281no—
168D4—281no—
169D4—281no—
170D4—281no—
171D4—281no—
172C2×C4—41no—
173C2×C4—71no—
174C2×C4—71no—
175C2×C4—71no—
176C2×C4—71no—
177C2×C4—71no—
178C2×C4—281no—
179C2×C4—281no—
180C2×C4—281no—
181C2×C4C22×D711yescharacteristic
182C2×C4C22×D711yescharacteristic
183C2×C4C22×D711yescharacteristic
184C2×C4C22×D711yescharacteristic
185C2×C4C22×D711yescharacteristic
186C2×C4C22×D711yescharacteristic
187C7#56211yescharacteristic, complemented, Sylow
188C22—21no—
189C22—21no—
190C22—21no—
191C22—141no—
192C22—141no—
193C22—141no—
194C22—141no—
195C22—141no—
196C22—141no—
197C22—141no—
198C22—141no—
199C22—141no—
200C22C23×D711yescharacteristic
201C4—21no—
202C4—21no—
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—141no—
208C4—141no—
209C4—141no—
210C4—141no—
211C4—141no—
212C2—41no—
213C2—281no—
214C2—281no—
215C2—281no—
216C2#654811yescharacteristic
217C2#655111yescharacteristic
218C2#655311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D7
commutator, socle, cosocleC22×C7C24
FittingC7×C22⋊Q8C2
hypercenterC22⋊Q8D7
hypocenterC7#562
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. #6493C2semidirect
5. #6492C2semidirect
6. #6460C2semidirect
7. #6459C2semidirect
8. #6458C2semidirect
9. #6457C2semidirect
10. #6457C2semidirect
11. #6452C2semidirect
12. #6448C2semidirect
13. #6447C2semidirect
14. #6447C2semidirect
15. #6446C2semidirect
89. C22⋊Q8D7semidirect
187. C7#562semidirect

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
#64932
#64922
#64602
#64592
#64582
#64572
#64522
#64482
#64472
#64462
#5627