Finite Group Databasebeta

← #28402021

Subgroups — #28402021

Order 14464.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#20299210C211yesmaximal, characteristic, complemented
2#20299203C211yesmaximal, characteristic
3#20298906C212yesmaximal, complemented
4#20298899C212yesmaximal, complemented
5#20298844C212yesmaximal, complemented
6#20298827C212yesmaximal, complemented
7#20298804C212yesmaximal, complemented
8#20298799C212yesmaximal, complemented
9#424×C113C211yesmaximal, characteristic, complemented
10C2×#14268636C2212yes—
11C2×#14268636C2212yes—
12#16813622C2211yescharacteristic
13#16813619C2211yescharacteristic
14#16813615—22no—
15#16813614C2211yescharacteristic
16#16813614C2211yescharacteristic
17#16813612—22no—
18#16813607—22no—
19#16813601C2212yes—
20#16813596C2212yes—
21#16813595—22no—
22C4×Dic226C2212yes—
23C4×D452C2212yes—
24#16813556C2212yes—
25#16813554C2212yes—
26#16813547C2212yes—
27#16813547C2212yes—
28#16813544C2212yes—
29#16813544C2212yes—
30C2×M16×C113C2211yescharacteristic
31C22×C8×C113C2211yescharacteristic
32#117×C113C2211yescharacteristic
33C4⋊C8×C113C2212yes—
34C4⋊C8×C113C2212yes—
35C2×Dic226C2312yes—
36C2×D452—22no—
37C2×D452C2312yes—
38C2×C22⋊D113—22no—
39C2×C4×D113—22no—
40#14268636—42no—
41#14268636—42no—
42C4⋊Dic113—22no—
43C4⋊Dic113C2312yes—
44C4⋊Dic113C2312yes—
45#14268626—22no—
46#14268626—22no—
47#14268626—22no—
48#14268626—22no—
49C22⋊Dic113—22no—
50#14268624—22no—
51#14268624—22no—
52C4×Dic113—22no—
53C22×C4×C113C2311yescharacteristic
54M16×C113—41no—
55C2×C8×C113—41no—
56C2×C8×C113C2312yes—
57C2×C8×C113C2312yes—
58C4⋊C4×C113D412yes—
59C22⋊C4×C113D412yes—
60C42×C113C2311yescharacteristic
61C42×C113C2311yescharacteristic
62C23×C113C2×D411yescharacteristic
63C22×D113—22no—
64C2×C4×C113—21no—
65C2×C4×C113—21no—
66C2×C4×C113C2×D412yes—
67C2×C4×C113C2×D412yes—
68C2×C4×C113C2×D411yescharacteristic
69C2×C4×C113C2411yescharacteristic
70C22⋊D113—82no—
71C2×Dic113—22no—
72C2×Dic113—22no—
73C2×Dic113—22no—
74D452—22no—
75D452—22no—
76D452—82no—
77C4×D113—82no—
78Dic226—22no—
79Dic226—22no—
80C8×C113—22no—
81C8×C113—22no—
82C22×C113—21no—
83C22×C113—22no—
84C22×C113C22×D411yescharacteristic
85C2×D113—42no—
86C2×D113—42no—
87C2×D113—42no—
88C4×C113—21no—
89C4×C113—22no—
90C4×C113—22no—
91C4×C113H(5,𝔽2)11yescharacteristic
92C4×C1132−1+411yescharacteristic
93Dic113—42no—
94Dic113—42no—
95Dic113—42no—
96C2×C113—41no—
97C2×C113#53711yescharacteristic
98C2×C113#57712yes—
99D113—82no—
100#3260—1131nomaximal, Sylow
101C113#326011yescharacteristic, complemented, Sylow
102#541—1131no—
103#540—1131no—
104#479—1132no—
105#478—1132no—
106#468—1132no—
107#465—1132no—
108#461—1132no—
109#460—1132no—
110#424D11311yescharacteristic, complemented
111C2×SD16—1132no—
112C2×SD16—1132no—
113C2×M16C2×D11311yescharacteristic
114C22×C8C2×D11311yescharacteristic
115C4⋊Q8—1131no—
116#127—1131no—
117#126—2262no—
118#125—1131no—
119#125—1131no—
120#125—2262no—
121#123—2262no—
122C22⋊Q8—1132no—
123#121—1132no—
124#121—2262no—
125C4×Q8—1132no—
126C4×D4—1132no—
127#117C2×D11311yescharacteristic
128#107—1132no—
129#106—1132no—
130C4⋊C8C2×D11312yes—
131C4⋊C8C2×D11312yes—
132#103—1132no—
133#103—1132no—
134#102—1132no—
135#102—1132no—
136C2×Q8—1132no—
137C2×D4—1132no—
138C2×D4—2262no—
139C2×D4—2262no—
140C22×C4—2262no—
141C22×C4C22×D11311yescharacteristic
142SD16—4522no—
143SD16—4522no—
144M16—41no—
145C2×C8—41no—
146C2×C8C22×D11312yes—
147C2×C8C22×D11312yes—
148C4⋊C4—1132no—
149C4⋊C4—1132no—
150C4⋊C4—2262no—
151C4⋊C4—2262no—
152C4⋊C4—2262no—
153C4⋊C4—2262no—
154C4⋊C4—2262no—
155C4⋊C4D45212yes—
156C22⋊C4—2262no—
157C22⋊C4—2262no—
158C22⋊C4—2262no—
159C22⋊C4D45212yes—
160C42—2262no—
161C42C22×D11311yescharacteristic
162C42C22×D11311yescharacteristic
163C23—2262no—
164C23C2×D45211yescharacteristic
165Q8—2262no—
166Q8—2262no—
167D4—9042no—
168D4—9042no—
169D4—2262no—
170D4—2262no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—9042no—
174C2×C4—2262no—
175C2×C4—2262no—
176C2×C4—2262no—
177C2×C4C2×D45212yes—
178C2×C4C2×D45212yes—
179C2×C4C2×D45211yescharacteristic
180C2×C4C23×D11311yescharacteristic
181C8—22no—
182C8—22no—
183C22—22no—
184C22—21no—
185C22—4522no—
186C22—4522no—
187C22—4522no—
188C22C22×D45211yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—4522no—
193C4—4522no—
194C4—4522no—
195C4#1681366711yescharacteristic
196C4#1681367011yescharacteristic
197C2—41no—
198C2—9042no—
199C2#2029917911yescharacteristic
200C2#2029941012yes—
subgroupG/H
centerC22C22×D452
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#424×C113C2
socleC22×C113C22×D4
prime radical#20299203C2
hypercenter#424D113
hypocenterC113#3260
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #20299210C2semidirect
3. #20298906C2semidirect
4. #20298899C2semidirect
5. #20298844C2semidirect
6. #20298827C2semidirect
7. #20298804C2semidirect
8. #20298799C2semidirect
9. #424×C113C2semidirect
101. C113#3260semidirect
110. #424D113semidirect

Maximal subgroups

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

subgroupindex
#202992102
#202992032
#202989062
#202988992
#202988442
#202988272
#202988042
#202987992
#424×C1132
#3260113