Finite Group Databasebeta

← #27144875

Subgroups — #27144875

Order 12416.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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×#15690779C211yesmaximal, characteristic
2#20260635C213yesmaximal, complemented
3#20260629C213yesmaximal, complemented
4#20260623C213yesmaximal, complemented
5#20260611C213yesmaximal, complemented
6#20260591C211yesmaximal, characteristic, complemented
7#379×C97C211yesmaximal, characteristic, complemented
8C22×C22⋊D97C2213yes—
9C22×C4×D97C2211yescharacteristic, complemented
10#15690779C418yescomplemented
11C2×#13095490C2213yescomplemented
12C2×C22⋊Dic97C2213yes—
13C2×C22⋊Dic97C2213yes—
14C2×C22⋊Dic97C2213yes—
15C2×#13095488C2213yescomplemented
16C2×C4×Dic97C2213yescomplemented
17#15690717C2213yescomplemented
18#15690717C2213yescomplemented
19C25×C97C2211yescharacteristic
20C2×C22⋊C4×C97C2213yes—
21C2×C22⋊C4×C97C2213yes—
22C23×D97C2311yescharacteristic
23C2×C22⋊D97—43no—
24C2×C22⋊D97—43no—
25C2×C22⋊D97C2×C4112yes—
26C22×Dic97C2313yes—
27C22×Dic97C2311yescharacteristic
28C22×Dic97C2313yes—
29C2×C4×D97—43no—
30#13095490—43no—
31C22⋊Dic97—43no—
32C22⋊Dic97—43no—
33C22⋊Dic97C2×C4112yescomplemented
34#13095488—43no—
35C4×Dic97—43no—
36C24×C97—43no—
37C24×C97—43no—
38C24×C97C2×C414yescomplemented
39C24×C97C2313yes—
40C22×C4×C97C2313yes—
41C22×C4×C97C2311yescharacteristic
42C22⋊C4×C97—43no—
43C22⋊C4×C97—43no—
44C23×C97—23no—
45C23×C97—23no—
46C23×C97—23no—
47C23×C97—23no—
48C23×C97—23no—
49C23×C97—23no—
50C23×C97—412no—
51C23×C97—412no—
52C23×C97—44no—
53C23×C97C22×C416yes—
54C23×C97C2411yescharacteristic
55C22×D97—23no—
56C22×D97—23no—
57C22×D97C22×C412yes—
58C2×C4×C97—23no—
59C2×C4×C97—23no—
60C2×C4×C97—23no—
61C2×C4×C97—23no—
62C22⋊D97—412no—
63C2×Dic97—23no—
64C2×Dic97—23no—
65C2×Dic97—23no—
66C2×Dic97—23no—
67C2×Dic97—23no—
68C2×Dic97—23no—
69C2×Dic97C22×C416yes—
70C4×D97—44no—
71C22×C97—26no—
72C22×C97—26no—
73C22×C97—26no—
74C22×C97—416no—
75C22×C97—43no—
76C22×C97—43no—
77C22×C97—43no—
78C22×C97—43no—
79C22×C97C23×C411yescharacteristic
80C22×C97H(5,𝔽2)13yes—
81C22×C97H(5,𝔽2)13yes—
82C2×D97—26no—
83C2×D97—43no—
84C2×D97—41no—
85C4×C97—43no—
86C4×C97—41no—
87Dic97—26no—
88Dic97—43no—
89Dic97—41no—
90C2×C97—46no—
91C2×C97#51813yes—
92C2×C97#56013yes—
93C2×C97H(3,𝔽4)11yescharacteristic
94D97—42no—
95#2375—971nomaximal, Sylow
96C97#237511yescharacteristic, complemented, Sylow
97C2×#120—971no—
98#390—973no—
99#388—973no—
100#388—973no—
101#386—973no—
102#379—971no—
103#379D9711yescharacteristic, complemented
104C25C2×D9711yescharacteristic
105C22×D4—973no—
106C23×C4—971no—
107#120—978no—
108C2×C4⋊C4—973no—
109C2×C22⋊C4—973no—
110C2×C22⋊C4—973no—
111C2×C22⋊C4—973no—
112C2×C22⋊C4—973no—
113C2×C22⋊C4C2×D9713yes—
114C2×C22⋊C4C2×D9713yes—
115C2×C42—973no—
116#95—973no—
117#95—973no—
118C24—43no—
119C24—43no—
120C24—971no—
121C24C4×D9714yescomplemented
122C24C22×D9713yes—
123C2×D4—9712no—
124C2×D4—3883no—
125C2×D4—3883no—
126C22×C4—973no—
127C22×C4—971no—
128C22×C4—973no—
129C22×C4—3883no—
130C22×C4C22×D9711yescharacteristic
131C22×C4C22×D9713yes—
132C4⋊C4—3883no—
133C22⋊C4—43no—
134C22⋊C4—43no—
135C22⋊C4—9712no—
136C22⋊C4—3883no—
137C22⋊C4—3883no—
138C22⋊C4—3883no—
139C42—3883no—
140C23—23no—
141C23—23no—
142C23—23no—
143C23—23no—
144C23—23no—
145C23—23no—
146C23—412no—
147C23—412no—
148C23—44no—
149C23—972no—
150C23—1943no—
151C23—1943no—
152C23C2×C4×D9716yes—
153C23C23×D9711yescharacteristic
154D4—38812no—
155C2×C4—23no—
156C2×C4—23no—
157C2×C4—23no—
158C2×C4—23no—
159C2×C4—976no—
160C2×C4—3884no—
161C2×C4—1943no—
162C2×C4—1943no—
163C2×C4—1943no—
164C2×C4—1943no—
165C2×C4—1943no—
166C2×C4—1943no—
167C22—26no—
168C22—26no—
169C22—26no—
170C22—416no—
171C22—43no—
172C22—43no—
173C22—43no—
174C22—43no—
175C22—3883no—
176C22—3881no—
177C22—1946no—
178C22C22×C4×D9711yescharacteristic
179C22#1569085513yes—
180C22#1569085513yes—
181C4—43no—
182C4—41no—
183C4—3883no—
184C4—3881no—
185C4—1946no—
186C2—46no—
187C2—3882no—
188C2#2026120813yes—
189C2#2026147413yes—
190C2#2026147811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#379×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690779C2
hypercenter#379D97
hypocenterC97#2375
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #20260635C2semidirect
3. #20260629C2semidirect
4. #20260623C2semidirect
5. #20260611C2semidirect
6. #20260591C2semidirect
7. #379×C97C2semidirect
9. C22×C4×D97C22semidirect
10. #15690779C4semidirect
11. C2×#13095490C22semidirect
15. C2×#13095488C22semidirect
16. C2×C4×Dic97C22semidirect
17. #15690717C22semidirect
18. #15690717C22semidirect
33. C22⋊Dic97C2×C4semidirect
38. C24×C97C2×C4semidirect
96. C97#2375semidirect
103. #379D97semidirect
121. C24C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907792
#202606352
#202606292
#202606232
#202606112
#202605912
#379×C972
#237597