Finite Group Databasebeta

← #27174271

Subgroups — #27174271

Order 12928.

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×#15692224C211yesmaximal, characteristic
2#20265077C213yesmaximal, complemented
3#20265071C213yesmaximal, complemented
4#20265065C213yesmaximal, complemented
5#20265053C213yesmaximal, complemented
6#20265033C211yesmaximal, characteristic, complemented
7#379×C101C211yesmaximal, characteristic, complemented
8C22×C22⋊D101C2213yes—
9C22×C4×D101C2211yescharacteristic, complemented
10#15692224C418yescomplemented
11C2×#13108053C2213yescomplemented
12C2×C22⋊Dic101C2213yes—
13C2×C22⋊Dic101C2213yes—
14C2×C22⋊Dic101C2213yes—
15C2×#13108051C2213yescomplemented
16C2×C4×Dic101C2213yescomplemented
17#15692162C2213yescomplemented
18#15692162C2213yescomplemented
19C25×C101C2211yescharacteristic
20C2×C22⋊C4×C101C2213yes—
21C2×C22⋊C4×C101C2213yes—
22C23×D101C2311yescharacteristic
23C2×C22⋊D101—43no—
24C2×C22⋊D101—43no—
25C2×C22⋊D101C2×C4112yes—
26C22×Dic101C2311yescharacteristic
27C22×Dic101C2313yes—
28C22×Dic101C2313yes—
29C2×C4×D101—43no—
30#13108053—43no—
31C22⋊Dic101—43no—
32C22⋊Dic101—43no—
33C22⋊Dic101C2×C4112yescomplemented
34#13108051—43no—
35C4×Dic101—43no—
36C24×C101—43no—
37C24×C101—43no—
38C24×C101C2×C414yescomplemented
39C24×C101C2313yes—
40C22×C4×C101C2311yescharacteristic
41C22×C4×C101C2313yes—
42C22⋊C4×C101—43no—
43C22⋊C4×C101—43no—
44C23×C101—23no—
45C23×C101—23no—
46C23×C101—23no—
47C23×C101—23no—
48C23×C101—23no—
49C23×C101—23no—
50C23×C101—412no—
51C23×C101—412no—
52C23×C101—44no—
53C23×C101C22×C416yes—
54C23×C101C2411yescharacteristic
55C22×D101—23no—
56C22×D101—23no—
57C22×D101C22×C412yes—
58C2×C4×C101—23no—
59C2×C4×C101—23no—
60C2×C4×C101—23no—
61C2×C4×C101—23no—
62C22⋊D101—412no—
63C2×Dic101—23no—
64C2×Dic101—23no—
65C2×Dic101—23no—
66C2×Dic101—23no—
67C2×Dic101—23no—
68C2×Dic101—23no—
69C2×Dic101C22×C416yes—
70C4×D101—44no—
71C22×C101—26no—
72C22×C101—26no—
73C22×C101—26no—
74C22×C101—416no—
75C22×C101—43no—
76C22×C101—43no—
77C22×C101—43no—
78C22×C101—43no—
79C22×C101C23×C411yescharacteristic
80C22×C101H(5,𝔽2)13yes—
81C22×C101H(5,𝔽2)13yes—
82C2×D101—26no—
83C2×D101—43no—
84C2×D101—41no—
85C4×C101—43no—
86C4×C101—41no—
87Dic101—26no—
88Dic101—43no—
89Dic101—41no—
90C2×C101—46no—
91C2×C101#51813yes—
92C2×C101#56013yes—
93C2×C101H(3,𝔽4)11yescharacteristic
94D101—42no—
95#2375—1011nomaximal, Sylow
96C101#237511yescharacteristic, complemented, Sylow
97C2×#120—1011no—
98#390—1013no—
99#388—1013no—
100#388—1013no—
101#386—1013no—
102#379—1011no—
103#379D10111yescharacteristic, complemented
104C25C2×D10111yescharacteristic
105C22×D4—1013no—
106C23×C4—1011no—
107#120—1018no—
108C2×C4⋊C4—1013no—
109C2×C22⋊C4—1013no—
110C2×C22⋊C4—1013no—
111C2×C22⋊C4—1013no—
112C2×C22⋊C4—1013no—
113C2×C22⋊C4C2×D10113yes—
114C2×C22⋊C4C2×D10113yes—
115C2×C42—1013no—
116#95—1013no—
117#95—1013no—
118C24—43no—
119C24—43no—
120C24—1011no—
121C24C4×D10114yescomplemented
122C24C22×D10113yes—
123C2×D4—10112no—
124C2×D4—4043no—
125C2×D4—4043no—
126C22×C4—1013no—
127C22×C4—1011no—
128C22×C4—1013no—
129C22×C4—4043no—
130C22×C4C22×D10111yescharacteristic
131C22×C4C22×D10113yes—
132C4⋊C4—4043no—
133C22⋊C4—43no—
134C22⋊C4—43no—
135C22⋊C4—10112no—
136C22⋊C4—4043no—
137C22⋊C4—4043no—
138C22⋊C4—4043no—
139C42—4043no—
140C23—23no—
141C23—23no—
142C23—23no—
143C23—23no—
144C23—23no—
145C23—23no—
146C23—412no—
147C23—412no—
148C23—44no—
149C23—1012no—
150C23—2023no—
151C23—2023no—
152C23C2×C4×D10116yes—
153C23C23×D10111yescharacteristic
154D4—40412no—
155C2×C4—23no—
156C2×C4—23no—
157C2×C4—23no—
158C2×C4—23no—
159C2×C4—1016no—
160C2×C4—4044no—
161C2×C4—2023no—
162C2×C4—2023no—
163C2×C4—2023no—
164C2×C4—2023no—
165C2×C4—2023no—
166C2×C4—2023no—
167C22—26no—
168C22—26no—
169C22—26no—
170C22—416no—
171C22—43no—
172C22—43no—
173C22—43no—
174C22—43no—
175C22—4043no—
176C22—4041no—
177C22—2026no—
178C22C22×C4×D10111yescharacteristic
179C22#1569230013yes—
180C22#1569230013yes—
181C4—43no—
182C4—41no—
183C4—4043no—
184C4—4041no—
185C4—2026no—
186C2—46no—
187C2—4042no—
188C2#2026565013yes—
189C2#2026591613yes—
190C2#2026592011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#379×C101C2
socle, cosocleC23×C101C24
prime radicalC2×#15692224C2
hypercenter#379D101
hypocenterC101#2375
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #20265077C2semidirect
3. #20265071C2semidirect
4. #20265065C2semidirect
5. #20265053C2semidirect
6. #20265033C2semidirect
7. #379×C101C2semidirect
9. C22×C4×D101C22semidirect
10. #15692224C4semidirect
11. C2×#13108053C22semidirect
15. C2×#13108051C22semidirect
16. C2×C4×Dic101C22semidirect
17. #15692162C22semidirect
18. #15692162C22semidirect
33. C22⋊Dic101C2×C4semidirect
38. C24×C101C2×C4semidirect
96. C101#2375semidirect
103. #379D101semidirect
121. C24C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922242
#202650772
#202650712
#202650652
#202650532
#202650332
#379×C1012
#2375101