Finite Group Databasebeta

← #27187429

Subgroups — #27187429

Order 12928.

230 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 54 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×#15692316C212yesmaximal, complemented
2#20266088C212yesmaximal, complemented
3#20266086C212yesmaximal, complemented
4#20265753C211yesmaximal, characteristic
5C2×#13108083—22no—
6#15692316—22no—
7#15692316C2214yes—
8C2×#13108074C411yescharacteristic
9C22×Dic202C2211yescharacteristic
10#15692255C411yescharacteristic
11#15692255C2211yescharacteristic
12#15692243C411yescharacteristic
13#15692243C2211yescharacteristic
14#15692239—22no—
15#15692235—22no—
16#117×C101C411yescharacteristic
17#13108083—24no—
18#13108083—42no—
19C2×#11717581—42no—
20#13108074—42no—
21C2×Dic202—42no—
22C2×Dic202C2×C411yescharacteristic
23C2×Dic202C2×C411yescharacteristic
24C2×Dic202D412yes—
25C2×Dic202C2311yescharacteristic
26C2×D404C2×C411yescharacteristic
27C2×C22⋊D101—21no—
28C22×Dic101D412yes—
29C2×C4×D101—21no—
30C4⋊Dic101—22no—
31C4⋊Dic101—22no—
32#13108051—22no—
33#13108051—22no—
34C22×C4×C101C2×C411yescharacteristic
35C4⋊C4×C101—21no—
36C22⋊C4×C101—21no—
37C42×C101C2×C411yescharacteristic
38C42×C101C2×C411yescharacteristic
39C23×C101C22⋊C411yescharacteristic
40C22×D101—21no—
41C2×C4×C101—22no—
42C2×C4×C101—21no—
43C2×C4×C101—21no—
44C2×C4×C101C22⋊C411yescharacteristic
45C2×C4×C101C22×C411yescharacteristic
46C22⋊D101—42no—
47C2×Dic101—22no—
48C2×Dic101—22no—
49C2×Dic101—21no—
50C2×Dic101C22⋊C412yes—
51C2×Dic101C2×D412yes—
52D404—41no—
53C4×D101—42no—
54Dic202—41no—
55Dic202—42no—
56Dic202—41no—
57Dic202—41no—
58#11717581—44no—
59C22×C101—41no—
60C22×C101#9912yes—
61C22×C101C2×C22⋊C411yescharacteristic
62C2×D101—42no—
63C2×D101—41no—
64C4×C101—21no—
65C4×C101—21no—
66C4×C101—41no—
67C4×C101—41no—
68Dic101—22no—
69Dic101—22no—
70Dic101—41no—
71C2×C101—21no—
72C2×C101—22no—
73C2×C101C2×#9911yescharacteristic
74D101—81no—
75#2133—1011nomaximal, Sylow
76C101#213311yescharacteristic, complemented, Sylow
77#536—1011no—
78C2×#101—1012no—
79#356—1012no—
80#355—1012no—
81C2×#41—1011no—
82C22×Q8—1011no—
83C2×M16—2022no—
84C4⋊Q8—1011no—
85C4⋊Q8—1011no—
86#124—1011no—
87#124—1011no—
88#123—2022no—
89C22⋊Q8—2022no—
90#117#8793811yescharacteristic
91#101—1014no—
92#101—2022no—
93#41—4042no—
94C2×Q8—1012no—
95C2×Q8—1011no—
96C2×Q8—1011no—
97C2×Q8—1011no—
98C2×Q8—4042no—
99C2×D4—1011no—
100C2×D4—2021no—
101C22×C4—1012no—
102C22×C4—2021no—
103C22×C4C2×#8793811yescharacteristic
104M16—4042no—
105M16—2024no—
106C2×C8—4042no—
107C4⋊C4—21no—
108C4⋊C4—2022no—
109C4⋊C4—2022no—
110C22⋊C4—21no—
111C22⋊C4—2022no—
112C22⋊C4—2022no—
113C42C2×#8793811yescharacteristic
114C42C2×#8793811yescharacteristic
115C23—2021no—
116C23C22⋊#8793811yescharacteristic
117Q8—4041no—
118Q8—4042no—
119Q8—4041no—
120Q8—4041no—
121D4—4041no—
122D4—4042no—
123C2×C4—22no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—1012no—
127C2×C4—1012no—
128C2×C4—4042no—
129C2×C4—2021no—
130C2×C4—2022no—
131C2×C4—2022no—
132C2×C4C22⋊#8793811yescharacteristic
133C2×C4C22×#8793811yescharacteristic
134C8—4044no—
135C22—41no—
136C22—4042no—
137C22—4041no—
138C22#1569231412yes—
139C22C2×C22⋊#8793811yescharacteristic
140C4—21no—
141C4—21no—
142C4—41no—
143C4—41no—
144C4—4041no—
145C4—2022no—
146C4—2022no—
147C2—21no—
148C2—22no—
149C2—8081no—
150C2C2×#1569231411yescharacteristic
subgroupG/H
centerC2C2×#15692314
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#117×C101C4
socleC2×C101C2×#99
prime radicalC2×#13108074C4
cosocleC2×Dic202C23
hypercenter#117#87938
hypocenterC101#2133
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#15692316C2semidirect
2. #20266088C2semidirect
3. #20266086C2semidirect
76. C101#2133semidirect

Maximal subgroups

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

subgroupindex
C2×#156923162
#202660882
#202660862
#202657532
#2133101