Finite Group Databasebeta

← #27170142

Subgroups — #27170142

Order 12928.

210 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265808C211yesmaximal, characteristic
2#20265508C212yesmaximal
3#20265107C211yesmaximal, characteristic
4#20264915C212yesmaximal
5#324×C101C211yesmaximal, characteristic
6#15692254—22no—
7#15692246—22no—
8#15692234—21no—
9#15692234C2212yes—
10#15692214—21no—
11C2×C4⋊Dic101C2211yescharacteristic
12C2×#13108053—21no—
13#15692187—22no—
14#15692180—22no—
15#15692162—21no—
16#15692162C2211yescharacteristic
17C2×C4⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yes—
19C2×Dic202—21no—
20C2×Dic202D412yes—
21C22×Dic101—21no—
22C22×Dic101—21no—
23C4⋊Dic101—21no—
24C4⋊Dic101—21no—
25C4⋊Dic101—21no—
26C4⋊Dic101—22no—
27C4⋊Dic101D412yes—
28#13108053—22no—
29#13108053—22no—
30#13108053—42no—
31C22⋊Dic101—22no—
32C4×Dic101—22no—
33C22×C4×C101D411yescharacteristic
34C22×C4×C101D411yescharacteristic
35C22×C4×C101C2311yescharacteristic
36C2×C8×C101—22no—
37C4⋊C4×C101—22no—
38C23×C101C2×D411yescharacteristic
39C2×C4×C101—21no—
40C2×C4×C101—21no—
41C2×C4×C101—21no—
42C2×C4×C101—41no—
43C2×C4×C101—41no—
44C2×C4×C101SD1612yes—
45C2×C4×C101C2×D412yes—
46C2×Dic101—22no—
47C2×Dic101—22no—
48C2×Dic101—22no—
49C2×Dic101—41no—
50C2×Dic101—41no—
51C2×Dic101—41no—
52C2×Dic101—41no—
53C2×Dic101—41no—
54Dic202—42no—
55Dic202—81no—
56C8×C101—42no—
57C22×C101—21no—
58C22×C101—21no—
59C22×C101#12011yescharacteristic
60C22×C101C2×SD1611yescharacteristic
61C22×C101#13711yescharacteristic
62C4×C101—22no—
63C4×C101—22no—
64C4×C101—41no—
65Dic101—42no—
66Dic101—42no—
67Dic101—42no—
68Dic101—81no—
69C2×C101—21no—
70C2×C101—21no—
71C2×C101#44811yescharacteristic
72C2×C101#45611yescharacteristic
73C2×C101#45811yescharacteristic
74#1611—1011nomaximal, Sylow
75C101#161111yescharacteristic, complemented, Sylow
76#544—1011no—
77#483—1012no—
78#398—1011no—
79#328—1012no—
80#324D10111yescharacteristic
81C4⋊Q8—2022no—
82#125—2022no—
83C22⋊Q8—1012no—
84C22⋊Q8—2021no—
85#117—2021no—
86C2×C4⋊C4—1011no—
87C2×C4⋊C4—2021no—
88C2×C4⋊C4C2×D10111yescharacteristic
89#106—2022no—
90#103—2022no—
91C22⋊C8C2×D10112yes—
92#95—1011no—
93#95—2021no—
94C2×Q8—1012no—
95C2×Q8—2021no—
96C22×C4—2021no—
97C22×C4—2021no—
98C22×C4D40411yescharacteristic
99C22×C4D40411yescharacteristic
100C22×C4C22×D10111yescharacteristic
101C2×C8—22no—
102C4⋊C4—22no—
103C4⋊C4—1012no—
104C4⋊C4—4042no—
105C4⋊C4—2022no—
106C4⋊C4—2021no—
107C4⋊C4—2021no—
108C4⋊C4—2021no—
109C4⋊C4—2022no—
110C4⋊C4—2022no—
111C22⋊C4—2022no—
112C42—2022no—
113C23C2×D40411yescharacteristic
114Q8—8081no—
115Q8—4042no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—4041no—
122C2×C4—4041no—
123C2×C4—4041no—
124C2×C4—4041no—
125C2×C4—4041no—
126C2×C4—2022no—
127C2×C4—2022no—
128C2×C4—2022no—
129C2×C4#1310806312yes—
130C2×C4D4×D10112yes—
131C8—42no—
132C22—21no—
133C22—21no—
134C22#1569222511yescharacteristic
135C22C2×#1310806311yescharacteristic
136C22#1569228811yescharacteristic
137C4—22no—
138C4—22no—
139C4—41no—
140C4—8081no—
141C4—4042no—
142C4—4042no—
143C4—4042no—
144C2—21no—
145C2—21no—
146C2#2026531811yescharacteristic
147C2#2026536211yescharacteristic
148C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#324×C101C2
socleC22×C101#120
prime radicalC23×C101C2×D4
hypercenter#324D101
hypocenterC101#1611
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
75. C101#1611semidirect

Maximal subgroups

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

subgroupindex
#202658082
#202655082
#202651072
#202649152
#324×C1012
#1611101