Finite Group Databasebeta

← #27172290

Subgroups — #27172290

Order 12928.

246 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 71 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#20265185C212yesmaximal
2#20265082C211yesmaximal, characteristic
3#20264946C211yesmaximal, characteristic
4#20264936C212yesmaximal
5#544×C101C211yesmaximal, characteristic
6C22×Dic202C2211yescharacteristic
7C2×#13108058C2212yes—
8C2×C4×Dic101C2211yescharacteristic
9#15692178—22no—
10#15692178—22no—
11#15692162—22no—
12C4⋊Q8×C101—22no—
13#125×C101—22no—
14C22⋊Q8×C101—22no—
15#117×C101C2212yes—
16C2×C4⋊C4×C101C2211yescharacteristic
17C2×Dic202—22no—
18C2×Dic202—22no—
19C2×Dic202—21no—
20C2×Dic202Q812yescomplemented
21C22×Dic101—21no—
22C22×Dic101—21no—
23#13108058—22no—
24#13108058—22no—
25C2×#11717579—22no—
26C4×Dic101—22no—
27C2×Q8×C101—22no—
28C22×C4×C101D412yes—
29C22×C4×C101C2311yescharacteristic
30C4⋊C4×C101—22no—
31C4⋊C4×C101—24no—
32C4⋊C4×C101—22no—
33C4⋊C4×C101D412yes—
34C4⋊C4×C101D412yes—
35C22⋊C4×C101—22no—
36C42×C101—22no—
37C23×C101C2×D411yescharacteristic
38Q8×C101—82no—
39C2×C4×C101—22no—
40C2×C4×C101—22no—
41C2×C4×C101—22no—
42C2×C4×C101—42no—
43C2×C4×C101C2×D412yes—
44C2×C4×C101C2×Q811yescharacteristic
45C2×C4×C101#4112yes—
46C2×C4×C101#4111yescharacteristic
47C2×Dic101—22no—
48C2×Dic101—22no—
49C2×Dic101—41no—
50C2×Dic101—41no—
51C2×Dic101—41no—
52C2×Dic101—41no—
53Dic202—24no—
54Dic202—42no—
55#11717579—42no—
56C22×C101—21no—
57C22×C101—21no—
58C22×C101#12011yescharacteristic
59C22×C101C22⋊Q811yescharacteristic
60C22×C101#12411yescharacteristic
61C4×C101—42no—
62C4×C101—42no—
63C4×C101—42no—
64C4×C101C22⋊Q812yes—
65C4×C101#12412yes—
66Dic101—42no—
67Dic101—42no—
68C2×C101—21no—
69C2×C101—21no—
70C2×C101#39311yescharacteristic
71C2×C101#45511yescharacteristic
72C2×C101#45611yescharacteristic
73#2028—1011nomaximal, Sylow
74C101#202811yescharacteristic, complemented, Sylow
75#544D10111yescharacteristic
76#419—1012no—
77#391—1011no—
78#343—1011no—
79#339—1012no—
80C22×Q8—1011no—
81C2×M16—1012no—
82C4⋊Q8—22no—
83#125—22no—
84C22⋊Q8—22no—
85#117C2×D10112yes—
86C2×C4⋊C4C2×D10111yescharacteristic
87C2×C42—1011no—
88#103—2022no—
89#103—2022no—
90#95—2022no—
91C2×Q8—22no—
92C2×Q8—1012no—
93C2×Q8—2022no—
94C2×Q8—2022no—
95C2×Q8—2021no—
96C22×C4—2021no—
97C22×C4—2021no—
98C22×C4C22⋊D10112yes—
99C22×C4C22×D10111yescharacteristic
100M16—2022no—
101M16—2022no—
102C2×C8—2022no—
103C4⋊C4—22no—
104C4⋊C4—24no—
105C4⋊C4—22no—
106C4⋊C4C22⋊D10112yes—
107C4⋊C4C22⋊D10112yes—
108C22⋊C4—22no—
109C42—22no—
110C42—2022no—
111C23C2×C22⋊D10111yescharacteristic
112Q8—82no—
113Q8—4042no—
114Q8—2024no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—42no—
119C2×C4—4041no—
120C2×C4—4041no—
121C2×C4—4041no—
122C2×C4—4041no—
123C2×C4—2022no—
124C2×C4—2022no—
125C2×C4C2×C22⋊D10112yes—
126C2×C4Q8×D10111yescharacteristic
127C2×C4#1310807512yes—
128C2×C4#1310807511yescharacteristic
129C8—4042no—
130C22—21no—
131C22—21no—
132C22#1569222411yescharacteristic
133C22#1569223111yescharacteristic
134C22#1569224411yescharacteristic
135C4—42no—
136C4—42no—
137C4—42no—
138C4—4042no—
139C4—4042no—
140C4#1569223112yes—
141C4#1569224412yes—
142C2—21no—
143C2—21no—
144C2#2026508811yescharacteristic
145C2#2026535511yescharacteristic
146C2#2026536011yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#544×C101C2
socleC22×C101#120
prime radicalC23×C101C2×D4
hypercenter#544D101
hypocenterC101#2028
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
20. C2×Dic202Q8semidirect
74. C101#2028semidirect

Maximal subgroups

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

subgroupindex
#202651852
#202650822
#202649462
#202649362
#544×C1012
#2028101