Finite Group Databasebeta

← #27172293

Subgroups — #27172293

Order 12928.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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#20265186C212yesmaximal
3#20264946C211yesmaximal, characteristic
4#20264937C212yesmaximal
5#391×C101C211yesmaximal, characteristic
6C2×#13108058C2212yes—
7#15692254—22no—
8#15692246—22no—
9#15692234—22no—
10#15692214C2212yes—
11#15692179—22no—
12#15692179—22no—
13C22×Q8×C101C2211yescharacteristic
14C2×C4⋊C4×C101C2211yescharacteristic
15C2×C42×C101C2211yescharacteristic
16#95×C101—22no—
17C2×Dic202—22no—
18#13108058—22no—
19#13108058—22no—
20C2×#11717579—22no—
21C4⋊Dic101—22no—
22C4⋊Dic101D412yes—
23C4⋊Dic101D412yes—
24#13108053—24no—
25C22⋊Dic101—22no—
26C4×Dic101—22no—
27C2×Q8×C101—21no—
28C2×Q8×C101—22no—
29C2×Q8×C101—22no—
30C2×Q8×C101Q812yescomplemented
31C22×C4×C101—21no—
32C22×C4×C101—21no—
33C22×C4×C101D412yes—
34C22×C4×C101C2311yescharacteristic
35C4⋊C4×C101—22no—
36C42×C101—22no—
37C23×C101C2×D411yescharacteristic
38Q8×C101—24no—
39Q8×C101—42no—
40C2×C4×C101—22no—
41C2×C4×C101—22no—
42C2×C4×C101—22no—
43C2×C4×C101—41no—
44C2×C4×C101—41no—
45C2×C4×C101—41no—
46C2×C4×C101—41no—
47C2×C4×C101—42no—
48C2×C4×C101C2×D412yes—
49C2×C4×C101C2×Q811yescharacteristic
50C2×C4×C101#4112yes—
51C2×C4×C101#4111yescharacteristic
52C2×Dic101—22no—
53C2×Dic101—22no—
54Dic202—82no—
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#544—1011no—
76#419—1012no—
77#391D10111yescharacteristic
78#343—1011no—
79#339—1012no—
80C22×Q8C2×D10111yescharacteristic
81C2×M16—1012no—
82C4⋊Q8—2022no—
83#125—2022no—
84C22⋊Q8—2022no—
85#117—1012no—
86C2×C4⋊C4C2×D10111yescharacteristic
87C2×C42C2×D10111yescharacteristic
88#103—2022no—
89#103—2022no—
90#95—22no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—2022no—
95C2×Q8Dic20212yescomplemented
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C22⋊D10112yes—
99C22×C4C22×D10111yescharacteristic
100M16—2022no—
101M16—2022no—
102C2×C8—2022no—
103C4⋊C4—22no—
104C4⋊C4—1012no—
105C4⋊C4—1012no—
106C4⋊C4—2024no—
107C4⋊C4—2022no—
108C22⋊C4—2022no—
109C42—22no—
110C42—2022no—
111C23C2×C22⋊D10111yescharacteristic
112Q8—24no—
113Q8—42no—
114Q8—8082no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—2022no—
124C2×C4—2022no—
125C2×C4D4×D10112yes—
126C2×C4C2×Dic20211yescharacteristic
127C2×C4#1310807312yes—
128C2×C4#1310807411yescharacteristic
129C8—4042no—
130C22—21no—
131C22—21no—
132C22#1569222311yescharacteristic
133C22#1569223411yescharacteristic
134C22#1569224211yescharacteristic
135C4—42no—
136C4—42no—
137C4—42no—
138C4—4042no—
139C4—4042no—
140C4#1569223212yes—
141C4#1569224112yes—
142C2—21no—
143C2—21no—
144C2#2026508911yescharacteristic
145C2#2026535411yescharacteristic
146C2#2026535911yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#391×C101C2
socleC22×C101#120
prime radicalC23×C101C2×D4
hypercenter#391D101
hypocenterC101#2028
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
30. C2×Q8×C101Q8semidirect
74. C101#2028semidirect
95. C2×Q8Dic202semidirect

Maximal subgroups

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

subgroupindex
#202658082
#202651862
#202649462
#202649372
#391×C1012
#2028101