Finite Group Databasebeta

← #27170122

Subgroups — #27170122

Order 12928.

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

normal subgroups only

This group has 228 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#20265803C211yesmaximal, characteristic
2#20265507C212yesmaximal, complemented
3#20264913C212yesmaximal, complemented
4#20264901C211yesmaximal, characteristic
5#397×C101C211yesmaximal, characteristic
6#15692234—21no—
7#15692234C2212yes—
8#15692232—22no—
9#15692232—22no—
10C2×C22⋊Dic101—21no—
11#15692186—22no—
12#15692178—22no—
13C22⋊#11717579C2212yes—
14C2×C4⋊C4×C101C2211yescharacteristic
15C2×C22⋊C4×C101—21no—
16C2×C22⋊C4×C101C2211yescharacteristic
17#95×C101—21no—
18#95×C101C2211yescharacteristic
19C2×Dic202—21no—
20C2×Dic202D412yes—
21C22×Dic101—21no—
22C22×Dic101—21no—
23C2×#11717579—22no—
24C4⋊Dic101—21no—
25C4⋊Dic101D412yes—
26#13108053—22no—
27#13108053—22no—
28C22⋊Dic101—22no—
29C22⋊Dic101—22no—
30C24×C101D411yescharacteristic
31C22×C4×C101—21no—
32C22×C4×C101—21no—
33C22×C4×C101D411yescharacteristic
34C22×C4×C101C2311yescharacteristic
35C4⋊C4×C101—22no—
36C22⋊C4×C101—22no—
37C22⋊C4×C101—42no—
38C23×C101—21no—
39C23×C101—21no—
40C23×C101—41no—
41C23×C101—41no—
42C23×C101SD1612yes—
43C23×C101C2×D411yescharacteristic
44C2×C4×C101—21no—
45C2×C4×C101—41no—
46C2×C4×C101—41no—
47C2×C4×C101—41no—
48C2×C4×C101—41no—
49C2×C4×C101—41no—
50C2×C4×C101—41no—
51C2×C4×C101—41no—
52C2×C4×C101—41no—
53C2×C4×C101C2×D412yes—
54C2×Dic101—22no—
55C2×Dic101—22no—
56C2×Dic101—81no—
57C2×Dic101—81no—
58Dic202—42no—
59Dic202—81no—
60#11717579—42no—
61C22×C101—21no—
62C22×C101—21no—
63C22×C101—22no—
64C22×C101—42no—
65C22×C101—41no—
66C22×C101—41no—
67C22×C101—81no—
68C22×C101#12011yescharacteristic
69C22×C101C2×SD1611yescharacteristic
70C22×C101#13711yescharacteristic
71C4×C101—22no—
72C4×C101—41no—
73C4×C101—81no—
74C4×C101—81no—
75Dic101—42no—
76Dic101—42no—
77C2×C101—21no—
78C2×C101—21no—
79C2×C101—42no—
80C2×C101#44811yescharacteristic
81C2×C101#45511yescharacteristic
82C2×C101#45811yescharacteristic
83#1608—1011nomaximal, Sylow
84C101#160811yescharacteristic, complemented, Sylow
85#543—1011no—
86#483—1012no—
87#397D10111yescharacteristic
88#328—1012no—
89#323—1011no—
90C22⋊Q8—1012no—
91C22⋊Q8—2022no—
92C22⋊Q8—2022no—
93C22⋊Q8—2021no—
94C2×C4⋊C4C2×D10111yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—2021no—
97C2×C22⋊C4C2×D10111yescharacteristic
98#106—2022no—
99#103—2022no—
100C22⋊C8—1012no—
101#95—21no—
102#95C2×D10111yescharacteristic
103C24C22⋊D10111yescharacteristic
104C2×Q8—1012no—
105C2×Q8—2021no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—2021no—
109C22×C4—2021no—
110C22×C4C22⋊D10111yescharacteristic
111C22×C4C22×D10111yescharacteristic
112C2×C8—2022no—
113C4⋊C4—22no—
114C4⋊C4—1012no—
115C4⋊C4—2022no—
116C4⋊C4—2021no—
117C4⋊C4—2022no—
118C22⋊C4—22no—
119C22⋊C4—42no—
120C22⋊C4—2022no—
121C22⋊C4—2022no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23#1310806212yes—
127C23C2×C22⋊D10111yescharacteristic
128Q8—8081no—
129Q8—4042no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—8081no—
140C2×C4—8081no—
141C2×C4—2022no—
142C2×C4—2022no—
143C2×C4D4×D10112yes—
144C8—4042no—
145C22—21no—
146C22—21no—
147C22—22no—
148C22—42no—
149C22—41no—
150C22—41no—
151C22—81no—
152C22#1569222311yescharacteristic
153C22C2×#1310806211yescharacteristic
154C22#1569228511yescharacteristic
155C4—22no—
156C4—41no—
157C4—81no—
158C4—81no—
159C4—4042no—
160C4—4042no—
161C2—21no—
162C2—21no—
163C2—42no—
164C2#2026531511yescharacteristic
165C2#2026535411yescharacteristic
166C2#2026536611yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#397×C101C2
socleC22×C101#120
prime radicalC24×C101D4
hypercenter#397D101
hypocenterC101#1608
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #20265507C2semidirect
3. #20264913C2semidirect
84. C101#1608semidirect

Maximal subgroups

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

subgroupindex
#202658032
#202655072
#202649132
#202649012
#397×C1012
#1608101