Finite Group Databasebeta

← #27170067

Subgroups — #27170067

Order 12928.

310 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 57 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#20265310C212yesmaximal, complemented
2#20265094C211yesmaximal, characteristic, complemented
3#20264911C212yesmaximal, complemented
4#20264903C211yesmaximal, characteristic, complemented
5#537×C101C211yesmaximal, characteristic, complemented
6C22×D404C2211yescharacteristic
7C2×#13108059—22no—
8C2×#13108051—21no—
9C2×#13108051—21no—
10#15692175—22no—
11C22⋊#11717579C2212yes—
12#15692162C2211yescharacteristic
13C2×#41×C101—21no—
14C22⋊Q8×C101—22no—
15#121×C101—22no—
16#121×C101—21no—
17#121×C101C2212yes—
18C2×C4⋊C4×C101C2211yescharacteristic
19C23×D101—21no—
20C2×D404—22no—
21C2×D404—42no—
22C22×Dic101—21no—
23#13108059—44no—
24C2×#11717579—22no—
25#13108051—42no—
26#13108051—42no—
27#41×C101—42no—
28#41×C101—42no—
29C2×Q8×C101—21no—
30C2×D4×C101—21no—
31C2×D4×C101—22no—
32C2×D4×C101—21no—
33C2×D4×C101D412yes—
34C22×C4×C101—22no—
35C22×C4×C101D411yescharacteristic
36C22×C4×C101D411yescharacteristic
37C22×C4×C101C2311yescharacteristic
38C4⋊C4×C101—21no—
39C4⋊C4×C101—22no—
40C4⋊C4×C101D412yes—
41C22⋊C4×C101—22no—
42C22⋊C4×C101—22no—
43C23×C101—22no—
44C23×C101C2×D411yescharacteristic
45C22×D101—22no—
46C22×D101—41no—
47C22×D101—41no—
48C22×D101—41no—
49C22×D101—41no—
50C22×D101—41no—
51C22×D101—41no—
52Q8×C101—42no—
53D4×C101—42no—
54D4×C101—42no—
55D4×C101—42no—
56D4×C101—42no—
57D4×C101—81no—
58C2×C4×C101—21no—
59C2×C4×C101—21no—
60C2×C4×C101—21no—
61C2×C4×C101—22no—
62C2×C4×C101—41no—
63C2×C4×C101—44no—
64C2×C4×C101—41no—
65C2×C4×C101D812yescomplemented
66C2×C4×C101C2×D412yes—
67C2×Dic101—41no—
68C2×Dic101—41no—
69C2×Dic101—41no—
70D404—24no—
71D404—42no—
72#11717579—42no—
73C22×C101—21no—
74C22×C101—21no—
75C22×C101—42no—
76C22×C101—42no—
77C22×C101—42no—
78C22×C101#12011yescharacteristic
79C22×C101C2×D811yescharacteristic
80C22×C101AGL(1,ℤ/8)11yescharacteristic
81C2×D101—42no—
82C2×D101—42no—
83C2×D101—42no—
84C2×D101—81no—
85C2×D101—81no—
86C2×D101—81no—
87C2×D101—81no—
88C4×C101—22no—
89C4×C101—22no—
90C4×C101—42no—
91C4×C101—41no—
92Dic101—81no—
93C2×C101—21no—
94C2×C101—21no—
95C2×C101—82no—
96C2×C101#44711yescharacteristic
97C2×C101#45411yescharacteristic
98C2×C101B(4,𝔽2)11yescharacteristic
99D101—82no—
100#1598—1011nomaximal, Sylow
101C101#159811yescharacteristic, complemented, Sylow
102#537D10111yescharacteristic, complemented
103#447—1012no—
104#394—1011no—
105#327—1012no—
106#324—1011no—
107C2×#41—21no—
108C22×D4—1011no—
109C2×D8—2022no—
110C22⋊Q8—22no—
111#121—22no—
112#121—21no—
113#121C2×D10112yes—
114C2×C4⋊C4C2×D10111yescharacteristic
115C2×C22⋊C4—2021no—
116C2×C22⋊C4—2021no—
117#102—2022no—
118C22⋊C8—1012no—
119#95—1011no—
120C24—2021no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—4042no—
128C2×D4—2022no—
129C2×D4C22⋊D10112yes—
130C22×C4—22no—
131C22×C4—2021no—
132C22×C4C22⋊D10111yescharacteristic
133C22×C4C22⋊D10111yescharacteristic
134C22×C4C22×D10111yescharacteristic
135D8—4044no—
136C2×C8—2022no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4C22⋊D10112yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—4042no—
143C22⋊C4—4042no—
144C23—22no—
145C23—4041no—
146C23—4041no—
147C23—4041no—
148C23—4041no—
149C23—4041no—
150C23—4041no—
151C23—2022no—
152C23C2×C22⋊D10111yescharacteristic
153Q8—42no—
154D4—42no—
155D4—42no—
156D4—42no—
157D4—42no—
158D4—81no—
159D4—4042no—
160D4—2024no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—4041no—
169C2×C4—4041no—
170C2×C4—4041no—
171C2×C4#1310805912yescomplemented
172C2×C4C2×C22⋊D10112yes—
173C8—4042no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—8081no—
180C22—8081no—
181C22—8081no—
182C22—8081no—
183C22—4042no—
184C22—4042no—
185C22—4042no—
186C22#1569222411yescharacteristic
187C22C2×#1310805911yescharacteristic
188C22#1569228111yescharacteristic
189C4—22no—
190C4—22no—
191C4—42no—
192C4—41no—
193C4—8081no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—8082no—
198C2#2026531111yescharacteristic
199C2#2026535011yescharacteristic
200C2#2026536411yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#537×C101C2
socleC22×C101#120
hypercenter#537D101
hypocenterC101#1598
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #20265310C2semidirect
2. #20265094C2semidirect
3. #20264911C2semidirect
4. #20264903C2semidirect
5. #537×C101C2semidirect
65. C2×C4×C101D8semidirect
101. C101#1598semidirect
102. #537D101semidirect
171. C2×C4#13108059semidirect

Maximal subgroups

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

subgroupindex
#202653102
#202650942
#202649112
#202649032
#537×C1012
#1598101