Finite Group Databasebeta

← #27173128

Subgroups — #27173128

Order 12928.

246 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 45 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#20265366C212yesmaximal, complemented
2#20265344C211yesmaximal, characteristic
3#20264970C211yesmaximal, characteristic, complemented
4#20264967C212yesmaximal, complemented
5#559×C101C211yesmaximal, characteristic, complemented
6#15692300C2211yescharacteristic
7#15692281—21no—
8#15692238—22no—
9#15692183—21no—
10#15692172C2211yescharacteristic
11#15692170C2212yes—
12#15692169—22no—
13#127×C101C2211yescharacteristic
14#124×C101—22no—
15#123×C101—21no—
16#123×C101C2212yes—
17C4×Q8×C101—21no—
18C4×D4×C101—22no—
19#13108074—21no—
20#13108073—41no—
21C2×C22⋊D101—22no—
22D4×D101—41no—
23#13108062—41no—
24#13108059—41no—
25#13108058—21no—
26#13108053—42no—
27C22⋊Dic101—22no—
28#13108051—42no—
29C2×Q8×C101—21no—
30C2×D4×C101—22no—
31C2×D4×C101D411yescharacteristic
32C2×D4×C101C2311yescharacteristic
33C22×C4×C101—21no—
34C22×C4×C101D412yes—
35C4⋊C4×C101—21no—
36C4⋊C4×C101—22no—
37C22⋊C4×C101—22no—
38C22⋊C4×C101—21no—
39C22⋊C4×C101—22no—
40C22⋊C4×C101D412yes—
41C42×C101—22no—
42C42×C101D411yescharacteristic
43C23×C101—21no—
44C23×C101C2×D412yes—
45C22×D101—22no—
46Q8×C101—42no—
47D4×C101—41no—
48D4×C101—41no—
49D4×C101—44no—
50C2×C4×C101—21no—
51C2×C4×C101—22no—
52C2×C4×C101—22no—
53C2×C4×C101—42no—
54C2×C4×C101—81no—
55C2×C4×C101C2×D411yescharacteristic
56C22⋊D101—42no—
57C22⋊D101—41no—
58C2×Dic101—22no—
59C2×Dic101—42no—
60D404—21no—
61C4×D101—41no—
62Dic202—21no—
63#11717579—41no—
64C22×C101—22no—
65C22×C101—42no—
66C22×C101—42no—
67C22×C101—41no—
68C22×C101#12011yescharacteristic
69C2×D101—41no—
70C2×D101—82no—
71C4×C101—22no—
72C4×C101—21no—
73C4×C101—42no—
74C4×C101—42no—
75Dic101—41no—
76Dic101—82no—
77C2×C101—21no—
78C2×C101—42no—
79C2×C101—81no—
80C2×C101B(4,𝔽2)11yescharacteristic
81D101—81no—
82#2198—1011nomaximal, Sylow
83C101#219811yescharacteristic, complemented, Sylow
84#559D10111yescharacteristic, complemented
85#458—1012no—
86#453—1011no—
87#353—1011no—
88#352—1012no—
89H(5,𝔽2)—1011no—
90AGL(1,ℤ/8)—2021no—
91#127C2×D10111yescharacteristic
92#124—22no—
93#123—21no—
94#123—2022no—
95#123C2×D10112yes—
96C4×Q8—21no—
97C4×D4—22no—
98#104—2021no—
99#100—1011no—
100#99—1012no—
101#99—2022no—
102#41—4041no—
103#41—2021no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—4041no—
107C2×D4—2022no—
108C2×D4C22⋊D10111yescharacteristic
109C2×D4C22×D10111yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D10112yes—
112SD16—4041no—
113D8—4041no—
114M16—2021no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—4042no—
118C22⋊C4—22no—
119C22⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—4042no—
122C22⋊C4—2022no—
123C22⋊C4C22⋊D10112yes—
124C42—22no—
125C42C22⋊D10111yescharacteristic
126C23—21no—
127C23—2022no—
128C23C2×C22⋊D10112yes—
129Q8—42no—
130Q8—2021no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—4042no—
135D4—4041no—
136D4—2021no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—81no—
142C2×C4—4041no—
143C2×C4—4042no—
144C2×C4—2022no—
145C2×C4C2×C22⋊D10111yescharacteristic
146C8—4041no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—8082no—
152C22—4041no—
153C22#1569222411yescharacteristic
154C4—22no—
155C4—21no—
156C4—42no—
157C4—42no—
158C4—8082no—
159C4—4041no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—8081no—
164C2#2026536411yescharacteristic
subgroupG/H
centerC2#20265364
commutator, cosocleC2×D4×C101C23
FrattiniC2×D4C22×D101
Fitting#559×C101C2
socleC2×C101B(4,𝔽2)
prime radical#20265344C2
hypercenter#559D101
hypocenterC101#2198
LCSG ▷ C2×D4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #20265366C2semidirect
3. #20264970C2semidirect
4. #20264967C2semidirect
5. #559×C101C2semidirect
83. C101#2198semidirect
84. #559D101semidirect

Maximal subgroups

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

subgroupindex
#202653662
#202653442
#202649702
#202649672
#559×C1012
#2198101