Finite Group Databasebeta

← #27172353

Subgroups — #27172353

Order 12928.

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

normal subgroups only

This group has 306 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
1C2×#15692182C212yesmaximal, complemented
2#20265178C212yesmaximal, complemented
3#20265046C211yesmaximal, characteristic, complemented
4#20264946C211yesmaximal, characteristic, complemented
5#537×C101C211yesmaximal, characteristic
6C2×#13108058C2212yes—
7C2×C4⋊Dic101C2211yescharacteristic
8C2×#13108053—22no—
9C2×C4×Dic101C2211yescharacteristic
10#15692182—24no—
11#15692179—22no—
12#15692176—22no—
13C2×#41×C101C2212yes—
14C22⋊Q8×C101—22no—
15#121×C101—22no—
16#121×C101—22no—
17C2×C4⋊C4×C101C2211yescharacteristic
18C22×Dic101—21no—
19C22×Dic101—21no—
20#13108058—22no—
21#13108058—22no—
22C2×#11717579—22no—
23C4⋊Dic101—22no—
24#13108053—44no—
25C4×Dic101—21no—
26C4×Dic101D412yescomplemented
27#41×C101—22no—
28#41×C101—22no—
29#41×C101—22no—
30#41×C101—22no—
31C2×Q8×C101D412yes—
32C2×D4×C101—22no—
33C2×D4×C101—22no—
34C2×D4×C101D412yes—
35C22×C4×C101—22no—
36C22×C4×C101D412yes—
37C22×C4×C101C2311yescharacteristic
38C4⋊C4×C101—22no—
39C4⋊C4×C101—22no—
40C22⋊C4×C101—24no—
41C23×C101—22no—
42C23×C101C2×D411yescharacteristic
43Q8×C101—22no—
44Q8×C101—22no—
45D4×C101—22no—
46D4×C101—22no—
47D4×C101—42no—
48D4×C101—42no—
49D4×C101—82no—
50C2×C4×C101—22no—
51C2×C4×C101—22no—
52C2×C4×C101—42no—
53C2×C4×C101—42no—
54C2×C4×C101—42no—
55C2×C4×C101C2×D411yescharacteristic
56C2×C4×C101C2×D412yes—
57C2×C4×C101#4112yes—
58C2×C4×C101#4111yescharacteristic
59C2×Dic101—22no—
60C2×Dic101—22no—
61C2×Dic101—22no—
62C2×Dic101—42no—
63C2×Dic101—41no—
64#11717579—42no—
65C22×C101—21no—
66C22×C101—21no—
67C22×C101—42no—
68C22×C101—42no—
69C22×C101—42no—
70C22×C101#12011yescharacteristic
71C22×C101#12111yescharacteristic
72C22×C101#12411yescharacteristic
73C4×C101—42no—
74C4×C101—42no—
75C4×C101#12112yes—
76C4×C101#12312yes—
77Dic101—24no—
78Dic101—81no—
79C2×C101—21no—
80C2×C101—21no—
81C2×C101—82no—
82C2×C101#39411yescharacteristic
83C2×C101#45311yescharacteristic
84C2×C101#45611yescharacteristic
85#2039—1011nomaximal, Sylow
86C101#203911yescharacteristic, complemented, Sylow
87#537D10111yescharacteristic
88C2×#104—1012no—
89#417—1012no—
90#384—1011no—
91#343—1011no—
92C2×#41C2×D10112yes—
93C2×M16—1012no—
94C22⋊Q8—22no—
95#121—22no—
96#121—22no—
97C2×C4⋊C4—1011no—
98C2×C4⋊C4—2022no—
99C2×C4⋊C4C2×D10111yescharacteristic
100C2×C42—1011no—
101#104—2024no—
102#103—2022no—
103#102—2022no—
104#41—22no—
105#41—22no—
106#41—22no—
107#41—22no—
108C2×Q8C22⋊D10112yes—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4C22⋊D10112yes—
112C22×C4—22no—
113C22×C4—2021no—
114C22×C4—2021no—
115C22×C4C22⋊D10112yes—
116C22×C4C22×D10111yescharacteristic
117M16—2022no—
118M16—2022no—
119C2×C8—2022no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—4044no—
123C4⋊C4—2022no—
124C22⋊C4—24no—
125C42—1012no—
126C42—2021no—
127C23—22no—
128C23C2×C22⋊D10111yescharacteristic
129Q8—22no—
130Q8—22no—
131D4—22no—
132D4—22no—
133D4—42no—
134D4—42no—
135D4—82no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—4042no—
142C2×C4—4041no—
143C2×C4—2022no—
144C2×C4—2022no—
145C2×C4—2022no—
146C2×C4D4×D10111yescharacteristic
147C2×C4C2×C22⋊D10112yes—
148C2×C4#1310807312yes—
149C2×C4#1310807311yescharacteristic
150C8—4042no—
151C22—21no—
152C22—21no—
153C22—42no—
154C22—42no—
155C22—42no—
156C22#1569222411yescharacteristic
157C22#1569222611yescharacteristic
158C22#1569224211yescharacteristic
159C4—42no—
160C4—42no—
161C4—8081no—
162C4—2024no—
163C4#1569222712yes—
164C4#1569223612yes—
165C2—21no—
166C2—21no—
167C2—82no—
168C2#2026509111yescharacteristic
169C2#2026534511yescharacteristic
170C2#2026536011yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting, prime radical#537×C101C2
socleC22×C101#120
hypercenter#537D101
hypocenterC101#2039
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#15692182C2semidirect
2. #20265178C2semidirect
3. #20265046C2semidirect
4. #20264946C2semidirect
26. C4×Dic101D4semidirect
86. C101#2039semidirect

Maximal subgroups

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

subgroupindex
C2×#156921822
#202651782
#202650462
#202649462
#537×C1012
#2039101