Finite Group Databasebeta

← #27170071

Subgroups — #27170071

Order 12928.

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

normal subgroups only

This group has 266 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#20265808C211yesmaximal, characteristic, complemented
2#20265328C212yesmaximal
3#20264913C212yesmaximal, complemented
4#20264903C211yesmaximal, characteristic, complemented
5#394×C101C211yesmaximal, characteristic
6C2×#13108062—22no—
7#15692254—22no—
8#15692246—22no—
9#15692234—21no—
10#15692234C2212yes—
11#15692214—21no—
12#15692176—22no—
13C22⋊#11717579C2212yes—
14C22×D4×C101C2211yescharacteristic
15C2×C4⋊C4×C101C2211yescharacteristic
16C2×C22⋊C4×C101—21no—
17C2×C22⋊C4×C101—21no—
18#95×C101C2211yescharacteristic
19C2×Dic202—21no—
20C2×Dic202D412yes—
21#13108062—44no—
22C2×#11717579—22no—
23C4⋊Dic101—21no—
24C4⋊Dic101—21no—
25C4⋊Dic101—21no—
26C4⋊Dic101D412yes—
27#13108053—22no—
28#13108053—22no—
29C22⋊Dic101—22no—
30C4×Dic101—22no—
31C24×C101—21no—
32C2×D4×C101—22no—
33C2×D4×C101—42no—
34C22×C4×C101—21no—
35C22×C4×C101D411yescharacteristic
36C22×C4×C101D411yescharacteristic
37C22×C4×C101C2311yescharacteristic
38C4⋊C4×C101—22no—
39C22⋊C4×C101—42no—
40C22⋊C4×C101—42no—
41C23×C101—22no—
42C23×C101—41no—
43C23×C101—41no—
44C23×C101—41no—
45C23×C101—41no—
46C23×C101—41no—
47C23×C101—41no—
48C23×C101C2×D411yescharacteristic
49D4×C101—24no—
50D4×C101—42no—
51C2×C4×C101—21no—
52C2×C4×C101—21no—
53C2×C4×C101—21no—
54C2×C4×C101—41no—
55C2×C4×C101—41no—
56C2×C4×C101—41no—
57C2×C4×C101—41no—
58C2×C4×C101—41no—
59C2×C4×C101SD1612yescomplemented
60C2×C4×C101C2×D412yes—
61C2×Dic101—22no—
62C2×Dic101—22no—
63Dic202—42no—
64Dic202—81no—
65#11717579—42no—
66C22×C101—21no—
67C22×C101—21no—
68C22×C101—42no—
69C22×C101—42no—
70C22×C101—42no—
71C22×C101—81no—
72C22×C101—81no—
73C22×C101—81no—
74C22×C101—81no—
75C22×C101#12011yescharacteristic
76C22×C101C2×SD1611yescharacteristic
77C22×C101#13711yescharacteristic
78C4×C101—22no—
79C4×C101—22no—
80C4×C101—41no—
81C4×C101—81no—
82Dic101—42no—
83Dic101—42no—
84C2×C101—21no—
85C2×C101—21no—
86C2×C101—82no—
87C2×C101#44811yescharacteristic
88C2×C101#45411yescharacteristic
89C2×C101B(4,𝔽2)11yescharacteristic
90#1599—1011nomaximal, Sylow
91C101#159911yescharacteristic, complemented, Sylow
92#544—1011no—
93#450—1012no—
94#394D10111yescharacteristic
95#328—1012no—
96#324—1011no—
97C22×D4C2×D10111yescharacteristic
98C2×SD16—2022no—
99C4⋊Q8—2022no—
100#125—2022no—
101C22⋊Q8—1012no—
102C22⋊Q8—2021no—
103#117—2021no—
104C2×C4⋊C4C2×D10111yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107#102—2022no—
108C22⋊C8—1012no—
109#95C2×D10111yescharacteristic
110C24—21no—
111C2×Q8—1012no—
112C2×Q8—2021no—
113C2×D4—22no—
114C2×D4—42no—
115C22×C4—21no—
116C22×C4C22⋊D10111yescharacteristic
117C22×C4C22⋊D10111yescharacteristic
118C22×C4C22×D10111yescharacteristic
119SD16—4044no—
120C2×C8—2022no—
121C4⋊C4—22no—
122C4⋊C4—1012no—
123C4⋊C4—2022no—
124C4⋊C4—2021no—
125C4⋊C4—2021no—
126C4⋊C4—2021no—
127C4⋊C4—2022no—
128C22⋊C4—42no—
129C22⋊C4—42no—
130C22⋊C4—2022no—
131C42—2022no—
132C23—22no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23C2×C22⋊D10111yescharacteristic
140Q8—8081no—
141Q8—4042no—
142D4—24no—
143D4—42no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—2022no—
153C2×C4—2022no—
154C2×C4#1310806212yescomplemented
155C2×C4D4×D10112yes—
156C8—4042no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22—81no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22#1569222311yescharacteristic
167C22C2×#1310806211yescharacteristic
168C22#1569228511yescharacteristic
169C4—22no—
170C4—22no—
171C4—41no—
172C4—81no—
173C4—4042no—
174C4—4042no—
175C2—21no—
176C2—21no—
177C2—82no—
178C2#2026531511yescharacteristic
179C2#2026534911yescharacteristic
180C2#2026536311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#394×C101C2
socleC22×C101#120
prime radicalC22×D4×C101C22
hypercenter#394D101
hypocenterC101#1599
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #20265808C2semidirect
3. #20264913C2semidirect
4. #20264903C2semidirect
59. C2×C4×C101SD16semidirect
91. C101#1599semidirect
154. C2×C4#13108062semidirect

Maximal subgroups

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

subgroupindex
#202658082
#202653282
#202649132
#202649032
#394×C1012
#1599101