Finite Group Databasebeta

← #27169664

Subgroups — #27169664

Order 12928.

274 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 73 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×#15692234C211yesmaximal, characteristic
2#20265141C211yesmaximal, characteristic
3#20264913C214yesmaximal, complemented
4#381×C101C211yesmaximal, characteristic
5C22×Dic202C411yescharacteristic, complemented
6C2×#13108058—21no—
7#15692234—21no—
8#15692234—21no—
9#15692234C412yescomplemented
10#15692234C2212yes—
11C2×C4⋊Dic101C411yescharacteristic, complemented
12C2×#13108053—21no—
13C2×C22⋊Dic101—21no—
14C22⋊#11717579—21no—
15C22⋊#11717579C2212yes—
16C23×C4×C101C2211yescharacteristic
17C2×C22⋊C4×C101—21no—
18#95×C101—21no—
19#95×C101C2212yes—
20C2×Dic202—21no—
21C2×Dic202—42no—
22C2×Dic202C2×C412yes—
23C22×Dic101—21no—
24C22×Dic101—21no—
25#13108058—42no—
26C2×#11717579—22no—
27C4⋊Dic101—21no—
28C4⋊Dic101C2×C412yes—
29#13108053—22no—
30#13108053—22no—
31C22⋊Dic101—22no—
32C22⋊Dic101—22no—
33C24×C101D411yescharacteristic
34C22×C4×C101—22no—
35C22×C4×C101—42no—
36C22×C4×C101C2×C411yescharacteristic
37C22×C4×C101C2×C411yescharacteristic
38C22×C4×C101D411yescharacteristic
39C22×C4×C101D411yescharacteristic
40C22×C4×C101D411yescharacteristic
41C22×C4×C101C2311yescharacteristic
42C22⋊C4×C101—42no—
43C23×C101—22no—
44C23×C101—22no—
45C23×C101—41no—
46C23×C101C22⋊C411yescharacteristic
47C23×C101C22⋊C411yescharacteristic
48C23×C101C2×D411yescharacteristic
49C2×C4×C101—22no—
50C2×C4×C101—22no—
51C2×C4×C101—41no—
52C2×C4×C101—41no—
53C2×C4×C101—41no—
54C2×C4×C101—41no—
55C2×C4×C101—42no—
56C2×C4×C101—42no—
57C2×C4×C101—42no—
58C2×C4×C101C22⋊C411yescharacteristic
59C2×C4×C101C22⋊C411yescharacteristic
60C2×C4×C101C22×C411yescharacteristic
61C2×C4×C101C2×D411yescharacteristic
62C2×Dic101—22no—
63C2×Dic101—22no—
64C2×Dic101—42no—
65C2×Dic101—42no—
66Dic202—42no—
67Dic202—42no—
68#11717579—42no—
69C22×C101—21no—
70C22×C101—21no—
71C22×C101—21no—
72C22×C101—21no—
73C22×C101—21no—
74C22×C101—21no—
75C22×C101—21no—
76C22×C101—44no—
77C22×C101#9912yes—
78C22×C101C2×C22⋊C411yescharacteristic
79C22×C101#13712yes—
80C4×C101—21no—
81C4×C101—21no—
82C4×C101—21no—
83C4×C101—21no—
84C4×C101—82no—
85Dic101—42no—
86Dic101—42no—
87C2×C101—22no—
88C2×C101—22no—
89C2×C101—41no—
90C2×C101C2×#9911yescharacteristic
91C2×C101#41911yescharacteristic
92C2×C101#42111yescharacteristic
93#1511—1011nomaximal, Sylow
94C101#151111yescharacteristic, complemented, Sylow
95C2×C22⋊Q8—1011no—
96#407—1011no—
97#381D10111yescharacteristic
98#328—1014no—
99C22×Q8—1011no—
100C23×C4C2×D10111yescharacteristic
101C2×M16—2021no—
102C22⋊Q8—1012no—
103C22⋊Q8—1012no—
104C22⋊Q8—2021no—
105C22⋊Q8—2021no—
106C2×C4⋊C4—1011no—
107C2×C4⋊C4—2021no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—2021no—
110C22⋊C8—1012no—
111C22⋊C8—2021no—
112#95—21no—
113#95C2×D10112yes—
114C24C22⋊D10111yescharacteristic
115C2×Q8—1012no—
116C2×Q8—4042no—
117C2×Q8—2021no—
118C22×C4—22no—
119C22×C4—42no—
120C22×C4—2021no—
121C22×C4—2021no—
122C22×C4C4×D10111yescharacteristic
123C22×C4C4×D10111yescharacteristic
124C22×C4D40411yescharacteristic
125C22×C4D40411yescharacteristic
126C22×C4C22⋊D10111yescharacteristic
127C22×C4C22×D10111yescharacteristic
128M16—4042no—
129C2×C8—2022no—
130C4⋊C4—1012no—
131C4⋊C4—2022no—
132C4⋊C4—2021no—
133C4⋊C4—2022no—
134C22⋊C4—42no—
135C22⋊C4—2022no—
136C22⋊C4—2022no—
137C23—22no—
138C23—22no—
139C23—41no—
140C23#1310805111yescharacteristic
141C23#1310805111yescharacteristic
142C23C2×C22⋊D10111yescharacteristic
143Q8—4042no—
144Q8—4042no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—4042no—
155C2×C4—4042no—
156C2×C4—2022no—
157C2×C4—2022no—
158C2×C4#1310805111yescharacteristic
159C2×C4#1310805111yescharacteristic
160C2×C4C2×C4×D10111yescharacteristic
161C2×C4C2×D40411yescharacteristic
162C8—4042no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—44no—
171C22#1569216912yes—
172C22C2×#1310805111yescharacteristic
173C22#1569228512yes—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—82no—
179C4—4042no—
180C4—4042no—
181C2—22no—
182C2—22no—
183C2—41no—
184C2C2×#1569216911yescharacteristic
185C2#2026518511yescharacteristic
186C2#2026519511yescharacteristic
subgroupG/H
centerC22C2×#13108051
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#381×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC24×C101D4
cosocleC22×C4×C101C23
hypercenter#381D101
hypocenterC101#1511
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #20264913C2semidirect
5. C22×Dic202C4semidirect
9. #15692234C4semidirect
11. C2×C4⋊Dic101C4semidirect
94. C101#1511semidirect

Maximal subgroups

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

subgroupindex
C2×#156922342
#202651412
#202649132
#381×C1012
#1511101