Finite Group Databasebeta

← #27170454

Subgroups — #27170454

Order 12928.

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

normal subgroups only

This group has 196 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#20265921C211yesmaximal, characteristic
2#20265545C211yesmaximal, characteristic, complemented
3#20265539C211yesmaximal, characteristic, complemented
4#20264918C211yesmaximal, characteristic, complemented
5#20264917C211yesmaximal, characteristic, complemented
6#20264916C211yesmaximal, characteristic, complemented
7#489×C101C211yesmaximal, characteristic, complemented
8#15692247C2211yescharacteristic
9#15692243C2211yescharacteristic
10#15692239—21no—
11#15692237—21no—
12#15692232—21no—
13#15692231—21no—
14#15692228—21no—
15#15692227—21no—
16#15692179—21no—
17#15692179—21no—
18#15692178—21no—
19#15692175—21no—
20#15692166C2211yescharacteristic
21#15692166C2211yescharacteristic
22C4⋊Q8×C101C2211yescharacteristic
23#124×C101C2211yescharacteristic
24#103×C101—21no—
25#102×C101—21no—
26#97×C101C2211yescharacteristic
27C2×Dic202D411yescharacteristic
28C2×D404D411yescharacteristic
29C2×C22⋊D101—21no—
30C2×C22⋊D101—21no—
31C22×Dic101—21no—
32C2×C4×D101—21no—
33C2×#11717579—21no—
34C2×#11717579—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101D411yescharacteristic
37C4⋊Dic101D411yescharacteristic
38#13108053—21no—
39#13108053—21no—
40#13108053—21no—
41C22⋊Dic101—21no—
42#13108051—21no—
43#13108051—21no—
44#13108051—21no—
45C2×Q8×C101—21no—
46C2×Q8×C101D411yescharacteristic
47C2×D4×C101D411yescharacteristic
48C2×C8×C101—21no—
49C4⋊C4×C101—21no—
50C4⋊C4×C101—21no—
51C22⋊C4×C101—21no—
52C22⋊C4×C101—21no—
53C42×C101C2311yescharacteristic
54C23×C101—21no—
55C22×D101—21no—
56Q8×C101—41no—
57Q8×C101—42no—
58D4×C101—41no—
59C2×C4×C101—21no—
60C2×C4×C101—41no—
61C2×C4×C101C2×D411yescharacteristic
62C2×C4×C101C2×D411yescharacteristic
63C2×C4×C101C2×D411yescharacteristic
64C22⋊D101—81no—
65C22⋊D101—81no—
66C2×Dic101—21no—
67C2×Dic101—21no—
68C2×Dic101—21no—
69C2×Dic101—81no—
70D404—41no—
71C4×D101—81no—
72Dic202—41no—
73C8×C101—41no—
74#11717579—41no—
75#11717579—41no—
76C22×C101—41no—
77C22×C101—41no—
78C22×C101—41no—
79C22×C101#12011yescharacteristic
80C2×D101—41no—
81C2×D101—41no—
82C2×D101—41no—
83C4×C101—21no—
84C4×C101—21no—
85C4×C101—21no—
86C4×C101—41no—
87C4×C101—81no—
88Dic101—41no—
89Dic101—41no—
90Dic101—41no—
91C2×C101—81no—
92C2×C101#45411yescharacteristic
93C2×C101#45511yescharacteristic
94C2×C101#45511yescharacteristic
95D101—81no—
96#1662—1011nomaximal, Sylow
97C101#166211yescharacteristic, complemented, Sylow
98#562—1011no—
99#491—1011no—
100#489—1011no—
101#489D10111yescharacteristic, complemented
102#330—1011no—
103#329—1011no—
104#329—1011no—
105C4⋊Q8C2×D10111yescharacteristic
106#125—1011no—
107#124—1011no—
108#124C2×D10111yescharacteristic
109#123—2021no—
110#123—2021no—
111C22⋊Q8—2021no—
112C22⋊Q8—2021no—
113#121—2021no—
114#121—2021no—
115#103—21no—
116#103—2021no—
117#103—2021no—
118#103—2021no—
119#102—21no—
120#102—2021no—
121#97—1011no—
122#97—1011no—
123#97C2×D10111yescharacteristic
124C2×Q8—21no—
125C2×Q8—1011no—
126C2×Q8C22⋊D10111yescharacteristic
127C2×D4—1011no—
128C2×D4—2021no—
129C2×D4—2021no—
130C2×D4C22⋊D10111yescharacteristic
131C22×C4—2021no—
132C22×C4—2021no—
133C2×C8—21no—
134C2×C8—2021no—
135C2×C8—2021no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—1011no—
139C4⋊C4—1011no—
140C4⋊C4—2021no—
141C4⋊C4—2021no—
142C4⋊C4—2021no—
143C4⋊C4—2021no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—2021no—
147C22⋊C4—2021no—
148C22⋊C4—2021no—
149C22⋊C4—2021no—
150C42C22×D10111yescharacteristic
151C23—21no—
152C23—2021no—
153Q8—41no—
154Q8—42no—
155Q8—4041no—
156D4—41no—
157D4—8081no—
158D4—8081no—
159D4—4041no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—8081no—
163C2×C4—8081no—
164C2×C4—2021no—
165C2×C4—2021no—
166C2×C4—2021no—
167C2×C4D4×D10111yescharacteristic
168C2×C4D4×D10111yescharacteristic
169C2×C4C2×C22⋊D10111yescharacteristic
170C8—41no—
171C8—4041no—
172C8—4041no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—4041no—
177C22—4041no—
178C22—4041no—
179C22#1569222311yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—41no—
184C4—81no—
185C4—4041no—
186C4—4041no—
187C4—4041no—
188C2—81no—
189C2—8081no—
190C2#2026534811yescharacteristic
191C2#2026535311yescharacteristic
192C2#2026535411yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#489×C101C2
socleC22×C101#120
prime radical#20265921C2
hypercenter#489D101
hypocenterC101#1662
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #20265545C2semidirect
3. #20265539C2semidirect
4. #20264918C2semidirect
5. #20264917C2semidirect
6. #20264916C2semidirect
7. #489×C101C2semidirect
97. C101#1662semidirect
101. #489D101semidirect

Maximal subgroups

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

subgroupindex
#202659212
#202655452
#202655392
#202649182
#202649172
#202649162
#489×C1012
#1662101