Finite Group Databasebeta

← #27171534

Subgroups — #27171534

Order 12928.

344 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 83 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×#15692243C211yesmaximal, characteristic
2#20265654C211yesmaximal, characteristic, complemented
3#20265183C211yesmaximal, characteristic
4#20265181C211yesmaximal, characteristic
5C2×#15692173C211yesmaximal, characteristic, complemented
6#20264937C211yesmaximal, characteristic, complemented
7#339×C101C211yesmaximal, characteristic, complemented
8C22×Dic202C2211yescharacteristic
9C22×D404C2211yescharacteristic
10C2×#13108058C2211yescharacteristic
11#15692243—22no—
12#15692243C414yescomplemented
13C4×Dic202—22no—
14#15692220—22no—
15#15692214C2211yescharacteristic
16#15692211—22no—
17C2×#13108051—22no—
18#15692177—22no—
19#15692175—22no—
20#15692173—22no—
21C2×M16×C101C2211yescharacteristic
22#117×C101C2211yescharacteristic
23C2×C42×C101C2211yescharacteristic
24C23×D101—21no—
25C2×Dic202—21no—
26C2×Dic202—21no—
27C2×Dic202—21no—
28C2×Dic202—21no—
29C2×Dic202C2×C412yes—
30C2×Dic202D412yes—
31C2×D404—21no—
32C2×D404—21no—
33C2×D404—21no—
34C2×D404—21no—
35C2×D404—21no—
36C2×D404C2×C412yes—
37C22×Dic101D412yes—
38#13108058—21no—
39#13108058—21no—
40C2×#11717579—21no—
41C4⋊Dic101D412yes—
42#13108053—24no—
43C22⋊Dic101—21no—
44#13108051—28no—
45C4×Dic101—24no—
46C4×Dic101—21no—
47C22×C4×C101—21no—
48C22×C4×C101C2311yescharacteristic
49M16×C101—21no—
50M16×C101—21no—
51C2×C8×C101—21no—
52C4⋊C4×C101D412yes—
53C22⋊C4×C101—21no—
54C42×C101—21no—
55C42×C101—21no—
56C42×C101C2×C412yes—
57C23×C101C2×D411yescharacteristic
58C22×D101—22no—
59C22×D101—44no—
60C22×D101—41no—
61C22×D101—41no—
62C2×C4×C101—22no—
63C2×C4×C101—22no—
64C2×C4×C101—42no—
65C2×C4×C101C22×C411yescharacteristic
66C2×C4×C101C2×D411yescharacteristic
67C2×C4×C101C2×D411yescharacteristic
68C2×C4×C101C2×D411yescharacteristic
69C2×C4×C101#4111yescharacteristic
70C2×C4×C101#4111yescharacteristic
71C2×Dic101—22no—
72C2×Dic101—22no—
73C2×Dic101—22no—
74C2×Dic101C22⋊C414yes—
75D404—22no—
76D404—22no—
77D404—41no—
78D404—41no—
79Dic202—22no—
80Dic202—22no—
81Dic202—22no—
82Dic202—22no—
83C8×C101—41no—
84#11717579—41no—
85C22×C101—22no—
86C22×C101C2×C22⋊C411yescharacteristic
87C22×C101#12011yescharacteristic
88C22×C101#12311yescharacteristic
89C2×D101—44no—
90C2×D101—42no—
91C2×D101—82no—
92C2×D101—82no—
93C4×C101—42no—
94C4×C101—42no—
95C4×C101C4×D411yescharacteristic
96C4×C101C4×D411yescharacteristic
97C4×C101#12111yescharacteristic
98C4×C101#12111yescharacteristic
99Dic101—24no—
100Dic101—42no—
101C2×C101—21no—
102C2×C101—21no—
103C2×C101#38611yescharacteristic
104C2×C101#45412yes—
105D101—82no—
106#1888—1011nomaximal, Sylow
107C101#188811yescharacteristic, complemented, Sylow
108C2×#124—1011no—
109#519—1011no—
110#418—1011no—
111#418—1011no—
112C2×#101—1011no—
113#339—1011no—
114#339D10111yescharacteristic, complemented
115C22×Q8—1011no—
116C22×D4—1011no—
117C2×M16—1011no—
118C2×M16C2×D10111yescharacteristic
119#124—1014no—
120#124—2022no—
121C4×Q8—2022no—
122C4×Q8—2022no—
123#117—1011no—
124#117—2022no—
125#117C2×D10111yescharacteristic
126C2×C22⋊C4—2022no—
127C2×C42C2×D10111yescharacteristic
128#102—2022no—
129#102—2022no—
130#101—2022no—
131C24—2021no—
132C2×Q8—1012no—
133C2×Q8—1012no—
134C2×Q8—2021no—
135C2×Q8—2021no—
136C2×Q8—2021no—
137C2×Q8—2021no—
138C2×D4—1012no—
139C2×D4—2021no—
140C2×D4—2021no—
141C2×D4—2021no—
142C2×D4—2021no—
143C2×D4—2021no—
144C22×C4—21no—
145C22×C4—1012no—
146C22×C4C22×D10111yescharacteristic
147M16—21no—
148M16—21no—
149M16—2021no—
150M16—2021no—
151C2×C8—21no—
152C2×C8—2021no—
153C4⋊C4—1012no—
154C4⋊C4—2024no—
155C4⋊C4D40412yes—
156C22⋊C4—21no—
157C22⋊C4—2028no—
158C22⋊C4—2021no—
159C42—21no—
160C42—21no—
161C42—2024no—
162C42—2021no—
163C42C4×D10112yes—
164C23—4044no—
165C23—4041no—
166C23—4041no—
167C23—2022no—
168C23D4×D10111yescharacteristic
169Q8—2022no—
170Q8—2022no—
171Q8—2022no—
172Q8—2022no—
173D4—4041no—
174D4—4041no—
175D4—2022no—
176D4—2022no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—1014no—
181C2×C4—2022no—
182C2×C4—2022no—
183C2×C4—2022no—
184C2×C4C2×C4×D10111yescharacteristic
185C2×C4D4×D10111yescharacteristic
186C2×C4D4×D10111yescharacteristic
187C2×C4C2×D40411yescharacteristic
188C2×C4#1310807411yescharacteristic
189C2×C4#1310807511yescharacteristic
190C8—41no—
191C8—4041no—
192C22—22no—
193C22—8082no—
194C22—8082no—
195C22—4044no—
196C22—4042no—
197C22C22⋊C4×D10111yescharacteristic
198C22#1569222511yescharacteristic
199C22#1569224011yescharacteristic
200C4—42no—
201C4—42no—
202C4—4042no—
203C4—2024no—
204C4#1569221711yescharacteristic
205C4C4×D40411yescharacteristic
206C4#1569222811yescharacteristic
207C4#1569223011yescharacteristic
208C2—21no—
209C2—21no—
210C2—8082no—
211C2#2026505511yescharacteristic
212C2#2026535112yes—
subgroupG/H
centerC22#15692225
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#339×C101C2
socleC22×C101#120
prime radicalC22×D404C22
cosocleC22×C4×C101C23
hypercenter#339D101
hypocenterC101#1888
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #20265654C2semidirect
5. C2×#15692173C2semidirect
6. #20264937C2semidirect
7. #339×C101C2semidirect
12. #15692243C4semidirect
107. C101#1888semidirect
114. #339D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922432
#202656542
#202651832
#202651812
C2×#156921732
#202649372
#339×C1012
#1888101