Finite Group Databasebeta

← #27171590

Subgroups — #27171590

Order 12928.

378 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 93 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×#15692238C211yesmaximal, characteristic
2C2×#15692169C212yesmaximal
3#20265040C211yesmaximal, characteristic, complemented
4#20264943C212yesmaximal, complemented
5C2×#117×C101C211yesmaximal, characteristic, complemented
6C22×C22⋊D101C2211yescharacteristic
7#15692238—22no—
8#15692238—21no—
9#15692238—21no—
10C2×#13108053C412yescomplemented
11C2×C22⋊Dic101C2212yes—
12C2×C22⋊Dic101C2211yescharacteristic
13C2×#13108051C412yescomplemented
14#15692169—24no—
15#15692162—22no—
16C23×C4×C101C2211yescharacteristic
17#117×C101—24no—
18C2×C4⋊C4×C101—21no—
19C2×C22⋊C4×C101C2212yes—
20C2×C42×C101—21no—
21C23×D101C2×C411yescharacteristic
22C2×C22⋊D101—22no—
23C2×C22⋊D101—21no—
24C2×C22⋊D101—21no—
25C2×C22⋊D101—22no—
26C22×Dic101—22no—
27C22×Dic101C2×C411yescharacteristic
28C22×Dic101C2×C412yes—
29#13108053—22no—
30#13108053—22no—
31C22⋊Dic101—21no—
32C22⋊Dic101—21no—
33C22⋊Dic101—22no—
34C22⋊Dic101—22no—
35#13108051—22no—
36#13108051—22no—
37C24×C101C2311yescharacteristic
38C22×C4×C101—22no—
39C22×C4×C101—22no—
40C22×C4×C101—22no—
41C22×C4×C101C2×C412yes—
42C22×C4×C101D412yes—
43C22×C4×C101D411yescharacteristic
44C22×C4×C101D411yescharacteristic
45C4⋊C4×C101—24no—
46C22⋊C4×C101—22no—
47C22⋊C4×C101D414yes—
48C42×C101—24no—
49C23×C101—42no—
50C23×C101C22×C411yescharacteristic
51C23×C101C2×D411yescharacteristic
52C23×C101C2×D412yes—
53C23×C101C2×D411yescharacteristic
54C23×C101#4112yes—
55C22×D101—21no—
56C22×D101—22no—
57C22×D101—42no—
58C2×C4×C101—24no—
59C2×C4×C101—21no—
60C2×C4×C101—21no—
61C2×C4×C101—24no—
62C2×C4×C101—21no—
63C2×C4×C101—21no—
64C2×C4×C101—24no—
65C2×C4×C101—44no—
66C2×C4×C101C22⋊C412yes—
67C2×C4×C101C22⋊C412yes—
68C22⋊D101—41no—
69C22⋊D101—42no—
70C22⋊D101—41no—
71C2×Dic101—22no—
72C2×Dic101—21no—
73C2×Dic101—22no—
74C2×Dic101—42no—
75C2×Dic101—44no—
76C2×Dic101—42no—
77C22×C101—22no—
78C22×C101—22no—
79C22×C101—22no—
80C22×C101—42no—
81C22×C101—42no—
82C22×C101C2×C22⋊C411yescharacteristic
83C22×C101C4×D412yes—
84C22×C101#12011yescharacteristic
85C22×C101#12112yes—
86C22×C101#12311yescharacteristic
87C2×D101—41no—
88C2×D101—42no—
89C2×D101—82no—
90C4×C101—22no—
91C4×C101—22no—
92C4×C101—44no—
93Dic101—41no—
94Dic101—42no—
95Dic101—82no—
96C2×C101—22no—
97C2×C101—21no—
98C2×C101—22no—
99C2×C101—21no—
100C2×C101#38611yescharacteristic
101C2×C101#41012yes—
102D101—81no—
103#1899—1011nomaximal, Sylow
104C101#189911yescharacteristic, complemented, Sylow
105C2×#123—1011no—
106C2×#117D10111yescharacteristic, complemented
107C2×#99—1012no—
108#381—1011no—
109#342—1012no—
110C22×D4—1011no—
111C23×C4C2×D10111yescharacteristic
112#123—2022no—
113#123—2021no—
114#123—2021no—
115#117—24no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—1012no—
118C2×C22⋊C4—1011no—
119C2×C22⋊C4—1012no—
120C2×C22⋊C4—1012no—
121C2×C22⋊C4C2×D10112yes—
122C2×C42—21no—
123#99—2024no—
124#95—2022no—
125C24—1011no—
126C24C22×D10111yescharacteristic
127C2×D4—2022no—
128C2×D4—2021no—
129C2×D4—2021no—
130C2×D4—2022no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—1012no—
135C22×C4—1011no—
136C22×C4—2022no—
137C22×C4C4×D10112yes—
138C22×C4D40412yes—
139C22×C4C22⋊D10111yescharacteristic
140C22×C4C22⋊D10111yescharacteristic
141C4⋊C4—24no—
142C4⋊C4—2022no—
143C4⋊C4—2022no—
144C22⋊C4—22no—
145C22⋊C4—2022no—
146C22⋊C4—2022no—
147C22⋊C4—2021no—
148C22⋊C4—2021no—
149C22⋊C4—2022no—
150C22⋊C4—2022no—
151C22⋊C4C22⋊D10114yes—
152C42—24no—
153C23—42no—
154C23—4042no—
155C23—2022no—
156C23—2021no—
157C23C2×C4×D10111yescharacteristic
158C23C2×C22⋊D10111yescharacteristic
159C23C2×C22⋊D10112yes—
160C23C2×D40411yescharacteristic
161C23#1310807412yes—
162D4—4041no—
163D4—4042no—
164D4—4041no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—24no—
172C2×C4—44no—
173C2×C4—4044no—
174C2×C4—4042no—
175C2×C4—4042no—
176C2×C4—2022no—
177C2×C4—2021no—
178C2×C4—2022no—
179C2×C4#1310805112yes—
180C2×C4#1310805112yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—8082no—
187C22—4041no—
188C22—4042no—
189C22C2×#1310805111yescharacteristic
190C22C4×C22⋊D10112yes—
191C22#1569222411yescharacteristic
192C22#1569222912yes—
193C22#1569223811yescharacteristic
194C4—22no—
195C4—22no—
196C4—44no—
197C4—8082no—
198C4—4041no—
199C4—4042no—
200C2—21no—
201C2—22no—
202C2—22no—
203C2—21no—
204C2—8081no—
205C2#2026505611yescharacteristic
206C2#2026515012yes—
subgroupG/H
centerC22C2×#13108051
commutatorC23×C101C22×C4
FrattiniC24C22×D101
FittingC2×#117×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC22×C22⋊D101C22
cosocleC24×C101C23
hypercenterC2×#117D101
hypocenterC101#1899
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #20265040C2semidirect
4. #20264943C2semidirect
5. C2×#117×C101C2semidirect
10. C2×#13108053C4semidirect
13. C2×#13108051C4semidirect
104. C101#1899semidirect
106. C2×#117D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922382
C2×#156921692
#202650402
#202649432
C2×#117×C1012
#1899101