Finite Group Databasebeta

← #1899×C211 #41009519

Subgroups — #1899×C211 #41009519

Order 27008.

378 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 112 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×#123×C211C211yesmaximal, characteristic
2C2×#117×C211C211yesmaximal, characteristic, complemented
3C2×#99×C211C212yesmaximal
4#381×C211C211yesmaximal, characteristic, complemented
5#342×C211C212yesmaximal, complemented
6C22×D4×C211C2211yescharacteristic
7C23×C4×C211C2211yescharacteristic
8#123×C211—22no—
9#123×C211—21no—
10#123×C211—21no—
11#117×C211—24no—
12C2×C4⋊C4×C211—21no—
13C2×C4⋊C4×C211C412yescomplemented
14C2×C22⋊C4×C211C412yescomplemented
15C2×C22⋊C4×C211C2212yes—
16C2×C22⋊C4×C211C2211yescharacteristic
17C2×C22⋊C4×C211C2212yes—
18C2×C42×C211—21no—
19#99×C211—24no—
20#95×C211—22no—
21C24×C211C2×C411yescharacteristic
22C24×C211C2311yescharacteristic
23C2×D4×C211—22no—
24C2×D4×C211—21no—
25C2×D4×C211—21no—
26C2×D4×C211—22no—
27C22×C4×C211—22no—
28C22×C4×C211—22no—
29C22×C4×C211—22no—
30C22×C4×C211—22no—
31C22×C4×C211C2×C411yescharacteristic
32C22×C4×C211C2×C412yes—
33C22×C4×C211C2×C412yes—
34C22×C4×C211D412yes—
35C22×C4×C211D411yescharacteristic
36C22×C4×C211D411yescharacteristic
37C4⋊C4×C211—24no—
38C4⋊C4×C211—22no—
39C4⋊C4×C211—22no—
40C22⋊C4×C211—22no—
41C22⋊C4×C211—21no—
42C22⋊C4×C211—21no—
43C22⋊C4×C211—22no—
44C22⋊C4×C211—22no—
45C22⋊C4×C211—22no—
46C22⋊C4×C211—22no—
47C22⋊C4×C211D414yes—
48C42×C211—24no—
49C23×C211—21no—
50C23×C211—22no—
51C23×C211—42no—
52C23×C211—42no—
53C23×C211C22×C411yescharacteristic
54C23×C211C2×D411yescharacteristic
55C23×C211C2×D411yescharacteristic
56C23×C211C2×D412yes—
57C23×C211#4112yes—
58D4×C211—41no—
59D4×C211—42no—
60D4×C211—41no—
61C2×C4×C211—24no—
62C2×C4×C211—21no—
63C2×C4×C211—21no—
64C2×C4×C211—24no—
65C2×C4×C211—21no—
66C2×C4×C211—21no—
67C2×C4×C211—22no—
68C2×C4×C211—21no—
69C2×C4×C211—22no—
70C2×C4×C211—24no—
71C2×C4×C211—44no—
72C2×C4×C211—42no—
73C2×C4×C211—44no—
74C2×C4×C211—42no—
75C2×C4×C211C22⋊C412yes—
76C2×C4×C211C22⋊C412yes—
77C22×C211—22no—
78C22×C211—22no—
79C22×C211—22no—
80C22×C211—42no—
81C22×C211—42no—
82C22×C211—41no—
83C22×C211—42no—
84C22×C211—82no—
85C22×C211C2×C22⋊C411yescharacteristic
86C22×C211C4×D412yes—
87C22×C211#12011yescharacteristic
88C22×C211#12112yes—
89C22×C211#12311yescharacteristic
90C4×C211—22no—
91C4×C211—22no—
92C4×C211—44no—
93C4×C211—41no—
94C4×C211—42no—
95C4×C211—82no—
96C2×C211—21no—
97C2×C211—22no—
98C2×C211—22no—
99C2×C211—21no—
100C2×C211—81no—
101C2×C211#38611yescharacteristic
102C2×C211#41012yes—
103C211#189911yescharacteristic, direct product factor, Sylow
104#1899C21111yesmaximal, characteristic, direct product factor, Sylow
105C2×#123C2×C21111yescharacteristic
106C2×#117C2×C21111yescharacteristic, complemented
107C2×#99C2×C21112yes—
108#381C2×C21111yescharacteristic, complemented
109#342C2×C21112yescomplemented
110C22×D4C22×C21111yescharacteristic
111C23×C4C22×C21111yescharacteristic
112#123—22no—
113#123—21no—
114#123—21no—
115#117—24no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4C4×C21112yescomplemented
118C2×C22⋊C4C4×C21112yescomplemented
119C2×C22⋊C4C22×C21112yes—
120C2×C22⋊C4C22×C21111yescharacteristic
121C2×C22⋊C4C22×C21112yes—
122C2×C42—21no—
123#99—24no—
124#95—22no—
125C24C2×C4×C21111yescharacteristic
126C24C23×C21111yescharacteristic
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4C2×C4×C21111yescharacteristic
136C22×C4C2×C4×C21112yes—
137C22×C4C2×C4×C21112yes—
138C22×C4D4×C21112yes—
139C22×C4D4×C21111yescharacteristic
140C22×C4D4×C21111yescharacteristic
141C4⋊C4—24no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C22⋊C4—22no—
148C22⋊C4—22no—
149C22⋊C4—22no—
150C22⋊C4—22no—
151C22⋊C4D4×C21114yes—
152C42—24no—
153C23—21no—
154C23—22no—
155C23—42no—
156C23—42no—
157C23C22×C4×C21111yescharacteristic
158C23C2×D4×C21111yescharacteristic
159C23C2×D4×C21111yescharacteristic
160C23C2×D4×C21112yes—
161C23#41×C21112yes—
162D4—41no—
163D4—42no—
164D4—41no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4C22⋊C4×C21112yes—
180C2×C4C22⋊C4×C21112yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—42no—
188C22—82no—
189C22C2×C22⋊C4×C21111yescharacteristic
190C22C4×D4×C21112yes—
191C22#120×C21111yescharacteristic
192C22#121×C21112yes—
193C22#123×C21111yescharacteristic
194C4—22no—
195C4—22no—
196C4—44no—
197C4—41no—
198C4—42no—
199C4—82no—
200C2—21no—
201C2—22no—
202C2—22no—
203C2—21no—
204C2—81no—
205C2#386×C21111yescharacteristic
206C2#410×C21112yes—
subgroupG/H
center, socleC22×C211C2×C22⋊C4
commutatorC23C22×C4×C211
Frattini, cosocleC24C23×C211
prime radicalC22×D4×C211C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#117×C211C2semidirect
4. #381×C211C2semidirect
5. #342×C211C2semidirect
13. C2×C4⋊C4×C211C4semidirect
14. C2×C22⋊C4×C211C4semidirect
103. C211#1899direct (N × H)
104. #1899C211direct (N × H)
106. C2×#117C2×C211semidirect
108. #381C2×C211semidirect
109. #342C2×C211semidirect
117. C2×C4⋊C4C4×C211semidirect
118. C2×C22⋊C4C4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#123×C2112
C2×#117×C2112
C2×#99×C2112
#381×C2112
#342×C2112
#1899211