Finite Group Databasebeta

← #1875×C211 #41009495

Subgroups — #1875×C211 #41009495

Order 27008.

412 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 128 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×#121×C211C211yesmaximal, characteristic
2C2×#117×C211C211yesmaximal, characteristic, complemented
3C2×#102×C211C212yesmaximal, complemented
4C2×C22⋊C8×C211C211yesmaximal, characteristic, complemented
5#340×C211C212yesmaximal, complemented
6C22×D4×C211—21no—
7C22×D4×C211C2211yescharacteristic
8C23×C4×C211C2211yescharacteristic
9C22×C8×C211C2212yes—
10#121×C211—22no—
11#121×C211C414yescomplemented
12#117×C211—24no—
13C2×C4⋊C4×C211C2211yescharacteristic
14C2×C4⋊C4×C211C2212yes—
15C2×C22⋊C4×C211—21no—
16C2×C22⋊C4×C211—21no—
17C2×C42×C211—21no—
18#102×C211—24no—
19C22⋊C8×C211—22no—
20C24×C211—21no—
21C24×C211D411yescharacteristic
22C2×D4×C211—24no—
23C2×D4×C211—22no—
24C2×D4×C211—21no—
25C2×D4×C211—22no—
26C2×D4×C211—44no—
27C2×D4×C211C2×C412yes—
28C22×C4×C211—22no—
29C22×C4×C211—22no—
30C22×C4×C211—22no—
31C22×C4×C211—21no—
32C22×C4×C211C2×C412yes—
33C22×C4×C211D411yescharacteristic
34C22×C4×C211D412yes—
35C22×C4×C211C2311yescharacteristic
36C2×C8×C211—22no—
37C2×C8×C211—22no—
38C2×C8×C211—22no—
39C4⋊C4×C211—22no—
40C4⋊C4×C211—21no—
41C4⋊C4×C211C2×C412yes—
42C4⋊C4×C211D414yescomplemented
43C22⋊C4×C211—24no—
44C22⋊C4×C211—24no—
45C42×C211—24no—
46C23×C211—21no—
47C23×C211—21no—
48C23×C211—22no—
49C23×C211—22no—
50C23×C211—22no—
51C23×C211—41no—
52C23×C211—41no—
53C23×C211—42no—
54C23×C211—42no—
55C23×C211C22⋊C412yes—
56C23×C211C2×D411yescharacteristic
57D4×C211—24no—
58D4×C211—48no—
59D4×C211—42no—
60C2×C4×C211—21no—
61C2×C4×C211—21no—
62C2×C4×C211—24no—
63C2×C4×C211—24no—
64C2×C4×C211—22no—
65C2×C4×C211—24no—
66C2×C4×C211—42no—
67C2×C4×C211—42no—
68C2×C4×C211—42no—
69C2×C4×C211C22⋊C412yes—
70C2×C4×C211C22×C411yescharacteristic
71C2×C4×C211C2×D412yes—
72C2×C4×C211C2×D411yescharacteristic
73C2×C4×C211#4112yes—
74C8×C211—42no—
75C22×C211—22no—
76C22×C211—22no—
77C22×C211—22no—
78C22×C211—42no—
79C22×C211—42no—
80C22×C211—42no—
81C22×C211—42no—
82C22×C211—42no—
83C22×C211—82no—
84C22×C211—82no—
85C22×C211C2×C22⋊C411yescharacteristic
86C22×C211#12011yescharacteristic
87C22×C211#12311yescharacteristic
88C22×C211#13512yes—
89C22×C211AGL(1,ℤ/8)12yes—
90C4×C211—22no—
91C4×C211—42no—
92C4×C211—44no—
93C4×C211C4×D412yes—
94C4×C211#12112yes—
95C2×C211—42no—
96C2×C211—82no—
97C2×C211#38611yescharacteristic
98C2×C211#41611yescharacteristic
99C2×C211#41811yescharacteristic
100C2×C211#44912yes—
101C2×C211#48212yes—
102C211#187511yescharacteristic, direct product factor, Sylow
103#1875C21111yesmaximal, characteristic, direct product factor, Sylow
104C2×#121C2×C21111yescharacteristic
105C2×#117C2×C21111yescharacteristic, complemented
106C2×#102C2×C21112yescomplemented
107C2×C22⋊C8C2×C21111yescharacteristic, complemented
108#340C2×C21112yescomplemented
109C22×D4—21no—
110C22×D4C22×C21111yescharacteristic
111C23×C4C22×C21111yescharacteristic
112C22×C8C22×C21112yes—
113#121—22no—
114#121C4×C21114yescomplemented
115#117—24no—
116C2×C4⋊C4C22×C21111yescharacteristic
117C2×C4⋊C4C22×C21112yes—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C42—21no—
121#102—24no—
122C22⋊C8—22no—
123C24—21no—
124C24D4×C21111yescharacteristic
125C2×D4—24no—
126C2×D4—22no—
127C2×D4—21no—
128C2×D4—22no—
129C2×D4—44no—
130C2×D4C2×C4×C21112yes—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4C2×C4×C21112yes—
136C22×C4D4×C21111yescharacteristic
137C22×C4D4×C21112yes—
138C22×C4C23×C21111yescharacteristic
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4C2×C4×C21112yes—
145C4⋊C4D4×C21114yescomplemented
146C22⋊C4—24no—
147C22⋊C4—24no—
148C42—24no—
149C23—21no—
150C23—21no—
151C23—22no—
152C23—22no—
153C23—22no—
154C23—41no—
155C23—41no—
156C23—42no—
157C23—42no—
158C23C22⋊C4×C21112yes—
159C23C2×D4×C21111yescharacteristic
160D4—24no—
161D4—48no—
162D4—42no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4C22⋊C4×C21112yes—
173C2×C4C22×C4×C21111yescharacteristic
174C2×C4C2×D4×C21112yes—
175C2×C4C2×D4×C21111yescharacteristic
176C2×C4#41×C21112yes—
177C8—42no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—42no—
186C22—82no—
187C22—82no—
188C22C2×C22⋊C4×C21111yescharacteristic
189C22#120×C21111yescharacteristic
190C22#123×C21111yescharacteristic
191C22#135×C21112yes—
192C22AGL(1,ℤ/8)×C21112yes—
193C4—22no—
194C4—42no—
195C4—44no—
196C4C4×D4×C21112yes—
197C4#121×C21112yes—
198C2—42no—
199C2—82no—
200C2#386×C21111yescharacteristic
201C2#416×C21111yescharacteristic
202C2#418×C21111yescharacteristic
203C2#449×C21112yes—
204C2#482×C21112yes—
subgroupG/H
center, socleC23×C211C2×D4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC2×#121×C211C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#117×C211C2semidirect
3. C2×#102×C211C2semidirect
4. C2×C22⋊C8×C211C2semidirect
5. #340×C211C2semidirect
11. #121×C211C4semidirect
42. C4⋊C4×C211D4semidirect
102. C211#1875direct (N × H)
103. #1875C211direct (N × H)
105. C2×#117C2×C211semidirect
106. C2×#102C2×C211semidirect
107. C2×C22⋊C8C2×C211semidirect
108. #340C2×C211semidirect
114. #121C4×C211semidirect
145. C4⋊C4D4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#121×C2112
C2×#117×C2112
C2×#102×C2112
C2×C22⋊C8×C2112
#340×C2112
#1875211