Finite Group Databasebeta

← #1390×C211 #41009010

Subgroups — #1390×C211 #41009010

Order 27008.

234 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 64 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×#100×C211C211yesmaximal, characteristic
2C2×#99×C211C211yesmaximal, characteristic
3#386×C211C211yesmaximal, characteristic
4C22×D4×C211C2211yescharacteristic
5C23×C4×C211C411yescharacteristic, complemented
6C2×M16×C211—21no—
7C2×C22⋊C4×C211—21no—
8C2×C22⋊C4×C211—21no—
9C2×C22⋊C4×C211C411yescharacteristic, complemented
10#100×C211—21no—
11#100×C211C412yescomplemented
12#99×C211—21no—
13#99×C211C412yescomplemented
14#95×C211—21no—
15C24×C211C2×C411yescharacteristic
16C24×C211D411yescharacteristic
17C2×D4×C211—41no—
18C2×D4×C211—41no—
19C2×D4×C211C2×C411yescharacteristic
20C2×D4×C211C2×C411yescharacteristic
21C2×D4×C211D411yescharacteristic
22C2×D4×C211Q811yescharacteristic
23C22×C4×C211—21no—
24C22×C4×C211—21no—
25C22×C4×C211—42no—
26C22×C4×C211—41no—
27C22×C4×C211—41no—
28C22×C4×C211D411yescharacteristic
29M16×C211—22no—
30M16×C211—41no—
31C2×C8×C211—41no—
32C22⋊C4×C211—22no—
33C22⋊C4×C211—42no—
34C22⋊C4×C211—41no—
35C22⋊C4×C211—41no—
36C23×C211—21no—
37C23×C211—21no—
38C23×C211—21no—
39C23×C211—21no—
40C23×C211—41no—
41C23×C211—41no—
42C23×C211—41no—
43C23×C211—41no—
44C23×C211C4211yescharacteristic
45C23×C211C22⋊C411yescharacteristic
46C23×C211C22⋊C411yescharacteristic
47C23×C211C4⋊C411yescharacteristic
48C23×C211C4⋊C411yescharacteristic
49D4×C211—41no—
50D4×C211—41no—
51D4×C211—41no—
52D4×C211—41no—
53C2×C4×C211—21no—
54C2×C4×C211—21no—
55C2×C4×C211—22no—
56C2×C4×C211—41no—
57C2×C4×C211—42no—
58C2×C4×C211—41no—
59C2×C4×C211—42no—
60C2×C4×C211—42no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211—81no—
65C2×C4×C211—81no—
66C2×C4×C211C22⋊C411yescharacteristic
67C2×C4×C211C4⋊C411yescharacteristic
68C8×C211—42no—
69C22×C211—21no—
70C22×C211—21no—
71C22×C211—21no—
72C22×C211—21no—
73C22×C211—21no—
74C22×C211—21no—
75C22×C211—41no—
76C22×C211—41no—
77C22×C211—41no—
78C22×C211—41no—
79C22×C211—42no—
80C22×C211—41no—
81C22×C211—41no—
82C22×C211—41no—
83C22×C211—41no—
84C22×C211—81no—
85C22×C211#9511yescharacteristic
86C22×C211#9911yescharacteristic
87C22×C211#9911yescharacteristic
88C4×C211—21no—
89C4×C211—21no—
90C4×C211—42no—
91C4×C211—81no—
92C4×C211—82no—
93C2×C211—21no—
94C2×C211—21no—
95C2×C211—41no—
96C2×C211—41no—
97C2×C211—41no—
98C2×C211—41no—
99C2×C211#34211yescharacteristic
100C2×C211AΣL(1,𝔽16)11yescharacteristic
101C2×C211#35211yescharacteristic
102C211#139011yescharacteristic, direct product factor, Sylow
103#1390C21111yesmaximal, characteristic, direct product factor, Sylow
104C2×#100C2×C21111yescharacteristic
105C2×#99C2×C21111yescharacteristic
106#386C2×C21111yescharacteristic
107C22×D4C22×C21111yescharacteristic
108C23×C4C4×C21111yescharacteristic, complemented
109C2×M16—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4C4×C21111yescharacteristic, complemented
113#100—21no—
114#100C4×C21112yescomplemented
115#99—21no—
116#99C4×C21112yescomplemented
117#95—21no—
118C24C2×C4×C21111yescharacteristic
119C24D4×C21111yescharacteristic
120C2×D4—41no—
121C2×D4—41no—
122C2×D4C2×C4×C21111yescharacteristic
123C2×D4C2×C4×C21111yescharacteristic
124C2×D4D4×C21111yescharacteristic
125C2×D4Q8×C21111yescharacteristic
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—42no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4D4×C21111yescharacteristic
132M16—22no—
133M16—41no—
134C2×C8—41no—
135C22⋊C4—22no—
136C22⋊C4—42no—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23C42×C21111yescharacteristic
148C23C22⋊C4×C21111yescharacteristic
149C23C22⋊C4×C21111yescharacteristic
150C23C4⋊C4×C21111yescharacteristic
151C23C4⋊C4×C21111yescharacteristic
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4C22⋊C4×C21111yescharacteristic
170C2×C4C4⋊C4×C21111yescharacteristic
171C8—42no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—42no—
183C22—41no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—81no—
188C22#95×C21111yescharacteristic
189C22#99×C21111yescharacteristic
190C22#99×C21111yescharacteristic
191C4—21no—
192C4—21no—
193C4—42no—
194C4—81no—
195C4—82no—
196C2—21no—
197C2—21no—
198C2—41no—
199C2—41no—
200C2—41no—
201C2—41no—
202C2#342×C21111yescharacteristic
203C2AΣL(1,𝔽16)×C21111yescharacteristic
204C2#352×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#99
commutatorC23C42×C211
Frattini, cosocleC22×D4C22×C211
prime radicalC22×D4×C211C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C23×C211 ◁ C22×D4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
5. C23×C4×C211C4semidirect
9. C2×C22⋊C4×C211C4semidirect
11. #100×C211C4semidirect
13. #99×C211C4semidirect
102. C211#1390direct (N × H)
103. #1390C211direct (N × H)
108. C23×C4C4×C211semidirect
112. C2×C22⋊C4C4×C211semidirect
114. #100C4×C211semidirect
116. #99C4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#100×C2112
C2×#99×C2112
#386×C2112
#1390211