Finite Group Databasebeta

← C23×#12308 #19159934

Subgroups — C23×#12308 #19159934

Order 5888.

274 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C23×#3255C211yesmaximal, characteristic
2C23×#2239C212yesmaximal
3C23×#2102C211yesmaximal, characteristic
4C23×#1340C212yesmaximal
5C23×#1327C211yesmaximal, characteristic
6C23×#533—21no—
7C23×#475—22no—
8C23×#470—21no—
9C23×#470C2212yes—
10C23×#462—22no—
11C23×#430C2211yescharacteristic
12C23×#423C2211yescharacteristic
13C23×#365—22no—
14C23×#362—22no—
15C23×#349C2212yes—
16C23×#344—21no—
17C23×#339—21no—
18C23×H(3,ℤ/4)C2211yescharacteristic
19C2×Q16×C23—21no—
20C2×Q16×C23D412yes—
21C2×M16×C23—21no—
22C2×M16×C23D411yescharacteristic
23C2×M16×C23C2311yescharacteristic
24C23×C4⋊Q8—21no—
25C23×C4⋊Q8—21no—
26C23×#125—41no—
27C23×C22⋊Q8—22no—
28C23×C22⋊Q8—41no—
29C4×Q8×C23—22no—
30C4×Q8×C23—41no—
31C23×#117—21no—
32C23×#117—41no—
33C2×C42×C23D411yescharacteristic
34C23×M32—22no—
35C23×#108—22no—
36C23×#106—21no—
37C23×#106D412yes—
38C23×C4⋊C8—22no—
39C23×#103—22no—
40C23×#103—22no—
41C2×Q8×C23—22no—
42C2×Q8×C23—41no—
43C22×C4×C23—21no—
44C22×C4×C23C2×D411yescharacteristic
45Q16×C23—42no—
46Q16×C23—81no—
47M16×C23—41no—
48M16×C23—41no—
49M16×C23—41no—
50M16×C23—41no—
51C2×C8×C23—21no—
52C2×C8×C23—41no—
53C2×C8×C23C2×D412yes—
54C4⋊C4×C23—22no—
55C4⋊C4×C23—41no—
56C4⋊C4×C23—41no—
57C4⋊C4×C23—42no—
58C4⋊C4×C23—41no—
59C4⋊C4×C23—41no—
60C4⋊C4×C23—42no—
61C22⋊C4×C23—42no—
62C22⋊C4×C23—41no—
63C42×C23—21no—
64C42×C23—42no—
65C42×C23—41no—
66C42×C23SD1611yescharacteristic
67C42×C23SD1611yescharacteristic
68C16×C23—42no—
69#12308C2311yesmaximal, characteristic, direct product factor, Sylow
70C23×C23AGL(1,ℤ/8)11yescharacteristic
71Q8×C23—42no—
72Q8×C23—42no—
73Q8×C23—82no—
74C2×C4×C23—21no—
75C2×C4×C23—21no—
76C2×C4×C23—21no—
77C2×C4×C23—21no—
78C2×C4×C23—42no—
79C2×C4×C23—42no—
80C2×C4×C23—42no—
81C2×C4×C23—81no—
82C2×C4×C23#12011yescharacteristic
83C2×C4×C23C2×SD1611yescharacteristic
84C8×C23—22no—
85C8×C23—41no—
86C8×C23—81no—
87#3255C2×C2311yescharacteristic
88#2239C2×C2312yes—
89#2102C2×C2311yescharacteristic
90#1340C2×C2312yes—
91#1327C2×C2311yescharacteristic
92C22×C23—21no—
93C22×C23—41no—
94C22×C23#45011yescharacteristic
95C4×C23—22no—
96C4×C23—21no—
97C4×C23—41no—
98C4×C23—82no—
99C4×C23—82no—
100C4×C23—161no—
101C4×C23#45411yescharacteristic
102C4×C23#45811yescharacteristic
103#533—21no—
104#475—22no—
105#470—21no—
106#470C22×C2312yes—
107#462—22no—
108#430C22×C2311yescharacteristic
109#423C22×C2311yescharacteristic
110#365—22no—
111#362—22no—
112#349C22×C2312yes—
113#344—21no—
114#339—21no—
115H(3,ℤ/4)C22×C2311yescharacteristic
116C2×C23—21no—
117C2×C23—41no—
118C2×C23#161611yescharacteristic
119C2×Q16—21no—
120C2×Q16D4×C2312yes—
121C2×M16—21no—
122C2×M16D4×C2311yescharacteristic
123C2×M16C23×C2311yescharacteristic
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126#125—41no—
127C22⋊Q8—22no—
128C22⋊Q8—41no—
129C4×Q8—22no—
130C4×Q8—41no—
131#117—21no—
132#117—41no—
133C2×C42D4×C2311yescharacteristic
134M32—22no—
135#108—22no—
136#106—21no—
137#106D4×C2312yes—
138C4⋊C8—22no—
139#103—22no—
140#103—22no—
141C23#1230811yescharacteristic, direct product factor, Sylow
142C2×Q8—22no—
143C2×Q8—41no—
144C22×C4—21no—
145C22×C4C2×D4×C2311yescharacteristic
146Q16—42no—
147Q16—81no—
148M16—41no—
149M16—41no—
150M16—41no—
151M16—41no—
152C2×C8—21no—
153C2×C8—41no—
154C2×C8C2×D4×C2312yes—
155C4⋊C4—22no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—41no—
164C42—21no—
165C42—42no—
166C42—41no—
167C42SD16×C2311yescharacteristic
168C42SD16×C2311yescharacteristic
169C16—42no—
170C23C23×AGL(1,ℤ/8)11yescharacteristic
171Q8—42no—
172Q8—42no—
173Q8—82no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—81no—
182C2×C4C23×#12011yescharacteristic
183C2×C4C2×SD16×C2311yescharacteristic
184C8—22no—
185C8—41no—
186C8—81no—
187C22—21no—
188C22—41no—
189C22C23×#45011yescharacteristic
190C4—22no—
191C4—21no—
192C4—41no—
193C4—82no—
194C4—82no—
195C4—161no—
196C4C23×#45411yescharacteristic
197C4C23×#45811yescharacteristic
198C2—21no—
199C2—41no—
200C2C23×#161611yescharacteristic
subgroupG/H
center, socleC2×C23#1616
commutator, Frattini, cosocleC2×M16C23×C23
prime radicalC23×C23AGL(1,ℤ/8)
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C2×M16×C23 ◁ G

Complements

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

NH ≅ G/Ntype
69. #12308C23direct (N × H)
141. C23#12308direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32552
C23×#22392
C23×#21022
C23×#13402
C23×#13272
#1230823