Finite Group Databasebeta

← C23×#7406 #19155032

Subgroups — C23×#7406 #19155032

Order 5888.

244 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 42 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×#2017C211yesmaximal, characteristic
2C23×#1391C211yesmaximal, characteristic, complemented
3C23×#1315C211yesmaximal, characteristic, complemented
4C23×#575—21no—
5C23×#575C411yescharacteristic, complemented
6C23×#410—21no—
7C23×#408—21no—
8C23×#408C2211yescharacteristic
9C23×C22⋊C16—21no—
10C23×#338C411yescharacteristic, complemented
11C2×#41×C23—21no—
12C2×#41×C23D411yescharacteristic
13C23×#137—41no—
14C23×#137—41no—
15C23×#137—41no—
16C23×AGL(1,ℤ/8)—41no—
17C23×AGL(1,ℤ/8)—41no—
18C23×AGL(1,ℤ/8)—41no—
19C23×#135—41no—
20C23×#135—42no—
21C2×M16×C23—21no—
22C2×M16×C23C2×C411yescharacteristic
23C22×C8×C23D411yescharacteristic
24C23×#117—41no—
25C2×C16×C23—41no—
26C23×#99—22no—
27C23×#99—41no—
28C23×C22⋊C8—41no—
29C23×C22⋊C8—42no—
30#41×C23—41no—
31#41×C23—41no—
32#41×C23—41no—
33#41×C23—41no—
34#41×C23—41no—
35#41×C23—41no—
36C2×Q8×C23—21no—
37C2×Q8×C23SD1611yescharacteristic
38C2×D4×C23—21no—
39C2×D4×C23—21no—
40C2×D4×C23—41no—
41C2×D4×C23D811yescharacteristic
42C22×C4×C23—21no—
43C22×C4×C23—41no—
44C22×C4×C23C22⋊C411yescharacteristic
45Q16×C23—41no—
46Q16×C23—42no—
47SD16×C23—42no—
48SD16×C23—42no—
49SD16×C23—42no—
50D8×C23—41no—
51D8×C23—42no—
52M16×C23—41no—
53M16×C23—41no—
54M16×C23—41no—
55C2×C8×C23—22no—
56C2×C8×C23—21no—
57C2×C8×C23—21no—
58C2×C8×C23—41no—
59C4⋊C4×C23—81no—
60C22⋊C4×C23—42no—
61C42×C23—81no—
62C16×C23—81no—
63#7406C2311yesmaximal, characteristic, direct product factor, Sylow
64C23×C23—21no—
65C23×C23—41no—
66C23×C23#10411yescharacteristic
67Q8×C23—41no—
68Q8×C23—41no—
69Q8×C23—41no—
70D4×C23—41no—
71D4×C23—41no—
72D4×C23—41no—
73D4×C23—81no—
74D4×C23—81no—
75D4×C23—81no—
76C2×C4×C23—21no—
77C2×C4×C23—21no—
78C2×C4×C23—21no—
79C2×C4×C23—41no—
80C2×C4×C23—41no—
81C2×C4×C23—41no—
82C2×C4×C23—81no—
83C2×C4×C23—81no—
84C2×C4×C23—82no—
85C2×C4×C23#9911yescharacteristic
86C2×C4×C23#10211yescharacteristic
87C8×C23—42no—
88C8×C23—41no—
89C8×C23—41no—
90#2017C2×C2311yescharacteristic
91#1391C2×C2311yescharacteristic, complemented
92#1315C2×C2311yescharacteristic, complemented
93C22×C23—21no—
94C22×C23—41no—
95C22×C23—41no—
96C22×C23—81no—
97C22×C23—82no—
98C22×C23—81no—
99C22×C23#32711yescharacteristic
100C4×C23—21no—
101C4×C23—41no—
102C4×C23—81no—
103C4×C23—162no—
104C4×C23AΣL(1,𝔽16)11yescharacteristic
105C4×C23#35311yescharacteristic
106#575—21no—
107#575C4×C2311yescharacteristic, complemented
108#410—21no—
109#408—21no—
110#408C22×C2311yescharacteristic
111C22⋊C16—21no—
112#338C4×C2311yescharacteristic, complemented
113C2×C23—21no—
114C2×C23—41no—
115C2×C23—81no—
116C2×C23—161no—
117C2×C23#134311yescharacteristic
118C2×#41—21no—
119C2×#41D4×C2311yescharacteristic
120#137—41no—
121#137—41no—
122#137—41no—
123AGL(1,ℤ/8)—41no—
124AGL(1,ℤ/8)—41no—
125AGL(1,ℤ/8)—41no—
126#135—41no—
127#135—42no—
128C2×M16—21no—
129C2×M16C2×C4×C2311yescharacteristic
130C22×C8D4×C2311yescharacteristic
131#117—41no—
132C2×C16—41no—
133#99—22no—
134#99—41no—
135C22⋊C8—41no—
136C22⋊C8—42no—
137C23#740611yescharacteristic, direct product factor, Sylow
138#41—41no—
139#41—41no—
140#41—41no—
141#41—41no—
142#41—41no—
143#41—41no—
144C2×Q8—21no—
145C2×Q8SD16×C2311yescharacteristic
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—41no—
149C2×D4D8×C2311yescharacteristic
150C22×C4—21no—
151C22×C4—41no—
152C22×C4C22⋊C4×C2311yescharacteristic
153Q16—41no—
154Q16—42no—
155SD16—42no—
156SD16—42no—
157SD16—42no—
158D8—41no—
159D8—42no—
160M16—41no—
161M16—41no—
162M16—41no—
163C2×C8—22no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—41no—
167C4⋊C4—81no—
168C22⋊C4—42no—
169C42—81no—
170C16—81no—
171C23—21no—
172C23—41no—
173C23C23×#10411yescharacteristic
174Q8—41no—
175Q8—41no—
176Q8—41no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—81no—
181D4—81no—
182D4—81no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—82no—
192C2×C4C23×#9911yescharacteristic
193C2×C4C23×#10211yescharacteristic
194C8—42no—
195C8—41no—
196C8—41no—
197C22—21no—
198C22—41no—
199C22—41no—
200C22—81no—
201C22—82no—
202C22—81no—
203C22C23×#32711yescharacteristic
204C4—21no—
205C4—41no—
206C4—81no—
207C4—162no—
208C4C23×AΣL(1,𝔽16)11yescharacteristic
209C4C23×#35311yescharacteristic
210C2—21no—
211C2—41no—
212C2—81no—
213C2—161no—
214C2C23×#134311yescharacteristic
subgroupG/H
centerC4×C23#353
commutatorC2×M16C2×C4×C23
Frattini, cosocle#408C22×C23
socleC2×C23#1343
prime radicalC23×#2017C2
LCSG ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C23 ◁ C2×C4×C23 ◁ C22×C4×C23 ◁ C23×#408 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1391C2semidirect
3. C23×#1315C2semidirect
5. C23×#575C4semidirect
10. C23×#338C4semidirect
63. #7406C23direct (N × H)
91. #1391C2×C23semidirect
92. #1315C2×C23semidirect
107. #575C4×C23semidirect
112. #338C4×C23semidirect
137. C23#7406direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#20172
C23×#13912
C23×#13152
#740623