Finite Group Databasebeta

← C23×#19745 #19167371

Subgroups — C23×#19745 #19167371

Order 5888.

994 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 254 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×#3184C213yesmaximal, complemented
2C23×#3151C213yesmaximal, complemented
3C23×#1721C213yesmaximal, complemented
4C23×#1715C213yesmaximal, complemented
5C23×#1712C211yesmaximal, characteristic, complemented
6C23×#1705C211yesmaximal, characteristic, complemented
7C23×#1455C211yesmaximal, characteristic, complemented
8C2×C23×AGL(1,ℤ/8)—26no—
9C23×#532C2213yes—
10C2×C23×#127C2211yescharacteristic
11C23×#498C2213yescomplemented
12C23×#496—26no—
13C23×#493C2213yes—
14C23×#493C2213yes—
15C23×#493C2213yescomplemented
16C23×#492C2213yes—
17C23×#492C2213yes—
18C23×#486C2216yescomplemented
19C23×#469—26no—
20C23×#442—26no—
21C4×M16×C23C2213yes—
22C23×#322C2213yescomplemented
23C82×C23C2211yescharacteristic, complemented
24C2×#41×C23—23no—
25C22×D4×C23—23no—
26C23×AGL(1,ℤ/8)—424no—
27C2×SD16×C23—26no—
28C2×SD16×C23—26no—
29C2×D8×C23—26no—
30C2×D8×C23—26no—
31C2×D8×C23—26no—
32C2×M16×C23D416yes—
33C23×C4⋊Q8C2313yes—
34C23×#127—23no—
35C23×#127C2313yes—
36C23×#127C2311yescharacteristic
37C23×#127C2311yescharacteristic
38C23×#124—23no—
39C23×#121—26no—
40C4×D4×C23—26no—
41C2×C42×C23C2311yescharacteristic
42C23×#106—26no—
43C23×#102—212no—
44C23×#97D416yes—
45C4×C8×C23C2313yes—
46C4×C8×C23C2313yes—
47C24×C23—41no—
48#41×C23—412no—
49C2×Q8×C23—23no—
50C2×D4×C23—26no—
51C2×D4×C23—23no—
52C2×D4×C23—23no—
53C2×D4×C23—23no—
54C2×D4×C23—412no—
55C2×D4×C23—43no—
56C2×D4×C23—43no—
57C22×C4×C23—43no—
58C22×C4×C23C2×D413yes—
59SD16×C23—412no—
60SD16×C23—412no—
61D8×C23—412no—
62D8×C23—412no—
63D8×C23—412no—
64M16×C23—212no—
65C2×C8×C23C2×D416yes—
66C2×C8×C23C2×D416yes—
67C4⋊C4×C23—26no—
68C22⋊C4×C23—46no—
69C42×C23C2×D413yes—
70C42×C23C2411yescharacteristic
71#19745C2311yesmaximal, characteristic, direct product factor, Sylow
72C23×C23—43no—
73C23×C23—41no—
74C23×C23—41no—
75C23×C23—86no—
76C23×C23#12711yescharacteristic
77Q8×C23—46no—
78D4×C23—46no—
79D4×C23—46no—
80D4×C23—46no—
81D4×C23—412no—
82D4×C23—86no—
83D4×C23—86no—
84C2×C4×C23—26no—
85C2×C4×C23—43no—
86C2×C4×C23—86no—
87C2×C4×C23#12713yes—
88C2×C4×C23C22×D413yes—
89C8×C23—26no—
90C8×C23AGL(1,ℤ/8)112yescomplemented
91#3184C2×C2313yescomplemented
92#3151C2×C2313yescomplemented
93#1721C2×C2313yescomplemented
94#1715C2×C2313yescomplemented
95#1712C2×C2311yescharacteristic, complemented
96#1705C2×C2311yescharacteristic, complemented
97#1455C2×C2311yescharacteristic, complemented
98C22×C23—23no—
99C22×C23—83no—
100C22×C23—83no—
101C22×C23—83no—
102C22×C23—86no—
103C22×C23—164no—
104C22×C23C2×#12711yescharacteristic
105C4×C23—23no—
106C4×C23—83no—
107C4×C23C2×AGL(1,ℤ/8)16yes—
108C2×AGL(1,ℤ/8)—26no—
109#532C22×C2313yes—
110C2×#127C22×C2311yescharacteristic
111#498C22×C2313yescomplemented
112#496—26no—
113#493C22×C2313yes—
114#493C22×C2313yes—
115#493C22×C2313yescomplemented
116#492C22×C2313yes—
117#492C22×C2313yes—
118#486C22×C2316yescomplemented
119#469—26no—
120#442—26no—
121C4×M16C22×C2313yes—
122#322C22×C2313yescomplemented
123C82C22×C2311yescharacteristic, complemented
124C2×C23—41no—
125C2×C23—163no—
126C2×C23—161no—
127C2×C23—161no—
128C2×C23#315113yes—
129C2×#41—23no—
130C22×D4—23no—
131AGL(1,ℤ/8)—424no—
132C2×SD16—26no—
133C2×SD16—26no—
134C2×D8—26no—
135C2×D8—26no—
136C2×D8—26no—
137C2×M16D4×C2316yes—
138C4⋊Q8C23×C2313yes—
139#127—23no—
140#127C23×C2311yescharacteristic
141#127C23×C2311yescharacteristic
142#127C23×C2313yes—
143#124—23no—
144#121—26no—
145C4×D4—26no—
146C2×C42C23×C2311yescharacteristic
147#106—26no—
148#102—212no—
149#97D4×C2316yes—
150C4×C8C23×C2313yes—
151C4×C8C23×C2313yes—
152C23#1974511yescharacteristic, direct product factor, Sylow
153C24—41no—
154#41—412no—
155C2×Q8—23no—
156C2×D4—26no—
157C2×D4—23no—
158C2×D4—23no—
159C2×D4—23no—
160C2×D4—412no—
161C2×D4—43no—
162C2×D4—43no—
163C22×C4—43no—
164C22×C4C2×D4×C2313yes—
165SD16—412no—
166SD16—412no—
167D8—412no—
168D8—412no—
169D8—412no—
170M16—212no—
171C2×C8C2×D4×C2316yes—
172C2×C8C2×D4×C2316yes—
173C4⋊C4—26no—
174C22⋊C4—46no—
175C42C2×D4×C2313yes—
176C42C24×C2311yescharacteristic
177C23—43no—
178C23—41no—
179C23—41no—
180C23—86no—
181C23C23×#12711yescharacteristic
182Q8—46no—
183D4—46no—
184D4—46no—
185D4—46no—
186D4—412no—
187D4—86no—
188D4—86no—
189C2×C4—26no—
190C2×C4—43no—
191C2×C4—86no—
192C2×C4C23×#12713yes—
193C2×C4C22×D4×C2313yes—
194C8—26no—
195C8C23×AGL(1,ℤ/8)112yescomplemented
196C22—23no—
197C22—83no—
198C22—83no—
199C22—83no—
200C22—86no—
201C22—164no—
202C22C2×C23×#12711yescharacteristic
203C4—23no—
204C4—83no—
205C4C2×C23×AGL(1,ℤ/8)16yes—
206C2—41no—
207C2—161no—
208C2—161no—
209C2—163no—
210C2C23×#315113yes—
subgroupG/H
center, socleC22×C23C2×#127
commutator, Frattini, cosocleC42C24×C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3184C2semidirect
2. C23×#3151C2semidirect
3. C23×#1721C2semidirect
4. C23×#1715C2semidirect
5. C23×#1712C2semidirect
6. C23×#1705C2semidirect
7. C23×#1455C2semidirect
11. C23×#498C22semidirect
15. C23×#493C22semidirect
18. C23×#486C22semidirect
22. C23×#322C22semidirect
23. C82×C23C22semidirect
71. #19745C23direct (N × H)
90. C8×C23AGL(1,ℤ/8)semidirect
91. #3184C2×C23semidirect
92. #3151C2×C23semidirect
93. #1721C2×C23semidirect
94. #1715C2×C23semidirect
95. #1712C2×C23semidirect
96. #1705C2×C23semidirect
97. #1455C2×C23semidirect
111. #498C22×C23semidirect
115. #493C22×C23semidirect
118. #486C22×C23semidirect
122. #322C22×C23semidirect
123. C82C22×C23semidirect
152. C23#19745direct (N × H)
195. C8C23×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C23×#31842
C23×#31512
C23×#17212
C23×#17152
C23×#17122
C23×#17052
C23×#14552
#1974523