Finite Group Databasebeta

← C23×#7802 #19155428

Subgroups — C23×#7802 #19155428

Order 5888.

366 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2809C211yesmaximal, characteristic
2C23×#2096C211yesmaximal, characteristic
3C23×#2083C212yesmaximal
4C23×#1272C211yesmaximal, characteristic
5C23×#1271C212yesmaximal
6C2×C23×#107—22no—
7C2×C23×#106—21no—
8C2×C23×#103—21no—
9C23×#400—21no—
10C23×#398—22no—
11C23×#391—21no—
12C23×#391C2211yescharacteristic
13C23×#389C2212yes—
14C23×#387—22no—
15C23×#384C2211yescharacteristic
16C23×#340—22no—
17C23×#336C2211yescharacteristic
18C23×#336C2212yes—
19C22×Q8×C23D411yescharacteristic
20C22×C8×C23—22no—
21C22×C8×C23—21no—
22C2×C4⋊C4×C23—22no—
23C2×C4⋊C4×C23—22no—
24C2×C4⋊C4×C23—21no—
25C2×C4⋊C4×C23—21no—
26C2×C4⋊C4×C23—21no—
27C2×C4⋊C4×C23D412yes—
28C2×C4⋊C4×C23D412yes—
29C2×C4⋊C4×C23D411yescharacteristic
30C2×C42×C23—21no—
31C2×C42×C23C2311yescharacteristic
32C23×#107—44no—
33C23×#106—42no—
34C23×#103—42no—
35C23×#95—21no—
36C23×#95—22no—
37C23×#95—22no—
38C23×#95—22no—
39C23×#95—22no—
40C2×Q8×C23—21no—
41C2×Q8×C23—41no—
42C2×Q8×C23—41no—
43C2×Q8×C23Q1612yes—
44C22×C4×C23—21no—
45C22×C4×C23—22no—
46C22×C4×C23—22no—
47C22×C4×C23—21no—
48C22×C4×C23—41no—
49C22×C4×C23C2×D411yescharacteristic
50C22×C4×C23C2×D412yes—
51C2×C8×C23—22no—
52C2×C8×C23—24no—
53C2×C8×C23—42no—
54C2×C8×C23—41no—
55C2×C8×C23—42no—
56C2×C8×C23—41no—
57C4⋊C4×C23—22no—
58C4⋊C4×C23—22no—
59C4⋊C4×C23—41no—
60C4⋊C4×C23—42no—
61C4⋊C4×C23—41no—
62C4⋊C4×C23—41no—
63C4⋊C4×C23—42no—
64C4⋊C4×C23—42no—
65C4⋊C4×C23—42no—
66C4⋊C4×C23—42no—
67C4⋊C4×C23—81no—
68C4⋊C4×C23—81no—
69C42×C23—41no—
70C42×C23—81no—
71#7802C2311yesmaximal, characteristic, direct product factor, Sylow
72C23×C23#12011yescharacteristic
73Q8×C23—42no—
74Q8×C23—81no—
75C2×C4×C23—21no—
76C2×C4×C23—22no—
77C2×C4×C23—22no—
78C2×C4×C23—21no—
79C2×C4×C23—22no—
80C2×C4×C23—41no—
81C2×C4×C23—41no—
82C2×C4×C23—42no—
83C2×C4×C23—42no—
84C2×C4×C23—42no—
85C2×C4×C23—41no—
86C2×C4×C23—42no—
87C2×C4×C23—42no—
88C2×C4×C23—42no—
89C2×C4×C23—42no—
90C2×C4×C23—82no—
91C2×C4×C23—81no—
92C2×C4×C23C2×Q1611yescharacteristic
93C2×C4×C23#13512yes—
94C2×C4×C23AGL(1,ℤ/8)12yes—
95C2×C4×C23#13711yescharacteristic
96C8×C23—44no—
97C8×C23—42no—
98#2809C2×C2311yescharacteristic
99#2096C2×C2311yescharacteristic
100#2083C2×C2312yes—
101#1272C2×C2311yescharacteristic
102#1271C2×C2312yes—
103C22×C23#44912yes—
104C22×C23#45111yescharacteristic
105C22×C23#45311yescharacteristic
106C22×C23#45612yes—
107C22×C23#45811yescharacteristic
108C4×C23—22no—
109C4×C23—22no—
110C4×C23—21no—
111C4×C23—21no—
112C4×C23—42no—
113C4×C23—82no—
114C4×C23—82no—
115C4×C23—81no—
116C4×C23—161no—
117C2×#107—22no—
118C2×#106—21no—
119C2×#103—21no—
120#400—21no—
121#398—22no—
122#391—21no—
123#391C22×C2311yescharacteristic
124#389C22×C2312yes—
125#387—22no—
126#384C22×C2311yescharacteristic
127#340—22no—
128#336C22×C2312yes—
129#336C22×C2311yescharacteristic
130C2×C23#160912yes—
131C2×C23#161411yescharacteristic
132C2×C23#162411yescharacteristic
133C2×C23#164812yes—
134C2×C23#166011yescharacteristic
135C22×Q8D4×C2311yescharacteristic
136C22×C8—22no—
137C22×C8—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4D4×C2312yes—
144C2×C4⋊C4D4×C2312yes—
145C2×C4⋊C4D4×C2311yescharacteristic
146C2×C42—21no—
147C2×C42C23×C2311yescharacteristic
148#107—44no—
149#106—42no—
150#103—42no—
151#95—21no—
152#95—22no—
153#95—22no—
154#95—22no—
155#95—22no—
156C23#780211yescharacteristic, direct product factor, Sylow
157C2×Q8—21no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8Q16×C2312yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—41no—
166C22×C4C2×D4×C2311yescharacteristic
167C22×C4C2×D4×C2312yes—
168C2×C8—22no—
169C2×C8—24no—
170C2×C8—42no—
171C2×C8—41no—
172C2×C8—42no—
173C2×C8—41no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C42—41no—
187C42—81no—
188C23C23×#12011yescharacteristic
189Q8—42no—
190Q8—81no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—82no—
207C2×C4—81no—
208C2×C4C2×Q16×C2311yescharacteristic
209C2×C4C23×#13512yes—
210C2×C4C23×AGL(1,ℤ/8)12yes—
211C2×C4C23×#13711yescharacteristic
212C8—44no—
213C8—42no—
214C22C23×#44912yes—
215C22C23×#45111yescharacteristic
216C22C23×#45311yescharacteristic
217C22C23×#45612yes—
218C22C23×#45811yescharacteristic
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—81no—
225C4—82no—
226C4—82no—
227C4—161no—
228C2C23×#160912yes—
229C2C23×#161411yescharacteristic
230C2C23×#162411yescharacteristic
231C2C23×#164812yes—
232C2C23×#166011yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C23#120
commutator, Frattini, cosocleC2×C42C23×C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
71. #7802C23direct (N × H)
156. C23#7802direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28092
C23×#20962
C23×#20832
C23×#12722
C23×#12712
#780223