Finite Group Databasebeta

← C23×#12261 #19159887

Subgroups — C23×#12261 #19159887

Order 5888.

330 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2844C211yesmaximal, characteristic
2C23×#2061C211yesmaximal, characteristic
3C23×#1617C212yesmaximal
4C23×#1353C212yesmaximal
5C23×#1324C211yesmaximal, characteristic
6C23×#544C2211yescharacteristic
7C23×#484—22no—
8C23×#428—21no—
9C23×#400—22no—
10C23×#400C2212yes—
11C23×#395—22no—
12C23×#395—21no—
13C23×#382—21no—
14C23×#382C2211yescharacteristic
15C23×#343C2211yescharacteristic
16C23×#339—21no—
17C23×#328—22no—
18C23×#324C2212yes—
19C22×Q8×C23—21no—
20C2×M16×C23—21no—
21C23×C4⋊Q8—41no—
22C23×#125—41no—
23C23×C22⋊Q8—22no—
24C23×#117—21no—
25C2×C4⋊C4×C23—21no—
26C2×C4⋊C4×C23—21no—
27C2×C4⋊C4×C23—22no—
28C2×C4⋊C4×C23D411yescharacteristic
29C2×C4⋊C4×C23D412yes—
30C2×C4⋊C4×C23C2311yescharacteristic
31C2×C42×C23D411yescharacteristic
32C23×#107—41no—
33C23×#107—42no—
34C23×#106—41no—
35C23×#103—42no—
36C23×C22⋊C8—22no—
37C23×#95—22no—
38C23×#95—21no—
39C23×#95—22no—
40C23×#95—22no—
41C23×#95—22no—
42C23×#95D412yes—
43C2×Q8×C23—22no—
44C2×Q8×C23—82no—
45C2×Q8×C23—81no—
46C22×C4×C23—21no—
47C22×C4×C23—22no—
48C22×C4×C23—21no—
49C22×C4×C23—22no—
50C22×C4×C23C2×D411yescharacteristic
51C22×C4×C23C2×D412yes—
52M16×C23—22no—
53M16×C23—41no—
54C2×C8×C23—41no—
55C2×C8×C23—42no—
56C4⋊C4×C23—21no—
57C4⋊C4×C23—21no—
58C4⋊C4×C23—22no—
59C4⋊C4×C23—41no—
60C4⋊C4×C23—42no—
61C4⋊C4×C23—41no—
62C4⋊C4×C23—42no—
63C4⋊C4×C23—81no—
64C4⋊C4×C23—81no—
65C4⋊C4×C23—82no—
66C22⋊C4×C23—41no—
67C42×C23—21no—
68C42×C23—41no—
69C42×C23Q1612yes—
70#12261C2311yesmaximal, characteristic, direct product factor, Sylow
71C23×C23#12011yescharacteristic
72Q8×C23—82no—
73Q8×C23—84no—
74C2×C4×C23—21no—
75C2×C4×C23—21no—
76C2×C4×C23—22no—
77C2×C4×C23—22no—
78C2×C4×C23—41no—
79C2×C4×C23—41no—
80C2×C4×C23—42no—
81C2×C4×C23—42no—
82C2×C4×C23—42no—
83C2×C4×C23—42no—
84C2×C4×C23—42no—
85C2×C4×C23—42no—
86C2×C4×C23—42no—
87C2×C4×C23—41no—
88C2×C4×C23—41no—
89C2×C4×C23—42no—
90C2×C4×C23—41no—
91C2×C4×C23C2×Q1611yescharacteristic
92C2×C4×C23#13711yescharacteristic
93C8×C23—42no—
94C8×C23—82no—
95#2844C2×C2311yescharacteristic
96#2061C2×C2311yescharacteristic
97#1617C2×C2312yes—
98#1353C2×C2312yes—
99#1324C2×C2311yescharacteristic
100C22×C23—21no—
101C22×C23—21no—
102C22×C23#45111yescharacteristic
103C22×C23#45611yescharacteristic
104C22×C23B(4,𝔽2)11yescharacteristic
105C4×C23—21no—
106C4×C23—21no—
107C4×C23—42no—
108C4×C23—42no—
109C4×C23—82no—
110C4×C23—81no—
111C4×C23—81no—
112C4×C23—82no—
113C4×C23—81no—
114#544C22×C2311yescharacteristic
115#484—22no—
116#428—21no—
117#400—22no—
118#400C22×C2312yes—
119#395—22no—
120#395—21no—
121#382—21no—
122#382C22×C2311yescharacteristic
123#343C22×C2311yescharacteristic
124#339—21no—
125#328—22no—
126#324C22×C2312yes—
127C2×C23—21no—
128C2×C23—21no—
129C2×C23#160511yescharacteristic
130C2×C23#220111yescharacteristic
131C2×C23#220211yescharacteristic
132C22×Q8—21no—
133C2×M16—21no—
134C4⋊Q8—41no—
135#125—41no—
136C22⋊Q8—22no—
137#117—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4D4×C2311yescharacteristic
142C2×C4⋊C4D4×C2312yes—
143C2×C4⋊C4C23×C2311yescharacteristic
144C2×C42D4×C2311yescharacteristic
145#107—41no—
146#107—42no—
147#106—41no—
148#103—42no—
149C22⋊C8—22no—
150#95—22no—
151#95—21no—
152#95—22no—
153#95—22no—
154#95—22no—
155#95D4×C2312yes—
156C23#1226111yescharacteristic, direct product factor, Sylow
157C2×Q8—22no—
158C2×Q8—82no—
159C2×Q8—81no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4C2×D4×C2311yescharacteristic
165C22×C4C2×D4×C2312yes—
166M16—22no—
167M16—41no—
168C2×C8—41no—
169C2×C8—42no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—82no—
180C22⋊C4—41no—
181C42—21no—
182C42—41no—
183C42Q16×C2312yes—
184C23C23×#12011yescharacteristic
185Q8—82no—
186Q8—84no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4C2×Q16×C2311yescharacteristic
205C2×C4C23×#13711yescharacteristic
206C8—42no—
207C8—82no—
208C22—21no—
209C22—21no—
210C22C23×#45111yescharacteristic
211C22C23×#45611yescharacteristic
212C22C23×B(4,𝔽2)11yescharacteristic
213C4—21no—
214C4—21no—
215C4—42no—
216C4—42no—
217C4—82no—
218C4—81no—
219C4—82no—
220C4—81no—
221C4—81no—
222C2—21no—
223C2—21no—
224C2C23×#160511yescharacteristic
225C2C23×#220111yescharacteristic
226C2C23×#220211yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC23×C23#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C2×C4⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
70. #12261C23direct (N × H)
156. C23#12261direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28442
C23×#20612
C23×#16172
C23×#13532
C23×#13242
#1226123