Finite Group Databasebeta

← C23×#12257 #19159883

Subgroups — C23×#12257 #19159883

Order 5888.

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

normal subgroups only

This group has 342 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×#2571C211yesmaximal, characteristic
2C23×#2060C211yesmaximal, characteristic
3C23×#1617C212yesmaximal
4C23×#1353C212yesmaximal
5C23×#1325C211yesmaximal, characteristic
6C23×#544C2211yescharacteristic
7C23×#484—22no—
8C23×#428—21no—
9C23×#400—21no—
10C23×#400C2212yes—
11C23×#389—22no—
12C23×#384C2211yescharacteristic
13C23×#382—22no—
14C4×C4⋊C4×C23—21no—
15C23×#343C2211yescharacteristic
16C23×#339—21no—
17C23×#328—22no—
18C23×#324C2212yes—
19C2×M16×C23—21no—
20C23×C4⋊Q8—41no—
21C23×#125—41no—
22C23×C22⋊Q8—22no—
23C23×#117—21no—
24C2×C4⋊C4×C23—21no—
25C2×C4⋊C4×C23—21no—
26C2×C4⋊C4×C23—22no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23D411yescharacteristic
29C2×C4⋊C4×C23D412yes—
30C2×C4⋊C4×C23C2311yescharacteristic
31C2×C42×C23—22no—
32C2×C42×C23D411yescharacteristic
33C23×#107—41no—
34C23×#107—42no—
35C23×#106—41no—
36C23×#103—42no—
37C23×C22⋊C8—22no—
38C23×#95—21no—
39C23×#95—22no—
40C23×#95—22no—
41C23×#95—22no—
42C23×#95D412yes—
43C2×Q8×C23—22no—
44C22×C4×C23—21no—
45C22×C4×C23—22no—
46C22×C4×C23—22no—
47C22×C4×C23—21no—
48C22×C4×C23C2×D411yescharacteristic
49C22×C4×C23C2×D412yes—
50M16×C23—22no—
51M16×C23—41no—
52C2×C8×C23—41no—
53C2×C8×C23—42no—
54C4⋊C4×C23—21no—
55C4⋊C4×C23—21no—
56C4⋊C4×C23—22no—
57C4⋊C4×C23—41no—
58C4⋊C4×C23—42no—
59C4⋊C4×C23—41no—
60C4⋊C4×C23—42no—
61C4⋊C4×C23—44no—
62C4⋊C4×C23—42no—
63C4⋊C4×C23—42no—
64C4⋊C4×C23—81no—
65C22⋊C4×C23—41no—
66C42×C23—21no—
67C42×C23—44no—
68C42×C23—41no—
69C42×C23Q1612yes—
70#12257C2311yesmaximal, characteristic, direct product factor, Sylow
71C23×C23#12011yescharacteristic
72Q8×C23—82no—
73C2×C4×C23—22no—
74C2×C4×C23—22no—
75C2×C4×C23—22no—
76C2×C4×C23—21no—
77C2×C4×C23—21no—
78C2×C4×C23—22no—
79C2×C4×C23—42no—
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—41no—
87C2×C4×C23—41no—
88C2×C4×C23—41no—
89C2×C4×C23—42no—
90C2×C4×C23C2×Q1611yescharacteristic
91C2×C4×C23#13711yescharacteristic
92C8×C23—42no—
93C8×C23—82no—
94#2571C2×C2311yescharacteristic
95#2060C2×C2311yescharacteristic
96#1617C2×C2312yes—
97#1353C2×C2312yes—
98#1325C2×C2311yescharacteristic
99C22×C23—21no—
100C22×C23—21no—
101C22×C23#45111yescharacteristic
102C22×C23#45611yescharacteristic
103C22×C23B(4,𝔽2)11yescharacteristic
104C4×C23—24no—
105C4×C23—21no—
106C4×C23—21no—
107C4×C23—42no—
108C4×C23—82no—
109C4×C23—81no—
110C4×C23—81no—
111C4×C23—81no—
112C4×C23—82no—
113#544C22×C2311yescharacteristic
114#484—22no—
115#428—21no—
116#400—21no—
117#400C22×C2312yes—
118#389—22no—
119#384C22×C2311yescharacteristic
120#382—22no—
121C4×C4⋊C4—21no—
122#343C22×C2311yescharacteristic
123#339—21no—
124#328—22no—
125#324C22×C2312yes—
126C2×C23—21no—
127C2×C23—21no—
128C2×C23#160511yescharacteristic
129C2×C23#219811yescharacteristic
130C2×C23#220411yescharacteristic
131C2×M16—21no—
132C4⋊Q8—41no—
133#125—41no—
134C22⋊Q8—22no—
135#117—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4D4×C2311yescharacteristic
141C2×C4⋊C4D4×C2312yes—
142C2×C4⋊C4C23×C2311yescharacteristic
143C2×C42—22no—
144C2×C42D4×C2311yescharacteristic
145#107—41no—
146#107—42no—
147#106—41no—
148#103—42no—
149C22⋊C8—22no—
150#95—21no—
151#95—22no—
152#95—22no—
153#95—22no—
154#95D4×C2312yes—
155C23#1225711yescharacteristic, direct product factor, Sylow
156C2×Q8—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4C2×D4×C2311yescharacteristic
162C22×C4C2×D4×C2312yes—
163M16—22no—
164M16—41no—
165C2×C8—41no—
166C2×C8—42no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C4⋊C4—41no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—81no—
178C22⋊C4—41no—
179C42—21no—
180C42—44no—
181C42—41no—
182C42Q16×C2312yes—
183C23C23×#12011yescharacteristic
184Q8—82no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4C2×Q16×C2311yescharacteristic
203C2×C4C23×#13711yescharacteristic
204C8—42no—
205C8—82no—
206C22—21no—
207C22—21no—
208C22C23×#45111yescharacteristic
209C22C23×#45611yescharacteristic
210C22C23×B(4,𝔽2)11yescharacteristic
211C4—24no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—82no—
216C4—81no—
217C4—81no—
218C4—82no—
219C4—81no—
220C2—21no—
221C2—21no—
222C2C23×#160511yescharacteristic
223C2C23×#219811yescharacteristic
224C2C23×#220411yescharacteristic
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. #12257C23direct (N × H)
155. C23#12257direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25712
C23×#20602
C23×#16172
C23×#13532
C23×#13252
#1225723