Finite Group Databasebeta

← C23×#7216 #19154842

Subgroups — C23×#7216 #19154842

Order 5888.

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

normal subgroups only

This group has 276 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×#1822C211yesmaximal, characteristic
2C23×#1750C211yesmaximal, characteristic
3C23×#1464C211yesmaximal, characteristic
4C2×C23×#117C2211yescharacteristic
5C23×#432—21no—
6C23×#407—21no—
7C2×C23×C22⋊C8—21no—
8C2×C23×#97C411yescharacteristic
9C2×C4×C8×C23C411yescharacteristic
10C23×#343—21no—
11C23×#343C412yes—
12C23×#336—21no—
13C23×#324—21no—
14C23×#323C412yes—
15C23×C4×C23D411yescharacteristic
16C2×M16×C23—22no—
17C22×C8×C23—21no—
18C22×C8×C23—21no—
19C23×#117—41no—
20C23×#117—41no—
21C2×C4⋊C4×C23C2×C411yescharacteristic
22C2×C4⋊C4×C23D411yescharacteristic
23C2×C22⋊C4×C23C2×C411yescharacteristic
24C2×C22⋊C4×C23Q811yescharacteristic
25C2×C42×C23C2×C411yescharacteristic
26C2×C42×C23D411yescharacteristic
27C23×C4⋊C8—41no—
28C23×C22⋊C8—24no—
29C23×C22⋊C8—22no—
30C23×C22⋊C8—41no—
31C23×C22⋊C8—41no—
32C23×#97—41no—
33C23×#97—41no—
34C4×C8×C23—41no—
35C24×C23C4⋊C411yescharacteristic
36C22×C4×C23—21no—
37C22×C4×C23—21no—
38C22×C4×C23—22no—
39C22×C4×C23—41no—
40C22×C4×C23C4211yescharacteristic
41C22×C4×C23C22⋊C411yescharacteristic
42C22×C4×C23C22⋊C411yescharacteristic
43C22×C4×C23C22⋊C411yescharacteristic
44C22×C4×C23C4⋊C411yescharacteristic
45C22×C4×C23C4⋊C411yescharacteristic
46M16×C23—42no—
47M16×C23—42no—
48C2×C8×C23—22no—
49C2×C8×C23—22no—
50C2×C8×C23—41no—
51C2×C8×C23—41no—
52C2×C8×C23—41no—
53C2×C8×C23—41no—
54C2×C8×C23—42no—
55C2×C8×C23—42no—
56C4⋊C4×C23—41no—
57C4⋊C4×C23—41no—
58C22⋊C4×C23—41no—
59C22⋊C4×C23—41no—
60C42×C23—21no—
61C42×C23—21no—
62C42×C23—21no—
63C42×C23—21no—
64#7216C2311yesmaximal, characteristic, direct product factor, Sylow
65C23×C23—21no—
66C23×C23—21no—
67C23×C23—21no—
68C23×C23—41no—
69C23×C23—41no—
70C23×C23#9511yescharacteristic
71C2×C4×C23—21no—
72C2×C4×C23—21no—
73C2×C4×C23—21no—
74C2×C4×C23—21no—
75C2×C4×C23—21no—
76C2×C4×C23—21no—
77C2×C4×C23—21no—
78C2×C4×C23—21no—
79C2×C4×C23—42no—
80C2×C4×C23—42no—
81C2×C4×C23—41no—
82C2×C4×C23—41no—
83C2×C4×C23—41no—
84C2×C4×C23—41no—
85C2×C4×C23#9911yescharacteristic
86C2×C4×C23#9911yescharacteristic
87C2×C4×C23#10412yes—
88C8×C23—42no—
89C8×C23—42no—
90C8×C23—82no—
91C8×C23—82no—
92#1822C2×C2311yescharacteristic
93#1750C2×C2311yescharacteristic
94#1464C2×C2311yescharacteristic
95C22×C23—21no—
96C22×C23—21no—
97C22×C23—42no—
98C22×C23—41no—
99C22×C23—41no—
100C22×C23—41no—
101C22×C23—81no—
102C22×C23H(3,ℤ/4)11yescharacteristic
103C22×C23#33911yescharacteristic
104C22×C23#34211yescharacteristic
105C4×C23—21no—
106C4×C23—21no—
107C4×C23—22no—
108C4×C23—41no—
109C4×C23—41no—
110C4×C23—41no—
111C4×C23—41no—
112C2×#117C22×C2311yescharacteristic
113#432—21no—
114#407—21no—
115C2×C22⋊C8—21no—
116C2×#97C4×C2311yescharacteristic
117C2×C4×C8C4×C2311yescharacteristic
118#343—21no—
119#343C4×C2312yes—
120#336—21no—
121#324—21no—
122#323C4×C2312yes—
123C2×C23—21no—
124C2×C23—21no—
125C2×C23—81no—
126C2×C23#128411yescharacteristic
127C2×C23#139211yescharacteristic
128C2×C23#139511yescharacteristic
129C23×C4D4×C2311yescharacteristic
130C2×M16—22no—
131C22×C8—21no—
132C22×C8—21no—
133#117—41no—
134#117—41no—
135C2×C4⋊C4C2×C4×C2311yescharacteristic
136C2×C4⋊C4D4×C2311yescharacteristic
137C2×C22⋊C4C2×C4×C2311yescharacteristic
138C2×C22⋊C4Q8×C2311yescharacteristic
139C2×C42C2×C4×C2311yescharacteristic
140C2×C42D4×C2311yescharacteristic
141C4⋊C8—41no—
142C22⋊C8—24no—
143C22⋊C8—22no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146#97—41no—
147#97—41no—
148C4×C8—41no—
149C23#721611yescharacteristic, direct product factor, Sylow
150C24C4⋊C4×C2311yescharacteristic
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—41no—
155C22×C4C42×C2311yescharacteristic
156C22×C4C22⋊C4×C2311yescharacteristic
157C22×C4C22⋊C4×C2311yescharacteristic
158C22×C4C22⋊C4×C2311yescharacteristic
159C22×C4C4⋊C4×C2311yescharacteristic
160C22×C4C4⋊C4×C2311yescharacteristic
161M16—42no—
162M16—42no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—42no—
170C2×C8—42no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C22⋊C4—41no—
174C22⋊C4—41no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23C23×#9511yescharacteristic
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4C23×#9911yescharacteristic
200C2×C4C23×#9911yescharacteristic
201C2×C4C23×#10412yes—
202C8—42no—
203C8—42no—
204C8—82no—
205C8—82no—
206C22—21no—
207C22—21no—
208C22—42no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22—81no—
213C22C23×H(3,ℤ/4)11yescharacteristic
214C22C23×#33911yescharacteristic
215C22C23×#34211yescharacteristic
216C4—21no—
217C4—21no—
218C4—22no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—41no—
223C2—21no—
224C2—21no—
225C2—81no—
226C2C23×#128411yescharacteristic
227C2C23×#139211yescharacteristic
228C2C23×#139511yescharacteristic
subgroupG/H
center, socleC22×C23#342
commutatorC22×C4C42×C23
Frattini, cosocleC2×#117C22×C23
prime radicalC24×C23C4⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C23×#117 ◁ G

Complements

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

NH ≅ G/Ntype
64. #7216C23direct (N × H)
149. C23#7216direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#18222
C23×#17502
C23×#14642
#721623