Finite Group Databasebeta

← C23×#12710 #19160336

Subgroups — C23×#12710 #19160336

Order 5888.

952 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 146 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×#3486C211yesmaximal, characteristic
2C23×#2133C214yesmaximal, complemented
3C23×#1784C212yesmaximal, complemented
4C2×C23×2−1+4C411yescharacteristic
5C23×Q8×C23C2211yescharacteristic
6D4×Q8×C23—24no—
7C23×#548—24no—
8C23×#536—22no—
9C23×#536C2212yes—
10C23×#525—22no—
11C23×#525C412yes—
12C23×#519C411yescharacteristic
13C2×C23×#101—22no—
14C2×C23×#101C2214yes—
15C23×#407—22no—
16C23×#356—24no—
17C23×#355—24no—
18C23×2−1+4—44no—
19C2×#41×C23—22no—
20C2×#41×C23—22no—
21C2×#41×C23C2×C412yes—
22C22×Q8×C23—22no—
23C22×Q8×C23—24no—
24C22×Q8×C23—44no—
25C22×Q8×C23C2×C411yescharacteristic
26C22×Q8×C23D412yes—
27C22×Q8×C23C2311yescharacteristic
28C23×C4×C23C2×C411yescharacteristic
29C23×C4×C23D412yes—
30C2×M16×C23—24no—
31C23×C4⋊Q8—24no—
32C23×C4⋊Q8D414yes—
33C23×#124—24no—
34C23×#124D414yes—
35C23×#123—22no—
36C23×#123—24no—
37C23×C22⋊Q8—28no—
38C23×C22⋊Q8—24no—
39C23×C22⋊Q8—22no—
40C23×C22⋊Q8—24no—
41C23×#121—24no—
42C23×#121—22no—
43C23×#120—22no—
44C4×Q8×C23—24no—
45C4×D4×C23—28no—
46C4×D4×C23—24no—
47C23×#117—22no—
48C23×#117C2×C412yes—
49C23×#101—28no—
50C23×#101—44no—
51C23×C22⋊C8—44no—
52C24×C23C22⋊C411yescharacteristic
53#41×C23—44no—
54#41×C23—44no—
55#41×C23—48no—
56#41×C23—44no—
57C2×Q8×C23—22no—
58C2×Q8×C23—24no—
59C2×Q8×C23—24no—
60C2×Q8×C23—22no—
61C2×Q8×C23—24no—
62C2×Q8×C23—42no—
63C2×Q8×C23—42no—
64C2×Q8×C23—416no—
65C2×Q8×C23—48no—
66C2×Q8×C23C22⋊C412yes—
67C2×Q8×C23C2×D414yes—
68C2×D4×C23—24no—
69C2×D4×C23—22no—
70C2×D4×C23—22no—
71C2×D4×C23C22⋊C412yes—
72C22×C4×C23—24no—
73C22×C4×C23—24no—
74C22×C4×C23—22no—
75C22×C4×C23—22no—
76C22×C4×C23—24no—
77C22×C4×C23—42no—
78C22×C4×C23—42no—
79C22×C4×C23C22⋊C411yescharacteristic
80C22×C4×C23C22⋊C412yes—
81C22×C4×C23C22×C411yescharacteristic
82C22×C4×C23C2×D412yes—
83M16×C23—44no—
84M16×C23—44no—
85C2×C8×C23—44no—
86C4⋊C4×C23—24no—
87C4⋊C4×C23—24no—
88C4⋊C4×C23—22no—
89C4⋊C4×C23—28no—
90C22⋊C4×C23—24no—
91C22⋊C4×C23—24no—
92C22⋊C4×C23—22no—
93C22⋊C4×C23—28no—
94C42×C23—24no—
95C42×C23C22⋊C414yes—
96#12710C2311yesmaximal, characteristic, direct product factor, Sylow
97C23×C23—21no—
98C23×C23—22no—
99C23×C23—22no—
100C23×C23—42no—
101C23×C23C2×C22⋊C411yescharacteristic
102Q8×C23—28no—
103Q8×C23—44no—
104Q8×C23—48no—
105Q8×C23—42no—
106Q8×C23—44no—
107D4×C23—42no—
108D4×C23—44no—
109D4×C23—48no—
110D4×C23—44no—
111C2×C4×C23—22no—
112C2×C4×C23—22no—
113C2×C4×C23—22no—
114C2×C4×C23—22no—
115C2×C4×C23—22no—
116C2×C4×C23—24no—
117C2×C4×C23—21no—
118C2×C4×C23—22no—
119C2×C4×C23—22no—
120C2×C4×C23—48no—
121C2×C4×C23—44no—
122C2×C4×C23—48no—
123C2×C4×C23—44no—
124C2×C4×C23#9914yes—
125C2×C4×C23C2×C22⋊C412yes—
126C2×C4×C23#12014yes—
127C8×C23—84no—
128#3486C2×C2311yescharacteristic
129#2133C2×C2314yescomplemented
130#1784C2×C2312yescomplemented
131C22×C23—22no—
132C22×C23—24no—
133C22×C23—41no—
134C22×C23—41no—
135C22×C23—42no—
136C22×C23—44no—
137C22×C23—41no—
138C22×C23—42no—
139C22×C23#37911yescharacteristic
140C22×C23C2×#9912yes—
141C4×C23—24no—
142C4×C23—24no—
143C4×C23—22no—
144C4×C23—44no—
145C4×C23—41no—
146C4×C23—42no—
147C2×2−1+4C4×C2311yescharacteristic
148C23×Q8C22×C2311yescharacteristic
149D4×Q8—24no—
150#548—24no—
151#536—22no—
152#536C22×C2312yes—
153#525—22no—
154#525C4×C2312yes—
155#519C4×C2311yescharacteristic
156C2×#101—22no—
157C2×#101C22×C2314yes—
158#407—22no—
159#356—24no—
160#355—24no—
161C2×C23—21no—
162C2×C23—22no—
163C2×C23—42no—
164C2×C23—82no—
165C2×C23#178111yescharacteristic
1662−1+4—44no—
167C2×#41—22no—
168C2×#41—22no—
169C2×#41C2×C4×C2312yes—
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—44no—
173C22×Q8C2×C4×C2311yescharacteristic
174C22×Q8D4×C2312yes—
175C22×Q8C23×C2311yescharacteristic
176C23×C4C2×C4×C2311yescharacteristic
177C23×C4D4×C2312yes—
178C2×M16—24no—
179C4⋊Q8—24no—
180C4⋊Q8D4×C2314yes—
181#124—24no—
182#124D4×C2314yes—
183#123—22no—
184#123—24no—
185C22⋊Q8—24no—
186C22⋊Q8—28no—
187C22⋊Q8—22no—
188C22⋊Q8—24no—
189#121—24no—
190#121—22no—
191#120—22no—
192C4×Q8—24no—
193C4×D4—28no—
194C4×D4—24no—
195#117—22no—
196#117C2×C4×C2312yes—
197#101—28no—
198#101—44no—
199C22⋊C8—44no—
200C23#1271011yescharacteristic, direct product factor, Sylow
201C24C22⋊C4×C2311yescharacteristic
202#41—44no—
203#41—44no—
204#41—48no—
205#41—44no—
206C2×Q8—22no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—22no—
210C2×Q8—24no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—416no—
214C2×Q8—48no—
215C2×Q8C22⋊C4×C2312yes—
216C2×Q8C2×D4×C2314yes—
217C2×D4—24no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4C22⋊C4×C2312yes—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C22⋊C4×C2311yescharacteristic
229C22×C4C22⋊C4×C2312yes—
230C22×C4C22×C4×C2311yescharacteristic
231C22×C4C2×D4×C2312yes—
232M16—44no—
233M16—44no—
234C2×C8—44no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—28no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—22no—
242C22⋊C4—28no—
243C42—24no—
244C42C22⋊C4×C2314yes—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—42no—
249C23C2×C22⋊C4×C2311yescharacteristic
250Q8—28no—
251Q8—44no—
252Q8—48no—
253Q8—42no—
254Q8—44no—
255D4—42no—
256D4—44no—
257D4—48no—
258D4—44no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—48no—
269C2×C4—44no—
270C2×C4—48no—
271C2×C4—44no—
272C2×C4C23×#9914yes—
273C2×C4C2×C22⋊C4×C2312yes—
274C2×C4C23×#12014yes—
275C8—84no—
276C22—22no—
277C22—24no—
278C22—41no—
279C22—41no—
280C22—42no—
281C22—44no—
282C22—41no—
283C22—42no—
284C22C23×#37911yescharacteristic
285C22C2×C23×#9912yes—
286C4—24no—
287C4—24no—
288C4—22no—
289C4—44no—
290C4—41no—
291C4—42no—
292C2—21no—
293C2—22no—
294C2—42no—
295C2—82no—
296C2C23×#178111yescharacteristic
subgroupG/H
center, socleC2×C23#1781
commutatorC22×C4C22×C4×C23
Frattini, cosocleC22×Q8C23×C23
prime radicalC23×#3486C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C23×C23 ◁ C23×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2133C2semidirect
3. C23×#1784C2semidirect
96. #12710C23direct (N × H)
129. #2133C2×C23semidirect
130. #1784C2×C23semidirect
200. C23#12710direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34862
C23×#21332
C23×#17842
#1271023