Finite Group Databasebeta

← C23×#19078 #19166704

Subgroups — C23×#19078 #19166704

Order 5888.

944 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 230 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×#3550C211yesmaximal, characteristic
2C23×#3165C212yesmaximal, complemented
3C23×#3152C211yesmaximal, characteristic
4C23×#1660C218yesmaximal, complemented
5C23×#1526C212yesmaximal, complemented
6C23×#1525C211yesmaximal, characteristic
7C2×C23×#137—22no—
8C23×#563C2214yes—
9C23×#559—21no—
10Q82×C23—21no—
11C23×#556—22no—
12D4×Q8×C23—22no—
13C23×#542—24no—
14C23×#536—22no—
15C23×#533C2212yes—
16C23×#532C2211yescharacteristic
17C2×C23×C4⋊Q8C2211yescharacteristic
18C23×#501C2218yes—
19C23×#500—22no—
20C23×#498—22no—
21C23×#497C2214yes—
22C23×#494—21no—
23C23×#492—21no—
24C23×#428—24no—
25C4×M16×C23C2212yes—
26C4×M16×C23C2211yescharacteristic
27C23×#330C2218yes—
28C23×#329C2214yes—
29C2×#41×C23D412yes—
30C22×Q8×C23—22no—
31C22×Q8×C23—21no—
32C23×#137—48no—
33C2×Q16×C23—24no—
34C2×SD16×C23—24no—
35C2×M16×C23—21no—
36C2×M16×C23—22no—
37C23×C4⋊Q8—24no—
38C23×C4⋊Q8—24no—
39C23×C4⋊Q8—22no—
40C23×C4⋊Q8—21no—
41C23×C4⋊Q8D411yescharacteristic
42C23×C4⋊Q8D412yes—
43C23×C4⋊Q8D412yes—
44C23×C4⋊Q8C2312yes—
45C23×#127D411yescharacteristic
46C23×#126—28no—
47C23×#125C2314yes—
48C23×#124—24no—
49C23×#124—24no—
50C23×#124C2312yes—
51C23×#123—24no—
52C23×#123—24no—
53C23×C22⋊Q8—24no—
54C23×C22⋊Q8—22no—
55C23×C22⋊Q8—22no—
56C23×C22⋊Q8—28no—
57C23×#121—22no—
58C4×Q8×C23—24no—
59C4×Q8×C23—22no—
60C4×Q8×C23—22no—
61C4×D4×C23—24no—
62C4×D4×C23—22no—
63C23×#117—22no—
64C23×#117D414yes—
65C2×C4⋊C4×C23—22no—
66C2×C42×C23C2311yescharacteristic
67C23×#107—28no—
68C23×#106—28no—
69C23×#97C2314yes—
70C23×#97C2312yes—
71C4×C8×C23—21no—
72C4×C8×C23—22no—
73#41×C23—44no—
74C2×Q8×C23—24no—
75C2×Q8×C23—22no—
76C2×Q8×C23—22no—
77C2×Q8×C23—22no—
78C2×Q8×C23—44no—
79C2×Q8×C23—41no—
80C2×Q8×C23—84no—
81C2×Q8×C23C2×D412yes—
82C2×D4×C23—22no—
83C2×D4×C23—22no—
84C2×D4×C23C2×D412yes—
85C22×C4×C23—24no—
86C22×C4×C23—22no—
87C22×C4×C23—41no—
88C22×C4×C23C2×D412yes—
89C22×C4×C23C2×D411yescharacteristic
90Q16×C23—48no—
91SD16×C23—48no—
92M16×C23—28no—
93M16×C23—24no—
94C2×C8×C23—24no—
95C2×C8×C23—22no—
96C4⋊C4×C23—28no—
97C4⋊C4×C23—22no—
98C4⋊C4×C23—22no—
99C4⋊C4×C23—22no—
100C4⋊C4×C23—24no—
101C4⋊C4×C23—24no—
102C4⋊C4×C23—24no—
103C4⋊C4×C23—42no—
104C4⋊C4×C23C2×D414yes—
105C4⋊C4×C23C2×D414yes—
106C22⋊C4×C23—24no—
107C22⋊C4×C23—28no—
108C22⋊C4×C23—24no—
109C22⋊C4×C23—24no—
110C42×C23—24no—
111C42×C23—24no—
112C42×C23C2×D412yes—
113C42×C23C2×D411yescharacteristic
114C42×C23C2411yescharacteristic
115#19078C2311yesmaximal, characteristic, direct product factor, Sylow
116C23×C23—22no—
117C23×C23#12011yescharacteristic
118Q8×C23—42no—
119Q8×C23—44no—
120Q8×C23—44no—
121Q8×C23—44no—
122Q8×C23—84no—
123Q8×C23—82no—
124D4×C23—42no—
125D4×C23—44no—
126D4×C23—82no—
127C2×C4×C23—22no—
128C2×C4×C23—24no—
129C2×C4×C23—24no—
130C2×C4×C23—22no—
131C2×C4×C23—24no—
132C2×C4×C23—44no—
133C2×C4×C23—42no—
134C2×C4×C23—84no—
135C2×C4×C23—82no—
136C2×C4×C23#12011yescharacteristic
137C2×C4×C23#12012yes—
138C2×C4×C23C22×D412yes—
139C2×C4×C23C22×D411yescharacteristic
140C8×C23—28no—
141C8×C23—24no—
142#3550C2×C2311yescharacteristic
143#3165C2×C2312yescomplemented
144#3152C2×C2311yescharacteristic
145#1660C2×C2318yescomplemented
146#1526C2×C2312yescomplemented
147#1525C2×C2311yescharacteristic
148C22×C23—21no—
149C22×C23—22no—
150C22×C23—42no—
151C22×C23—44no—
152C22×C23C2×#12011yescharacteristic
153C4×C23—21no—
154C4×C23—22no—
155C4×C23—42no—
156C4×C23—44no—
157C4×C23—44no—
158C4×C23—82no—
159C4×C23#45312yes—
160C4×C23#45614yes—
161C2×#137—22no—
162#563C22×C2314yes—
163#559—21no—
164Q82—21no—
165#556—22no—
166D4×Q8—22no—
167#542—24no—
168#536—22no—
169#533C22×C2312yes—
170#532C22×C2311yescharacteristic
171C2×C4⋊Q8C22×C2311yescharacteristic
172#501C22×C2318yes—
173#500—22no—
174#498—22no—
175#497C22×C2314yes—
176#494—21no—
177#492—21no—
178#428—24no—
179C4×M16C22×C2311yescharacteristic
180C4×M16C22×C2312yes—
181#330C22×C2318yes—
182#329C22×C2314yes—
183C2×C23—41no—
184C2×C23—82no—
185C2×C23C2×#45311yescharacteristic
186C2×C23C2×#45612yes—
187C2×#41D4×C2312yes—
188C22×Q8—22no—
189C22×Q8—21no—
190#137—48no—
191C2×Q16—24no—
192C2×SD16—24no—
193C2×M16—21no—
194C2×M16—22no—
195C4⋊Q8—24no—
196C4⋊Q8—24no—
197C4⋊Q8—22no—
198C4⋊Q8—21no—
199C4⋊Q8D4×C2311yescharacteristic
200C4⋊Q8D4×C2312yes—
201C4⋊Q8D4×C2312yes—
202C4⋊Q8C23×C2312yes—
203#127D4×C2311yescharacteristic
204#126—28no—
205#125C23×C2314yes—
206#124—24no—
207#124—24no—
208#124C23×C2312yes—
209#123—24no—
210#123—24no—
211C22⋊Q8—24no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—28no—
215#121—22no—
216C4×Q8—24no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×D4—24no—
220C4×D4—22no—
221#117—22no—
222#117D4×C2314yes—
223C2×C4⋊C4—22no—
224C2×C42C23×C2311yescharacteristic
225#107—28no—
226#106—28no—
227#97C23×C2314yes—
228#97C23×C2312yes—
229C4×C8—21no—
230C4×C8—22no—
231C23#1907811yescharacteristic, direct product factor, Sylow
232#41—44no—
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—44no—
238C2×Q8—41no—
239C2×Q8—84no—
240C2×Q8C2×D4×C2312yes—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4C2×D4×C2312yes—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4C2×D4×C2312yes—
248C22×C4C2×D4×C2311yescharacteristic
249Q16—48no—
250SD16—48no—
251M16—24no—
252M16—28no—
253C2×C8—24no—
254C2×C8—22no—
255C4⋊C4—28no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—42no—
263C4⋊C4C2×D4×C2314yes—
264C4⋊C4C2×D4×C2314yes—
265C22⋊C4—24no—
266C22⋊C4—28no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C42—24no—
270C42—24no—
271C42C2×D4×C2312yes—
272C42C2×D4×C2311yescharacteristic
273C42C24×C2311yescharacteristic
274C23—22no—
275C23C23×#12011yescharacteristic
276Q8—42no—
277Q8—44no—
278Q8—44no—
279Q8—44no—
280Q8—84no—
281Q8—82no—
282D4—42no—
283D4—44no—
284D4—82no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—84no—
293C2×C4—82no—
294C2×C4C23×#12011yescharacteristic
295C2×C4C23×#12012yes—
296C2×C4C22×D4×C2312yes—
297C2×C4C22×D4×C2311yescharacteristic
298C8—24no—
299C8—28no—
300C22—21no—
301C22—22no—
302C22—42no—
303C22—44no—
304C22C2×C23×#12011yescharacteristic
305C4—21no—
306C4—22no—
307C4—42no—
308C4—44no—
309C4—44no—
310C4—82no—
311C4C23×#45312yes—
312C4C23×#45614yes—
313C2—41no—
314C2—82no—
315C2C2×C23×#45311yescharacteristic
316C2C2×C23×#45612yes—
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3165C2semidirect
4. C23×#1660C2semidirect
5. C23×#1526C2semidirect
115. #19078C23direct (N × H)
143. #3165C2×C23semidirect
145. #1660C2×C23semidirect
146. #1526C2×C23semidirect
231. C23#19078direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35502
C23×#31652
C23×#31522
C23×#16602
C23×#15262
C23×#15252
#1907823