Finite Group Databasebeta

← C23×#19058 #19166684

Subgroups — C23×#19058 #19166684

Order 5888.

840 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 222 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×#3237C212yesmaximal, complemented
3C23×#3135C211yesmaximal, characteristic
4C23×#1664C218yesmaximal, complemented
5C23×#1536C211yesmaximal, characteristic
6C23×#1531C212yesmaximal, complemented
7C23×#563C2214yes—
8C23×#559—21no—
9Q82×C23—21no—
10C23×#556—22no—
11D4×Q8×C23—22no—
12C23×#542—24no—
13C23×#536—22no—
14C23×#533C2211yescharacteristic
15C23×#533C2212yes—
16C23×#532C2211yescharacteristic
17C23×#501C2214yes—
18C23×#491—21no—
19C23×#491C2214yes—
20C23×#490—21no—
21C23×#489C2214yes—
22C23×#485—24no—
23C23×#477—22no—
24C23×#475—22no—
25C23×#432C2212yes—
26C23×#416—22no—
27C2×C23×#97C2211yescharacteristic
28C23×#330C2218yes—
29C23×#329C2214yes—
30C2×#41×C23D412yes—
31C22×Q8×C23—22no—
32C22×C8×C23—21no—
33C23×C4⋊Q8—24no—
34C23×C4⋊Q8—24no—
35C23×C4⋊Q8D411yescharacteristic
36C23×C4⋊Q8D412yes—
37C23×C4⋊Q8D412yes—
38C23×C4⋊Q8C2312yes—
39C23×#127D411yescharacteristic
40C23×#126—28no—
41C23×#125—21no—
42C23×#125C2314yes—
43C23×#124—24no—
44C23×#124—24no—
45C23×#124C2312yes—
46C23×#123—24no—
47C23×#123—24no—
48C23×C22⋊Q8—24no—
49C23×C22⋊Q8—22no—
50C23×C22⋊Q8—22no—
51C23×C22⋊Q8—28no—
52C23×C22⋊Q8—22no—
53C23×#121—22no—
54C4×Q8×C23—24no—
55C4×Q8×C23—22no—
56C4×Q8×C23—22no—
57C4×Q8×C23—22no—
58C4×D4×C23—24no—
59C4×D4×C23—22no—
60C23×#117—22no—
61C23×#117—21no—
62C23×#117D414yes—
63C2×C42×C23C2311yescharacteristic
64C23×#107—24no—
65C23×#106—24no—
66C23×C4⋊C8—22no—
67C23×#103—24no—
68C23×#103—24no—
69C23×#103—24no—
70C23×#102—24no—
71C23×C22⋊C8—22no—
72C23×#97—21no—
73C23×#97C2314yes—
74C23×#97C2312yes—
75#41×C23—44no—
76C2×Q8×C23—24no—
77C2×Q8×C23—22no—
78C2×Q8×C23—22no—
79C2×Q8×C23—22no—
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×C23C2×D411yescharacteristic
88C22×C4×C23C2×D412yes—
89C2×C8×C23—22no—
90C2×C8×C23—24no—
91C2×C8×C23—42no—
92C4⋊C4×C23—28no—
93C4⋊C4×C23—22no—
94C4⋊C4×C23—22no—
95C4⋊C4×C23—22no—
96C4⋊C4×C23—24no—
97C4⋊C4×C23—24no—
98C4⋊C4×C23—22no—
99C4⋊C4×C23—22no—
100C4⋊C4×C23—42no—
101C4⋊C4×C23C2×D414yes—
102C4⋊C4×C23C2×D414yes—
103C22⋊C4×C23—24no—
104C22⋊C4×C23—28no—
105C22⋊C4×C23—24no—
106C22⋊C4×C23—24no—
107C22⋊C4×C23—41no—
108C42×C23—24no—
109C42×C23—24no—
110C42×C23—41no—
111C42×C23C2×D412yes—
112C42×C23C2×D411yescharacteristic
113C42×C23C2411yescharacteristic
114#19058C2311yesmaximal, characteristic, direct product factor, Sylow
115C23×C23—22no—
116C23×C23#12011yescharacteristic
117Q8×C23—42no—
118Q8×C23—44no—
119Q8×C23—44no—
120Q8×C23—44no—
121Q8×C23—84no—
122D4×C23—42no—
123D4×C23—44no—
124D4×C23—82no—
125C2×C4×C23—22no—
126C2×C4×C23—24no—
127C2×C4×C23—22no—
128C2×C4×C23—24no—
129C2×C4×C23—42no—
130C2×C4×C23—44no—
131C2×C4×C23—42no—
132C2×C4×C23—84no—
133C2×C4×C23#12012yes—
134C2×C4×C23#12011yescharacteristic
135C2×C4×C23C22×D412yes—
136C2×C4×C23C22×D411yescharacteristic
137C8×C23—44no—
138C8×C23—42no—
139#3550C2×C2311yescharacteristic
140#3237C2×C2312yescomplemented
141#3135C2×C2311yescharacteristic
142#1664C2×C2318yescomplemented
143#1536C2×C2311yescharacteristic
144#1531C2×C2312yescomplemented
145C22×C23—21no—
146C22×C23—22no—
147C22×C23—42no—
148C22×C23—44no—
149C22×C23C2×#12011yescharacteristic
150C4×C23—21no—
151C4×C23—22no—
152C4×C23—22no—
153C4×C23—42no—
154C4×C23—44no—
155C4×C23—44no—
156C4×C23—82no—
157C4×C23#45512yes—
158#563C22×C2314yes—
159#559—21no—
160Q82—21no—
161#556—22no—
162D4×Q8—22no—
163#542—24no—
164#536—22no—
165#533C22×C2311yescharacteristic
166#533C22×C2312yes—
167#532C22×C2311yescharacteristic
168#501C22×C2314yes—
169#491—21no—
170#491C22×C2314yes—
171#490—21no—
172#489C22×C2314yes—
173#485—24no—
174#477—22no—
175#475—22no—
176#432C22×C2312yes—
177#416—22no—
178C2×#97C22×C2311yescharacteristic
179#330C22×C2318yes—
180#329C22×C2314yes—
181C2×C23—41no—
182C2×C23—82no—
183C2×C23C2×#45511yescharacteristic
184C2×C23#302012yes—
185C2×#41D4×C2312yes—
186C22×Q8—22no—
187C22×C8—21no—
188C4⋊Q8—24no—
189C4⋊Q8—24no—
190C4⋊Q8D4×C2311yescharacteristic
191C4⋊Q8D4×C2312yes—
192C4⋊Q8D4×C2312yes—
193C4⋊Q8C23×C2312yes—
194#127D4×C2311yescharacteristic
195#126—28no—
196#125—21no—
197#125C23×C2314yes—
198#124—24no—
199#124—24no—
200#124C23×C2312yes—
201#123—24no—
202#123—24no—
203C22⋊Q8—24no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—28no—
207C22⋊Q8—22no—
208#121—22no—
209C4×Q8—24no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×D4—24no—
214C4×D4—22no—
215#117—22no—
216#117—21no—
217#117D4×C2314yes—
218C2×C42C23×C2311yescharacteristic
219#107—24no—
220#106—24no—
221C4⋊C8—22no—
222#103—24no—
223#103—24no—
224#103—24no—
225#102—24no—
226C22⋊C8—22no—
227#97—21no—
228#97C23×C2314yes—
229#97C23×C2312yes—
230C23#1905811yescharacteristic, direct product factor, Sylow
231#41—44no—
232C2×Q8—24no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—84no—
237C2×Q8C2×D4×C2312yes—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4C2×D4×C2312yes—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4C2×D4×C2311yescharacteristic
244C22×C4C2×D4×C2312yes—
245C2×C8—22no—
246C2×C8—24no—
247C2×C8—42no—
248C4⋊C4—28no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—42no—
257C4⋊C4C2×D4×C2314yes—
258C4⋊C4C2×D4×C2314yes—
259C22⋊C4—24no—
260C22⋊C4—28no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—41no—
264C42—24no—
265C42—24no—
266C42—41no—
267C42C2×D4×C2311yescharacteristic
268C42C2×D4×C2312yes—
269C42C24×C2311yescharacteristic
270C23—22no—
271C23C23×#12011yescharacteristic
272Q8—42no—
273Q8—44no—
274Q8—44no—
275Q8—44no—
276Q8—84no—
277D4—42no—
278D4—44no—
279D4—82no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—84no—
288C2×C4C23×#12012yes—
289C2×C4C23×#12011yescharacteristic
290C2×C4C22×D4×C2312yes—
291C2×C4C22×D4×C2311yescharacteristic
292C8—42no—
293C8—44no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—44no—
298C22C2×C23×#12011yescharacteristic
299C4—21no—
300C4—22no—
301C4—22no—
302C4—42no—
303C4—44no—
304C4—44no—
305C4—82no—
306C4C23×#45512yes—
307C2—41no—
308C2—82no—
309C2C2×C23×#45511yescharacteristic
310C2C23×#302012yes—
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×#3237C2semidirect
4. C23×#1664C2semidirect
6. C23×#1531C2semidirect
114. #19058C23direct (N × H)
140. #3237C2×C23semidirect
142. #1664C2×C23semidirect
144. #1531C2×C23semidirect
230. C23#19058direct (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×#32372
C23×#31352
C23×#16642
C23×#15362
C23×#15312
#1905823