Finite Group Databasebeta

← C23×#8456 #19156082

Subgroups — C23×#8456 #19156082

Order 5888.

1026 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 154 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×#3496C211yesmaximal, characteristic
2C23×#1955C211yesmaximal, characteristic, complemented
3C23×#1551C212yesmaximal, complemented
4C23×#1530C212yesmaximal, complemented
5C23×#1466C211yesmaximal, characteristic, complemented
6C2×C23×H(5,𝔽2)C412yes—
7C23×#547—22no—
8C23×#546—24no—
9D42×C23—22no—
10C23×#537C412yes—
11C23×#534—22no—
12C23×#532C2212yes—
13C2×C23×#121—22no—
14C2×C4×D4×C23C2211yescharacteristic
15C23×#432C2212yescomplemented
16C2×C23×C4⋊C8C2211yescharacteristic, complemented
17C2×C23×C22⋊C8—22no—
18C2×C23×#97C2211yescharacteristic, complemented
19C23×#333—22no—
20C23×#331—22no—
21C23×#329—24no—
22C23×#323—22no—
23C23×H(5,𝔽2)—48no—
24C2×#41×C23—24no—
25C2×#41×C23C2×C414yes—
26C22×D4×C23—24no—
27C22×D4×C23—22no—
28C22×D4×C23—24no—
29C22×D4×C23C2×C411yescharacteristic
30C23×C4×C23D412yes—
31C22×C8×C23—21no—
32C22×C8×C23—21no—
33C23×C4⋊Q8D412yes—
34C23×#127D412yes—
35C23×#124D414yes—
36C23×#123—28no—
37C23×C22⋊Q8—24no—
38C23×#121—24no—
39C23×#121—24no—
40C23×#121—24no—
41C23×#121—24no—
42C23×#121—22no—
43C23×#121—22no—
44C23×#120—28no—
45C4×D4×C23—22no—
46C4×D4×C23—22no—
47C4×D4×C23—22no—
48C4×D4×C23—22no—
49C2×C4⋊C4×C23—22no—
50C2×C4⋊C4×C23C2×C411yescharacteristic
51C2×C22⋊C4×C23—24no—
52C2×C22⋊C4×C23D412yes—
53C2×C42×C23C2311yescharacteristic
54C23×C4⋊C8—22no—
55C23×C4⋊C8—22no—
56C23×C22⋊C8—22no—
57C23×C22⋊C8—44no—
58C23×#97—22no—
59C23×#97—22no—
60C24×C23—24no—
61C24×C23C22⋊C412yes—
62#41×C23—48no—
63#41×C23—416no—
64C2×Q8×C23C22⋊C414yes—
65C2×D4×C23—24no—
66C2×D4×C23—28no—
67C2×D4×C23—24no—
68C2×D4×C23—24no—
69C2×D4×C23—22no—
70C2×D4×C23—22no—
71C2×D4×C23—22no—
72C2×D4×C23—416no—
73C2×D4×C23—48no—
74C2×D4×C23—48no—
75C2×D4×C23—82no—
76C2×D4×C23—82no—
77C2×D4×C23C22⋊C414yes—
78C22×C4×C23—24no—
79C22×C4×C23—24no—
80C22×C4×C23—22no—
81C22×C4×C23—22no—
82C22×C4×C23—42no—
83C22×C4×C23—42no—
84C22×C4×C23C22⋊C412yes—
85C22×C4×C23C22×C411yescharacteristic
86C22×C4×C23C2×D411yescharacteristic
87C22×C4×C23C2×D411yescharacteristic
88C2×C8×C23—22no—
89C2×C8×C23—41no—
90C2×C8×C23—41no—
91C2×C8×C23—42no—
92C2×C8×C23—41no—
93C2×C8×C23—42no—
94C4⋊C4×C23—24no—
95C4⋊C4×C23—22no—
96C4⋊C4×C23—22no—
97C4⋊C4×C23—22no—
98C22⋊C4×C23—28no—
99C22⋊C4×C23—24no—
100C22⋊C4×C23—24no—
101C22⋊C4×C23—24no—
102C42×C23C2×D412yes—
103C42×C23C2×D412yes—
104#8456C2311yesmaximal, characteristic, direct product factor, Sylow
105C23×C23—22no—
106C23×C23—22no—
107C23×C23—24no—
108C23×C23—42no—
109C23×C23—42no—
110C23×C23—48no—
111C23×C23—84no—
112C23×C23—84no—
113C23×C23#10014yes—
114C23×C23C2×C22⋊C411yescharacteristic
115Q8×C23—44no—
116D4×C23—28no—
117D4×C23—44no—
118D4×C23—416no—
119D4×C23—48no—
120D4×C23—42no—
121D4×C23—42no—
122D4×C23—82no—
123D4×C23—84no—
124C2×C4×C23—22no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—22no—
128C2×C4×C23—24no—
129C2×C4×C23—41no—
130C2×C4×C23—48no—
131C2×C4×C23—48no—
132C2×C4×C23—42no—
133C2×C4×C23—42no—
134C2×C4×C23—42no—
135C2×C4×C23—82no—
136C2×C4×C23C2×C22⋊C412yes—
137C2×C4×C23#12012yes—
138C2×C4×C23#12011yescharacteristic
139C2×C4×C23#12011yescharacteristic
140C8×C23—42no—
141C8×C23—82no—
142C8×C23—81no—
143#3496C2×C2311yescharacteristic
144#1955C2×C2311yescharacteristic, complemented
145#1551C2×C2312yescomplemented
146#1530C2×C2312yescomplemented
147#1466C2×C2311yescharacteristic, complemented
148C22×C23—21no—
149C22×C23—21no—
150C22×C23—24no—
151C22×C23—44no—
152C22×C23—42no—
153C22×C23—42no—
154C22×C23—44no—
155C22×C23—44no—
156C22×C23—44no—
157C22×C23—82no—
158C22×C23—168no—
159C22×C23#37911yescharacteristic
160C22×C23C2×#10012yes—
161C4×C23—21no—
162C4×C23—22no—
163C4×C23—22no—
164C4×C23—42no—
165C4×C23—44no—
166C4×C23#45311yescharacteristic
167C4×C23#45511yescharacteristic
168C2×H(5,𝔽2)C4×C2312yes—
169#547—22no—
170#546—24no—
171D42—22no—
172#537C4×C2312yes—
173#534—22no—
174#532C22×C2312yes—
175C2×#121—22no—
176C2×C4×D4C22×C2311yescharacteristic
177#432C22×C2312yescomplemented
178C2×C4⋊C8C22×C2311yescharacteristic, complemented
179C2×C22⋊C8—22no—
180C2×#97C22×C2311yescharacteristic, complemented
181#333—22no—
182#331—22no—
183#329—24no—
184#323—22no—
185C2×C23—22no—
186C2×C23—44no—
187C2×C23—84no—
188C2×C23#178311yescharacteristic
189C2×C23#179211yescharacteristic
190C2×C23#179411yescharacteristic
191H(5,𝔽2)—48no—
192C2×#41—24no—
193C2×#41C2×C4×C2314yes—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4C2×C4×C2311yescharacteristic
198C23×C4D4×C2312yes—
199C22×C8—21no—
200C22×C8—21no—
201C4⋊Q8D4×C2312yes—
202#127D4×C2312yes—
203#124D4×C2314yes—
204#123—28no—
205C22⋊Q8—24no—
206#121—24no—
207#121—24no—
208#121—24no—
209#121—24no—
210#121—22no—
211#121—22no—
212#120—28no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4C2×C4×C2311yescharacteristic
219C2×C22⋊C4—24no—
220C2×C22⋊C4D4×C2312yes—
221C2×C42C23×C2311yescharacteristic
222C4⋊C8—22no—
223C4⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—44no—
226#97—22no—
227#97—22no—
228C23#845611yescharacteristic, direct product factor, Sylow
229C24—24no—
230C24C22⋊C4×C2312yes—
231#41—48no—
232#41—416no—
233C2×Q8C22⋊C4×C2314yes—
234C2×D4—24no—
235C2×D4—28no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—416no—
242C2×D4—48no—
243C2×D4—48no—
244C2×D4—82no—
245C2×D4—82no—
246C2×D4C22⋊C4×C2314yes—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4C22⋊C4×C2312yes—
254C22×C4C22×C4×C2311yescharacteristic
255C22×C4C2×D4×C2311yescharacteristic
256C22×C4C2×D4×C2311yescharacteristic
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—41no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C22⋊C4—28no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C42C2×D4×C2312yes—
272C42C2×D4×C2312yes—
273C23—22no—
274C23—22no—
275C23—24no—
276C23—42no—
277C23—42no—
278C23—48no—
279C23—84no—
280C23—84no—
281C23C23×#10014yes—
282C23C2×C22⋊C4×C2311yescharacteristic
283Q8—44no—
284D4—28no—
285D4—44no—
286D4—416no—
287D4—48no—
288D4—42no—
289D4—42no—
290D4—82no—
291D4—84no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—41no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4C2×C22⋊C4×C2312yes—
305C2×C4C23×#12012yes—
306C2×C4C23×#12011yescharacteristic
307C2×C4C23×#12011yescharacteristic
308C8—42no—
309C8—82no—
310C8—81no—
311C22—21no—
312C22—21no—
313C22—24no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22—82no—
321C22—168no—
322C22C23×#37911yescharacteristic
323C22C2×C23×#10012yes—
324C4—22no—
325C4—21no—
326C4—22no—
327C4—44no—
328C4—42no—
329C4C23×#45311yescharacteristic
330C4C23×#45511yescharacteristic
331C2—22no—
332C2—44no—
333C2—84no—
334C2C23×#178311yescharacteristic
335C2C23×#179211yescharacteristic
336C2C23×#179411yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3496C2
LCSG ▷ C22×C4 ▷ 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×#1955C2semidirect
3. C23×#1551C2semidirect
4. C23×#1530C2semidirect
5. C23×#1466C2semidirect
15. C23×#432C22semidirect
16. C2×C23×C4⋊C8C22semidirect
18. C2×C23×#97C22semidirect
104. #8456C23direct (N × H)
144. #1955C2×C23semidirect
145. #1551C2×C23semidirect
146. #1530C2×C23semidirect
147. #1466C2×C23semidirect
177. #432C22×C23semidirect
178. C2×C4⋊C8C22×C23semidirect
180. C2×#97C22×C23semidirect
228. C23#8456direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34962
C23×#19552
C23×#15512
C23×#15302
C23×#14662
#845623