Subgroups — C₂₃×#29155 #19176781

Order 5888.

608 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3162	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#444	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2031	C₂	1	4	yes	maximal, complemented
4	C₂₃×#1976	C₂	1	2	yes	maximal
5	C₂₃×#1833	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1814	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1812	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1773	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#131	C₂²	1	2	yes	—
11	C₂₃×#533	C₂²	1	2	yes	—
12	C₂₃×#501	—	2	4	no	—
13	C₂₃×#444	—	2	4	no	—
14	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
16	C₂₃×#427	C₂²	1	2	yes	—
17	C₂₃×#427	C₂²	1	1	yes	characteristic
18	C₂₃×#427	C₂²	1	2	yes	—
19	C₂₃×#421	C₂²	1	4	yes	—
20	C₂×C₂₃×#104	C₂²	1	2	yes	—
21	C₂₃×#410	C₂²	1	4	yes	—
22	C₂₃×#405	C₂²	1	2	yes	—
23	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
24	C₂₃×#344	C₂²	1	4	yes	—
25	C₂₃×#339	C₂²	1	2	yes	—
26	C₂₃×H(3,ℤ/4)	C₂²	1	4	yes	—
27	C₂×#41×C₂₃	C₂³	1	2	yes	—
28	C₂₃×#135	—	2	2	no	—
29	C₂₃×#135	—	2	2	no	—
30	C₂×Q₁₆×C₂₃	Q₈	1	1	yes	characteristic, complemented
31	C₂×SD₁₆×C₂₃	Q₈	1	2	yes	complemented
32	C₂×D₈×C₂₃	Q₈	1	1	yes	characteristic, complemented
33	C₂₃×#131	—	2	2	no	—
34	C₂₃×#131	—	2	2	no	—
35	C₂₃×#131	—	2	2	no	—
36	C₂₃×#131	—	2	2	no	—
37	C₂×M₁₆×C₂₃	—	2	2	no	—
38	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
39	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₂₃	—	2	2	no	—
41	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
43	C₂₃×C₄⋊Q₈	—	2	2	no	—
44	C₂₃×#125	—	2	2	no	—
45	C₂₃×C₂²⋊Q₈	—	4	2	no	—
46	C₄×Q₈×C₂₃	—	4	2	no	—
47	C₂₃×#117	C₂³	1	2	yes	—
48	C₂₃×#117	C₂³	1	4	yes	—
49	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
50	C₂₃×#108	—	2	2	no	—
51	C₂₃×#108	—	2	2	no	—
52	C₂₃×#107	—	2	2	no	—
53	C₂₃×#107	—	2	2	no	—
54	C₂₃×#107	—	2	1	no	—
55	C₂₃×#106	—	2	2	no	—
56	C₂₃×#106	—	2	2	no	—
57	C₂₃×#106	—	2	1	no	—
58	C₂₃×#104	—	2	4	no	—
59	C₂₃×#104	—	4	4	no	—
60	C₂₃×#99	—	4	4	no	—
61	C₂₃×#97	—	2	2	no	—
62	C₂₃×#97	D₄	1	4	yes	—
63	C₄×C₈×C₂₃	—	2	2	no	—
64	#41×C₂₃	—	2	2	no	—
65	#41×C₂₃	—	2	2	no	—
66	#41×C₂₃	—	2	2	no	—
67	#41×C₂₃	—	2	2	no	—
68	C₂×Q₈×C₂₃	—	4	2	no	—
69	C₂×Q₈×C₂₃	C₂×Q₈	1	2	yes	—
70	C₂×D₄×C₂₃	—	2	2	no	—
71	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₂₃	—	2	1	no	—
73	C₂²×C₄×C₂₃	—	2	2	no	—
74	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
75	Q₁₆×C₂₃	—	2	2	no	—
76	SD₁₆×C₂₃	—	2	4	no	—
77	D₈×C₂₃	—	2	2	no	—
78	M₁₆×C₂₃	—	2	2	no	—
79	M₁₆×C₂₃	—	2	2	no	—
80	M₁₆×C₂₃	—	2	2	no	—
81	M₁₆×C₂₃	—	2	2	no	—
82	M₁₆×C₂₃	—	4	2	no	—
83	M₁₆×C₂₃	—	4	2	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	2	no	—
86	C₂×C₈×C₂₃	—	4	2	no	—
87	C₂×C₈×C₂₃	—	4	2	no	—
88	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
91	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₈×C₂₃	#41	1	2	yes	—
93	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
94	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₂₃	#41	1	2	yes	—
96	C₄⋊C₄×C₂₃	—	2	4	no	—
97	C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₄⋊C₄×C₂₃	—	4	4	no	—
99	C₄⋊C₄×C₂₃	—	4	2	no	—
100	C₂²⋊C₄×C₂₃	—	2	2	no	—
101	C₂²⋊C₄×C₂₃	—	2	4	no	—
102	C₄²×C₂₃	—	2	1	no	—
103	C₄²×C₂₃	—	2	2	no	—
104	C₄²×C₂₃	—	2	4	no	—
105	C₄²×C₂₃	C₂×D₄	1	2	yes	—
106	#29155	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₂₃	—	2	2	no	—
108	C₂³×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
109	Q₈×C₂₃	—	2	2	no	—
110	Q₈×C₂₃	—	2	2	no	—
111	Q₈×C₂₃	—	8	4	no	—
112	D₄×C₂₃	—	2	2	no	—
113	D₄×C₂₃	—	2	2	no	—
114	D₄×C₂₃	—	4	2	no	—
115	D₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	4	2	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	4	4	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	—	8	1	no	—
123	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
129	C₈×C₂₃	—	4	2	no	—
130	C₈×C₂₃	—	4	2	no	—
131	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
132	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
133	C₈×C₂₃	#124	1	2	yes	—
134	C₈×C₂₃	#124	1	2	yes	—
135	#3162	C₂×C₂₃	1	1	yes	characteristic, complemented
136	C₂×#444	C₂×C₂₃	1	1	yes	characteristic
137	#2031	C₂×C₂₃	1	4	yes	complemented
138	#1976	C₂×C₂₃	1	2	yes	—
139	#1833	C₂×C₂₃	1	2	yes	complemented
140	#1814	C₂×C₂₃	1	2	yes	complemented
141	#1812	C₂×C₂₃	1	2	yes	complemented
142	#1773	C₂×C₂₃	1	1	yes	characteristic, complemented
143	C₂²×C₂₃	—	4	1	no	—
144	C₂²×C₂₃	—	4	2	no	—
145	C₂²×C₂₃	—	8	2	no	—
146	C₂²×C₂₃	#535	1	1	yes	characteristic
147	C₂²×C₂₃	#554	1	1	yes	characteristic
148	C₂²×C₂₃	#555	1	1	yes	characteristic
149	C₄×C₂₃	—	4	2	no	—
150	C₄×C₂₃	—	4	2	no	—
151	C₄×C₂₃	—	8	2	no	—
152	C₄×C₂₃	—	8	4	no	—
153	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
154	C₄×C₂₃	C₂×#124	1	1	yes	characteristic
155	C₄×C₂₃	#554	1	1	yes	characteristic
156	C₄×C₂₃	#555	1	1	yes	characteristic
157	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×#131	C₂²×C₂₃	1	2	yes	—
159	#533	C₂²×C₂₃	1	2	yes	—
160	#501	—	2	4	no	—
161	#444	—	2	4	no	—
162	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
163	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
164	#427	C₂²×C₂₃	1	1	yes	characteristic
165	#427	C₂²×C₂₃	1	2	yes	—
166	#427	C₂²×C₂₃	1	2	yes	—
167	#421	C₂²×C₂₃	1	4	yes	—
168	C₂×#104	C₂²×C₂₃	1	2	yes	—
169	#410	C₂²×C₂₃	1	4	yes	—
170	#405	C₂²×C₂₃	1	2	yes	—
171	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
172	#344	C₂²×C₂₃	1	4	yes	—
173	#339	C₂²×C₂₃	1	2	yes	—
174	H(3,ℤ/4)	C₂²×C₂₃	1	4	yes	—
175	C₂×C₂₃	—	2	1	no	—
176	C₂×C₂₃	—	2	1	no	—
177	C₂×C₂₃	—	2	1	no	—
178	C₂×C₂₃	—	8	2	no	—
179	C₂×C₂₃	#2563	1	1	yes	characteristic
180	C₂×#41	C₂³×C₂₃	1	2	yes	—
181	#135	—	2	2	no	—
182	#135	—	2	2	no	—
183	C₂×Q₁₆	Q₈×C₂₃	1	1	yes	characteristic, complemented
184	C₂×SD₁₆	Q₈×C₂₃	1	2	yes	complemented
185	C₂×D₈	Q₈×C₂₃	1	1	yes	characteristic, complemented
186	#131	—	2	2	no	—
187	#131	—	2	2	no	—
188	#131	—	2	2	no	—
189	#131	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
192	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
196	C₄⋊Q₈	—	2	2	no	—
197	#125	—	2	2	no	—
198	C₂²⋊Q₈	—	4	2	no	—
199	C₄×Q₈	—	4	2	no	—
200	#117	C₂³×C₂₃	1	4	yes	—
201	#117	C₂³×C₂₃	1	2	yes	—
202	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
203	#108	—	2	2	no	—
204	#108	—	2	2	no	—
205	#107	—	2	2	no	—
206	#107	—	2	2	no	—
207	#107	—	2	1	no	—
208	#106	—	2	2	no	—
209	#106	—	2	2	no	—
210	#106	—	2	1	no	—
211	#104	—	2	4	no	—
212	#104	—	4	4	no	—
213	#99	—	4	4	no	—
214	#97	—	2	2	no	—
215	#97	D₄×C₂₃	1	4	yes	—
216	C₄×C₈	—	2	2	no	—
217	C₂₃	#29155	1	1	yes	characteristic, direct product factor, Sylow
218	#41	—	2	2	no	—
219	#41	—	2	2	no	—
220	#41	—	2	2	no	—
221	#41	—	2	2	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂×Q₈	C₂×Q₈×C₂₃	1	2	yes	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
229	Q₁₆	—	2	2	no	—
230	SD₁₆	—	2	4	no	—
231	D₈	—	2	2	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	4	2	no	—
237	M₁₆	—	4	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
243	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
244	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
245	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
246	C₂×C₈	#41×C₂₃	1	2	yes	—
247	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
248	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
249	C₂×C₈	#41×C₂₃	1	2	yes	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	4	no	—
259	C₄²	C₂×D₄×C₂₃	1	2	yes	—
260	C₂³	—	2	2	no	—
261	C₂³	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
262	Q₈	—	2	2	no	—
263	Q₈	—	2	2	no	—
264	Q₈	—	8	4	no	—
265	D₄	—	2	2	no	—
266	D₄	—	2	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
278	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
280	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
281	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
282	C₈	—	4	2	no	—
283	C₈	—	4	2	no	—
284	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
285	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
286	C₈	C₂₃×#124	1	2	yes	—
287	C₈	C₂₃×#124	1	2	yes	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	C₂₃×#535	1	1	yes	characteristic
292	C₂²	C₂₃×#554	1	1	yes	characteristic
293	C₂²	C₂₃×#555	1	1	yes	characteristic
294	C₄	—	4	2	no	—
295	C₄	—	4	2	no	—
296	C₄	—	8	4	no	—
297	C₄	—	8	2	no	—
298	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
299	C₄	C₂×C₂₃×#124	1	1	yes	characteristic
300	C₄	C₂₃×#554	1	1	yes	characteristic
301	C₄	C₂₃×#555	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	8	2	no	—
306	C₂	C₂₃×#2563	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂×C₂₃	#2563
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3162	C₂	semidirect
3. C₂₃×#2031	C₂	semidirect
5. C₂₃×#1833	C₂	semidirect
6. C₂₃×#1814	C₂	semidirect
7. C₂₃×#1812	C₂	semidirect
8. C₂₃×#1773	C₂	semidirect
30. C₂×Q₁₆×C₂₃	Q₈	semidirect
31. C₂×SD₁₆×C₂₃	Q₈	semidirect
32. C₂×D₈×C₂₃	Q₈	semidirect
106. #29155	C₂₃	direct (N × H)
135. #3162	C₂×C₂₃	semidirect
137. #2031	C₂×C₂₃	semidirect
139. #1833	C₂×C₂₃	semidirect
140. #1814	C₂×C₂₃	semidirect
141. #1812	C₂×C₂₃	semidirect
142. #1773	C₂×C₂₃	semidirect
183. C₂×Q₁₆	Q₈×C₂₃	semidirect
184. C₂×SD₁₆	Q₈×C₂₃	semidirect
185. C₂×D₈	Q₈×C₂₃	semidirect
217. C₂₃	#29155	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3162	2
C₂×C₂₃×#444	2
C₂₃×#2031	2
C₂₃×#1976	2
C₂₃×#1833	2
C₂₃×#1814	2
C₂₃×#1812	2
C₂₃×#1773	2
#29155	23

Subgroups — C23×#29155 #19176781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29155 #19176781