Subgroups — C₂₃×#33583 #19181209

Order 5888.

702 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#510	C₂	1	2	yes	maximal
2	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×#135	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2209	C₂	1	4	yes	maximal, complemented
5	C₂₃×#2208	C₂	1	4	yes	maximal, complemented
6	C₂₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2141	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆×C₂₃	—	2	1	no	—
10	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
12	C₄×#41×C₂₃	—	2	1	no	—
13	C₂₃×#510	—	2	8	no	—
14	C₂×C₂₃×Q₃₂	C₂²	1	4	yes	—
15	C₂×C₂₃×SD₃₂	C₂²	1	4	yes	—
16	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
17	C₂₃×#494	—	2	2	no	—
18	C₂₃×#494	C₂²	1	4	yes	—
19	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
20	C₄×SD₁₆×C₂₃	—	2	2	no	—
21	C₄×D₈×C₂₃	C₂²	1	2	yes	—
22	C₂₃×#427	C₂²	1	1	yes	characteristic
23	C₂₃×#416	—	2	1	no	—
24	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×C₄⋊C₁₆	C₂²	1	4	yes	—
26	C₂₃×#358	C₂²	1	8	yes	—
27	C₂×#41×C₂₃	—	2	1	no	—
28	C₂²×Q₈×C₂₃	—	2	2	no	—
29	C₂₃×#135	—	2	2	no	—
30	C₂₃×#135	D₄	1	4	yes	—
31	C₂×Q₁₆×C₂₃	—	2	4	no	—
32	C₂×Q₁₆×C₂₃	—	2	4	no	—
33	C₂×Q₁₆×C₂₃	—	2	2	no	—
34	C₂×Q₁₆×C₂₃	—	2	4	no	—
35	C₂×Q₁₆×C₂₃	—	4	4	no	—
36	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
37	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
38	C₂×SD₁₆×C₂₃	—	2	1	no	—
39	C₂×D₈×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×C₄⋊Q₈	—	2	4	no	—
43	C₂₃×C₄⋊Q₈	—	4	1	no	—
44	C₂₃×C₄⋊Q₈	—	4	1	no	—
45	C₄×Q₈×C₂₃	—	2	2	no	—
46	C₄×D₄×C₂₃	—	2	2	no	—
47	C₄×D₄×C₂₃	—	4	2	no	—
48	C₂₃×#117	—	2	1	no	—
49	C₂₃×#117	—	4	1	no	—
50	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
51	C₂×C₄²×C₂₃	—	4	1	no	—
52	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
53	C₂₃×Q₃₂	—	2	8	no	—
54	C₂₃×SD₃₂	—	2	8	no	—
55	C₂₃×M₃₂	—	2	4	no	—
56	C₂×C₁₆×C₂₃	C₂³	1	4	yes	—
57	C₂₃×#107	C₂³	1	2	yes	—
58	C₂₃×#106	—	2	1	no	—
59	C₂₃×#103	—	2	2	no	—
60	C₂₃×#102	—	2	2	no	—
61	C₄×C₈×C₂₃	D₄	1	2	yes	—
62	C₄×C₈×C₂₃	C₂³	1	2	yes	—
63	#41×C₂₃	—	2	4	no	—
64	#41×C₂₃	—	4	2	no	—
65	C₂×Q₈×C₂₃	—	2	1	no	—
66	C₂×Q₈×C₂₃	—	2	4	no	—
67	C₂×Q₈×C₂₃	—	4	2	no	—
68	C₂×Q₈×C₂₃	—	4	4	no	—
69	C₂×Q₈×C₂₃	—	4	4	no	—
70	C₂×D₄×C₂₃	—	2	1	no	—
71	C₂×D₄×C₂₃	—	4	1	no	—
72	C₂²×C₄×C₂₃	—	4	1	no	—
73	C₂²×C₄×C₂₃	—	4	2	no	—
74	C₂²×C₄×C₂₃	—	4	2	no	—
75	C₂²×C₄×C₂₃	D₈	1	2	yes	—
76	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₂₃	—	2	4	no	—
78	Q₁₆×C₂₃	—	2	1	no	—
79	Q₁₆×C₂₃	—	2	4	no	—
80	Q₁₆×C₂₃	—	4	8	no	—
81	Q₁₆×C₂₃	—	4	4	no	—
82	Q₁₆×C₂₃	C₂×D₄	1	2	yes	—
83	SD₁₆×C₂₃	—	2	4	no	—
84	D₈×C₂₃	—	2	1	no	—
85	D₈×C₂₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₂₃	—	2	2	no	—
87	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₂₃	#41	1	2	yes	—
91	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₂₃	—	2	2	no	—
93	C₄⋊C₄×C₂₃	—	4	4	no	—
94	C₄⋊C₄×C₂₃	—	4	4	no	—
95	C₄⋊C₄×C₂₃	—	4	2	no	—
96	C₂²⋊C₄×C₂₃	—	4	1	no	—
97	C₂²⋊C₄×C₂₃	—	4	2	no	—
98	C₄²×C₂₃	—	4	1	no	—
99	C₄²×C₂₃	—	4	2	no	—
100	C₄²×C₂₃	D₈	1	1	yes	characteristic
101	C₄²×C₂₃	D₈	1	1	yes	characteristic
102	C₄²×C₂₃	C₂×D₄	1	2	yes	—
103	C₁₆×C₂₃	—	2	4	no	—
104	#33583	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₂₃	—	4	1	no	—
106	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
107	Q₈×C₂₃	—	2	2	no	—
108	Q₈×C₂₃	—	4	1	no	—
109	Q₈×C₂₃	—	4	4	no	—
110	Q₈×C₂₃	—	4	4	no	—
111	Q₈×C₂₃	—	8	4	no	—
112	D₄×C₂₃	—	2	2	no	—
113	D₄×C₂₃	—	4	4	no	—
114	D₄×C₂₃	—	4	1	no	—
115	C₂×C₄×C₂₃	—	2	4	no	—
116	C₂×C₄×C₂₃	—	4	1	no	—
117	C₂×C₄×C₂₃	—	4	4	no	—
118	C₂×C₄×C₂₃	—	4	2	no	—
119	C₂×C₄×C₂₃	—	4	4	no	—
120	C₂×C₄×C₂₃	—	8	4	no	—
121	C₂×C₄×C₂₃	—	8	4	no	—
122	C₂×C₄×C₂₃	#121	1	2	yes	—
123	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
128	C₈×C₂₃	—	2	2	no	—
129	C₈×C₂₃	#121	1	2	yes	—
130	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
131	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
132	C₂×#510	C₂×C₂₃	1	2	yes	—
133	C₂×#494	C₂×C₂₃	1	1	yes	characteristic, complemented
134	C₄×#135	C₂×C₂₃	1	1	yes	characteristic
135	#2209	C₂×C₂₃	1	4	yes	complemented
136	#2208	C₂×C₂₃	1	4	yes	complemented
137	#2150	C₂×C₂₃	1	1	yes	characteristic, complemented
138	#2141	C₂×C₂₃	1	2	yes	complemented
139	C₂²×C₂₃	—	2	2	no	—
140	C₂²×C₂₃	—	4	2	no	—
141	C₂²×C₂₃	—	8	2	no	—
142	C₂²×C₂₃	#459	1	2	yes	—
143	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₂₃	—	2	2	no	—
145	C₄×C₂₃	—	4	2	no	—
146	C₄×C₂₃	—	8	2	no	—
147	C₄×C₂₃	—	8	4	no	—
148	C₄×C₂₃	#459	1	2	yes	—
149	C₄×C₂₃	#510	1	4	yes	complemented
150	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
151	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
153	C₂²×Q₁₆	—	2	1	no	—
154	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
155	C₂×C₄⋊Q₈	—	2	1	no	—
156	C₄×#41	—	2	1	no	—
157	#510	—	2	8	no	—
158	C₂×Q₃₂	C₂²×C₂₃	1	4	yes	—
159	C₂×SD₃₂	C₂²×C₂₃	1	4	yes	—
160	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
161	#494	—	2	2	no	—
162	#494	C₂²×C₂₃	1	4	yes	—
163	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
164	C₄×SD₁₆	—	2	2	no	—
165	C₄×D₈	C₂²×C₂₃	1	2	yes	—
166	#427	C₂²×C₂₃	1	1	yes	characteristic
167	#416	—	2	1	no	—
168	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
169	C₄⋊C₁₆	C₂²×C₂₃	1	4	yes	—
170	#358	C₂²×C₂₃	1	8	yes	—
171	C₂×C₂₃	—	2	2	no	—
172	C₂×C₂₃	—	8	2	no	—
173	C₂×C₂₃	C₂×#459	1	1	yes	characteristic
174	C₂×C₂₃	C₂×#510	1	2	yes	—
175	C₂×#41	—	2	1	no	—
176	C₂²×Q₈	—	2	2	no	—
177	#135	—	2	2	no	—
178	#135	D₄×C₂₃	1	4	yes	—
179	C₂×Q₁₆	—	2	4	no	—
180	C₂×Q₁₆	—	2	4	no	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	4	no	—
183	C₂×Q₁₆	—	4	4	no	—
184	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
185	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
186	C₂×SD₁₆	—	2	1	no	—
187	C₂×D₈	C₂³×C₂₃	1	1	yes	characteristic
188	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
190	C₄⋊Q₈	—	2	4	no	—
191	C₄⋊Q₈	—	4	1	no	—
192	C₄⋊Q₈	—	4	1	no	—
193	C₄×Q₈	—	2	2	no	—
194	C₄×D₄	—	2	2	no	—
195	C₄×D₄	—	4	2	no	—
196	#117	—	2	1	no	—
197	#117	—	4	1	no	—
198	C₂×C₄⋊C₄	—	4	2	no	—
199	C₂×C₄²	—	4	1	no	—
200	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
201	Q₃₂	—	2	8	no	—
202	SD₃₂	—	2	8	no	—
203	M₃₂	—	2	4	no	—
204	C₂×C₁₆	C₂³×C₂₃	1	4	yes	—
205	#107	C₂³×C₂₃	1	2	yes	—
206	#106	—	2	1	no	—
207	#103	—	2	2	no	—
208	#102	—	2	2	no	—
209	C₄×C₈	D₄×C₂₃	1	2	yes	—
210	C₄×C₈	C₂³×C₂₃	1	2	yes	—
211	C₂₃	#33583	1	1	yes	characteristic, direct product factor, Sylow
212	#41	—	2	4	no	—
213	#41	—	4	2	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×Q₈	—	4	4	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	D₈×C₂₃	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
226	Q₁₆	—	2	4	no	—
227	Q₁₆	—	2	1	no	—
228	Q₁₆	—	2	4	no	—
229	Q₁₆	—	4	8	no	—
230	Q₁₆	—	4	4	no	—
231	Q₁₆	C₂×D₄×C₂₃	1	2	yes	—
232	SD₁₆	—	2	4	no	—
233	D₈	—	2	1	no	—
234	D₈	C₂×D₄×C₂₃	1	2	yes	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
237	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
239	C₂×C₈	#41×C₂₃	1	2	yes	—
240	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	1	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	2	no	—
249	C₄²	D₈×C₂₃	1	1	yes	characteristic
250	C₄²	D₈×C₂₃	1	1	yes	characteristic
251	C₄²	C₂×D₄×C₂₃	1	2	yes	—
252	C₁₆	—	2	4	no	—
253	C₂³	—	4	1	no	—
254	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	Q₈	—	4	1	no	—
257	Q₈	—	4	4	no	—
258	Q₈	—	4	4	no	—
259	Q₈	—	8	4	no	—
260	D₄	—	2	2	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	1	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	C₂₃×#121	1	2	yes	—
271	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
272	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
273	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
274	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
275	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	C₂₃×#121	1	2	yes	—
278	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
279	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
280	C₂²	—	2	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	C₂₃×#459	1	2	yes	—
284	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	4	2	no	—
287	C₄	—	8	4	no	—
288	C₄	—	8	2	no	—
289	C₄	C₂₃×#459	1	2	yes	—
290	C₄	C₂₃×#510	1	4	yes	complemented
291	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
292	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	8	2	no	—
295	C₂	C₂×C₂₃×#459	1	1	yes	characteristic
296	C₂	C₂×C₂₃×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#494	C₂	semidirect
4. C₂₃×#2209	C₂	semidirect
5. C₂₃×#2208	C₂	semidirect
6. C₂₃×#2150	C₂	semidirect
7. C₂₃×#2141	C₂	semidirect
104. #33583	C₂₃	direct (N × H)
133. C₂×#494	C₂×C₂₃	semidirect
135. #2209	C₂×C₂₃	semidirect
136. #2208	C₂×C₂₃	semidirect
137. #2150	C₂×C₂₃	semidirect
138. #2141	C₂×C₂₃	semidirect
149. C₄×C₂₃	#510	semidirect
211. C₂₃	#33583	direct (N × H)
290. C₄	C₂₃×#510	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#510	2
C₂×C₂₃×#494	2
C₄×C₂₃×#135	2
C₂₃×#2209	2
C₂₃×#2208	2
C₂₃×#2150	2
C₂₃×#2141	2
#33583	23

Subgroups — C23×#33583 #19181209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33583 #19181209