Subgroups — C₂₃×#33459 #19181085

Order 5888.

674 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#508	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2951	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2175	C₂	1	4	yes	maximal, complemented
4	C₄×C₂₃×Q₃₂	C₂	1	2	yes	maximal, complemented
5	C₄×C₂₃×D₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2139	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#135	C₂²	1	2	yes	—
10	C₂₃×#520	—	2	2	no	—
11	C₂₃×#508	C₄	1	8	yes	complemented
12	C₂×C₂₃×Q₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₂₃×D₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#440	—	2	4	no	—
18	C₄×Q₁₆×C₂₃	C₂²	1	4	yes	—
19	C₄×D₈×C₂₃	C₂²	1	4	yes	—
20	C₂×C₂₃×#107	C₂²	1	2	yes	—
21	C₂₃×#417	—	2	2	no	—
22	C₂₃×#405	C₂²	1	1	yes	characteristic
23	C₂₃×#366	C₂²	1	2	yes	—
24	C₂₃×#366	C₂²	1	2	yes	—
25	C₂₃×#358	C₂²	1	4	yes	—
26	C₂₃×#357	C₂²	1	4	yes	—
27	C₂₃×#346	C₂²	1	2	yes	—
28	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
29	C₂×#41×C₂₃	—	2	2	no	—
30	C₂₃×#135	—	2	4	no	—
31	C₂₃×#135	C₂×C₄	1	8	yes	complemented
32	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
33	C₂×SD₁₆×C₂₃	—	2	2	no	—
34	C₂×D₈×C₂₃	C₂³	1	2	yes	—
35	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₂₃	—	2	4	no	—
38	C₄×D₄×C₂₃	—	2	4	no	—
39	C₄×D₄×C₂₃	—	4	4	no	—
40	C₂₃×#117	—	4	2	no	—
41	C₂₃×#117	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
44	C₂₃×Q₃₂	C₂×C₄	1	4	yes	complemented
45	C₂₃×SD₃₂	C₂×C₄	1	8	yes	complemented
46	C₂₃×D₁₆	C₂×C₄	1	4	yes	complemented
47	C₂₃×M₃₂	—	2	2	no	—
48	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
50	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×#107	—	2	2	no	—
53	C₂₃×#107	C₂³	1	4	yes	—
54	C₂₃×#103	—	2	4	no	—
55	C₂₃×#102	—	2	4	no	—
56	C₂₃×#97	D₄	1	2	yes	—
57	C₄×C₈×C₂₃	C₂³	1	2	yes	—
58	#41×C₂₃	—	2	8	no	—
59	#41×C₂₃	—	4	4	no	—
60	C₂×Q₈×C₂₃	—	2	2	no	—
61	C₂×D₄×C₂₃	—	2	2	no	—
62	C₂×D₄×C₂₃	—	4	2	no	—
63	C₂²×C₄×C₂₃	—	4	2	no	—
64	C₂²×C₄×C₂₃	—	4	2	no	—
65	C₂²×C₄×C₂₃	—	4	4	no	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₂₃	—	2	2	no	—
68	Q₁₆×C₂₃	C₂²×C₄	1	4	yes	—
69	SD₁₆×C₂₃	—	2	8	no	—
70	D₈×C₂₃	—	2	2	no	—
71	D₈×C₂₃	C₂²×C₄	1	4	yes	—
72	M₁₆×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₂₃	#41	1	2	yes	—
77	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₂₃	—	2	4	no	—
79	C₄⋊C₄×C₂₃	—	4	2	no	—
80	C₄⋊C₄×C₂₃	—	4	4	no	—
81	C₄⋊C₄×C₂₃	D₈	1	1	yes	characteristic
82	C₄⋊C₄×C₂₃	D₈	1	1	yes	characteristic
83	C₂²⋊C₄×C₂₃	—	4	4	no	—
84	C₂²⋊C₄×C₂₃	D₈	1	2	yes	—
85	C₄²×C₂₃	—	4	4	no	—
86	C₄²×C₂₃	C₂×D₄	1	2	yes	—
87	C₁₆×C₂₃	—	2	2	no	—
88	C₁₆×C₂₃	C₂²×C₄	1	2	yes	—
89	C₁₆×C₂₃	C₂²×C₄	1	2	yes	—
90	#33459	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	—	4	2	no	—
92	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
93	Q₈×C₂₃	—	2	4	no	—
94	Q₈×C₂₃	—	4	2	no	—
95	D₄×C₂₃	—	2	4	no	—
96	D₄×C₂₃	—	4	8	no	—
97	D₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	4	8	no	—
100	C₂×C₄×C₂₃	—	4	4	no	—
101	C₂×C₄×C₂₃	—	16	4	no	—
102	C₂×C₄×C₂₃	—	16	2	no	—
103	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#135	1	2	yes	—
108	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
109	C₈×C₂₃	—	2	2	no	—
110	C₈×C₂₃	C₄×D₄	1	2	yes	—
111	C₈×C₂₃	C₂³×C₄	1	1	yes	characteristic
112	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
113	C₂×#508	C₂×C₂₃	1	1	yes	characteristic
114	#2951	C₂×C₂₃	1	2	yes	complemented
115	#2175	C₂×C₂₃	1	4	yes	complemented
116	C₄×Q₃₂	C₂×C₂₃	1	2	yes	complemented
117	C₄×D₁₆	C₂×C₂₃	1	2	yes	complemented
118	C₂×#366	C₂×C₂₃	1	1	yes	characteristic, complemented
119	#2139	C₂×C₂₃	1	2	yes	complemented
120	#2108	C₂×C₂₃	1	1	yes	characteristic, complemented
121	C₂²×C₂₃	—	2	1	no	—
122	C₂²×C₂₃	—	2	1	no	—
123	C₂²×C₂₃	—	4	4	no	—
124	C₂²×C₂₃	—	8	2	no	—
125	C₂²×C₂₃	—	8	2	no	—
126	C₂²×C₂₃	C₄×D₈	1	2	yes	—
127	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
128	C₄×C₂₃	—	2	2	no	—
129	C₄×C₂₃	—	2	2	no	—
130	C₄×C₂₃	—	4	4	no	—
131	C₄×C₂₃	—	8	4	no	—
132	C₄×C₂₃	C₄×D₈	1	2	yes	—
133	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
134	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
135	C₂×#135	C₂²×C₂₃	1	2	yes	—
136	#520	—	2	2	no	—
137	#508	C₄×C₂₃	1	8	yes	complemented
138	C₂×Q₃₂	C₂²×C₂₃	1	1	yes	characteristic
139	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	—
140	C₂×D₁₆	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
142	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
143	#440	—	2	4	no	—
144	C₄×Q₁₆	C₂²×C₂₃	1	4	yes	—
145	C₄×D₈	C₂²×C₂₃	1	4	yes	—
146	C₂×#107	C₂²×C₂₃	1	2	yes	—
147	#417	—	2	2	no	—
148	#405	C₂²×C₂₃	1	1	yes	characteristic
149	#366	C₂²×C₂₃	1	2	yes	—
150	#366	C₂²×C₂₃	1	2	yes	—
151	#358	C₂²×C₂₃	1	4	yes	—
152	#357	C₂²×C₂₃	1	4	yes	—
153	#346	C₂²×C₂₃	1	2	yes	—
154	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
155	C₂×C₂₃	—	2	2	no	—
156	C₂×C₂₃	—	8	4	no	—
157	C₂×C₂₃	C₂×C₄×D₈	1	1	yes	characteristic
158	C₂×C₂₃	#3415	1	1	yes	characteristic
159	C₂×C₂₃	#3417	1	1	yes	characteristic
160	C₂×#41	—	2	2	no	—
161	#135	—	2	4	no	—
162	#135	C₂×C₄×C₂₃	1	8	yes	complemented
163	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
164	C₂×SD₁₆	—	2	2	no	—
165	C₂×D₈	C₂³×C₂₃	1	2	yes	—
166	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
168	C₄×Q₈	—	2	4	no	—
169	C₄×D₄	—	2	4	no	—
170	C₄×D₄	—	4	4	no	—
171	#117	—	4	2	no	—
172	#117	D₄×C₂₃	1	1	yes	characteristic
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	4	2	no	—
175	Q₃₂	C₂×C₄×C₂₃	1	4	yes	complemented
176	SD₃₂	C₂×C₄×C₂₃	1	8	yes	complemented
177	D₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
178	M₃₂	—	2	2	no	—
179	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
180	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
181	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
182	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
183	#107	—	2	2	no	—
184	#107	C₂³×C₂₃	1	4	yes	—
185	#103	—	2	4	no	—
186	#102	—	2	4	no	—
187	#97	D₄×C₂₃	1	2	yes	—
188	C₄×C₈	C₂³×C₂₃	1	2	yes	—
189	C₂₃	#33459	1	1	yes	characteristic, direct product factor, Sylow
190	#41	—	2	8	no	—
191	#41	—	4	4	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
199	Q₁₆	—	2	2	no	—
200	Q₁₆	C₂²×C₄×C₂₃	1	4	yes	—
201	SD₁₆	—	2	8	no	—
202	D₈	—	2	2	no	—
203	D₈	C₂²×C₄×C₂₃	1	4	yes	—
204	M₁₆	—	2	2	no	—
205	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
206	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
207	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
208	C₂×C₈	#41×C₂₃	1	2	yes	—
209	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	D₈×C₂₃	1	1	yes	characteristic
214	C₄⋊C₄	D₈×C₂₃	1	1	yes	characteristic
215	C₂²⋊C₄	—	4	4	no	—
216	C₂²⋊C₄	D₈×C₂₃	1	2	yes	—
217	C₄²	—	4	4	no	—
218	C₄²	C₂×D₄×C₂₃	1	2	yes	—
219	C₁₆	—	2	2	no	—
220	C₁₆	C₂²×C₄×C₂₃	1	2	yes	—
221	C₁₆	C₂²×C₄×C₂₃	1	2	yes	—
222	C₂³	—	4	2	no	—
223	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	4	2	no	—
226	D₄	—	2	4	no	—
227	D₄	—	4	8	no	—
228	D₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	16	4	no	—
233	C₂×C₄	—	16	2	no	—
234	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
235	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
236	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
237	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂₃×#135	1	2	yes	—
239	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	C₄×D₄×C₂₃	1	2	yes	—
242	C₈	C₂³×C₄×C₂₃	1	1	yes	characteristic
243	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	C₄×D₈×C₂₃	1	2	yes	—
250	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	4	no	—
255	C₄	C₄×D₈×C₂₃	1	2	yes	—
256	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
257	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	8	4	no	—
260	C₂	C₂×C₄×D₈×C₂₃	1	1	yes	characteristic
261	C₂	C₂₃×#3415	1	1	yes	characteristic
262	C₂	C₂₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator	C₈	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ C₂₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2951	C₂	semidirect
3. C₂₃×#2175	C₂	semidirect
4. C₄×C₂₃×Q₃₂	C₂	semidirect
5. C₄×C₂₃×D₁₆	C₂	semidirect
6. C₂×C₂₃×#366	C₂	semidirect
7. C₂₃×#2139	C₂	semidirect
8. C₂₃×#2108	C₂	semidirect
11. C₂₃×#508	C₄	semidirect
31. C₂₃×#135	C₂×C₄	semidirect
44. C₂₃×Q₃₂	C₂×C₄	semidirect
45. C₂₃×SD₃₂	C₂×C₄	semidirect
46. C₂₃×D₁₆	C₂×C₄	semidirect
48. C₂×C₁₆×C₂₃	C₂×C₄	semidirect
90. #33459	C₂₃	direct (N × H)
114. #2951	C₂×C₂₃	semidirect
115. #2175	C₂×C₂₃	semidirect
116. C₄×Q₃₂	C₂×C₂₃	semidirect
117. C₄×D₁₆	C₂×C₂₃	semidirect
118. C₂×#366	C₂×C₂₃	semidirect
119. #2139	C₂×C₂₃	semidirect
120. #2108	C₂×C₂₃	semidirect
137. #508	C₄×C₂₃	semidirect
162. #135	C₂×C₄×C₂₃	semidirect
175. Q₃₂	C₂×C₄×C₂₃	semidirect
176. SD₃₂	C₂×C₄×C₂₃	semidirect
177. D₁₆	C₂×C₄×C₂₃	semidirect
179. C₂×C₁₆	C₂×C₄×C₂₃	semidirect
189. C₂₃	#33459	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#508	2
C₂₃×#2951	2
C₂₃×#2175	2
C₄×C₂₃×Q₃₂	2
C₄×C₂₃×D₁₆	2
C₂×C₂₃×#366	2
C₂₃×#2139	2
C₂₃×#2108	2
#33459	23

Subgroups — C23×#33459 #19181085

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33459 #19181085