Subgroups — C₂×C₂₃×#2232 #19181265

Order 5888.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#466	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2232	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#366	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#357	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#121	—	2	1	no	—
10	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
11	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#466	—	2	2	no	—
13	C₂₃×#466	C₂²	1	4	yes	—
14	C₂×C₂₃×#107	—	2	2	no	—
15	C₂×C₂₃×#107	—	2	1	no	—
16	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#107	C₂²	1	2	yes	—
18	C₂×C₂₃×#107	C₂²	1	4	yes	complemented
19	C₂×C₂₃×#102	—	2	1	no	—
20	C₂₃×#366	C₂²	1	8	yes	complemented
21	C₂₃×#357	C₂²	1	8	yes	complemented
22	C₂₃×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
23	C₂²×D₄×C₂₃	—	2	1	no	—
24	C₂²×D₄×C₂₃	—	4	1	no	—
25	C₂³×C₄×C₂₃	—	4	1	no	—
26	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
27	C₂×D₈×C₂₃	—	2	1	no	—
28	C₂×D₈×C₂₃	—	2	4	no	—
29	C₂×D₈×C₂₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₂₃	—	2	4	no	—
31	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	D₄	1	2	yes	—
33	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×#121	—	2	4	no	—
36	C₂₃×#121	—	4	2	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
41	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
42	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
43	C₂×C₁₆×C₂₃	—	2	4	no	—
44	C₂×C₁₆×C₂₃	C₂³	1	4	yes	—
45	C₂₃×#107	—	2	2	no	—
46	C₂₃×#107	—	2	4	no	—
47	C₂₃×#107	—	2	1	no	—
48	C₂₃×#107	C₂³	1	4	yes	—
49	C₂₃×#107	C₂³	1	2	yes	—
50	C₂₃×#102	—	2	4	no	—
51	C₂⁴×C₂₃	—	4	1	no	—
52	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
53	C₂×D₄×C₂₃	—	2	2	no	—
54	C₂×D₄×C₂₃	—	4	4	no	—
55	C₂×D₄×C₂₃	—	4	2	no	—
56	C₂×D₄×C₂₃	—	4	1	no	—
57	C₂×D₄×C₂₃	—	4	2	no	—
58	C₂×D₄×C₂₃	—	8	4	no	—
59	C₂²×C₄×C₂₃	—	2	2	no	—
60	C₂²×C₄×C₂₃	—	2	2	no	—
61	C₂²×C₄×C₂₃	—	4	4	no	—
62	C₂²×C₄×C₂₃	—	4	1	no	—
63	C₂²×C₄×C₂₃	—	4	2	no	—
64	C₂²×C₄×C₂₃	—	8	4	no	—
65	C₂²×C₄×C₂₃	D₈	1	2	yes	—
66	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	D₈×C₂₃	—	2	4	no	—
70	D₈×C₂₃	—	4	2	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	4	no	—
73	C₂×C₈×C₂₃	—	2	4	no	—
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	4	yes	—
77	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₂₃	—	2	2	no	—
79	C₄⋊C₄×C₂₃	—	2	4	no	—
80	C₄⋊C₄×C₂₃	—	4	4	no	—
81	C₄⋊C₄×C₂₃	—	4	2	no	—
82	C₄⋊C₄×C₂₃	—	4	1	no	—
83	C₂²⋊C₄×C₂₃	—	4	4	no	—
84	C₁₆×C₂₃	—	2	4	no	—
85	C₂×#2232	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	—	2	2	no	—
87	C₂³×C₂₃	—	2	2	no	—
88	C₂³×C₂₃	—	4	2	no	—
89	C₂³×C₂₃	—	8	2	no	—
90	C₂³×C₂₃	—	8	2	no	—
91	C₂³×C₂₃	—	8	2	no	—
92	C₂³×C₂₃	D₁₆	1	4	yes	—
93	C₂³×C₂₃	C₂×D₈	1	2	yes	—
94	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
95	D₄×C₂₃	—	4	2	no	—
96	D₄×C₂₃	—	4	2	no	—
97	D₄×C₂₃	—	8	8	no	—
98	D₄×C₂₃	—	8	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	—	8	8	no	—
106	C₂×C₄×C₂₃	—	8	4	no	—
107	C₂×C₄×C₂₃	—	8	2	no	—
108	C₂×C₄×C₂₃	#123	1	2	yes	—
109	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₂₃	#137	1	2	yes	—
112	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
113	C₈×C₂₃	—	2	2	no	—
114	C₈×C₂₃	#123	1	2	yes	—
115	C₈×C₂₃	C₂×#41	1	2	yes	—
116	C₂×#466	C₂×C₂₃	1	1	yes	characteristic
117	C₂²×#107	C₂×C₂₃	1	1	yes	characteristic, complemented
118	#2232	C₂×C₂₃	1	8	yes	direct product factor
119	C₂×#366	C₂×C₂₃	1	2	yes	complemented
120	C₂×#357	C₂×C₂₃	1	2	yes	complemented
121	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
122	C₂²×C₂₃	—	2	2	no	—
123	C₂²×C₂₃	—	2	2	no	—
124	C₂²×C₂₃	—	2	8	no	—
125	C₂²×C₂₃	—	8	2	no	—
126	C₂²×C₂₃	—	8	2	no	—
127	C₂²×C₂₃	—	8	2	no	—
128	C₂²×C₂₃	—	16	4	no	—
129	C₂²×C₂₃	#480	1	2	yes	—
130	C₂²×C₂₃	C₂×D₁₆	1	4	yes	—
131	C₂²×C₂₃	C₂×D₁₆	1	2	yes	—
132	C₂²×C₂₃	#510	1	2	yes	—
133	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
134	C₄×C₂₃	—	2	2	no	—
135	C₄×C₂₃	—	8	4	no	—
136	C₄×C₂₃	—	8	2	no	—
137	C₄×C₂₃	#480	1	2	yes	—
138	C₄×C₂₃	C₂×#123	1	1	yes	characteristic
139	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
140	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
141	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×#121	—	2	1	no	—
143	C₂²×C₄⋊C₄	—	2	1	no	—
144	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
145	#466	—	2	2	no	—
146	#466	C₂²×C₂₃	1	4	yes	—
147	C₂×#107	—	2	2	no	—
148	C₂×#107	—	2	1	no	—
149	C₂×#107	C₂²×C₂₃	1	2	yes	—
150	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×#107	C₂²×C₂₃	1	4	yes	complemented
152	C₂×#102	—	2	1	no	—
153	#366	C₂²×C₂₃	1	8	yes	complemented
154	#357	C₂²×C₂₃	1	8	yes	complemented
155	C₂²⋊C₁₆	C₂²×C₂₃	1	4	yes	complemented
156	C₂×C₂₃	—	2	4	no	—
157	C₂×C₂₃	—	16	2	no	—
158	C₂×C₂₃	#2232	1	4	yes	direct product factor
159	C₂×C₂₃	C₂×#480	1	1	yes	characteristic
160	C₂×C₂₃	C₂²×D₁₆	1	1	yes	characteristic
161	C₂×C₂₃	C₂×#510	1	1	yes	characteristic
162	C₂²×D₄	—	2	1	no	—
163	C₂²×D₄	—	4	1	no	—
164	C₂³×C₄	—	4	1	no	—
165	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	—	2	4	no	—
168	C₂×D₈	C₂³×C₂₃	1	2	yes	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₈	D₄×C₂₃	1	2	yes	—
172	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
173	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
174	#121	—	2	4	no	—
175	#121	—	4	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	4	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	4	1	no	—
180	C₂×C₄⋊C₄	—	4	2	no	—
181	C₂×C₂²⋊C₄	—	4	1	no	—
182	C₂×C₁₆	—	2	4	no	—
183	C₂×C₁₆	C₂³×C₂₃	1	4	yes	—
184	#107	—	2	2	no	—
185	#107	—	2	4	no	—
186	#107	—	2	1	no	—
187	#107	C₂³×C₂₃	1	4	yes	—
188	#107	C₂³×C₂₃	1	2	yes	—
189	#102	—	2	4	no	—
190	C₂₃	C₂×#2232	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁴	—	4	1	no	—
192	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	4	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	—	8	4	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	4	4	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	8	4	no	—
205	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	D₈×C₂₃	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
208	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
209	D₈	—	2	4	no	—
210	D₈	—	4	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
215	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂×C₈	#41×C₂₃	1	4	yes	—
217	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	1	no	—
223	C₂²⋊C₄	—	4	4	no	—
224	C₁₆	—	2	4	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	2	no	—
230	C₂³	—	8	2	no	—
231	C₂³	C₂₃×D₁₆	1	4	yes	—
232	C₂³	C₂×D₈×C₂₃	1	2	yes	—
233	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	8	8	no	—
237	D₄	—	8	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	8	8	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	C₂₃×#123	1	2	yes	—
248	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
249	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂₃×#137	1	2	yes	—
251	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	C₂₃×#123	1	2	yes	—
254	C₈	C₂×#41×C₂₃	1	2	yes	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	16	4	no	—
262	C₂²	C₂₃×#480	1	2	yes	—
263	C₂²	C₂×C₂₃×D₁₆	1	4	yes	—
264	C₂²	C₂×C₂₃×D₁₆	1	2	yes	—
265	C₂²	C₂₃×#510	1	2	yes	—
266	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	8	4	no	—
269	C₄	—	8	2	no	—
270	C₄	C₂₃×#480	1	2	yes	—
271	C₄	C₂×C₂₃×#123	1	1	yes	characteristic
272	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
273	C₂	—	2	4	no	—
274	C₂	—	16	2	no	—
275	C₂	C₂₃×#2232	1	4	yes	direct product factor
276	C₂	C₂×C₂₃×#480	1	1	yes	characteristic
277	C₂	C₂²×C₂₃×D₁₆	1	1	yes	characteristic
278	C₂	C₂×C₂₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂₃×#107	C₂	semidirect
3. C₂₃×#2232	C₂	direct (N × H)
4. C₂×C₂₃×#366	C₂	semidirect
5. C₂×C₂₃×#357	C₂	semidirect
6. C₂×C₂₃×C₂²⋊C₁₆	C₂	semidirect
18. C₂×C₂₃×#107	C₂²	semidirect
20. C₂₃×#366	C₂²	semidirect
21. C₂₃×#357	C₂²	semidirect
22. C₂₃×C₂²⋊C₁₆	C₂²	semidirect
85. C₂×#2232	C₂₃	direct (N × H)
117. C₂²×#107	C₂×C₂₃	semidirect
118. #2232	C₂×C₂₃	direct (N × H)
119. C₂×#366	C₂×C₂₃	semidirect
120. C₂×#357	C₂×C₂₃	semidirect
121. C₂×C₂²⋊C₁₆	C₂×C₂₃	semidirect
151. C₂×#107	C₂²×C₂₃	semidirect
153. #366	C₂²×C₂₃	semidirect
154. #357	C₂²×C₂₃	semidirect
155. C₂²⋊C₁₆	C₂²×C₂₃	semidirect
158. C₂×C₂₃	#2232	direct (N × H)
190. C₂₃	C₂×#2232	direct (N × H)
275. C₂	C₂₃×#2232	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×#466	2
C₂²×C₂₃×#107	2
C₂₃×#2232	2
C₂×C₂₃×#366	2
C₂×C₂₃×#357	2
C₂×C₂₃×C₂²⋊C₁₆	2
C₂×#2232	23

Subgroups — C2×C23×#2232 #19181265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2232 #19181265