Subgroups — C₂²×C₂₃×#361 #19181015

Order 5888.

1390 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₂₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#361	C₂	1	12	yes	maximal, direct product factor
5	C₂³×D₄×C₂₃	—	2	1	no	—
6	C₂²×D₈×C₂₃	—	2	3	no	—
7	C₂²×D₈×C₂₃	—	2	1	no	—
8	C₂²×D₈×C₂₃	—	2	3	no	—
9	C₂²×D₈×C₂₃	C₄	1	6	yes	—
10	C₂²×D₈×C₂₃	C₄	1	2	yes	—
11	C₂²×D₈×C₂₃	C₂²	1	6	yes	—
12	C₂²×M₁₆×C₂₃	—	2	1	no	—
13	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×M₃₂	—	2	3	no	—
15	C₂×C₂₃×M₃₂	C₂²	1	6	yes	complemented
16	C₂²×C₁₆×C₂₃	—	2	1	no	—
17	C₂×C₂₃×#108	—	2	3	no	—
18	C₂×C₂₃×#108	C₂²	1	6	yes	complemented
19	C₂₃×#361	C₂²	1	16	yes	direct product factor
20	C₂⁵×C₂₃	—	4	1	no	—
21	C₂²×D₄×C₂₃	—	2	6	no	—
22	C₂²×D₄×C₂₃	—	2	2	no	—
23	C₂²×D₄×C₂₃	—	2	6	no	—
24	C₂²×D₄×C₂₃	—	4	3	no	—
25	C₂²×D₄×C₂₃	—	4	1	no	—
26	C₂²×D₄×C₂₃	—	4	3	no	—
27	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
28	C₂×D₈×C₂₃	—	2	6	no	—
29	C₂×D₈×C₂₃	—	2	12	no	—
30	C₂×D₈×C₂₃	—	2	6	no	—
31	C₂×D₈×C₂₃	—	2	6	no	—
32	C₂×D₈×C₂₃	—	2	6	no	—
33	C₂×D₈×C₂₃	—	2	6	no	—
34	C₂×D₈×C₂₃	C₂×C₄	1	12	yes	—
35	C₂×D₈×C₂₃	C₂×C₄	1	12	yes	—
36	C₂×D₈×C₂₃	C₂³	1	4	yes	—
37	C₂×M₁₆×C₂₃	—	2	6	no	—
38	C₂×M₁₆×C₂₃	—	4	3	no	—
39	C₂²×C₈×C₂₃	—	4	1	no	—
40	C₂²×C₈×C₂₃	C₂×C₄	1	3	yes	—
41	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₂₃	D₄	1	3	yes	—
43	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₂₃	D₄	1	3	yes	—
45	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
46	C₂₃×M₃₂	—	2	6	no	—
47	C₂₃×M₃₂	C₂³	1	4	yes	complemented
48	C₂×C₁₆×C₂₃	—	2	6	no	—
49	C₂₃×#108	—	2	6	no	—
50	C₂₃×#108	C₂³	1	4	yes	complemented
51	C₂⁴×C₂₃	—	4	6	no	—
52	C₂⁴×C₂₃	—	4	2	no	—
53	C₂⁴×C₂₃	—	4	6	no	—
54	C₂⁴×C₂₃	—	8	8	no	—
55	C₂⁴×C₂₃	SD₁₆	1	1	yes	characteristic
56	C₂×D₄×C₂₃	—	2	12	no	—
57	C₂×D₄×C₂₃	—	2	12	no	—
58	C₂×D₄×C₂₃	—	2	4	no	—
59	C₂×D₄×C₂₃	—	4	12	no	—
60	C₂×D₄×C₂₃	—	4	6	no	—
61	C₂×D₄×C₂₃	—	4	6	no	—
62	C₂×D₄×C₂₃	—	4	6	no	—
63	C₂×D₄×C₂₃	—	4	6	no	—
64	C₂×D₄×C₂₃	—	4	6	no	—
65	C₂²×C₄×C₂₃	C₂²⋊C₄	1	3	yes	—
66	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	D₈	1	3	yes	—
68	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	SD₁₆	1	3	yes	—
70	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
71	D₈×C₂₃	—	2	12	no	—
72	D₈×C₂₃	—	2	12	no	—
73	D₈×C₂₃	—	2	4	no	—
74	D₈×C₂₃	C₂²×C₄	1	8	yes	—
75	M₁₆×C₂₃	—	2	4	no	—
76	M₁₆×C₂₃	—	4	6	no	—
77	C₂×C₈×C₂₃	—	4	6	no	—
78	C₂×C₈×C₂₃	C₂²⋊C₄	1	3	yes	—
79	C₂×C₈×C₂₃	C₂²⋊C₄	1	6	yes	—
80	C₂×C₈×C₂₃	C₂²⋊C₄	1	3	yes	—
81	C₂×C₈×C₂₃	C₂²×C₄	1	3	yes	—
82	C₂×C₈×C₂₃	C₂²×C₄	1	3	yes	—
83	C₂×C₈×C₂₃	C₂×D₄	1	3	yes	—
84	C₂×C₈×C₂₃	C₂×D₄	1	3	yes	—
85	C₂×C₈×C₂₃	C₂×D₄	1	3	yes	—
86	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
87	C₁₆×C₂₃	—	2	4	no	—
88	C₂²×#361	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₂₃	—	2	4	no	—
90	C₂³×C₂₃	—	4	4	no	—
91	C₂³×C₂₃	—	4	12	no	—
92	C₂³×C₂₃	—	4	12	no	—
93	C₂³×C₂₃	—	8	24	no	—
94	C₂³×C₂₃	—	8	24	no	—
95	C₂³×C₂₃	—	8	8	no	—
96	C₂³×C₂₃	#102	1	3	yes	—
97	C₂³×C₂₃	#102	1	1	yes	characteristic
98	C₂³×C₂₃	C₂×SD₁₆	1	3	yes	—
99	D₄×C₂₃	—	2	8	no	—
100	D₄×C₂₃	—	4	4	no	—
101	D₄×C₂₃	—	4	12	no	—
102	D₄×C₂₃	—	4	12	no	—
103	C₂×C₄×C₂₃	#102	1	6	yes	—
104	C₂×C₄×C₂₃	#102	1	3	yes	—
105	C₂×C₄×C₂₃	#102	1	3	yes	—
106	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
107	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
108	C₂×C₄×C₂₃	C₂×D₈	1	3	yes	—
109	C₂×C₄×C₂₃	C₂×D₈	1	3	yes	—
110	C₂×C₄×C₂₃	C₂×SD₁₆	1	3	yes	—
111	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₂₃	—	4	4	no	—
113	C₈×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
114	C₈×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
115	C₈×C₂₃	C₂³×C₄	1	1	yes	characteristic
116	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
117	C₂³×D₈	C₂×C₂₃	1	1	yes	characteristic
118	C₂²×M₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
119	C₂²×#108	C₂×C₂₃	1	1	yes	characteristic, complemented
120	C₂×#361	C₂×C₂₃	1	12	yes	direct product factor
121	C₂²×C₂₃	—	2	6	no	—
122	C₂²×C₂₃	—	2	6	no	—
123	C₂²×C₂₃	—	4	8	no	—
124	C₂²×C₂₃	—	8	8	no	—
125	C₂²×C₂₃	—	8	24	no	—
126	C₂²×C₂₃	—	8	24	no	—
127	C₂²×C₂₃	#361	1	4	yes	direct product factor
128	C₂²×C₂₃	C₂×#102	1	3	yes	—
129	C₂²×C₂₃	C₂×#102	1	3	yes	—
130	C₂²×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
131	C₄×C₂₃	C₂×#102	1	3	yes	—
132	C₄×C₂₃	C₂×#102	1	3	yes	—
133	C₄×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
135	C₂³×D₄	—	2	1	no	—
136	C₂²×D₈	—	2	3	no	—
137	C₂²×D₈	—	2	1	no	—
138	C₂²×D₈	—	2	3	no	—
139	C₂²×D₈	C₄×C₂₃	1	6	yes	—
140	C₂²×D₈	C₄×C₂₃	1	2	yes	—
141	C₂²×D₈	C₂²×C₂₃	1	6	yes	—
142	C₂²×M₁₆	—	2	1	no	—
143	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×M₃₂	—	2	3	no	—
145	C₂×M₃₂	C₂²×C₂₃	1	6	yes	complemented
146	C₂²×C₁₆	—	2	1	no	—
147	C₂×#108	—	2	3	no	—
148	C₂×#108	C₂²×C₂₃	1	6	yes	complemented
149	#361	C₂²×C₂₃	1	16	yes	direct product factor
150	C₂×C₂₃	—	2	1	no	—
151	C₂×C₂₃	—	2	3	no	—
152	C₂×C₂₃	—	8	8	no	—
153	C₂×C₂₃	C₂×#361	1	6	yes	direct product factor
154	C₂×C₂₃	C₂²×#102	1	1	yes	characteristic
155	C₂⁵	—	4	1	no	—
156	C₂²×D₄	—	2	6	no	—
157	C₂²×D₄	—	2	2	no	—
158	C₂²×D₄	—	2	6	no	—
159	C₂²×D₄	—	4	1	no	—
160	C₂²×D₄	—	4	3	no	—
161	C₂²×D₄	—	4	3	no	—
162	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
163	C₂×D₈	—	2	6	no	—
164	C₂×D₈	—	2	12	no	—
165	C₂×D₈	—	2	6	no	—
166	C₂×D₈	—	2	6	no	—
167	C₂×D₈	—	2	6	no	—
168	C₂×D₈	—	2	6	no	—
169	C₂×D₈	C₂×C₄×C₂₃	1	12	yes	—
170	C₂×D₈	C₂×C₄×C₂₃	1	12	yes	—
171	C₂×D₈	C₂³×C₂₃	1	4	yes	—
172	C₂×M₁₆	—	2	6	no	—
173	C₂×M₁₆	—	4	3	no	—
174	C₂²×C₈	—	4	1	no	—
175	C₂²×C₈	C₂×C₄×C₂₃	1	3	yes	—
176	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
177	C₂²×C₈	D₄×C₂₃	1	3	yes	—
178	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
179	C₂²×C₈	D₄×C₂₃	1	3	yes	—
180	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
181	M₃₂	—	2	6	no	—
182	M₃₂	C₂³×C₂₃	1	4	yes	complemented
183	C₂×C₁₆	—	2	6	no	—
184	#108	—	2	6	no	—
185	#108	C₂³×C₂₃	1	4	yes	complemented
186	C₂₃	C₂²×#361	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	—	4	6	no	—
188	C₂⁴	—	4	2	no	—
189	C₂⁴	—	4	6	no	—
190	C₂⁴	—	8	8	no	—
191	C₂⁴	SD₁₆×C₂₃	1	1	yes	characteristic
192	C₂×D₄	—	2	12	no	—
193	C₂×D₄	—	2	12	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	4	12	no	—
196	C₂×D₄	—	4	6	no	—
197	C₂×D₄	—	4	6	no	—
198	C₂×D₄	—	4	6	no	—
199	C₂×D₄	—	4	6	no	—
200	C₂×D₄	—	4	6	no	—
201	C₂²×C₄	C₂²⋊C₄×C₂₃	1	3	yes	—
202	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	D₈×C₂₃	1	3	yes	—
205	C₂²×C₄	SD₁₆×C₂₃	1	3	yes	—
206	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
207	D₈	—	2	12	no	—
208	D₈	—	2	12	no	—
209	D₈	—	2	4	no	—
210	D₈	C₂²×C₄×C₂₃	1	8	yes	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	6	no	—
213	C₂×C₈	—	4	6	no	—
214	C₂×C₈	C₂²⋊C₄×C₂₃	1	3	yes	—
215	C₂×C₈	C₂²⋊C₄×C₂₃	1	6	yes	—
216	C₂×C₈	C₂²⋊C₄×C₂₃	1	3	yes	—
217	C₂×C₈	C₂²×C₄×C₂₃	1	3	yes	—
218	C₂×C₈	C₂²×C₄×C₂₃	1	3	yes	—
219	C₂×C₈	C₂×D₄×C₂₃	1	3	yes	—
220	C₂×C₈	C₂×D₄×C₂₃	1	3	yes	—
221	C₂×C₈	C₂×D₄×C₂₃	1	3	yes	—
222	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
223	C₁₆	—	2	4	no	—
224	C₂³	—	2	4	no	—
225	C₂³	—	4	4	no	—
226	C₂³	—	4	12	no	—
227	C₂³	—	4	12	no	—
228	C₂³	—	8	24	no	—
229	C₂³	—	8	24	no	—
230	C₂³	—	8	8	no	—
231	C₂³	C₂₃×#102	1	3	yes	—
232	C₂³	C₂₃×#102	1	1	yes	characteristic
233	C₂³	C₂×SD₁₆×C₂₃	1	3	yes	—
234	D₄	—	2	8	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	12	no	—
237	D₄	—	4	12	no	—
238	C₂×C₄	C₂₃×#102	1	6	yes	—
239	C₂×C₄	C₂₃×#102	1	3	yes	—
240	C₂×C₄	C₂₃×#102	1	3	yes	—
241	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
243	C₂×C₄	C₂×D₈×C₂₃	1	3	yes	—
244	C₂×C₄	C₂×D₈×C₂₃	1	3	yes	—
245	C₂×C₄	C₂×SD₁₆×C₂₃	1	3	yes	—
246	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
247	C₈	—	4	4	no	—
248	C₈	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
249	C₈	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
250	C₈	C₂³×C₄×C₂₃	1	1	yes	characteristic
251	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
252	C₂²	—	2	6	no	—
253	C₂²	—	2	6	no	—
254	C₂²	—	4	8	no	—
255	C₂²	—	8	8	no	—
256	C₂²	—	8	24	no	—
257	C₂²	—	8	24	no	—
258	C₂²	C₂₃×#361	1	4	yes	direct product factor
259	C₂²	C₂×C₂₃×#102	1	3	yes	—
260	C₂²	C₂×C₂₃×#102	1	3	yes	—
261	C₂²	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
262	C₄	C₂×C₂₃×#102	1	3	yes	—
263	C₄	C₂×C₂₃×#102	1	3	yes	—
264	C₄	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
265	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	3	no	—
268	C₂	—	8	8	no	—
269	C₂	C₂×C₂₃×#361	1	6	yes	direct product factor
270	C₂	C₂²×C₂₃×#102	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#102
commutator	C₈	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂³×D₈×C₂₃	C₂
LCS	G ▷ 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×M₃₂	C₂	semidirect
3. C₂²×C₂₃×#108	C₂	semidirect
4. C₂×C₂₃×#361	C₂	direct (N × H)
15. C₂×C₂₃×M₃₂	C₂²	semidirect
18. C₂×C₂₃×#108	C₂²	semidirect
19. C₂₃×#361	C₂²	direct (N × H)
47. C₂₃×M₃₂	C₂³	semidirect
50. C₂₃×#108	C₂³	semidirect
88. C₂²×#361	C₂₃	direct (N × H)
118. C₂²×M₃₂	C₂×C₂₃	semidirect
119. C₂²×#108	C₂×C₂₃	semidirect
120. C₂×#361	C₂×C₂₃	direct (N × H)
127. C₂²×C₂₃	#361	direct (N × H)
145. C₂×M₃₂	C₂²×C₂₃	semidirect
148. C₂×#108	C₂²×C₂₃	semidirect
149. #361	C₂²×C₂₃	direct (N × H)
153. C₂×C₂₃	C₂×#361	direct (N × H)
182. M₃₂	C₂³×C₂₃	semidirect
185. #108	C₂³×C₂₃	semidirect
186. C₂₃	C₂²×#361	direct (N × H)
258. C₂²	C₂₃×#361	direct (N × H)
269. C₂	C₂×C₂₃×#361	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×D₈×C₂₃	2
C₂²×C₂₃×M₃₂	2
C₂²×C₂₃×#108	2
C₂×C₂₃×#361	2
C₂²×#361	23

Subgroups — C22×C23×#361 #19181015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#361 #19181015