Subgroups — C₂₃×#2432 #15670051

Order 2944.

836 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#394	C₂	1	4	yes	maximal, complemented
6	C₂₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#385	C₂	1	4	yes	maximal, complemented
9	C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₃	—	2	4	no	—
11	C₂×#41×C₂₃	—	2	4	no	—
12	C₂×#41×C₂₃	—	2	4	no	—
13	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
16	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
18	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×C₄⋊Q₈	—	2	4	no	—
20	C₂₃×#124	—	2	4	no	—
21	C₂₃×C₂²⋊Q₈	—	2	4	no	—
22	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
24	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
27	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
28	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
29	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#95	C₂²	1	2	yes	—
32	C₂₃×#95	C₂²	1	4	yes	—
33	C₂⁴×C₂₃	C₂³	1	2	yes	—
34	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
35	#41×C₂₃	—	2	32	no	—
36	C₂×Q₈×C₂₃	—	2	2	no	—
37	C₂×Q₈×C₂₃	—	2	2	no	—
38	C₂×Q₈×C₂₃	—	2	2	no	—
39	C₂×Q₈×C₂₃	—	2	2	no	—
40	C₂×Q₈×C₂₃	—	2	2	no	—
41	C₂×Q₈×C₂₃	D₄	1	4	yes	—
42	C₂×D₄×C₂₃	—	2	4	no	—
43	C₂×D₄×C₂₃	—	2	4	no	—
44	C₂×D₄×C₂₃	—	2	4	no	—
45	C₂×D₄×C₂₃	—	2	2	no	—
46	C₂×D₄×C₂₃	—	2	2	no	—
47	C₂×D₄×C₂₃	D₄	1	4	yes	—
48	C₂²×C₄×C₂₃	—	2	4	no	—
49	C₂²×C₄×C₂₃	—	2	4	no	—
50	C₂²×C₄×C₂₃	—	2	4	no	—
51	C₂²×C₄×C₂₃	—	2	2	no	—
52	C₂²×C₄×C₂₃	—	2	2	no	—
53	C₂²×C₄×C₂₃	D₄	1	4	yes	—
54	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	—	2	4	no	—
63	C₄⋊C₄×C₂₃	—	2	8	no	—
64	C₂²⋊C₄×C₂₃	—	2	4	no	—
65	C₂²⋊C₄×C₂₃	—	2	8	no	—
66	C₂²⋊C₄×C₂₃	—	2	4	no	—
67	C₂²⋊C₄×C₂₃	—	2	8	no	—
68	C₄²×C₂₃	—	2	2	no	—
69	C₄²×C₂₃	—	2	2	no	—
70	C₂³×C₂₃	—	2	2	no	—
71	C₂³×C₂₃	—	2	1	no	—
72	C₂³×C₂₃	—	2	2	no	—
73	C₂³×C₂₃	—	2	1	no	—
74	C₂³×C₂₃	—	2	2	no	—
75	C₂³×C₂₃	—	2	4	no	—
76	C₂³×C₂₃	—	2	2	no	—
77	C₂³×C₂₃	—	2	4	no	—
78	C₂³×C₂₃	C₂×D₄	1	4	yes	—
79	C₂³×C₂₃	C₂×D₄	1	2	yes	—
80	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
81	Q₈×C₂₃	—	2	8	no	—
82	Q₈×C₂₃	—	4	4	no	—
83	D₄×C₂₃	—	2	16	no	—
84	D₄×C₂₃	—	2	8	no	—
85	C₂×C₄×C₂₃	—	2	16	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	8	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	4	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
103	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₂₃	#41	1	4	yes	complemented
107	#2432	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₂₃	—	2	4	no	—
109	C₂²×C₂₃	—	2	2	no	—
110	C₂²×C₂₃	—	2	2	no	—
111	C₂²×C₂₃	—	2	2	no	—
112	C₂²×C₂₃	—	2	8	no	—
113	C₂²×C₂₃	—	4	2	no	—
114	C₂²×C₂₃	—	4	2	no	—
115	C₂²×C₂₃	—	4	2	no	—
116	C₂²×C₂₃	—	4	2	no	—
117	C₂²×C₂₃	—	4	4	no	—
118	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
119	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
120	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
121	C₂²×C₂₃	C₂×#41	1	2	yes	—
122	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
123	C₄×C₂₃	—	2	2	no	—
124	C₄×C₂₃	—	2	4	no	—
125	C₄×C₂₃	—	2	4	no	—
126	C₄×C₂₃	—	4	2	no	—
127	C₄×C₂₃	—	4	2	no	—
128	C₄×C₂₃	—	4	2	no	—
129	C₄×C₂₃	#120	1	4	yes	—
130	C₂²×#41	C₂×C₂₃	1	1	yes	characteristic
131	C₂×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
132	C₂×#124	C₂×C₂₃	1	1	yes	characteristic, complemented
133	C₂×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
134	#394	C₂×C₂₃	1	4	yes	complemented
135	#391	C₂×C₂₃	1	1	yes	characteristic, complemented
136	#390	C₂×C₂₃	1	1	yes	characteristic, complemented
137	#385	C₂×C₂₃	1	4	yes	complemented
138	C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
139	C₂×C₂₃	—	4	2	no	—
140	C₂×C₂₃	—	4	4	no	—
141	C₂×C₂₃	C₂×#120	1	1	yes	characteristic
142	C₂×C₂₃	#532	1	1	yes	characteristic
143	C₂×C₂₃	#536	1	1	yes	characteristic
144	C₂×C₂₃	#547	1	2	yes	—
145	C₂×C₂₃	#548	1	2	yes	—
146	C₂×#41	—	2	4	no	—
147	C₂×#41	—	2	4	no	—
148	C₂×#41	—	2	4	no	—
149	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
151	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
152	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
153	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
154	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
155	C₄⋊Q₈	—	2	4	no	—
156	#124	—	2	4	no	—
157	C₂²⋊Q₈	—	2	4	no	—
158	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
159	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
160	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
161	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
162	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
163	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
164	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
165	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
167	#95	C₂²×C₂₃	1	2	yes	—
168	#95	C₂²×C₂₃	1	4	yes	—
169	C₂₃	#2432	1	1	yes	characteristic, direct product factor, Sylow
170	C₂⁴	C₂³×C₂₃	1	2	yes	—
171	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
172	#41	—	2	32	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	D₄×C₂₃	1	4	yes	—
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	2	4	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	D₄×C₂₃	1	4	yes	—
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	D₄×C₂₃	1	4	yes	—
191	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
192	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
193	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
194	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
195	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
196	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
197	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	8	no	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₂²⋊C₄	—	2	8	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	8	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	4	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	4	no	—
214	C₂³	—	2	2	no	—
215	C₂³	C₂×D₄×C₂₃	1	4	yes	—
216	C₂³	C₂×D₄×C₂₃	1	2	yes	—
217	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
218	Q₈	—	2	8	no	—
219	Q₈	—	4	4	no	—
220	D₄	—	2	16	no	—
221	D₄	—	2	8	no	—
222	C₂×C₄	—	2	16	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
239	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
240	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
241	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
242	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
243	C₂×C₄	#41×C₂₃	1	4	yes	complemented
244	C₂²	—	2	8	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
255	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
256	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
257	C₂²	C₂×#41×C₂₃	1	2	yes	—
258	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	C₂₃×#120	1	4	yes	—
266	C₂	—	4	4	no	—
267	C₂	—	4	2	no	—
268	C₂	C₂×C₂₃×#120	1	1	yes	characteristic
269	C₂	C₂₃×#532	1	1	yes	characteristic
270	C₂	C₂₃×#536	1	1	yes	characteristic
271	C₂	C₂₃×#547	1	2	yes	—
272	C₂	C₂₃×#548	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂²×#41×C₂₃	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂×C₂₃×#124	C₂	semidirect
4. C₂×C₂₃×C₂²⋊Q₈	C₂	semidirect
5. C₂₃×#394	C₂	semidirect
6. C₂₃×#391	C₂	semidirect
7. C₂₃×#390	C₂	semidirect
8. C₂₃×#385	C₂	semidirect
9. C₄×C₂²⋊C₄×C₂₃	C₂	semidirect
106. C₂×C₄×C₂₃	#41	semidirect
107. #2432	C₂₃	direct (N × H)
131. C₂×C₄⋊Q₈	C₂×C₂₃	semidirect
132. C₂×#124	C₂×C₂₃	semidirect
133. C₂×C₂²⋊Q₈	C₂×C₂₃	semidirect
134. #394	C₂×C₂₃	semidirect
135. #391	C₂×C₂₃	semidirect
136. #390	C₂×C₂₃	semidirect
137. #385	C₂×C₂₃	semidirect
138. C₄×C₂²⋊C₄	C₂×C₂₃	semidirect
169. C₂₃	#2432	direct (N × H)
243. C₂×C₄	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₂₃	2
C₂×C₂₃×C₄⋊Q₈	2
C₂×C₂₃×#124	2
C₂×C₂₃×C₂²⋊Q₈	2
C₂₃×#394	2
C₂₃×#391	2
C₂₃×#390	2
C₂₃×#385	2
C₄×C₂²⋊C₄×C₂₃	2
#2432	23

Subgroups — C23×#2432 #15670051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2432 #15670051