Subgroups — C₂₃×#2712 #15670331

Order 2944.

556 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#124	C₂	1	2	yes	maximal, complemented
2	C₂₃×#397	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#394	C₂	1	2	yes	maximal, complemented
4	C₂₃×#393	C₂	1	2	yes	maximal, complemented
5	C₂₃×#392	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#391	C₂	1	2	yes	maximal, complemented
7	C₂₃×#388	C₂	1	2	yes	maximal, complemented
8	C₂₃×#386	C₂	1	2	yes	maximal, complemented
9	C₂₃×#385	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
11	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
12	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#124	—	2	8	no	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#95	C₂²	1	2	yes	—
27	C₂₃×#95	C₂²	1	2	yes	—
28	C₂₃×#95	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	1	yes	characteristic
30	C₂₃×#95	C₂²	1	2	yes	—
31	C₂₃×#95	C₂²	1	1	yes	characteristic
32	C₂⁴×C₂₃	C₂³	1	2	yes	—
33	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
34	C₂×Q₈×C₂₃	—	2	4	no	—
35	C₂×Q₈×C₂₃	—	2	4	no	—
36	C₂×Q₈×C₂₃	—	4	2	no	—
37	C₂×D₄×C₂₃	—	2	4	no	—
38	C₂×D₄×C₂₃	—	2	4	no	—
39	C₂×D₄×C₂₃	—	4	2	no	—
40	C₂×D₄×C₂₃	—	4	1	no	—
41	C₂×D₄×C₂₃	—	4	2	no	—
42	C₂²×C₄×C₂₃	—	4	1	no	—
43	C₂²×C₄×C₂₃	—	4	2	no	—
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
51	C₄⋊C₄×C₂₃	—	4	2	no	—
52	C₄⋊C₄×C₂₃	—	4	1	no	—
53	C₂²⋊C₄×C₂₃	—	2	4	no	—
54	C₂²⋊C₄×C₂₃	—	2	4	no	—
55	C₂²⋊C₄×C₂₃	—	2	4	no	—
56	C₂²⋊C₄×C₂₃	—	2	4	no	—
57	C₂²⋊C₄×C₂₃	—	4	2	no	—
58	C₂²⋊C₄×C₂₃	—	4	1	no	—
59	C₂²⋊C₄×C₂₃	—	4	2	no	—
60	C₂²⋊C₄×C₂₃	D₄	1	4	yes	complemented
61	C₄²×C₂₃	—	2	4	no	—
62	C₂³×C₂₃	—	2	2	no	—
63	C₂³×C₂₃	—	2	1	no	—
64	C₂³×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	—	2	1	no	—
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	2	2	no	—
69	C₂³×C₂₃	—	2	2	no	—
70	C₂³×C₂₃	—	2	2	no	—
71	C₂³×C₂₃	—	2	2	no	—
72	C₂³×C₂₃	—	2	2	no	—
73	C₂³×C₂₃	C₂×D₄	1	2	yes	—
74	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
75	Q₈×C₂₃	—	4	8	no	—
76	D₄×C₂₃	—	4	8	no	—
77	D₄×C₂₃	—	4	4	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	4	4	no	—
96	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
97	C₂×C₄×C₂₃	#41	1	4	yes	—
98	#2712	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₂₃	—	2	2	no	—
100	C₂²×C₂₃	—	2	4	no	—
101	C₂²×C₂₃	—	4	2	no	—
102	C₂²×C₂₃	—	4	1	no	—
103	C₂²×C₂₃	—	4	1	no	—
104	C₂²×C₂₃	—	4	2	no	—
105	C₂²×C₂₃	—	4	2	no	—
106	C₂²×C₂₃	—	4	2	no	—
107	C₂²×C₂₃	—	4	2	no	—
108	C₂²×C₂₃	—	4	2	no	—
109	C₂²×C₂₃	—	4	2	no	—
110	C₂²×C₂₃	—	4	2	no	—
111	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
112	C₂²×C₂₃	C₂×#41	1	2	yes	—
113	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
115	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
116	C₄×C₂₃	—	2	4	no	—
117	C₄×C₂₃	—	2	4	no	—
118	C₄×C₂₃	—	4	2	no	—
119	C₄×C₂₃	—	4	2	no	—
120	C₄×C₂₃	—	4	1	no	—
121	C₄×C₂₃	—	4	2	no	—
122	C₄×C₂₃	—	4	1	no	—
123	C₂×#124	C₂×C₂₃	1	2	yes	complemented
124	#397	C₂×C₂₃	1	1	yes	characteristic, complemented
125	#394	C₂×C₂₃	1	2	yes	complemented
126	#393	C₂×C₂₃	1	2	yes	complemented
127	#392	C₂×C₂₃	1	1	yes	characteristic
128	#391	C₂×C₂₃	1	2	yes	complemented
129	#388	C₂×C₂₃	1	2	yes	complemented
130	#386	C₂×C₂₃	1	2	yes	complemented
131	#385	C₂×C₂₃	1	1	yes	characteristic, complemented
132	C₂×C₂₃	—	4	2	no	—
133	C₂×C₂₃	—	8	2	no	—
134	C₂×C₂₃	#537	1	1	yes	characteristic
135	C₂×C₂₃	#542	1	2	yes	—
136	C₂×C₂₃	#546	1	2	yes	—
137	C₂×C₂₃	#559	1	1	yes	characteristic
138	C₂×C₂₃	H(3,𝔽₄)	1	1	yes	characteristic
139	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
140	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
141	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
142	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
143	#124	—	2	8	no	—
144	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
145	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
148	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
149	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
155	#95	C₂²×C₂₃	1	2	yes	—
156	#95	C₂²×C₂₃	1	2	yes	—
157	#95	C₂²×C₂₃	1	2	yes	—
158	#95	C₂²×C₂₃	1	1	yes	characteristic
159	#95	C₂²×C₂₃	1	2	yes	—
160	#95	C₂²×C₂₃	1	1	yes	characteristic
161	C₂₃	#2712	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴	C₂³×C₂₃	1	2	yes	—
163	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	—	2	4	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	1	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂²×C₄	—	4	1	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
176	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
177	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
179	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
180	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₂²⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	4	1	no	—
189	C₂²⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	D₄×C₂₃	1	4	yes	complemented
191	C₄²	—	2	4	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	2	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	C₂×D₄×C₂₃	1	2	yes	—
204	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
205	Q₈	—	4	8	no	—
206	D₄	—	4	8	no	—
207	D₄	—	4	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
227	C₂×C₄	#41×C₂₃	1	4	yes	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	1	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
241	C₂²	C₂×#41×C₂₃	1	2	yes	—
242	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
244	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	2	no	—
252	C₂	—	4	2	no	—
253	C₂	—	8	2	no	—
254	C₂	C₂₃×#537	1	1	yes	characteristic
255	C₂	C₂₃×#542	1	2	yes	—
256	C₂	C₂₃×#546	1	2	yes	—
257	C₂	C₂₃×#559	1	1	yes	characteristic
258	C₂	C₂₃×H(3,𝔽₄)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂₃×#392	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
1. C₂×C₂₃×#124	C₂	semidirect
2. C₂₃×#397	C₂	semidirect
3. C₂₃×#394	C₂	semidirect
4. C₂₃×#393	C₂	semidirect
6. C₂₃×#391	C₂	semidirect
7. C₂₃×#388	C₂	semidirect
8. C₂₃×#386	C₂	semidirect
9. C₂₃×#385	C₂	semidirect
60. C₂²⋊C₄×C₂₃	D₄	semidirect
98. #2712	C₂₃	direct (N × H)
123. C₂×#124	C₂×C₂₃	semidirect
124. #397	C₂×C₂₃	semidirect
125. #394	C₂×C₂₃	semidirect
126. #393	C₂×C₂₃	semidirect
128. #391	C₂×C₂₃	semidirect
129. #388	C₂×C₂₃	semidirect
130. #386	C₂×C₂₃	semidirect
131. #385	C₂×C₂₃	semidirect
161. C₂₃	#2712	direct (N × H)
190. C₂²⋊C₄	D₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#124	2
C₂₃×#397	2
C₂₃×#394	2
C₂₃×#393	2
C₂₃×#392	2
C₂₃×#391	2
C₂₃×#388	2
C₂₃×#386	2
C₂₃×#385	2
#2712	23

Subgroups — C23×#2712 #15670331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2712 #15670331