Subgroups — C₂₃×#3213 #15670832

Order 2944.

444 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#496	C₂	1	2	yes	maximal, complemented
4	C₂₃×#489	C₂	1	2	yes	maximal, complemented
5	C₂₃×#463	C₂	1	4	yes	maximal, complemented
6	C₂₃×#448	C₂	1	2	yes	maximal, complemented
7	C₂₃×#447	C₂	1	2	yes	maximal, complemented
8	C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₃	—	2	1	no	—
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×SD₁₆×C₂₃	C₂²	1	4	yes	complemented
13	C₂×D₈×C₂₃	C₂²	1	4	yes	—
14	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₂₃×#127	C₂²	1	1	yes	characteristic, complemented
16	C₂₃×#124	—	2	1	no	—
17	C₂₃×#124	—	2	1	no	—
18	C₂₃×#124	—	2	1	no	—
19	C₂₃×#124	C₂²	1	2	yes	—
20	C₂₃×#124	C₂²	1	2	yes	—
21	C₂₃×#121	—	2	1	no	—
22	C₂₃×#121	C₂²	1	2	yes	—
23	C₄×D₄×C₂₃	—	2	1	no	—
24	C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
25	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
26	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
27	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
29	C₂₃×#103	C₂²	1	4	yes	complemented
30	C₂₃×#102	C₂²	1	4	yes	—
31	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₂₃×#97	C₂²	1	2	yes	complemented
33	C₂⁴×C₂₃	—	2	1	no	—
34	#41×C₂₃	—	4	4	no	—
35	C₂×Q₈×C₂₃	—	2	1	no	—
36	C₂×Q₈×C₂₃	—	2	1	no	—
37	C₂×Q₈×C₂₃	—	4	2	no	—
38	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
39	C₂×D₄×C₂₃	—	2	2	no	—
40	C₂×D₄×C₂₃	—	2	1	no	—
41	C₂×D₄×C₂₃	—	2	1	no	—
42	C₂×D₄×C₂₃	—	2	1	no	—
43	C₂×D₄×C₂₃	—	4	2	no	—
44	C₂×D₄×C₂₃	C₂³	1	2	yes	—
45	C₂×D₄×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	—	2	1	no	—
47	C₂²×C₄×C₂₃	—	2	2	no	—
48	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	SD₁₆×C₂₃	—	4	4	no	—
52	D₈×C₂₃	—	4	4	no	—
53	C₂×C₈×C₂₃	C₂³	1	4	yes	—
54	C₄⋊C₄×C₂₃	—	2	1	no	—
55	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₂₃	—	2	2	no	—
57	C₂²⋊C₄×C₂₃	—	2	2	no	—
58	C₂²⋊C₄×C₂₃	—	2	2	no	—
59	C₂²⋊C₄×C₂₃	—	2	2	no	—
60	C₂²⋊C₄×C₂₃	—	2	2	no	—
61	C₂²⋊C₄×C₂₃	—	2	2	no	—
62	C₄²×C₂₃	D₄	1	1	yes	characteristic
63	C₄²×C₂₃	D₄	1	1	yes	characteristic
64	C₄²×C₂₃	C₂³	1	1	yes	characteristic
65	C₄²×C₂₃	C₂³	1	1	yes	characteristic
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	4	2	no	—
69	C₂³×C₂₃	—	4	2	no	—
70	C₂³×C₂₃	—	4	2	no	—
71	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
72	Q₈×C₂₃	—	2	4	no	—
73	Q₈×C₂₃	—	4	2	no	—
74	Q₈×C₂₃	—	4	2	no	—
75	D₄×C₂₃	—	2	4	no	—
76	D₄×C₂₃	—	2	4	no	—
77	D₄×C₂₃	—	4	2	no	—
78	D₄×C₂₃	—	4	2	no	—
79	D₄×C₂₃	—	4	2	no	—
80	D₄×C₂₃	—	4	4	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	—	4	1	no	—
86	C₂×C₄×C₂₃	—	4	4	no	—
87	C₂×C₄×C₂₃	—	4	2	no	—
88	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₃	—	2	4	no	—
95	#3213	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂²×C₂₃	—	2	1	no	—
97	C₂²×C₂₃	—	2	1	no	—
98	C₂²×C₂₃	—	4	2	no	—
99	C₂²×C₂₃	—	4	2	no	—
100	C₂²×C₂₃	—	4	2	no	—
101	C₂²×C₂₃	—	4	2	no	—
102	C₂²×C₂₃	—	4	2	no	—
103	C₂²×C₂₃	—	4	2	no	—
104	C₂²×C₂₃	—	8	4	no	—
105	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
106	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	2	1	no	—
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	2	1	no	—
111	C₄×C₂₃	—	2	1	no	—
112	C₄×C₂₃	—	4	2	no	—
113	C₄×C₂₃	—	4	2	no	—
114	C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
115	#532	C₂×C₂₃	1	1	yes	characteristic, complemented
116	C₂×#124	C₂×C₂₃	1	1	yes	characteristic, complemented
117	#496	C₂×C₂₃	1	2	yes	complemented
118	#489	C₂×C₂₃	1	2	yes	complemented
119	#463	C₂×C₂₃	1	4	yes	complemented
120	#448	C₂×C₂₃	1	2	yes	complemented
121	#447	C₂×C₂₃	1	2	yes	complemented
122	#432	C₂×C₂₃	1	1	yes	characteristic, complemented
123	C₂×C₂₃	—	2	2	no	—
124	C₂×C₂₃	—	8	2	no	—
125	C₂×C₂₃	—	8	2	no	—
126	C₂×C₂₃	#535	1	1	yes	characteristic
127	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₂₃	#575	1	1	yes	characteristic
129	C₂×#41	—	2	1	no	—
130	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
131	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
132	C₂×SD₁₆	C₂²×C₂₃	1	4	yes	complemented
133	C₂×D₈	C₂²×C₂₃	1	4	yes	—
134	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
135	#127	C₂²×C₂₃	1	1	yes	characteristic, complemented
136	#124	—	2	1	no	—
137	#124	—	2	1	no	—
138	#124	—	2	1	no	—
139	#124	C₂²×C₂₃	1	2	yes	—
140	#124	C₂²×C₂₃	1	2	yes	—
141	#121	—	2	1	no	—
142	#121	C₂²×C₂₃	1	2	yes	—
143	C₄×D₄	—	2	1	no	—
144	C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
145	C₂×C₂²⋊C₄	—	2	1	no	—
146	C₂×C₂²⋊C₄	—	2	1	no	—
147	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
148	C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
149	#103	C₂²×C₂₃	1	4	yes	complemented
150	#102	C₂²×C₂₃	1	4	yes	—
151	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
152	#97	C₂²×C₂₃	1	2	yes	complemented
153	C₂₃	#3213	1	1	yes	characteristic, direct product factor, Sylow
154	C₂⁴	—	2	1	no	—
155	#41	—	4	4	no	—
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	—	2	1	no	—
162	C₂×D₄	—	2	1	no	—
163	C₂×D₄	—	2	1	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	C₂³×C₂₃	1	2	yes	—
166	C₂×D₄	C₂³×C₂₃	1	2	yes	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
170	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
172	SD₁₆	—	4	4	no	—
173	D₈	—	4	4	no	—
174	C₂×C₈	C₂³×C₂₃	1	4	yes	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₄²	D₄×C₂₃	1	1	yes	characteristic
184	C₄²	D₄×C₂₃	1	1	yes	characteristic
185	C₄²	C₂³×C₂₃	1	1	yes	characteristic
186	C₄²	C₂³×C₂₃	1	1	yes	characteristic
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	4	2	no	—
190	C₂³	—	4	2	no	—
191	C₂³	—	4	2	no	—
192	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	Q₈	—	4	2	no	—
195	Q₈	—	4	2	no	—
196	D₄	—	2	4	no	—
197	D₄	—	2	4	no	—
198	D₄	—	4	2	no	—
199	D₄	—	4	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	4	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
212	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	—	8	4	no	—
225	C₂²	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
226	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
235	C₂	—	2	2	no	—
236	C₂	—	8	2	no	—
237	C₂	—	8	2	no	—
238	C₂	C₂₃×#535	1	1	yes	characteristic
239	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
240	C₂	C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#532	C₂	semidirect
2. C₂×C₂₃×#124	C₂	semidirect
3. C₂₃×#496	C₂	semidirect
4. C₂₃×#489	C₂	semidirect
5. C₂₃×#463	C₂	semidirect
6. C₂₃×#448	C₂	semidirect
7. C₂₃×#447	C₂	semidirect
8. C₂₃×#432	C₂	semidirect
12. C₂×SD₁₆×C₂₃	C₂²	semidirect
14. C₂₃×C₄⋊Q₈	C₂²	semidirect
15. C₂₃×#127	C₂²	semidirect
24. C₄×D₄×C₂₃	C₂²	semidirect
28. C₂₃×C₄⋊C₈	C₂²	semidirect
29. C₂₃×#103	C₂²	semidirect
32. C₂₃×#97	C₂²	semidirect
95. #3213	C₂₃	direct (N × H)
115. #532	C₂×C₂₃	semidirect
116. C₂×#124	C₂×C₂₃	semidirect
117. #496	C₂×C₂₃	semidirect
118. #489	C₂×C₂₃	semidirect
119. #463	C₂×C₂₃	semidirect
120. #448	C₂×C₂₃	semidirect
121. #447	C₂×C₂₃	semidirect
122. #432	C₂×C₂₃	semidirect
132. C₂×SD₁₆	C₂²×C₂₃	semidirect
134. C₄⋊Q₈	C₂²×C₂₃	semidirect
135. #127	C₂²×C₂₃	semidirect
144. C₄×D₄	C₂²×C₂₃	semidirect
148. C₄⋊C₈	C₂²×C₂₃	semidirect
149. #103	C₂²×C₂₃	semidirect
152. #97	C₂²×C₂₃	semidirect
153. C₂₃	#3213	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×#532	2
C₂×C₂₃×#124	2
C₂₃×#496	2
C₂₃×#489	2
C₂₃×#463	2
C₂₃×#448	2
C₂₃×#447	2
C₂₃×#432	2
#3213	23

Subgroups — C23×#3213 #15670832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3213 #15670832