Subgroups — C₂₃×#2677 #15670296

Order 2944.

544 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
2	C₂×C₂₃×C₂²⋊Q₈	C₂	1	2	yes	maximal
3	C₂₃×#399	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#398	C₂	1	2	yes	maximal, complemented
5	C₂₃×#395	C₂	1	2	yes	maximal, complemented
6	C₂₃×#393	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#388	C₂	1	2	yes	maximal
8	C₂₃×#387	C₂	1	2	yes	maximal, complemented
9	C₂₃×#385	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
12	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×C₄⋊Q₈	—	2	8	no	—
14	C₂₃×C₂²⋊Q₈	—	2	8	no	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
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×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
27	C₂₃×#95	C₂²	1	2	yes	—
28	C₂₃×#95	C₂²	1	1	yes	characteristic
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃×#95	C₂²	1	2	yes	—
31	C₂₃×#95	C₂²	1	2	yes	—
32	C₂₃×#95	C₂²	1	1	yes	characteristic
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	—	2	4	no	—
37	C₂×Q₈×C₂₃	—	4	2	no	—
38	C₂×Q₈×C₂₃	—	4	1	no	—
39	C₂²×C₄×C₂₃	—	2	4	no	—
40	C₂²×C₄×C₂₃	—	4	1	no	—
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
49	C₄⋊C₄×C₂₃	—	2	4	no	—
50	C₄⋊C₄×C₂₃	—	2	4	no	—
51	C₄⋊C₄×C₂₃	—	2	4	no	—
52	C₄⋊C₄×C₂₃	—	2	4	no	—
53	C₄⋊C₄×C₂₃	—	2	4	no	—
54	C₄⋊C₄×C₂₃	—	4	1	no	—
55	C₄⋊C₄×C₂₃	D₄	1	8	yes	—
56	C₂²⋊C₄×C₂₃	—	2	4	no	—
57	C₂²⋊C₄×C₂₃	—	2	4	no	—
58	C₂²⋊C₄×C₂₃	—	4	1	no	—
59	C₂²⋊C₄×C₂₃	Q₈	1	4	yes	complemented
60	C₄²×C₂₃	—	2	4	no	—
61	C₂³×C₂₃	—	2	2	no	—
62	C₂³×C₂₃	—	2	1	no	—
63	C₂³×C₂₃	—	2	1	no	—
64	C₂³×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
66	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
67	Q₈×C₂₃	—	4	8	no	—
68	Q₈×C₂₃	—	4	4	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	2	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	1	no	—
82	C₂×C₄×C₂₃	—	2	1	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	—	4	4	no	—
88	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
89	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
90	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
91	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
92	#2677	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₂₃	—	2	2	no	—
94	C₂²×C₂₃	—	2	4	no	—
95	C₂²×C₂₃	—	4	2	no	—
96	C₂²×C₂₃	—	4	1	no	—
97	C₂²×C₂₃	—	4	1	no	—
98	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
99	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
100	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
103	C₄×C₂₃	—	2	4	no	—
104	C₄×C₂₃	—	2	4	no	—
105	C₄×C₂₃	—	2	4	no	—
106	C₄×C₂₃	—	2	4	no	—
107	C₄×C₂₃	—	4	2	no	—
108	C₄×C₂₃	—	4	2	no	—
109	C₄×C₂₃	—	4	1	no	—
110	C₄×C₂₃	—	4	1	no	—
111	C₂×C₄⋊Q₈	C₂×C₂₃	1	2	yes	complemented
112	C₂×C₂²⋊Q₈	C₂×C₂₃	1	2	yes	—
113	#399	C₂×C₂₃	1	1	yes	characteristic
114	#398	C₂×C₂₃	1	2	yes	complemented
115	#395	C₂×C₂₃	1	2	yes	complemented
116	#393	C₂×C₂₃	1	1	yes	characteristic
117	#388	C₂×C₂₃	1	2	yes	—
118	#387	C₂×C₂₃	1	2	yes	complemented
119	#385	C₂×C₂₃	1	1	yes	characteristic
120	C₂×C₂₃	—	4	2	no	—
121	C₂×C₂₃	#536	1	2	yes	—
122	C₂×C₂₃	#544	1	1	yes	characteristic
123	C₂×C₂₃	D₄²	1	1	yes	characteristic
124	C₂×C₂₃	D₄×Q₈	1	2	yes	—
125	C₂×C₂₃	#563	1	1	yes	characteristic
126	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
127	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
128	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
129	C₄⋊Q₈	—	2	8	no	—
130	C₂²⋊Q₈	—	2	8	no	—
131	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
132	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
133	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
134	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
135	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
136	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
138	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
139	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
140	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
143	#95	C₂²×C₂₃	1	2	yes	—
144	#95	C₂²×C₂₃	1	1	yes	characteristic
145	#95	C₂²×C₂₃	1	2	yes	—
146	#95	C₂²×C₂₃	1	2	yes	—
147	#95	C₂²×C₂₃	1	2	yes	—
148	#95	C₂²×C₂₃	1	1	yes	characteristic
149	C₂₃	#2677	1	1	yes	characteristic, direct product factor, Sylow
150	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
151	C₂×Q₈	—	2	4	no	—
152	C₂×Q₈	—	2	4	no	—
153	C₂×Q₈	—	2	4	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	—	4	1	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	4	1	no	—
158	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
159	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
160	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
161	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
163	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
164	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
165	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	4	1	no	—
172	C₄⋊C₄	D₄×C₂₃	1	8	yes	—
173	C₂²⋊C₄	—	2	4	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	—	4	1	no	—
176	C₂²⋊C₄	Q₈×C₂₃	1	4	yes	complemented
177	C₄²	—	2	4	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	1	no	—
182	C₂³	C₂×Q₈×C₂₃	1	2	yes	—
183	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
184	Q₈	—	4	8	no	—
185	Q₈	—	4	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
206	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
207	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
208	C₂×C₄	C₂×Q₈×C₂₃	1	4	yes	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	4	1	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	1	no	—
214	C₂²	C₂²×D₄×C₂₃	1	2	yes	—
215	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
216	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
217	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
218	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
219	C₄	—	2	4	no	—
220	C₄	—	2	4	no	—
221	C₄	—	2	4	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	1	no	—
227	C₂	—	4	2	no	—
228	C₂	C₂₃×#536	1	2	yes	—
229	C₂	C₂₃×#544	1	1	yes	characteristic
230	C₂	D₄²×C₂₃	1	1	yes	characteristic
231	C₂	D₄×Q₈×C₂₃	1	2	yes	—
232	C₂	C₂₃×#563	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂³
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×C₄⋊Q₈	C₂	semidirect
4. C₂₃×#398	C₂	semidirect
5. C₂₃×#395	C₂	semidirect
8. C₂₃×#387	C₂	semidirect
59. C₂²⋊C₄×C₂₃	Q₈	semidirect
92. #2677	C₂₃	direct (N × H)
111. C₂×C₄⋊Q₈	C₂×C₂₃	semidirect
114. #398	C₂×C₂₃	semidirect
115. #395	C₂×C₂₃	semidirect
118. #387	C₂×C₂₃	semidirect
149. C₂₃	#2677	direct (N × H)
176. C₂²⋊C₄	Q₈×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×C₄⋊Q₈	2
C₂×C₂₃×C₂²⋊Q₈	2
C₂₃×#399	2
C₂₃×#398	2
C₂₃×#395	2
C₂₃×#393	2
C₂₃×#388	2
C₂₃×#387	2
C₂₃×#385	2
#2677	23

Subgroups — C23×#2677 #15670296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2677 #15670296