Subgroups — C₂₃×#2639 #15670258

Order 2944.

576 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#121	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#400	C₂	1	2	yes	maximal, complemented
4	C₂₃×#397	C₂	1	2	yes	maximal, complemented
5	C₂₃×#394	C₂	1	2	yes	maximal, complemented
6	C₂₃×#393	C₂	1	2	yes	maximal, complemented
7	C₂₃×#386	C₂	1	4	yes	maximal, complemented
8	C₂×C₂₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
11	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₂²⋊Q₈	—	2	4	no	—
15	C₂₃×#121	—	2	4	no	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
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	4	yes	—
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
23	C₂₃×#95	C₂²	1	2	yes	—
24	C₂₃×#95	C₂²	1	2	yes	—
25	C₂₃×#95	C₂²	1	4	yes	—
26	C₂₃×#95	C₂²	1	4	yes	—
27	C₂⁴×C₂₃	C₂³	1	2	yes	—
28	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₂₃	—	2	2	no	—
30	C₂×Q₈×C₂₃	—	4	2	no	—
31	C₂×D₄×C₂₃	—	2	4	no	—
32	C₂×D₄×C₂₃	—	2	8	no	—
33	C₂×D₄×C₂₃	—	2	2	no	—
34	C₂×D₄×C₂₃	—	4	2	no	—
35	C₂²×C₄×C₂₃	—	2	4	no	—
36	C₂²×C₄×C₂₃	—	4	2	no	—
37	C₂²×C₄×C₂₃	—	4	2	no	—
38	C₂²×C₄×C₂₃	—	4	2	no	—
39	C₂²×C₄×C₂₃	D₄	1	4	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₄⋊C₄×C₂₃	—	2	2	no	—
48	C₄⋊C₄×C₂₃	—	2	4	no	—
49	C₂²⋊C₄×C₂₃	—	2	4	no	—
50	C₂²⋊C₄×C₂₃	—	2	4	no	—
51	C₂²⋊C₄×C₂₃	—	4	4	no	—
52	C₂²⋊C₄×C₂₃	—	4	4	no	—
53	C₂³×C₂₃	—	2	2	no	—
54	C₂³×C₂₃	—	2	1	no	—
55	C₂³×C₂₃	—	2	1	no	—
56	C₂³×C₂₃	—	2	2	no	—
57	C₂³×C₂₃	—	2	4	no	—
58	C₂³×C₂₃	—	2	2	no	—
59	C₂³×C₂₃	—	2	4	no	—
60	C₂³×C₂₃	—	2	2	no	—
61	C₂³×C₂₃	C₂×D₄	1	2	yes	—
62	C₂³×C₂₃	#41	1	4	yes	—
63	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
64	Q₈×C₂₃	—	4	4	no	—
65	D₄×C₂₃	—	4	4	no	—
66	D₄×C₂₃	—	4	8	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	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	4	no	—
73	C₂×C₄×C₂₃	—	2	4	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	4	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	4	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	4	4	no	—
80	C₂×C₄×C₂₃	—	4	8	no	—
81	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
83	#2639	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₂₃	—	2	2	no	—
85	C₂²×C₂₃	—	2	2	no	—
86	C₂²×C₂₃	—	2	4	no	—
87	C₂²×C₂₃	—	2	4	no	—
88	C₂²×C₂₃	—	4	2	no	—
89	C₂²×C₂₃	—	4	2	no	—
90	C₂²×C₂₃	—	4	2	no	—
91	C₂²×C₂₃	—	4	4	no	—
92	C₂²×C₂₃	—	4	4	no	—
93	C₂²×C₂₃	#124	1	4	yes	—
94	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
95	C₂²×C₂₃	C₂×#41	1	2	yes	—
96	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
97	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
98	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
99	C₄×C₂₃	—	2	2	no	—
100	C₄×C₂₃	—	2	2	no	—
101	C₄×C₂₃	—	4	2	no	—
102	C₄×C₂₃	—	4	2	no	—
103	C₄×C₂₃	—	4	2	no	—
104	C₄×C₂₃	—	4	4	no	—
105	C₂×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
106	C₂×#121	C₂×C₂₃	1	1	yes	characteristic
107	#400	C₂×C₂₃	1	2	yes	complemented
108	#397	C₂×C₂₃	1	2	yes	complemented
109	#394	C₂×C₂₃	1	2	yes	complemented
110	#393	C₂×C₂₃	1	2	yes	complemented
111	#386	C₂×C₂₃	1	4	yes	complemented
112	C₂×#95	C₂×C₂₃	1	1	yes	characteristic, complemented
113	C₂×C₂₃	—	2	4	no	—
114	C₂×C₂₃	—	8	2	no	—
115	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
116	C₂×C₂₃	#534	1	1	yes	characteristic
117	C₂×C₂₃	#537	1	1	yes	characteristic
118	C₂×C₂₃	#538	1	2	yes	—
119	C₂×C₂₃	#539	1	2	yes	—
120	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
121	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
122	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
123	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
124	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
125	C₂²⋊Q₈	—	2	4	no	—
126	#121	—	2	4	no	—
127	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
129	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
130	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
134	#95	C₂²×C₂₃	1	2	yes	—
135	#95	C₂²×C₂₃	1	2	yes	—
136	#95	C₂²×C₂₃	1	4	yes	—
137	#95	C₂²×C₂₃	1	4	yes	—
138	C₂₃	#2639	1	1	yes	characteristic, direct product factor, Sylow
139	C₂⁴	C₂³×C₂₃	1	2	yes	—
140	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×D₄	—	2	4	no	—
144	C₂×D₄	—	2	8	no	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	4	2	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	D₄×C₂₃	1	4	yes	—
152	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
153	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
154	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
155	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
156	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₂²⋊C₄	—	2	4	no	—
162	C₂²⋊C₄	—	2	4	no	—
163	C₂²⋊C₄	—	4	4	no	—
164	C₂²⋊C₄	—	4	4	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	1	no	—
167	C₂³	—	2	1	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	4	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	4	no	—
172	C₂³	—	2	2	no	—
173	C₂³	C₂×D₄×C₂₃	1	2	yes	—
174	C₂³	#41×C₂₃	1	4	yes	—
175	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
176	Q₈	—	4	4	no	—
177	D₄	—	4	4	no	—
178	D₄	—	4	8	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	4	8	no	—
193	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
194	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	4	no	—
203	C₂²	—	4	4	no	—
204	C₂²	C₂₃×#124	1	4	yes	—
205	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
206	C₂²	C₂×#41×C₂₃	1	2	yes	—
207	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
208	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
209	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	4	2	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	4	no	—
216	C₂	—	2	4	no	—
217	C₂	—	8	2	no	—
218	C₂	C₂×C₂₃×#124	1	1	yes	characteristic
219	C₂	C₂₃×#534	1	1	yes	characteristic
220	C₂	C₂₃×#537	1	1	yes	characteristic
221	C₂	C₂₃×#538	1	2	yes	—
222	C₂	C₂₃×#539	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#121	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×C₂²⋊Q₈	C₂	semidirect
3. C₂₃×#400	C₂	semidirect
4. C₂₃×#397	C₂	semidirect
5. C₂₃×#394	C₂	semidirect
6. C₂₃×#393	C₂	semidirect
7. C₂₃×#386	C₂	semidirect
8. C₂×C₂₃×#95	C₂	semidirect
83. #2639	C₂₃	direct (N × H)
105. C₂×C₂²⋊Q₈	C₂×C₂₃	semidirect
107. #400	C₂×C₂₃	semidirect
108. #397	C₂×C₂₃	semidirect
109. #394	C₂×C₂₃	semidirect
110. #393	C₂×C₂₃	semidirect
111. #386	C₂×C₂₃	semidirect
112. C₂×#95	C₂×C₂₃	semidirect
138. C₂₃	#2639	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×C₂²⋊Q₈	2
C₂×C₂₃×#121	2
C₂₃×#400	2
C₂₃×#397	2
C₂₃×#394	2
C₂₃×#393	2
C₂₃×#386	2
C₂×C₂₃×#95	2
#2639	23

Subgroups — C23×#2639 #15670258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2639 #15670258