Subgroups — C₂₃×#36644 #19184270

Order 5888.

4086 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#532	C₂	1	6	yes	maximal, complemented
2	C₂₃×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2867	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2858	C₂	1	6	yes	maximal, complemented
5	C₂₃×#2856	C₂	1	8	yes	maximal, complemented
6	C₄×C₂₃×#127	C₂	1	8	yes	maximal, complemented
7	C₂×C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₂₃	C₂²	1	4	yes	—
9	C₂₃×#532	—	2	48	no	—
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	6	yes	—
11	C₂×C₂₃×#127	C₂²	1	6	yes	—
12	C₂×C₂₃×#127	C₂²	1	8	yes	—
13	C₂×C₂₃×#124	C₂²	1	12	yes	—
14	C₂×C₂₃×#121	C₂²	1	24	yes	—
15	C₂×C₄×D₄×C₂₃	C₂²	1	24	yes	—
16	C₂²×C₄²×C₂₃	C₂²	1	4	yes	—
17	C₂²×C₄²×C₂₃	C₂²	1	3	yes	—
18	C₂₃×#390	C₂²	1	48	yes	—
19	C₂₃×#384	C₂²	1	8	yes	—
20	C₄³×C₂₃	C₂²	1	4	yes	—
21	C₄³×C₂₃	C₂²	1	3	yes	—
22	C₄³×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×#41×C₂₃	—	2	48	no	—
24	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
25	C₂²×D₄×C₂₃	C₂³	1	8	yes	—
26	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
27	C₂²×D₄×C₂₃	C₂³	1	24	yes	—
28	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
29	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
30	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂₃×C₄⋊Q₈	—	2	24	no	—
32	C₂₃×#127	—	2	24	no	—
33	C₂₃×#127	—	2	32	no	—
34	C₂₃×#124	—	2	48	no	—
35	C₂₃×#121	—	2	96	no	—
36	C₄×D₄×C₂₃	—	2	96	no	—
37	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
38	C₂×C₂²⋊C₄×C₂₃	C₂³	1	48	yes	—
39	C₂×C₄²×C₂₃	—	2	16	no	—
40	C₂×C₄²×C₂₃	D₄	1	24	yes	complemented
41	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
42	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
43	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
44	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
45	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
46	C₂⁴×C₂₃	C₂⁴	1	8	yes	—
47	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
48	#41×C₂₃	—	4	64	no	—
49	C₂×Q₈×C₂₃	—	2	24	no	—
50	C₂×D₄×C₂₃	—	2	48	no	—
51	C₂×D₄×C₂₃	—	2	48	no	—
52	C₂×D₄×C₂₃	—	2	24	no	—
53	C₂×D₄×C₂₃	—	2	96	no	—
54	C₂²×C₄×C₂₃	—	2	24	no	—
55	C₂²×C₄×C₂₃	—	2	48	no	—
56	C₂²×C₄×C₂₃	C₂×D₄	1	24	yes	—
57	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
58	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
59	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
60	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
62	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₂₃	—	2	48	no	—
64	C₂²⋊C₄×C₂₃	—	2	96	no	—
65	C₄²×C₂₃	—	2	24	no	—
66	C₄²×C₂₃	C₂×D₄	1	24	yes	—
67	C₄²×C₂₃	C₂×D₄	1	12	yes	—
68	C₄²×C₂₃	C₂×D₄	1	12	yes	—
69	C₄²×C₂₃	#41	1	16	yes	complemented
70	#36644	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₂₃	—	2	4	no	—
72	C₂³×C₂₃	—	2	24	no	—
73	C₂³×C₂₃	—	2	32	no	—
74	C₂³×C₂₃	C₂²×D₄	1	6	yes	—
75	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
76	Q₈×C₂₃	—	4	16	no	—
77	D₄×C₂₃	—	4	32	no	—
78	D₄×C₂₃	—	4	16	no	—
79	D₄×C₂₃	—	4	96	no	—
80	C₂×C₄×C₂₃	—	2	12	no	—
81	C₂×C₄×C₂₃	—	2	48	no	—
82	C₂×C₄×C₂₃	—	2	24	no	—
83	C₂×C₄×C₂₃	—	4	32	no	—
84	C₂×C₄×C₂₃	#127	1	16	yes	complemented
85	C₂×C₄×C₂₃	C₂²×D₄	1	12	yes	—
86	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
87	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
88	C₂×C₄×C₂₃	C₂²×D₄	1	12	yes	—
89	C₂×C₄×C₂₃	C₂×#41	1	24	yes	—
90	C₂×#532	C₂×C₂₃	1	6	yes	complemented
91	#2868	C₂×C₂₃	1	1	yes	characteristic, complemented
92	#2867	C₂×C₂₃	1	1	yes	characteristic, complemented
93	#2858	C₂×C₂₃	1	6	yes	complemented
94	#2856	C₂×C₂₃	1	8	yes	complemented
95	C₄×#127	C₂×C₂₃	1	8	yes	complemented
96	C₂×C₄³	C₂×C₂₃	1	1	yes	characteristic, complemented
97	C₂²×C₂₃	—	2	12	no	—
98	C₂²×C₂₃	—	4	8	no	—
99	C₂²×C₂₃	—	4	48	no	—
100	C₂²×C₂₃	C₂×#127	1	4	yes	—
101	C₂²×C₂₃	C₂³×D₄	1	3	yes	—
102	C₂²×C₂₃	C₂²×#41	1	4	yes	—
103	C₄×C₂₃	—	2	12	no	—
104	C₄×C₂₃	—	4	8	no	—
105	C₄×C₂₃	C₂×#127	1	4	yes	—
106	C₄×C₂₃	C₂×#127	1	4	yes	—
107	C₄×C₂₃	#532	1	24	yes	complemented
108	C₂²×#41	C₂²×C₂₃	1	4	yes	—
109	#532	—	2	48	no	—
110	C₂×C₄⋊Q₈	C₂²×C₂₃	1	6	yes	—
111	C₂×#127	C₂²×C₂₃	1	8	yes	—
112	C₂×#127	C₂²×C₂₃	1	6	yes	—
113	C₂×#124	C₂²×C₂₃	1	12	yes	—
114	C₂×#121	C₂²×C₂₃	1	24	yes	—
115	C₂×C₄×D₄	C₂²×C₂₃	1	24	yes	—
116	C₂²×C₄²	C₂²×C₂₃	1	4	yes	—
117	C₂²×C₄²	C₂²×C₂₃	1	3	yes	—
118	#390	C₂²×C₂₃	1	48	yes	—
119	#384	C₂²×C₂₃	1	8	yes	—
120	C₄³	C₂²×C₂₃	1	4	yes	—
121	C₄³	C₂²×C₂₃	1	3	yes	—
122	C₄³	C₂²×C₂₃	1	1	yes	characteristic
123	C₂×C₂₃	—	2	4	no	—
124	C₂×C₂₃	—	8	8	no	—
125	C₂×C₂₃	C₂²×#127	1	1	yes	characteristic
126	C₂×C₂₃	C₂×#532	1	6	yes	—
127	C₂×#41	—	2	48	no	—
128	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
129	C₂²×D₄	C₂³×C₂₃	1	24	yes	—
130	C₂²×D₄	C₂³×C₂₃	1	8	yes	—
131	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
132	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
133	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
134	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	24	no	—
136	#127	—	2	24	no	—
137	#127	—	2	32	no	—
138	#124	—	2	48	no	—
139	#121	—	2	96	no	—
140	C₄×D₄	—	2	96	no	—
141	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
142	C₂×C₂²⋊C₄	C₂³×C₂₃	1	48	yes	—
143	C₂×C₄²	—	2	16	no	—
144	C₂×C₄²	D₄×C₂₃	1	24	yes	complemented
145	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
146	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
147	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
148	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
149	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
150	C₂₃	#36644	1	1	yes	characteristic, direct product factor, Sylow
151	C₂⁴	C₂⁴×C₂₃	1	8	yes	—
152	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
153	#41	—	4	64	no	—
154	C₂×Q₈	—	2	24	no	—
155	C₂×D₄	—	2	48	no	—
156	C₂×D₄	—	2	48	no	—
157	C₂×D₄	—	2	24	no	—
158	C₂×D₄	—	2	96	no	—
159	C₂²×C₄	—	2	24	no	—
160	C₂²×C₄	—	2	48	no	—
161	C₂²×C₄	C₂×D₄×C₂₃	1	24	yes	—
162	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
163	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
164	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
165	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
166	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
167	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
168	C₄⋊C₄	—	2	48	no	—
169	C₂²⋊C₄	—	2	96	no	—
170	C₄²	—	2	24	no	—
171	C₄²	C₂×D₄×C₂₃	1	24	yes	—
172	C₄²	C₂×D₄×C₂₃	1	12	yes	—
173	C₄²	C₂×D₄×C₂₃	1	12	yes	—
174	C₄²	#41×C₂₃	1	16	yes	complemented
175	C₂³	—	2	4	no	—
176	C₂³	—	2	24	no	—
177	C₂³	—	2	32	no	—
178	C₂³	C₂²×D₄×C₂₃	1	6	yes	—
179	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
180	Q₈	—	4	16	no	—
181	D₄	—	4	32	no	—
182	D₄	—	4	16	no	—
183	D₄	—	4	96	no	—
184	C₂×C₄	—	2	12	no	—
185	C₂×C₄	—	2	48	no	—
186	C₂×C₄	—	2	24	no	—
187	C₂×C₄	—	4	32	no	—
188	C₂×C₄	C₂₃×#127	1	16	yes	complemented
189	C₂×C₄	C₂²×D₄×C₂₃	1	12	yes	—
190	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
191	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
192	C₂×C₄	C₂²×D₄×C₂₃	1	12	yes	—
193	C₂×C₄	C₂×#41×C₂₃	1	24	yes	—
194	C₂²	—	2	12	no	—
195	C₂²	—	4	8	no	—
196	C₂²	—	4	48	no	—
197	C₂²	C₂×C₂₃×#127	1	4	yes	—
198	C₂²	C₂³×D₄×C₂₃	1	3	yes	—
199	C₂²	C₂²×#41×C₂₃	1	4	yes	—
200	C₄	—	2	12	no	—
201	C₄	—	4	8	no	—
202	C₄	C₂×C₂₃×#127	1	4	yes	—
203	C₄	C₂×C₂₃×#127	1	4	yes	—
204	C₄	C₂₃×#532	1	24	yes	complemented
205	C₂	—	2	4	no	—
206	C₂	—	8	8	no	—
207	C₂	C₂²×C₂₃×#127	1	1	yes	characteristic
208	C₂	C₂×C₂₃×#532	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#532	C₂	semidirect
2. C₂₃×#2868	C₂	semidirect
3. C₂₃×#2867	C₂	semidirect
4. C₂₃×#2858	C₂	semidirect
5. C₂₃×#2856	C₂	semidirect
6. C₄×C₂₃×#127	C₂	semidirect
7. C₂×C₄³×C₂₃	C₂	semidirect
40. C₂×C₄²×C₂₃	D₄	semidirect
69. C₄²×C₂₃	#41	semidirect
70. #36644	C₂₃	direct (N × H)
84. C₂×C₄×C₂₃	#127	semidirect
90. C₂×#532	C₂×C₂₃	semidirect
91. #2868	C₂×C₂₃	semidirect
92. #2867	C₂×C₂₃	semidirect
93. #2858	C₂×C₂₃	semidirect
94. #2856	C₂×C₂₃	semidirect
95. C₄×#127	C₂×C₂₃	semidirect
96. C₂×C₄³	C₂×C₂₃	semidirect
107. C₄×C₂₃	#532	semidirect
144. C₂×C₄²	D₄×C₂₃	semidirect
150. C₂₃	#36644	direct (N × H)
174. C₄²	#41×C₂₃	semidirect
188. C₂×C₄	C₂₃×#127	semidirect
204. C₄	C₂₃×#532	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#532	2
C₂₃×#2868	2
C₂₃×#2867	2
C₂₃×#2858	2
C₂₃×#2856	2
C₄×C₂₃×#127	2
C₂×C₄³×C₂₃	2
#36644	23

Subgroups — C23×#36644 #19184270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#36644 #19184270