Subgroups — C₂₃×#2542 #15670161

Order 2944.

676 conjugacy classes of subgroups.

The subgroup lattice has 186 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	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#399	C₂	1	4	yes	maximal, complemented
5	C₂₃×#390	C₂	1	2	yes	maximal
6	C₂₃×#384	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
9	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
11	C₂₃×C₄⋊Q₈	—	2	4	no	—
12	C₂₃×C₂²⋊Q₈	—	2	16	no	—
13	C₄×D₄×C₂₃	—	2	4	no	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	8	yes	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	—
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#95	C₂²	1	2	yes	—
24	C₂⁴×C₂₃	C₂³	1	2	yes	—
25	C₂×Q₈×C₂₃	—	2	8	no	—
26	C₂×Q₈×C₂₃	—	2	4	no	—
27	C₂×D₄×C₂₃	—	2	2	no	—
28	C₂×D₄×C₂₃	—	2	2	no	—
29	C₂×D₄×C₂₃	Q₈	1	4	yes	complemented
30	C₂²×C₄×C₂₃	—	2	4	no	—
31	C₂²×C₄×C₂₃	—	2	8	no	—
32	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
33	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
35	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₂₃	—	2	2	no	—
39	C₄⋊C₄×C₂₃	—	2	16	no	—
40	C₄⋊C₄×C₂₃	—	2	4	no	—
41	C₄⋊C₄×C₂₃	—	2	4	no	—
42	C₄⋊C₄×C₂₃	D₄	1	8	yes	complemented
43	C₂²⋊C₄×C₂₃	—	2	4	no	—
44	C₂²⋊C₄×C₂₃	—	2	16	no	—
45	C₄²×C₂₃	—	2	4	no	—
46	C₄²×C₂₃	D₄	1	4	yes	—
47	C₂³×C₂₃	—	2	4	no	—
48	C₂³×C₂₃	—	2	2	no	—
49	C₂³×C₂₃	—	2	2	no	—
50	C₂³×C₂₃	—	2	4	no	—
51	C₂³×C₂₃	C₂×Q₈	1	4	yes	—
52	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
53	Q₈×C₂₃	—	4	8	no	—
54	D₄×C₂₃	—	2	8	no	—
55	C₂×C₄×C₂₃	—	2	4	no	—
56	C₂×C₄×C₂₃	—	2	2	no	—
57	C₂×C₄×C₂₃	—	2	4	no	—
58	C₂×C₄×C₂₃	—	2	2	no	—
59	C₂×C₄×C₂₃	—	2	8	no	—
60	C₂×C₄×C₂₃	—	2	4	no	—
61	C₂×C₄×C₂₃	—	2	4	no	—
62	C₂×C₄×C₂₃	—	4	8	no	—
63	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₂₃	C₂×D₄	1	8	yes	—
65	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
68	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
69	C₂×C₄×C₂₃	#41	1	2	yes	—
70	#2542	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₂₃	—	2	4	no	—
72	C₂²×C₂₃	—	2	8	no	—
73	C₂²×C₂₃	—	4	2	no	—
74	C₂²×C₂₃	—	4	2	no	—
75	C₂²×C₂₃	—	4	4	no	—
76	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
78	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
79	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
80	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
81	C₄×C₂₃	—	2	2	no	—
82	C₄×C₂₃	—	2	8	no	—
83	C₄×C₂₃	—	4	2	no	—
84	C₄×C₂₃	—	4	4	no	—
85	C₄×C₂₃	C₂²⋊Q₈	1	4	yes	complemented
86	C₄×C₂₃	#127	1	4	yes	—
87	C₂×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
88	C₂×C₂²⋊Q₈	C₂×C₂₃	1	4	yes	complemented
89	C₂×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic
90	#399	C₂×C₂₃	1	4	yes	complemented
91	#390	C₂×C₂₃	1	2	yes	—
92	#384	C₂×C₂₃	1	2	yes	complemented
93	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
94	C₂×C₂₃	—	4	4	no	—
95	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₂₃	C₂×#127	1	1	yes	characteristic
97	C₂×C₂₃	#536	1	1	yes	characteristic
98	C₂×C₂₃	#547	1	2	yes	—
99	C₂×C₂₃	D₄×Q₈	1	2	yes	—
100	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
101	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
102	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
103	C₄⋊Q₈	—	2	4	no	—
104	C₂²⋊Q₈	—	2	16	no	—
105	C₄×D₄	—	2	4	no	—
106	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
107	C₂×C₄⋊C₄	C₂²×C₂₃	1	8	yes	—
108	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
109	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
111	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
112	C₂×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	—
113	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
114	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
115	#95	C₂²×C₂₃	1	2	yes	—
116	C₂₃	#2542	1	1	yes	characteristic, direct product factor, Sylow
117	C₂⁴	C₂³×C₂₃	1	2	yes	—
118	C₂×Q₈	—	2	8	no	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×D₄	—	2	2	no	—
121	C₂×D₄	—	2	2	no	—
122	C₂×D₄	Q₈×C₂₃	1	4	yes	complemented
123	C₂²×C₄	—	2	4	no	—
124	C₂²×C₄	—	2	8	no	—
125	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
126	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
127	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
128	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
129	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
130	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	16	no	—
133	C₄⋊C₄	—	2	4	no	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	D₄×C₂₃	1	8	yes	complemented
136	C₂²⋊C₄	—	2	4	no	—
137	C₂²⋊C₄	—	2	16	no	—
138	C₄²	—	2	4	no	—
139	C₄²	D₄×C₂₃	1	4	yes	—
140	C₂³	—	2	4	no	—
141	C₂³	—	2	2	no	—
142	C₂³	—	2	2	no	—
143	C₂³	—	2	4	no	—
144	C₂³	C₂×Q₈×C₂₃	1	4	yes	—
145	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
146	Q₈	—	4	8	no	—
147	D₄	—	2	8	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	8	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	4	8	no	—
156	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
157	C₂×C₄	C₂×D₄×C₂₃	1	8	yes	—
158	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
159	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
160	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
161	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
162	C₂×C₄	#41×C₂₃	1	2	yes	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	4	4	no	—
166	C₂²	—	4	2	no	—
167	C₂²	—	4	2	no	—
168	C₂²	C₂²×D₄×C₂₃	1	2	yes	—
169	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
170	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
171	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
172	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
173	C₄	—	2	8	no	—
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	4	no	—
177	C₄	C₂₃×C₂²⋊Q₈	1	4	yes	complemented
178	C₄	C₂₃×#127	1	4	yes	—
179	C₂	—	4	4	no	—
180	C₂	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
181	C₂	C₂×C₂₃×#127	1	1	yes	characteristic
182	C₂	C₂₃×#536	1	1	yes	characteristic
183	C₂	C₂₃×#547	1	2	yes	—
184	C₂	D₄×Q₈×C₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
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
2. C₂×C₂₃×C₂²⋊Q₈	C₂	semidirect
4. C₂₃×#399	C₂	semidirect
6. C₂₃×#384	C₂	semidirect
7. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
29. C₂×D₄×C₂₃	Q₈	semidirect
42. C₄⋊C₄×C₂₃	D₄	semidirect
70. #2542	C₂₃	direct (N × H)
85. C₄×C₂₃	C₂²⋊Q₈	semidirect
87. C₂×C₄⋊Q₈	C₂×C₂₃	semidirect
88. C₂×C₂²⋊Q₈	C₂×C₂₃	semidirect
90. #399	C₂×C₂₃	semidirect
92. #384	C₂×C₂₃	semidirect
93. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
116. C₂₃	#2542	direct (N × H)
122. C₂×D₄	Q₈×C₂₃	semidirect
135. C₄⋊C₄	D₄×C₂₃	semidirect
177. C₄	C₂₃×C₂²⋊Q₈	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
C₂×C₄×D₄×C₂₃	2
C₂₃×#399	2
C₂₃×#390	2
C₂₃×#384	2
C₄×C₄⋊C₄×C₂₃	2
#2542	23

Subgroups — C23×#2542 #15670161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2542 #15670161