Subgroups — C₂₃×#2962 #15670581

Order 2944.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#446	C₂	1	4	yes	maximal, complemented
3	C₂₃×#446	C₂	1	2	yes	maximal, complemented
4	C₈×Q₈×C₂₃	C₂	1	2	yes	maximal, complemented
5	C₂₃×#424	C₂	1	2	yes	maximal
6	C₂×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#405	C₂	1	2	yes	maximal
8	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
12	C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
13	C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
14	C₂₃×#125	C₄	1	2	yes	—
15	C₂₃×C₂²⋊Q₈	C₄	1	2	yes	—
16	C₂₃×C₂²⋊Q₈	C₄	1	2	yes	—
17	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
18	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
19	C₂₃×#117	C₂²	1	2	yes	—
20	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
25	C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
26	C₂₃×#97	C₂²	1	2	yes	—
27	C₂₃×#97	C₂²	1	4	yes	—
28	C₄×C₈×C₂₃	C₂²	1	2	yes	—
29	C₄×C₈×C₂₃	C₂²	1	4	yes	—
30	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	M₁₆×C₂₃	—	2	2	no	—
36	M₁₆×C₂₃	Q₈	1	4	yes	complemented
37	C₂×C₈×C₂₃	—	2	4	no	—
38	C₂×C₈×C₂₃	C₂³	1	2	yes	—
39	C₂×C₈×C₂₃	C₂³	1	2	yes	—
40	C₂×C₈×C₂₃	C₂³	1	2	yes	—
41	C₂×C₈×C₂₃	C₂³	1	2	yes	—
42	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
45	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
46	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
47	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
48	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
49	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₂₃	C₂³	1	1	yes	characteristic
52	C₄²×C₂₃	C₂³	1	1	yes	characteristic
53	C₄²×C₂₃	C₂³	1	4	yes	—
54	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	Q₈×C₂₃	—	2	4	no	—
56	Q₈×C₂₃	—	4	2	no	—
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	—	4	1	no	—
59	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₂₃	#41	1	2	yes	—
68	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
69	C₈×C₂₃	—	2	2	no	—
70	C₈×C₂₃	—	2	2	no	—
71	C₈×C₂₃	—	2	2	no	—
72	C₈×C₂₃	C₂×Q₈	1	4	yes	—
73	#2962	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂²×C₂₃	—	2	1	no	—
75	C₂²×C₂₃	—	2	1	no	—
76	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
77	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₂₃	—	2	1	no	—
79	C₄×C₂₃	—	2	4	no	—
80	C₄×C₂₃	—	2	1	no	—
81	C₄×C₂₃	—	2	1	no	—
82	C₄×C₂₃	—	2	4	no	—
83	C₄×C₂₃	C₄×Q₈	1	2	yes	—
84	C₄×C₂₃	#131	1	2	yes	—
85	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
86	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
87	#533	C₂×C₂₃	1	1	yes	characteristic
88	#446	C₂×C₂₃	1	2	yes	complemented
89	#446	C₂×C₂₃	1	4	yes	complemented
90	C₈×Q₈	C₂×C₂₃	1	2	yes	complemented
91	#424	C₂×C₂₃	1	2	yes	—
92	C₂×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
93	#405	C₂×C₂₃	1	2	yes	—
94	C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
95	C₂×C₂₃	—	2	2	no	—
96	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
98	C₂×C₂₃	#568	1	1	yes	characteristic
99	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
100	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
101	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
102	C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
103	C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
104	#125	C₄×C₂₃	1	2	yes	—
105	C₂²⋊Q₈	C₄×C₂₃	1	2	yes	—
106	C₂²⋊Q₈	C₄×C₂₃	1	2	yes	—
107	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
108	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
109	#117	C₂²×C₂₃	1	2	yes	—
110	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
111	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
112	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
113	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
114	C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
115	C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
116	#97	C₂²×C₂₃	1	2	yes	—
117	#97	C₂²×C₂₃	1	4	yes	—
118	C₄×C₈	C₂²×C₂₃	1	2	yes	—
119	C₄×C₈	C₂²×C₂₃	1	4	yes	—
120	C₂₃	#2962	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×Q₈	C₂×C₄×C₂₃	1	2	yes	—
122	C₂×Q₈	C₂×C₄×C₂₃	1	2	yes	—
123	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
124	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
125	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
126	M₁₆	—	2	2	no	—
127	M₁₆	Q₈×C₂₃	1	4	yes	complemented
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	C₂³×C₂₃	1	2	yes	—
130	C₂×C₈	C₂³×C₂₃	1	2	yes	—
131	C₂×C₈	C₂³×C₂₃	1	2	yes	—
132	C₂×C₈	C₂³×C₂₃	1	2	yes	—
133	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
134	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
135	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
136	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
137	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
138	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
139	C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
140	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
141	C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
142	C₄²	C₂³×C₂₃	1	1	yes	characteristic
143	C₄²	C₂³×C₂₃	1	4	yes	—
144	C₄²	C₂³×C₂₃	1	1	yes	characteristic
145	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
146	Q₈	—	2	4	no	—
147	Q₈	—	4	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
151	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
152	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
153	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
154	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
155	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
156	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
157	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
158	C₂×C₄	#41×C₂₃	1	2	yes	—
159	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	C₂×Q₈×C₂₃	1	4	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₄×Q₈×C₂₃	1	2	yes	—
167	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	4	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	4	no	—
173	C₄	C₄×Q₈×C₂₃	1	2	yes	—
174	C₄	C₂₃×#131	1	2	yes	—
175	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
176	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
179	C₂	C₂×C₂₃×#131	1	1	yes	characteristic
180	C₂	C₂₃×#568	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂₃×#446	C₂	semidirect
3. C₂₃×#446	C₂	semidirect
4. C₈×Q₈×C₂₃	C₂	semidirect
36. M₁₆×C₂₃	Q₈	semidirect
73. #2962	C₂₃	direct (N × H)
88. #446	C₂×C₂₃	semidirect
89. #446	C₂×C₂₃	semidirect
90. C₈×Q₈	C₂×C₂₃	semidirect
120. C₂₃	#2962	direct (N × H)
127. M₁₆	Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#533	2
C₂₃×#446	2
C₈×Q₈×C₂₃	2
C₂₃×#424	2
C₂×C₂₃×C₄⋊C₈	2
C₂₃×#405	2
C₄×M₁₆×C₂₃	2
#2962	23

Subgroups — C23×#2962 #15670581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2962 #15670581