Subgroups — C₂₃×#2977 #15670596

Order 2944.

422 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#525	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#435	C₂	1	4	yes	maximal, complemented
3	C₈×D₄×C₂₃	C₂	1	4	yes	maximal, complemented
4	C₂₃×#433	C₂	1	2	yes	maximal, complemented
5	C₂₃×#408	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₂₃	C₂²	1	2	yes	complemented
10	C₂²×C₈×C₂₃	C₂²	1	4	yes	—
11	C₂²×C₈×C₂₃	C₂²	1	2	yes	complemented
12	C₂₃×#123	C₄	1	2	yes	—
13	C₂₃×C₂²⋊Q₈	C₄	1	2	yes	—
14	C₂₃×#121	C₄	1	2	yes	—
15	C₂₃×#120	C₄	1	2	yes	—
16	C₄×D₄×C₂₃	C₂²	1	4	yes	—
17	C₂₃×#117	C₂²	1	1	yes	characteristic
18	C₂₃×C₄⋊C₈	C₂²	1	4	yes	complemented
19	C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
20	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂₃×C₂²⋊C₈	C₂²	1	4	yes	complemented
23	C₂₃×#97	C₂²	1	2	yes	complemented
24	C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
25	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
26	#41×C₂₃	—	4	2	no	—
27	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	—	4	2	no	—
32	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
37	M₁₆×C₂₃	—	4	2	no	—
38	C₂×C₈×C₂₃	—	4	4	no	—
39	C₂×C₈×C₂₃	—	4	2	no	—
40	C₂×C₈×C₂₃	C₂³	1	4	yes	—
41	C₂×C₈×C₂₃	C₂³	1	4	yes	—
42	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
44	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
45	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
46	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
47	C₄²×C₂₃	C₂³	1	2	yes	—
48	C₂³×C₂₃	—	2	4	no	—
49	C₂³×C₂₃	—	4	1	no	—
50	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
52	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
53	Q₈×C₂₃	—	4	1	no	—
54	D₄×C₂₃	—	2	8	no	—
55	D₄×C₂₃	—	4	1	no	—
56	D₄×C₂₃	—	4	2	no	—
57	C₂×C₄×C₂₃	—	2	4	no	—
58	C₂×C₄×C₂₃	—	2	2	no	—
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	4	1	no	—
62	C₂×C₄×C₂₃	—	4	2	no	—
63	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
68	C₈×C₂₃	—	2	4	no	—
69	C₈×C₂₃	—	2	4	no	—
70	#2977	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₂₃	—	2	2	no	—
72	C₂²×C₂₃	—	2	2	no	—
73	C₂²×C₂₃	—	2	1	no	—
74	C₂²×C₂₃	—	2	2	no	—
75	C₂²×C₂₃	—	2	4	no	—
76	C₂²×C₂₃	—	2	2	no	—
77	C₂²×C₂₃	—	4	4	no	—
78	C₂²×C₂₃	#131	1	4	yes	—
79	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
80	C₄×C₂₃	—	2	2	no	—
81	C₄×C₂₃	—	2	1	no	—
82	C₄×C₂₃	—	2	2	no	—
83	C₄×C₂₃	—	2	4	no	—
84	C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
85	#525	C₂×C₂₃	1	1	yes	characteristic
86	#435	C₂×C₂₃	1	4	yes	complemented
87	C₈×D₄	C₂×C₂₃	1	4	yes	complemented
88	#433	C₂×C₂₃	1	2	yes	complemented
89	#408	C₂×C₂₃	1	2	yes	complemented
90	C₂×C₂²⋊C₈	C₂×C₂₃	1	2	yes	complemented
91	C₂×C₂₃	—	2	4	no	—
92	C₂×C₂₃	—	4	2	no	—
93	C₂×C₂₃	—	4	1	no	—
94	C₂×C₂₃	#518	1	1	yes	characteristic
95	C₂×C₂₃	C₂×#131	1	2	yes	—
96	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
97	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
98	C₂×M₁₆	C₂²×C₂₃	1	2	yes	complemented
99	C₂²×C₈	C₂²×C₂₃	1	4	yes	—
100	C₂²×C₈	C₂²×C₂₃	1	2	yes	complemented
101	#123	C₄×C₂₃	1	2	yes	—
102	C₂²⋊Q₈	C₄×C₂₃	1	2	yes	—
103	#121	C₄×C₂₃	1	2	yes	—
104	#120	C₄×C₂₃	1	2	yes	—
105	C₄×D₄	C₂²×C₂₃	1	4	yes	—
106	#117	C₂²×C₂₃	1	1	yes	characteristic
107	C₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
108	C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
109	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
110	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
111	C₂²⋊C₈	C₂²×C₂₃	1	4	yes	complemented
112	#97	C₂²×C₂₃	1	2	yes	complemented
113	C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
114	C₂₃	#2977	1	1	yes	characteristic, direct product factor, Sylow
115	C₂⁴	C₂×C₄×C₂₃	1	1	yes	characteristic
116	#41	—	4	2	no	—
117	C₂×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
118	C₂×D₄	C₂×C₄×C₂₃	1	4	yes	—
119	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
120	C₂×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
123	C₂²×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
124	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
125	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
126	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
127	M₁₆	—	4	2	no	—
128	C₂×C₈	—	4	4	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	C₂³×C₂₃	1	4	yes	—
131	C₂×C₈	C₂³×C₂₃	1	4	yes	—
132	C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
133	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
134	C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
135	C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
136	C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
137	C₄²	C₂³×C₂₃	1	2	yes	—
138	C₂³	—	2	4	no	—
139	C₂³	—	4	1	no	—
140	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
141	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
142	C₂³	C₂²×C₄×C₂₃	1	2	yes	—
143	Q₈	—	4	1	no	—
144	D₄	—	2	8	no	—
145	D₄	—	4	1	no	—
146	D₄	—	4	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₂²×C₄×C₂₃	1	4	yes	—
154	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
155	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
156	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
157	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	2	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	4	4	no	—
167	C₂²	C₂₃×#131	1	4	yes	—
168	C₂²	C₂³×C₄×C₂₃	1	1	yes	characteristic
169	C₄	—	2	4	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	2	no	—
172	C₄	—	2	2	no	—
173	C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
174	C₂	—	2	4	no	—
175	C₂	—	4	1	no	—
176	C₂	—	4	2	no	—
177	C₂	C₂₃×#518	1	1	yes	characteristic
178	C₂	C₂×C₂₃×#131	1	2	yes	—

	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×#525	C₂
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×#435	C₂	semidirect
3. C₈×D₄×C₂₃	C₂	semidirect
4. C₂₃×#433	C₂	semidirect
5. C₂₃×#408	C₂	semidirect
6. C₂×C₂₃×C₂²⋊C₈	C₂	semidirect
9. C₂×M₁₆×C₂₃	C₂²	semidirect
11. C₂²×C₈×C₂₃	C₂²	semidirect
18. C₂₃×C₄⋊C₈	C₂²	semidirect
22. C₂₃×C₂²⋊C₈	C₂²	semidirect
23. C₂₃×#97	C₂²	semidirect
24. C₄×C₈×C₂₃	C₂²	semidirect
70. #2977	C₂₃	direct (N × H)
86. #435	C₂×C₂₃	semidirect
87. C₈×D₄	C₂×C₂₃	semidirect
88. #433	C₂×C₂₃	semidirect
89. #408	C₂×C₂₃	semidirect
90. C₂×C₂²⋊C₈	C₂×C₂₃	semidirect
98. C₂×M₁₆	C₂²×C₂₃	semidirect
100. C₂²×C₈	C₂²×C₂₃	semidirect
107. C₄⋊C₈	C₂²×C₂₃	semidirect
111. C₂²⋊C₈	C₂²×C₂₃	semidirect
112. #97	C₂²×C₂₃	semidirect
113. C₄×C₈	C₂²×C₂₃	semidirect
114. C₂₃	#2977	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#525	2
C₂₃×#435	2
C₈×D₄×C₂₃	2
C₂₃×#433	2
C₂₃×#408	2
C₂×C₂₃×C₂²⋊C₈	2
#2977	23

Subgroups — C23×#2977 #15670596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2977 #15670596