Subgroups — C₂₃×#3174 #15670793

Order 2944.

362 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#540	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#490	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#468	C₂	1	2	yes	maximal, complemented
8	C₂₃×#465	C₂	1	2	yes	maximal, complemented
9	C₂₃×#440	C₂	1	2	yes	maximal, complemented
10	C₄×SD₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
11	C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂₃×#127	C₂²	1	1	yes	characteristic
18	C₂₃×#126	—	2	2	no	—
19	C₂₃×#125	—	2	2	no	—
20	C₂₃×#125	C₂²	1	1	yes	characteristic
21	C₂₃×#125	C₂²	1	1	yes	characteristic
22	C₂₃×#123	—	2	2	no	—
23	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₂₃×#121	—	2	2	no	—
25	C₂₃×#121	C₂²	1	2	yes	—
26	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
27	C₄×D₄×C₂₃	C₂²	1	2	yes	—
28	C₂₃×#117	C₂²	1	1	yes	characteristic
29	C₂₃×#107	C₂²	1	2	yes	—
30	C₂₃×#106	C₂²	1	2	yes	—
31	C₂₃×#103	C₂²	1	2	yes	—
32	C₂₃×#103	C₂²	1	2	yes	—
33	C₂₃×#102	C₂²	1	2	yes	—
34	C₂₃×#102	C₂²	1	2	yes	—
35	C₂₃×#97	C₂²	1	1	yes	characteristic
36	C₂₃×#97	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
39	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
40	C₂×D₄×C₂₃	—	2	2	no	—
41	C₂×D₄×C₂₃	—	2	2	no	—
42	C₂×D₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	—	2	2	no	—
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	SD₁₆×C₂₃	—	2	4	no	—
46	SD₁₆×C₂₃	—	2	4	no	—
47	M₁₆×C₂₃	—	2	2	no	—
48	C₂×C₈×C₂₃	—	2	2	no	—
49	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	2	2	no	—
58	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
59	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
60	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₂₃	—	2	2	no	—
62	C₂²⋊C₄×C₂₃	—	2	2	no	—
63	C₂²⋊C₄×C₂₃	—	2	2	no	—
64	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
65	C₄²×C₂₃	—	2	2	no	—
66	C₄²×C₂₃	C₂³	1	1	yes	characteristic
67	C₄²×C₂₃	C₂³	1	1	yes	characteristic
68	C₂³×C₂₃	—	2	2	no	—
69	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
70	Q₈×C₂₃	—	2	2	no	—
71	Q₈×C₂₃	—	2	2	no	—
72	D₄×C₂₃	—	2	2	no	—
73	D₄×C₂₃	—	2	2	no	—
74	D₄×C₂₃	—	8	2	no	—
75	D₄×C₂₃	—	8	2	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	8	2	no	—
82	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
86	C₈×C₂₃	—	2	1	no	—
87	C₈×C₂₃	—	2	1	no	—
88	C₈×C₂₃	#41	1	2	yes	—
89	C₈×C₂₃	#41	1	2	yes	—
90	#3174	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₂₃	—	2	1	no	—
92	C₂²×C₂₃	—	2	2	no	—
93	C₂²×C₂₃	—	4	2	no	—
94	C₂²×C₂₃	—	4	2	no	—
95	C₂²×C₂₃	—	4	2	no	—
96	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₂₃	—	2	1	no	—
98	C₄×C₂₃	—	2	2	no	—
99	C₄×C₂₃	—	2	2	no	—
100	C₄×C₂₃	—	4	2	no	—
101	C₄×C₂₃	—	4	2	no	—
102	C₄×C₂₃	—	4	2	no	—
103	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
104	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
105	#541	C₂×C₂₃	1	1	yes	characteristic, complemented
106	#540	C₂×C₂₃	1	1	yes	characteristic
107	#499	C₂×C₂₃	1	1	yes	characteristic, complemented
108	#492	C₂×C₂₃	1	1	yes	characteristic, complemented
109	#491	C₂×C₂₃	1	1	yes	characteristic, complemented
110	#490	C₂×C₂₃	1	1	yes	characteristic, complemented
111	#468	C₂×C₂₃	1	2	yes	complemented
112	#465	C₂×C₂₃	1	2	yes	complemented
113	#440	C₂×C₂₃	1	2	yes	complemented
114	C₄×SD₁₆	C₂×C₂₃	1	2	yes	complemented
115	#405	C₂×C₂₃	1	1	yes	characteristic, complemented
116	C₂×C₂₃	—	4	1	no	—
117	C₂×C₂₃	—	8	2	no	—
118	C₂×C₂₃	#532	1	1	yes	characteristic
119	C₂×C₂₃	#577	1	2	yes	—
120	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
121	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
122	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
123	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
124	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
125	#127	C₂²×C₂₃	1	1	yes	characteristic
126	#126	—	2	2	no	—
127	#125	—	2	2	no	—
128	#125	C₂²×C₂₃	1	1	yes	characteristic
129	#125	C₂²×C₂₃	1	1	yes	characteristic
130	#123	—	2	2	no	—
131	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
132	#121	—	2	2	no	—
133	#121	C₂²×C₂₃	1	2	yes	—
134	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
135	C₄×D₄	C₂²×C₂₃	1	2	yes	—
136	#117	C₂²×C₂₃	1	1	yes	characteristic
137	#107	C₂²×C₂₃	1	2	yes	—
138	#106	C₂²×C₂₃	1	2	yes	—
139	#103	C₂²×C₂₃	1	2	yes	—
140	#103	C₂²×C₂₃	1	2	yes	—
141	#102	C₂²×C₂₃	1	2	yes	—
142	#102	C₂²×C₂₃	1	2	yes	—
143	#97	C₂²×C₂₃	1	1	yes	characteristic
144	#97	C₂²×C₂₃	1	1	yes	characteristic
145	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
146	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
147	C₂₃	#3174	1	1	yes	characteristic, direct product factor, Sylow
148	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	C₂³×C₂₃	1	2	yes	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
154	SD₁₆	—	2	4	no	—
155	SD₁₆	—	2	4	no	—
156	M₁₆	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
159	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
160	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
161	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
168	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
169	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
174	C₄²	—	2	2	no	—
175	C₄²	C₂³×C₂₃	1	1	yes	characteristic
176	C₄²	C₂³×C₂₃	1	1	yes	characteristic
177	C₂³	—	2	2	no	—
178	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
179	Q₈	—	2	2	no	—
180	Q₈	—	2	2	no	—
181	D₄	—	2	2	no	—
182	D₄	—	2	2	no	—
183	D₄	—	8	2	no	—
184	D₄	—	8	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
192	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
193	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
194	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
195	C₈	—	2	1	no	—
196	C₈	—	2	1	no	—
197	C₈	#41×C₂₃	1	2	yes	—
198	C₈	#41×C₂₃	1	2	yes	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	2	1	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
212	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
213	C₂	—	4	1	no	—
214	C₂	—	8	2	no	—
215	C₂	C₂₃×#532	1	1	yes	characteristic
216	C₂	C₂₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#540	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#541	C₂	semidirect
3. C₂₃×#499	C₂	semidirect
4. C₂₃×#492	C₂	semidirect
5. C₂₃×#491	C₂	semidirect
6. C₂₃×#490	C₂	semidirect
7. C₂₃×#468	C₂	semidirect
8. C₂₃×#465	C₂	semidirect
9. C₂₃×#440	C₂	semidirect
10. C₄×SD₁₆×C₂₃	C₂	semidirect
11. C₂₃×#405	C₂	semidirect
90. #3174	C₂₃	direct (N × H)
105. #541	C₂×C₂₃	semidirect
107. #499	C₂×C₂₃	semidirect
108. #492	C₂×C₂₃	semidirect
109. #491	C₂×C₂₃	semidirect
110. #490	C₂×C₂₃	semidirect
111. #468	C₂×C₂₃	semidirect
112. #465	C₂×C₂₃	semidirect
113. #440	C₂×C₂₃	semidirect
114. C₄×SD₁₆	C₂×C₂₃	semidirect
115. #405	C₂×C₂₃	semidirect
147. C₂₃	#3174	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×#541	2
C₂₃×#540	2
C₂₃×#499	2
C₂₃×#492	2
C₂₃×#491	2
C₂₃×#490	2
C₂₃×#468	2
C₂₃×#465	2
C₂₃×#440	2
C₄×SD₁₆×C₂₃	2
C₂₃×#405	2
#3174	23

Subgroups — C23×#3174 #15670793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3174 #15670793