Subgroups — C₂₃×#3283 #15670902

Order 2944.

484 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#135	C₂	1	2	yes	maximal, complemented
2	C₂₃×#547	C₂	1	2	yes	maximal, complemented
3	C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#467	C₂	1	2	yes	maximal, complemented
5	C₂₃×#461	C₂	1	2	yes	maximal, complemented
6	C₂₃×#452	C₂	1	4	yes	maximal, complemented
7	C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₃	C₂²	1	4	yes	—
10	C₂₃×#135	—	2	8	no	—
11	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₂₃	C₂²	1	4	yes	—
14	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
16	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₂₃×#123	—	2	4	no	—
18	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
19	C₂₃×#121	—	2	2	no	—
20	C₄×D₄×C₂₃	C₂²	1	2	yes	—
21	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₂₃×#107	C₂²	1	1	yes	characteristic
24	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂₃×#103	C₂²	1	4	yes	—
26	C₂₃×#102	C₂²	1	2	yes	—
27	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
29	#41×C₂₃	—	2	8	no	—
30	#41×C₂₃	—	4	4	no	—
31	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
32	C₂×D₄×C₂₃	—	2	4	no	—
33	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
34	C₂×D₄×C₂₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₂₃	—	2	2	no	—
36	C₂²×C₄×C₂₃	—	2	4	no	—
37	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
38	Q₁₆×C₂₃	—	2	4	no	—
39	Q₁₆×C₂₃	—	4	2	no	—
40	SD₁₆×C₂₃	—	2	8	no	—
41	D₈×C₂₃	D₄	1	4	yes	complemented
42	C₂×C₈×C₂₃	—	2	4	no	—
43	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₂₃	C₂³	1	2	yes	—
45	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₄⋊C₄×C₂₃	—	2	4	no	—
48	C₄⋊C₄×C₂₃	—	2	2	no	—
49	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
51	C₂²⋊C₄×C₂₃	—	2	4	no	—
52	C₂²⋊C₄×C₂₃	—	2	2	no	—
53	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
54	C₄²×C₂₃	C₂³	1	1	yes	characteristic
55	C₂³×C₂₃	—	2	2	no	—
56	C₂³×C₂₃	C₂×D₄	1	2	yes	—
57	Q₈×C₂₃	—	2	4	no	—
58	Q₈×C₂₃	—	2	4	no	—
59	D₄×C₂₃	—	2	2	no	—
60	D₄×C₂₃	—	2	2	no	—
61	D₄×C₂₃	—	4	8	no	—
62	D₄×C₂₃	C₂×D₄	1	4	yes	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	4	no	—
67	C₂×C₄×C₂₃	—	4	8	no	—
68	C₂×C₄×C₂₃	—	4	4	no	—
69	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
73	C₈×C₂₃	—	2	2	no	—
74	C₈×C₂₃	—	2	1	no	—
75	C₈×C₂₃	C₂×D₄	1	2	yes	—
76	#3283	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₂₃	—	2	2	no	—
78	C₂²×C₂₃	—	2	4	no	—
79	C₂²×C₂₃	—	2	2	no	—
80	C₂²×C₂₃	—	2	2	no	—
81	C₂²×C₂₃	—	4	2	no	—
82	C₂²×C₂₃	—	4	2	no	—
83	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
84	C₄×C₂₃	—	2	2	no	—
85	C₄×C₂₃	—	2	1	no	—
86	C₄×C₂₃	—	4	4	no	—
87	C₄×C₂₃	—	4	2	no	—
88	C₄×C₂₃	#135	1	2	yes	complemented
89	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
90	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
91	C₂×#135	C₂×C₂₃	1	2	yes	complemented
92	#547	C₂×C₂₃	1	2	yes	complemented
93	#494	C₂×C₂₃	1	1	yes	characteristic, complemented
94	#467	C₂×C₂₃	1	2	yes	complemented
95	#461	C₂×C₂₃	1	2	yes	complemented
96	#452	C₂×C₂₃	1	4	yes	complemented
97	C₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
98	C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
99	C₂×C₂₃	—	4	2	no	—
100	C₂×C₂₃	—	4	4	no	—
101	C₂×C₂₃	D₄²	1	1	yes	characteristic
102	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
103	C₂×C₂₃	#578	1	1	yes	characteristic
104	C₂×#41	C₂²×C₂₃	1	4	yes	—
105	#135	—	2	8	no	—
106	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
107	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
108	C₂×SD₁₆	C₂²×C₂₃	1	4	yes	—
109	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
110	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
111	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
112	#123	—	2	4	no	—
113	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
114	#121	—	2	2	no	—
115	C₄×D₄	C₂²×C₂₃	1	2	yes	—
116	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	2	no	—
118	#107	C₂²×C₂₃	1	1	yes	characteristic
119	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
120	#103	C₂²×C₂₃	1	4	yes	—
121	#102	C₂²×C₂₃	1	2	yes	—
122	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
123	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
124	C₂₃	#3283	1	1	yes	characteristic, direct product factor, Sylow
125	#41	—	2	8	no	—
126	#41	—	4	4	no	—
127	C₂×Q₈	C₂³×C₂₃	1	4	yes	—
128	C₂×D₄	—	2	4	no	—
129	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
130	C₂×D₄	C₂³×C₂₃	1	2	yes	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
134	Q₁₆	—	2	4	no	—
135	Q₁₆	—	4	2	no	—
136	SD₁₆	—	2	8	no	—
137	D₈	D₄×C₂₃	1	4	yes	complemented
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
140	C₂×C₈	C₂³×C₂₃	1	2	yes	—
141	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
146	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
147	C₂²⋊C₄	—	2	4	no	—
148	C₂²⋊C₄	—	2	2	no	—
149	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
150	C₄²	C₂³×C₂₃	1	1	yes	characteristic
151	C₂³	—	2	2	no	—
152	C₂³	C₂×D₄×C₂₃	1	2	yes	—
153	Q₈	—	2	4	no	—
154	Q₈	—	2	4	no	—
155	D₄	—	2	2	no	—
156	D₄	—	2	2	no	—
157	D₄	—	4	8	no	—
158	D₄	C₂×D₄×C₂₃	1	4	yes	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
166	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
167	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
168	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	2	1	no	—
171	C₈	C₂×D₄×C₂₃	1	2	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	4	no	—
176	C₂²	—	4	2	no	—
177	C₂²	—	4	2	no	—
178	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	2	2	no	—
181	C₄	—	4	2	no	—
182	C₄	—	4	4	no	—
183	C₄	C₂₃×#135	1	2	yes	complemented
184	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
185	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
186	C₂	—	4	2	no	—
187	C₂	—	4	4	no	—
188	C₂	D₄²×C₂₃	1	1	yes	characteristic
189	C₂	C₂×C₂₃×#135	1	1	yes	characteristic
190	C₂	C₂₃×#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#135	C₂	semidirect
2. C₂₃×#547	C₂	semidirect
3. C₂₃×#494	C₂	semidirect
4. C₂₃×#467	C₂	semidirect
5. C₂₃×#461	C₂	semidirect
6. C₂₃×#452	C₂	semidirect
7. C₄×D₈×C₂₃	C₂	semidirect
8. C₈×D₄×C₂₃	C₂	semidirect
41. D₈×C₂₃	D₄	semidirect
76. #3283	C₂₃	direct (N × H)
88. C₄×C₂₃	#135	semidirect
91. C₂×#135	C₂×C₂₃	semidirect
92. #547	C₂×C₂₃	semidirect
93. #494	C₂×C₂₃	semidirect
94. #467	C₂×C₂₃	semidirect
95. #461	C₂×C₂₃	semidirect
96. #452	C₂×C₂₃	semidirect
97. C₄×D₈	C₂×C₂₃	semidirect
98. C₈×D₄	C₂×C₂₃	semidirect
124. C₂₃	#3283	direct (N × H)
137. D₈	D₄×C₂₃	semidirect
183. C₄	C₂₃×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#135	2
C₂₃×#547	2
C₂₃×#494	2
C₂₃×#467	2
C₂₃×#461	2
C₂₃×#452	2
C₄×D₈×C₂₃	2
C₈×D₄×C₂₃	2
#3283	23

Subgroups — C23×#3283 #15670902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3283 #15670902