Subgroups — C₂₃×#3334 #15670953

Order 2944.

420 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×#547	C₂	1	2	yes	maximal, complemented
2	C₂₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#480	C₂	1	4	yes	maximal, complemented
4	C₂₃×#474	C₂	1	2	yes	maximal, complemented
5	C₂₃×#466	C₂	1	2	yes	maximal, complemented
6	C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#107	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₂₃	—	2	2	no	—
10	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
12	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂₃×#123	—	2	4	no	—
14	C₂₃×C₂²⋊Q₈	—	2	2	no	—
15	C₂₃×#121	C₂²	1	4	yes	—
16	C₄×D₄×C₂₃	C₂²	1	2	yes	—
17	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
19	C₂₃×#107	C₂²	1	1	yes	characteristic
20	C₂₃×#107	C₂²	1	2	yes	—
21	C₂₃×#107	C₂²	1	4	yes	—
22	C₂₃×#107	C₂²	1	2	yes	—
23	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂₃×#102	C₂²	1	2	yes	—
25	C₂₃×#102	C₂²	1	4	yes	—
26	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	#41×C₂₃	—	4	8	no	—
29	C₂×Q₈×C₂₃	—	2	2	no	—
30	C₂×D₄×C₂₃	—	2	4	no	—
31	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
32	C₂×D₄×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₃	—	2	4	no	—
34	C₂²×C₄×C₂₃	—	2	2	no	—
35	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
36	D₈×C₂₃	—	2	2	no	—
37	C₂×C₈×C₂₃	—	2	4	no	—
38	C₂×C₈×C₂₃	C₂³	1	2	yes	—
39	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₂₃	—	2	4	no	—
42	C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
44	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
45	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
46	C₂²⋊C₄×C₂₃	—	2	2	no	—
47	C₂²⋊C₄×C₂₃	—	2	4	no	—
48	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
49	C₄²×C₂₃	C₂³	1	1	yes	characteristic
50	C₂³×C₂₃	—	2	2	no	—
51	C₂³×C₂₃	C₂×D₄	1	2	yes	—
52	Q₈×C₂₃	—	4	4	no	—
53	D₄×C₂₃	—	2	2	no	—
54	D₄×C₂₃	—	2	2	no	—
55	D₄×C₂₃	—	4	8	no	—
56	D₄×C₂₃	D₈	1	4	yes	complemented
57	C₂×C₄×C₂₃	—	2	2	no	—
58	C₂×C₄×C₂₃	—	2	2	no	—
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	—	2	4	no	—
61	C₂×C₄×C₂₃	—	4	8	no	—
62	C₂×C₄×C₂₃	—	4	4	no	—
63	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
67	C₈×C₂₃	—	2	2	no	—
68	C₈×C₂₃	—	2	1	no	—
69	C₈×C₂₃	#41	1	2	yes	complemented
70	#3334	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	—	4	2	no	—
74	C₂²×C₂₃	—	4	2	no	—
75	C₂²×C₂₃	—	4	2	no	—
76	C₂²×C₂₃	C₂×D₈	1	4	yes	—
77	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₂₃	—	2	2	no	—
79	C₄×C₂₃	—	2	1	no	—
80	C₄×C₂₃	—	4	2	no	—
81	C₄×C₂₃	—	4	4	no	—
82	C₄×C₂₃	C₂×D₈	1	2	yes	—
83	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
84	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
85	#547	C₂×C₂₃	1	2	yes	complemented
86	#500	C₂×C₂₃	1	1	yes	characteristic, complemented
87	#480	C₂×C₂₃	1	4	yes	complemented
88	#474	C₂×C₂₃	1	2	yes	complemented
89	#466	C₂×C₂₃	1	2	yes	complemented
90	C₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
91	C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
92	C₂×#107	C₂×C₂₃	1	2	yes	complemented
93	C₂×C₂₃	—	2	4	no	—
94	C₂×C₂₃	—	8	2	no	—
95	C₂×C₂₃	#547	1	1	yes	characteristic
96	C₂×C₂₃	C₂²×D₈	1	1	yes	characteristic
97	C₂×C₂₃	#578	1	1	yes	characteristic
98	C₂×#41	—	2	2	no	—
99	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
100	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
101	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
102	#123	—	2	4	no	—
103	C₂²⋊Q₈	—	2	2	no	—
104	#121	C₂²×C₂₃	1	4	yes	—
105	C₄×D₄	C₂²×C₂₃	1	2	yes	—
106	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
107	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
108	#107	C₂²×C₂₃	1	1	yes	characteristic
109	#107	C₂²×C₂₃	1	2	yes	—
110	#107	C₂²×C₂₃	1	4	yes	—
111	#107	C₂²×C₂₃	1	2	yes	—
112	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
113	#102	C₂²×C₂₃	1	2	yes	—
114	#102	C₂²×C₂₃	1	4	yes	—
115	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
116	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
117	C₂₃	#3334	1	1	yes	characteristic, direct product factor, Sylow
118	#41	—	4	8	no	—
119	C₂×Q₈	—	2	2	no	—
120	C₂×D₄	—	2	4	no	—
121	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
122	C₂×D₄	C₂³×C₂₃	1	2	yes	—
123	C₂²×C₄	—	2	4	no	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
126	D₈	—	2	2	no	—
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
129	C₂×C₈	C₂³×C₂₃	1	2	yes	—
130	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
131	C₄⋊C₄	—	2	4	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
134	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
135	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
136	C₂²⋊C₄	—	2	2	no	—
137	C₂²⋊C₄	—	2	4	no	—
138	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
139	C₄²	C₂³×C₂₃	1	1	yes	characteristic
140	C₂³	—	2	2	no	—
141	C₂³	C₂×D₄×C₂₃	1	2	yes	—
142	Q₈	—	4	4	no	—
143	D₄	—	2	2	no	—
144	D₄	—	2	2	no	—
145	D₄	—	4	8	no	—
146	D₄	D₈×C₂₃	1	4	yes	complemented
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	4	8	no	—
152	C₂×C₄	—	4	4	no	—
153	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
154	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
155	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
156	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
157	C₈	—	2	2	no	—
158	C₈	—	2	1	no	—
159	C₈	#41×C₂₃	1	2	yes	complemented
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	4	2	no	—
164	C₂²	—	4	2	no	—
165	C₂²	C₂×D₈×C₂₃	1	4	yes	—
166	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	4	no	—
171	C₄	C₂×D₈×C₂₃	1	2	yes	—
172	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
173	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
174	C₂	—	2	4	no	—
175	C₂	—	8	2	no	—
176	C₂	C₂₃×#547	1	1	yes	characteristic
177	C₂	C₂²×D₈×C₂₃	1	1	yes	characteristic
178	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. C₂₃×#547	C₂	semidirect
2. C₂₃×#500	C₂	semidirect
3. C₂₃×#480	C₂	semidirect
4. C₂₃×#474	C₂	semidirect
5. C₂₃×#466	C₂	semidirect
6. C₄×D₈×C₂₃	C₂	semidirect
7. C₈×D₄×C₂₃	C₂	semidirect
8. C₂×C₂₃×#107	C₂	semidirect
56. D₄×C₂₃	D₈	semidirect
69. C₈×C₂₃	#41	semidirect
70. #3334	C₂₃	direct (N × H)
85. #547	C₂×C₂₃	semidirect
86. #500	C₂×C₂₃	semidirect
87. #480	C₂×C₂₃	semidirect
88. #474	C₂×C₂₃	semidirect
89. #466	C₂×C₂₃	semidirect
90. C₄×D₈	C₂×C₂₃	semidirect
91. C₈×D₄	C₂×C₂₃	semidirect
92. C₂×#107	C₂×C₂₃	semidirect
117. C₂₃	#3334	direct (N × H)
146. D₄	D₈×C₂₃	semidirect
159. C₈	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#547	2
C₂₃×#500	2
C₂₃×#480	2
C₂₃×#474	2
C₂₃×#466	2
C₄×D₈×C₂₃	2
C₈×D₄×C₂₃	2
C₂×C₂₃×#107	2
#3334	23

Subgroups — C23×#3334 #15670953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3334 #15670953