Subgroups — C₂₃×#3078 #15670697

Order 2944.

376 conjugacy classes of subgroups.

The subgroup lattice has 214 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	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#479	C₂	1	4	yes	maximal, complemented
4	C₂₃×#477	C₂	1	2	yes	maximal, complemented
5	C₂₃×#475	C₂	1	2	yes	maximal, complemented
6	C₂₃×#428	C₂	1	2	yes	maximal
7	C₂₃×#423	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#419	C₂	1	2	yes	maximal
9	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂₃×#125	C₂²	1	2	yes	—
14	C₂₃×C₂²⋊Q₈	—	2	1	no	—
15	C₂₃×C₂²⋊Q₈	—	2	1	no	—
16	C₄×Q₈×C₂₃	—	2	1	no	—
17	C₄×Q₈×C₂₃	—	2	1	no	—
18	C₄×Q₈×C₂₃	—	2	1	no	—
19	C₄×Q₈×C₂₃	—	2	1	no	—
20	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
21	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
22	C₂₃×#117	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
24	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₃	—	2	1	no	—
26	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#107	C₂²	1	4	yes	—
28	C₂₃×#106	C₂²	1	4	yes	—
29	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₂₃×#103	C₂²	1	4	yes	—
32	C₂₃×#103	C₂²	1	4	yes	—
33	C₂×Q₈×C₂₃	—	2	1	no	—
34	C₂×Q₈×C₂₃	—	2	1	no	—
35	C₂×Q₈×C₂₃	—	2	1	no	—
36	C₂×Q₈×C₂₃	—	2	2	no	—
37	C₂×Q₈×C₂₃	Q₈	1	4	yes	complemented
38	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	—	2	1	no	—
40	C₂²×C₄×C₂₃	—	2	1	no	—
41	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	M₁₆×C₂₃	—	2	4	no	—
45	C₂×C₈×C₂₃	C₂³	1	4	yes	—
46	C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₄⋊C₄×C₂₃	—	2	2	no	—
48	C₄⋊C₄×C₂₃	—	2	1	no	—
49	C₄⋊C₄×C₂₃	—	2	2	no	—
50	C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₄⋊C₄×C₂₃	—	2	1	no	—
52	C₄⋊C₄×C₂₃	—	2	1	no	—
53	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
54	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
55	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₂₃	—	2	2	no	—
57	C₄²×C₂₃	—	2	2	no	—
58	C₄²×C₂₃	—	2	2	no	—
59	C₄²×C₂₃	—	2	2	no	—
60	C₄²×C₂₃	D₄	1	1	yes	characteristic
61	C₄²×C₂₃	D₄	1	1	yes	characteristic
62	C₄²×C₂₃	C₂³	1	1	yes	characteristic
63	C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
65	Q₈×C₂₃	—	2	4	no	—
66	Q₈×C₂₃	—	2	2	no	—
67	Q₈×C₂₃	—	4	2	no	—
68	Q₈×C₂₃	—	4	2	no	—
69	Q₈×C₂₃	C₂×Q₈	1	4	yes	—
70	C₂×C₄×C₂₃	—	2	4	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	4	1	no	—
77	C₂×C₄×C₂₃	—	4	2	no	—
78	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
84	C₂×C₄×C₂₃	#41	1	2	yes	—
85	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
86	C₈×C₂₃	—	2	4	no	—
87	#3078	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂²×C₂₃	—	2	1	no	—
89	C₂²×C₂₃	—	2	1	no	—
90	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
91	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₂₃	—	2	4	no	—
93	C₄×C₂₃	—	2	1	no	—
94	C₄×C₂₃	—	2	1	no	—
95	C₄×C₂₃	—	2	1	no	—
96	C₄×C₂₃	—	4	2	no	—
97	C₄×C₂₃	—	4	2	no	—
98	C₄×C₂₃	—	4	2	no	—
99	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
100	C₄×C₂₃	#137	1	2	yes	—
101	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
102	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
103	#533	C₂×C₂₃	1	1	yes	characteristic
104	C₂×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic
105	#479	C₂×C₂₃	1	4	yes	complemented
106	#477	C₂×C₂₃	1	2	yes	complemented
107	#475	C₂×C₂₃	1	2	yes	complemented
108	#428	C₂×C₂₃	1	2	yes	—
109	#423	C₂×C₂₃	1	1	yes	characteristic
110	#419	C₂×C₂₃	1	2	yes	—
111	C₂×C₂₃	—	2	2	no	—
112	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
114	C₂×C₂₃	#575	1	1	yes	characteristic
115	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
116	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
117	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
118	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
119	#125	C₂²×C₂₃	1	2	yes	—
120	C₂²⋊Q₈	—	2	1	no	—
121	C₂²⋊Q₈	—	2	1	no	—
122	C₄×Q₈	—	2	1	no	—
123	C₄×Q₈	—	2	1	no	—
124	C₄×Q₈	—	2	1	no	—
125	C₄×Q₈	—	2	1	no	—
126	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
127	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
128	#117	C₂²×C₂₃	1	2	yes	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
131	C₂×C₄²	—	2	1	no	—
132	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
133	#107	C₂²×C₂₃	1	4	yes	—
134	#106	C₂²×C₂₃	1	4	yes	—
135	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
136	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
137	#103	C₂²×C₂₃	1	4	yes	—
138	#103	C₂²×C₂₃	1	4	yes	—
139	C₂₃	#3078	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×Q₈	—	2	1	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	2	1	no	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	Q₈×C₂₃	1	4	yes	complemented
145	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
149	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
150	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
151	M₁₆	—	2	4	no	—
152	C₂×C₈	C₂³×C₂₃	1	4	yes	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	1	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	—	2	1	no	—
160	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
161	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
162	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	2	2	no	—
166	C₄²	—	2	2	no	—
167	C₄²	D₄×C₂₃	1	1	yes	characteristic
168	C₄²	D₄×C₂₃	1	1	yes	characteristic
169	C₄²	C₂³×C₂₃	1	1	yes	characteristic
170	C₄²	C₂³×C₂₃	1	1	yes	characteristic
171	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
172	Q₈	—	2	4	no	—
173	Q₈	—	2	2	no	—
174	Q₈	—	4	2	no	—
175	Q₈	—	4	2	no	—
176	Q₈	C₂×Q₈×C₂₃	1	4	yes	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
186	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
187	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
188	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
189	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
190	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
191	C₂×C₄	#41×C₂₃	1	2	yes	—
192	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
193	C₈	—	2	4	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	C₂₃×C₂²⋊Q₈	1	2	yes	—
197	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	4	no	—
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
206	C₄	C₂₃×#137	1	2	yes	—
207	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
208	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
209	C₂	—	2	2	no	—
210	C₂	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
211	C₂	C₂×C₂₃×#137	1	1	yes	characteristic
212	C₂	C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂³×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#479	C₂	semidirect
4. C₂₃×#477	C₂	semidirect
5. C₂₃×#475	C₂	semidirect
37. C₂×Q₈×C₂₃	Q₈	semidirect
87. #3078	C₂₃	direct (N × H)
105. #479	C₂×C₂₃	semidirect
106. #477	C₂×C₂₃	semidirect
107. #475	C₂×C₂₃	semidirect
139. C₂₃	#3078	direct (N × H)
144. C₂×Q₈	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
C₂×C₄×Q₈×C₂₃	2
C₂₃×#479	2
C₂₃×#477	2
C₂₃×#475	2
C₂₃×#428	2
C₂₃×#423	2
C₂₃×#419	2
#3078	23

Subgroups — C23×#3078 #15670697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3078 #15670697