Subgroups — C₂₃×#3375 #15670994

Order 2944.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#559	C₂	1	1	yes	maximal, characteristic
2	Q₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#489	C₂	1	2	yes	maximal, complemented
4	C₂₃×#486	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#474	C₂	1	2	yes	maximal, complemented
7	C₂₃×#464	C₂	1	2	yes	maximal, complemented
8	C₂₃×#460	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#440	C₂	1	2	yes	maximal, complemented
11	C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂₃×C₄⋊Q₈	—	2	1	no	—
15	C₂₃×C₄⋊Q₈	—	2	2	no	—
16	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂₃×#127	C₂²	1	1	yes	characteristic
19	C₂₃×#124	—	2	1	no	—
20	C₂₃×#124	C₂²	1	2	yes	—
21	C₂₃×#123	—	2	2	no	—
22	C₄×Q₈×C₂₃	—	2	1	no	—
23	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
25	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₂₃	—	2	1	no	—
28	C₄×D₄×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#107	C₂²	1	1	yes	characteristic
30	C₂₃×#107	C₂²	1	2	yes	—
31	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
33	C₂₃×#103	C₂²	1	1	yes	characteristic
34	C₂₃×#103	C₂²	1	1	yes	characteristic
35	C₂₃×#103	C₂²	1	2	yes	—
36	C₂₃×#102	C₂²	1	2	yes	—
37	C₂₃×#102	C₂²	1	2	yes	—
38	C₂₃×#102	C₂²	1	2	yes	—
39	C₂₃×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₂₃	—	2	2	no	—
42	C₂×Q₈×C₂₃	—	2	1	no	—
43	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	—	2	1	no	—
48	C₂×D₄×C₂₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	Q₁₆×C₂₃	—	2	2	no	—
51	SD₁₆×C₂₃	—	2	4	no	—
52	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₂₃	C₂³	1	2	yes	—
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄⋊C₄×C₂₃	—	2	1	no	—
57	C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
62	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
63	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₂₃	—	2	2	no	—
66	C₂²⋊C₄×C₂₃	—	2	2	no	—
67	C₂²⋊C₄×C₂₃	—	2	2	no	—
68	C₄²×C₂₃	—	2	1	no	—
69	C₄²×C₂₃	—	2	2	no	—
70	C₄²×C₂₃	—	2	1	no	—
71	C₄²×C₂₃	C₂³	1	2	yes	—
72	C₄²×C₂₃	C₂³	1	1	yes	characteristic
73	C₂³×C₂₃	—	2	2	no	—
74	Q₈×C₂₃	—	2	2	no	—
75	Q₈×C₂₃	—	2	1	no	—
76	Q₈×C₂₃	—	2	1	no	—
77	Q₈×C₂₃	—	2	1	no	—
78	Q₈×C₂₃	—	8	2	no	—
79	Q₈×C₂₃	—	8	1	no	—
80	Q₈×C₂₃	#41	1	2	yes	—
81	D₄×C₂₃	—	2	2	no	—
82	D₄×C₂₃	—	2	2	no	—
83	D₄×C₂₃	—	4	2	no	—
84	D₄×C₂₃	—	8	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	8	2	no	—
91	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
96	C₈×C₂₃	—	2	2	no	—
97	C₈×C₂₃	—	2	1	no	—
98	C₈×C₂₃	—	2	1	no	—
99	#3375	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₂₃	—	4	2	no	—
101	C₂²×C₂₃	—	4	2	no	—
102	C₂²×C₂₃	—	4	2	no	—
103	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	—	2	2	no	—
106	C₄×C₂₃	—	2	2	no	—
107	C₄×C₂₃	—	2	1	no	—
108	C₄×C₂₃	—	4	1	no	—
109	C₄×C₂₃	—	4	1	no	—
110	C₄×C₂₃	—	4	1	no	—
111	C₄×C₂₃	—	4	2	no	—
112	C₄×C₂₃	#137	1	2	yes	—
113	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
114	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
115	#559	C₂×C₂₃	1	1	yes	characteristic
116	Q₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
117	#489	C₂×C₂₃	1	2	yes	complemented
118	#486	C₂×C₂₃	1	1	yes	characteristic
119	#477	C₂×C₂₃	1	1	yes	characteristic, complemented
120	#474	C₂×C₂₃	1	2	yes	complemented
121	#464	C₂×C₂₃	1	2	yes	complemented
122	#460	C₂×C₂₃	1	1	yes	characteristic
123	#446	C₂×C₂₃	1	1	yes	characteristic, complemented
124	#440	C₂×C₂₃	1	2	yes	complemented
125	C₄×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
126	C₂×C₂₃	—	8	2	no	—
127	C₂×C₂₃	#548	1	1	yes	characteristic
128	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
129	C₂×C₂₃	#576	1	1	yes	characteristic
130	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
131	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
132	C₄⋊Q₈	—	2	1	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
135	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
136	#127	C₂²×C₂₃	1	1	yes	characteristic
137	#124	—	2	1	no	—
138	#124	C₂²×C₂₃	1	2	yes	—
139	#123	—	2	2	no	—
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
142	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
143	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
144	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
145	C₄×D₄	—	2	1	no	—
146	C₄×D₄	C₂²×C₂₃	1	2	yes	—
147	#107	C₂²×C₂₃	1	1	yes	characteristic
148	#107	C₂²×C₂₃	1	2	yes	—
149	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
151	#103	C₂²×C₂₃	1	1	yes	characteristic
152	#103	C₂²×C₂₃	1	1	yes	characteristic
153	#103	C₂²×C₂₃	1	2	yes	—
154	#102	C₂²×C₂₃	1	2	yes	—
155	#102	C₂²×C₂₃	1	2	yes	—
156	#102	C₂²×C₂₃	1	2	yes	—
157	#97	C₂²×C₂₃	1	2	yes	—
158	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
159	C₂₃	#3375	1	1	yes	characteristic, direct product factor, Sylow
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
163	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	C₂³×C₂₃	1	2	yes	—
168	C₂²×C₄	—	2	2	no	—
169	Q₁₆	—	2	2	no	—
170	SD₁₆	—	2	4	no	—
171	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
172	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₂₃	1	2	yes	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
180	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
181	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
182	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
183	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	1	no	—
190	C₄²	C₂³×C₂₃	1	1	yes	characteristic
191	C₄²	C₂³×C₂₃	1	2	yes	—
192	C₂³	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	2	1	no	—
195	Q₈	—	2	1	no	—
196	Q₈	—	2	1	no	—
197	Q₈	—	8	2	no	—
198	Q₈	—	8	1	no	—
199	Q₈	#41×C₂₃	1	2	yes	—
200	D₄	—	2	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	4	2	no	—
203	D₄	—	8	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
211	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
212	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	2	no	—
230	C₄	C₂₃×#137	1	2	yes	—
231	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
232	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
233	C₂	—	8	2	no	—
234	C₂	C₂₃×#548	1	1	yes	characteristic
235	C₂	C₂×C₂₃×#137	1	1	yes	characteristic
236	C₂	C₂₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈²×C₂₃	C₂	semidirect
3. C₂₃×#489	C₂	semidirect
5. C₂₃×#477	C₂	semidirect
6. C₂₃×#474	C₂	semidirect
7. C₂₃×#464	C₂	semidirect
9. C₂₃×#446	C₂	semidirect
10. C₂₃×#440	C₂	semidirect
11. C₄×Q₁₆×C₂₃	C₂	semidirect
99. #3375	C₂₃	direct (N × H)
116. Q₈²	C₂×C₂₃	semidirect
117. #489	C₂×C₂₃	semidirect
119. #477	C₂×C₂₃	semidirect
120. #474	C₂×C₂₃	semidirect
121. #464	C₂×C₂₃	semidirect
123. #446	C₂×C₂₃	semidirect
124. #440	C₂×C₂₃	semidirect
125. C₄×Q₁₆	C₂×C₂₃	semidirect
159. C₂₃	#3375	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×#559	2
Q₈²×C₂₃	2
C₂₃×#489	2
C₂₃×#486	2
C₂₃×#477	2
C₂₃×#474	2
C₂₃×#464	2
C₂₃×#460	2
C₂₃×#446	2
C₂₃×#440	2
C₄×Q₁₆×C₂₃	2
#3375	23

Subgroups — C23×#3375 #15670994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3375 #15670994