Subgroups — C₂₃×#17782 #19165408

Order 5888.

1244 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3498	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2991	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1560	C₂	1	3	yes	maximal, complemented
4	C₂₃×#1545	C₂	1	6	yes	maximal, complemented
5	C₂₃×#1543	C₂	1	3	yes	maximal, complemented
6	C₂×C₂₃×H(5,𝔽₂)	C₄	1	2	yes	—
7	C₂₃×#550	—	2	2	no	—
8	D₄²×C₂₃	—	2	6	no	—
9	C₂₃×#535	C₄	1	6	yes	—
10	C₂₃×#532	C₂²	1	3	yes	—
11	C₂×C₂₃×#127	C₂²	1	3	yes	—
12	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₈×Q₈×C₂₃	—	2	2	no	—
14	C₂₃×#436	—	2	6	no	—
15	C₂₃×#431	C₂²	1	6	yes	complemented
16	C₂₃×#423	C₂²	1	6	yes	complemented
17	C₂₃×#333	C₂²	1	1	yes	characteristic, complemented
18	C₂₃×#333	C₂²	1	3	yes	complemented
19	C₂₃×#331	C₂²	1	6	yes	complemented
20	C₂₃×#331	C₂²	1	6	yes	complemented
21	C₂₃×H(5,𝔽₂)	—	4	8	no	—
22	C₂×#41×C₂₃	—	2	2	no	—
23	C₂×#41×C₂₃	C₂×C₄	1	6	yes	—
24	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×D₄×C₂₃	—	2	6	no	—
26	C₂²×D₄×C₂₃	—	2	6	no	—
27	C₂²×D₄×C₂₃	C₂×C₄	1	6	yes	—
28	C₂×M₁₆×C₂₃	—	2	6	no	—
29	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
30	C₂₃×#127	—	2	6	no	—
31	C₂₃×#127	C₂×C₄	1	6	yes	—
32	C₂₃×#127	C₂³	1	3	yes	—
33	C₂₃×#124	C₂×C₄	1	6	yes	—
34	C₂₃×#121	—	2	6	no	—
35	C₂₃×#121	—	2	12	no	—
36	C₂₃×#120	—	2	12	no	—
37	C₄×Q₈×C₂₃	D₄	1	2	yes	—
38	C₄×D₄×C₂₃	—	2	6	no	—
39	C₄×D₄×C₂₃	D₄	1	6	yes	—
40	C₂₃×#117	C₂×C₄	1	3	yes	—
41	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
42	C₂₃×C₄⋊C₈	—	2	6	no	—
43	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
44	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
45	C₂₃×C₂²⋊C₈	—	2	6	no	—
46	C₄×C₈×C₂₃	—	2	6	no	—
47	C₂⁴×C₂₃	—	2	6	no	—
48	#41×C₂₃	—	2	4	no	—
49	#41×C₂₃	—	4	12	no	—
50	#41×C₂₃	—	4	8	no	—
51	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
53	C₂×D₄×C₂₃	—	2	6	no	—
54	C₂×D₄×C₂₃	—	2	6	no	—
55	C₂×D₄×C₂₃	—	2	12	no	—
56	C₂×D₄×C₂₃	—	2	6	no	—
57	C₂×D₄×C₂₃	—	4	12	no	—
58	C₂×D₄×C₂₃	—	4	24	no	—
59	C₂×D₄×C₂₃	—	8	6	no	—
60	C₂×D₄×C₂₃	—	8	6	no	—
61	C₂×D₄×C₂₃	C₂²⋊C₄	1	3	yes	—
62	C₂×D₄×C₂₃	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₂₃	—	2	6	no	—
64	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
66	M₁₆×C₂₃	—	8	6	no	—
67	C₂×C₈×C₂₃	—	2	6	no	—
68	C₂×C₈×C₂₃	—	2	2	no	—
69	C₄⋊C₄×C₂₃	—	2	2	no	—
70	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	6	yes	—
71	C₂²⋊C₄×C₂₃	—	2	12	no	—
72	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	6	yes	—
73	C₄²×C₂₃	C₂²×C₄	1	3	yes	—
74	C₄²×C₂₃	C₂×D₄	1	6	yes	—
75	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
76	#17782	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₂₃	—	2	6	no	—
78	C₂³×C₂₃	—	4	12	no	—
79	C₂³×C₂₃	—	8	12	no	—
80	C₂³×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
81	Q₈×C₂₃	—	4	2	no	—
82	Q₈×C₂₃	#100	1	4	yes	—
83	D₄×C₂₃	—	2	6	no	—
84	D₄×C₂₃	—	4	6	no	—
85	D₄×C₂₃	—	4	12	no	—
86	D₄×C₂₃	—	4	6	no	—
87	D₄×C₂₃	—	4	6	no	—
88	D₄×C₂₃	—	4	12	no	—
89	D₄×C₂₃	—	8	12	no	—
90	C₂×C₄×C₂₃	—	2	6	no	—
91	C₂×C₄×C₂₃	—	2	6	no	—
92	C₂×C₄×C₂₃	—	2	6	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	4	12	no	—
95	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	6	yes	—
96	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
97	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
99	C₈×C₂₃	—	4	6	no	—
100	C₈×C₂₃	—	4	2	no	—
101	#3498	C₂×C₂₃	1	1	yes	characteristic
102	#2991	C₂×C₂₃	1	2	yes	complemented
103	#1560	C₂×C₂₃	1	3	yes	complemented
104	#1545	C₂×C₂₃	1	6	yes	complemented
105	#1543	C₂×C₂₃	1	3	yes	complemented
106	C₂²×C₂₃	—	2	3	no	—
107	C₂²×C₂₃	—	2	6	no	—
108	C₂²×C₂₃	—	4	6	no	—
109	C₂²×C₂₃	—	4	12	no	—
110	C₂²×C₂₃	—	16	12	no	—
111	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
112	C₄×C₂₃	—	2	1	no	—
113	C₄×C₂₃	—	2	6	no	—
114	C₄×C₂₃	—	4	2	no	—
115	C₄×C₂₃	C₂×#100	1	6	yes	—
116	C₄×C₂₃	#576	1	2	yes	—
117	C₄×C₂₃	#577	1	2	yes	—
118	C₂×H(5,𝔽₂)	C₄×C₂₃	1	2	yes	—
119	#550	—	2	2	no	—
120	D₄²	—	2	6	no	—
121	#535	C₄×C₂₃	1	6	yes	—
122	#532	C₂²×C₂₃	1	3	yes	—
123	C₂×#127	C₂²×C₂₃	1	3	yes	—
124	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
125	C₈×Q₈	—	2	2	no	—
126	#436	—	2	6	no	—
127	#431	C₂²×C₂₃	1	6	yes	complemented
128	#423	C₂²×C₂₃	1	6	yes	complemented
129	#333	C₂²×C₂₃	1	1	yes	characteristic, complemented
130	#333	C₂²×C₂₃	1	3	yes	complemented
131	#331	C₂²×C₂₃	1	6	yes	complemented
132	#331	C₂²×C₂₃	1	6	yes	complemented
133	C₂×C₂₃	—	4	3	no	—
134	C₂×C₂₃	—	8	6	no	—
135	C₂×C₂₃	C₂²×#100	1	1	yes	characteristic
136	C₂×C₂₃	#2897	1	2	yes	—
137	H(5,𝔽₂)	—	4	8	no	—
138	C₂×#41	—	2	2	no	—
139	C₂×#41	C₂×C₄×C₂₃	1	6	yes	—
140	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
141	C₂²×D₄	—	2	6	no	—
142	C₂²×D₄	—	2	6	no	—
143	C₂²×D₄	C₂×C₄×C₂₃	1	6	yes	—
144	C₂×M₁₆	—	2	6	no	—
145	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
146	#127	—	2	6	no	—
147	#127	C₂×C₄×C₂₃	1	6	yes	—
148	#127	C₂³×C₂₃	1	3	yes	—
149	#124	C₂×C₄×C₂₃	1	6	yes	—
150	#121	—	2	6	no	—
151	#121	—	2	12	no	—
152	#120	—	2	12	no	—
153	C₄×Q₈	D₄×C₂₃	1	2	yes	—
154	C₄×D₄	—	2	6	no	—
155	C₄×D₄	D₄×C₂₃	1	6	yes	—
156	#117	C₂×C₄×C₂₃	1	3	yes	—
157	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
158	C₄⋊C₈	—	2	6	no	—
159	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
160	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
161	C₂²⋊C₈	—	2	6	no	—
162	C₄×C₈	—	2	6	no	—
163	C₂₃	#17782	1	1	yes	characteristic, direct product factor, Sylow
164	C₂⁴	—	2	6	no	—
165	#41	—	2	4	no	—
166	#41	—	4	12	no	—
167	#41	—	4	8	no	—
168	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
169	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
170	C₂×D₄	—	2	6	no	—
171	C₂×D₄	—	2	6	no	—
172	C₂×D₄	—	2	12	no	—
173	C₂×D₄	—	2	6	no	—
174	C₂×D₄	—	4	12	no	—
175	C₂×D₄	—	4	24	no	—
176	C₂×D₄	—	8	6	no	—
177	C₂×D₄	—	8	6	no	—
178	C₂×D₄	C₂²⋊C₄×C₂₃	1	3	yes	—
179	C₂×D₄	C₂²×C₄×C₂₃	1	6	yes	—
180	C₂²×C₄	—	2	6	no	—
181	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
182	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
183	M₁₆	—	8	6	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	6	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	6	yes	—
188	C₂²⋊C₄	—	2	12	no	—
189	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	6	yes	—
190	C₄²	C₂²×C₄×C₂₃	1	3	yes	—
191	C₄²	C₂×D₄×C₂₃	1	6	yes	—
192	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
193	C₂³	—	2	6	no	—
194	C₂³	—	4	12	no	—
195	C₂³	—	8	12	no	—
196	C₂³	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
197	Q₈	—	4	2	no	—
198	Q₈	C₂₃×#100	1	4	yes	—
199	D₄	—	2	6	no	—
200	D₄	—	4	6	no	—
201	D₄	—	4	12	no	—
202	D₄	—	4	6	no	—
203	D₄	—	4	6	no	—
204	D₄	—	4	12	no	—
205	D₄	—	8	12	no	—
206	C₂×C₄	—	2	6	no	—
207	C₂×C₄	—	2	6	no	—
208	C₂×C₄	—	2	6	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	12	no	—
211	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	6	yes	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
213	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
215	C₈	—	4	6	no	—
216	C₈	—	4	2	no	—
217	C₂²	—	2	3	no	—
218	C₂²	—	2	6	no	—
219	C₂²	—	4	6	no	—
220	C₂²	—	4	12	no	—
221	C₂²	—	16	12	no	—
222	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	6	no	—
225	C₄	—	4	2	no	—
226	C₄	C₂×C₂₃×#100	1	6	yes	—
227	C₄	C₂₃×#576	1	2	yes	—
228	C₄	C₂₃×#577	1	2	yes	—
229	C₂	—	4	3	no	—
230	C₂	—	8	6	no	—
231	C₂	C₂²×C₂₃×#100	1	1	yes	characteristic
232	C₂	C₂₃×#2897	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2991	C₂	semidirect
3. C₂₃×#1560	C₂	semidirect
4. C₂₃×#1545	C₂	semidirect
5. C₂₃×#1543	C₂	semidirect
15. C₂₃×#431	C₂²	semidirect
16. C₂₃×#423	C₂²	semidirect
17. C₂₃×#333	C₂²	semidirect
18. C₂₃×#333	C₂²	semidirect
19. C₂₃×#331	C₂²	semidirect
20. C₂₃×#331	C₂²	semidirect
76. #17782	C₂₃	direct (N × H)
102. #2991	C₂×C₂₃	semidirect
103. #1560	C₂×C₂₃	semidirect
104. #1545	C₂×C₂₃	semidirect
105. #1543	C₂×C₂₃	semidirect
127. #431	C₂²×C₂₃	semidirect
128. #423	C₂²×C₂₃	semidirect
129. #333	C₂²×C₂₃	semidirect
130. #333	C₂²×C₂₃	semidirect
131. #331	C₂²×C₂₃	semidirect
132. #331	C₂²×C₂₃	semidirect
163. C₂₃	#17782	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×#3498	2
C₂₃×#2991	2
C₂₃×#1560	2
C₂₃×#1545	2
C₂₃×#1543	2
#17782	23

Subgroups — C23×#17782 #19165408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17782 #19165408