Subgroups — C₂₃×#17618 #19165244

Order 5888.

1202 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3497	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2351	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1517	C₂	1	6	yes	maximal, complemented
5	C₂₃×#1502	C₂	1	6	yes	maximal, complemented
6	C₂×C₂₃×2₋¹⁺⁴	—	2	1	no	—
7	C₂₃×#550	C₄	1	4	yes	complemented
8	D₄×Q₈×C₂₃	C₄	1	4	yes	complemented
9	C₂₃×#548	—	2	4	no	—
10	C₂₃×#537	—	2	3	no	—
11	C₂₃×#532	C₂²	1	6	yes	—
12	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
13	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₂₃	—	2	4	no	—
15	C₂₃×#431	C₂²	1	6	yes	—
16	C₂₃×#389	—	2	3	no	—
17	C₂₃×#382	C₂²	1	3	yes	—
18	C₄×C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
19	C₂₃×#327	C₂²	1	4	yes	—
20	C₂₃×#327	C₂²	1	12	yes	—
21	C₂₃×2₋¹⁺⁴	—	4	4	no	—
22	C₂₃×2₋¹⁺⁴	—	4	4	no	—
23	C₂×#41×C₂₃	—	2	4	no	—
24	C₂×#41×C₂₃	—	2	6	no	—
25	C₂²×Q₈×C₂₃	—	2	4	no	—
26	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
27	C₂₃×C₄⋊Q₈	C₂×C₄	1	6	yes	—
28	C₂₃×#127	C₂×C₄	1	6	yes	—
29	C₂₃×#124	—	2	6	no	—
30	C₂₃×C₂²⋊Q₈	—	2	12	no	—
31	C₂₃×#121	—	2	6	no	—
32	C₂₃×#121	—	2	12	no	—
33	C₂₃×#121	C₂³	1	4	yes	—
34	C₄×Q₈×C₂₃	—	4	4	no	—
35	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
36	C₄×Q₈×C₂₃	D₄	1	4	yes	—
37	C₄×D₄×C₂₃	—	2	6	no	—
38	C₄×D₄×C₂₃	C₂×C₄	1	12	yes	—
39	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
41	C₂×C₄⋊C₄×C₂₃	D₄	1	3	yes	—
42	C₂×C₄²×C₂₃	—	2	3	no	—
43	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
44	C₂₃×C₄⋊C₈	—	2	6	no	—
45	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
46	C₄×C₈×C₂₃	—	2	6	no	—
47	C₂₃×#95	—	2	3	no	—
48	C₂₃×#95	C₂³	1	1	yes	characteristic
49	C₂₃×#95	C₂³	1	3	yes	—
50	#41×C₂₃	—	4	16	no	—
51	#41×C₂₃	—	4	12	no	—
52	#41×C₂₃	—	4	12	no	—
53	C₂×Q₈×C₂₃	—	2	6	no	—
54	C₂×Q₈×C₂₃	—	4	16	no	—
55	C₂×Q₈×C₂₃	C₂²⋊C₄	1	4	yes	—
56	C₂×Q₈×C₂₃	D₈	1	4	yes	—
57	C₂×Q₈×C₂₃	SD₁₆	1	4	yes	—
58	C₂×D₄×C₂₃	—	2	4	no	—
59	C₂×D₄×C₂₃	—	2	6	no	—
60	C₂×D₄×C₂₃	—	2	6	no	—
61	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₂₃	—	2	1	no	—
63	C₂²×C₄×C₂₃	—	2	12	no	—
64	C₂²×C₄×C₂₃	—	2	3	no	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
67	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₂₃	—	2	4	no	—
69	C₄⋊C₄×C₂₃	—	2	6	no	—
70	C₄⋊C₄×C₂₃	—	4	6	no	—
71	C₄⋊C₄×C₂₃	—	4	6	no	—
72	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	12	yes	—
73	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
74	C₂²⋊C₄×C₂₃	—	2	12	no	—
75	C₂²⋊C₄×C₂₃	—	2	4	no	—
76	C₄²×C₂₃	—	4	6	no	—
77	C₄²×C₂₃	C₂²×C₄	1	6	yes	—
78	C₄²×C₂₃	C₂×D₄	1	6	yes	—
79	#17618	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₂₃	—	2	4	no	—
81	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
82	Q₈×C₂₃	—	4	12	no	—
83	Q₈×C₂₃	#102	1	16	yes	—
84	D₄×C₂₃	—	4	4	no	—
85	D₄×C₂₃	—	4	4	no	—
86	D₄×C₂₃	—	4	4	no	—
87	D₄×C₂₃	—	4	12	no	—
88	D₄×C₂₃	—	8	6	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	4	no	—
91	C₂×C₄×C₂₃	—	4	3	no	—
92	C₂×C₄×C₂₃	—	4	1	no	—
93	C₂×C₄×C₂₃	—	4	1	no	—
94	C₂×C₄×C₂₃	—	4	1	no	—
95	C₂×C₄×C₂₃	—	4	24	no	—
96	C₂×C₄×C₂₃	—	4	3	no	—
97	C₂×C₄×C₂₃	—	4	3	no	—
98	C₂×C₄×C₂₃	—	4	3	no	—
99	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	6	yes	—
100	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	6	yes	—
101	C₂×C₄×C₂₃	C₂×D₈	1	6	yes	—
102	C₂×C₄×C₂₃	C₂×SD₁₆	1	6	yes	—
103	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₂₃	—	4	4	no	—
106	#3497	C₂×C₂₃	1	1	yes	characteristic
107	#2982	C₂×C₂₃	1	1	yes	characteristic, complemented
108	#2351	C₂×C₂₃	1	1	yes	characteristic, complemented
109	#1517	C₂×C₂₃	1	6	yes	complemented
110	#1502	C₂×C₂₃	1	6	yes	complemented
111	C₂²×C₂₃	—	2	1	no	—
112	C₂²×C₂₃	—	2	1	no	—
113	C₂²×C₂₃	—	4	4	no	—
114	C₂²×C₂₃	—	4	4	no	—
115	C₂²×C₂₃	—	4	4	no	—
116	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
118	C₂²×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
119	C₄×C₂₃	—	2	1	no	—
120	C₄×C₂₃	—	2	1	no	—
121	C₄×C₂₃	—	2	6	no	—
122	C₄×C₂₃	—	4	2	no	—
123	C₄×C₂₃	—	4	4	no	—
124	C₄×C₂₃	—	8	3	no	—
125	C₄×C₂₃	C₂×#102	1	12	yes	—
126	C₂×2₋¹⁺⁴	—	2	1	no	—
127	#550	C₄×C₂₃	1	4	yes	complemented
128	D₄×Q₈	C₄×C₂₃	1	4	yes	complemented
129	#548	—	2	4	no	—
130	#537	—	2	3	no	—
131	#532	C₂²×C₂₃	1	6	yes	—
132	C₂×C₄×Q₈	—	2	1	no	—
133	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
134	C₈×Q₈	—	2	4	no	—
135	#431	C₂²×C₂₃	1	6	yes	—
136	#389	—	2	3	no	—
137	#382	C₂²×C₂₃	1	3	yes	—
138	C₄×C₄⋊C₄	C₂²×C₂₃	1	3	yes	—
139	#327	C₂²×C₂₃	1	4	yes	—
140	#327	C₂²×C₂₃	1	12	yes	—
141	C₂×C₂₃	—	2	1	no	—
142	C₂×C₂₃	—	2	1	no	—
143	C₂×C₂₃	—	8	4	no	—
144	C₂×C₂₃	#2884	1	1	yes	characteristic
145	C₂×C₂₃	C₂²×#102	1	1	yes	characteristic
146	C₂×C₂₃	#2901	1	1	yes	characteristic
147	2₋¹⁺⁴	—	4	4	no	—
148	2₋¹⁺⁴	—	4	4	no	—
149	C₂×#41	—	2	4	no	—
150	C₂×#41	—	2	6	no	—
151	C₂²×Q₈	—	2	4	no	—
152	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
153	C₄⋊Q₈	C₂×C₄×C₂₃	1	6	yes	—
154	#127	C₂×C₄×C₂₃	1	6	yes	—
155	#124	—	2	6	no	—
156	C₂²⋊Q₈	—	2	12	no	—
157	#121	—	2	6	no	—
158	#121	—	2	12	no	—
159	#121	C₂³×C₂₃	1	4	yes	—
160	C₄×Q₈	—	4	4	no	—
161	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
162	C₄×Q₈	D₄×C₂₃	1	4	yes	—
163	C₄×D₄	—	2	6	no	—
164	C₄×D₄	C₂×C₄×C₂₃	1	12	yes	—
165	C₂×C₄⋊C₄	—	2	3	no	—
166	C₂×C₄⋊C₄	—	2	3	no	—
167	C₂×C₄⋊C₄	D₄×C₂₃	1	3	yes	—
168	C₂×C₄²	—	2	3	no	—
169	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
170	C₄⋊C₈	—	2	6	no	—
171	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
172	C₄×C₈	—	2	6	no	—
173	#95	—	2	3	no	—
174	#95	C₂³×C₂₃	1	1	yes	characteristic
175	#95	C₂³×C₂₃	1	3	yes	—
176	C₂₃	#17618	1	1	yes	characteristic, direct product factor, Sylow
177	#41	—	4	16	no	—
178	#41	—	4	12	no	—
179	#41	—	4	12	no	—
180	C₂×Q₈	—	2	6	no	—
181	C₂×Q₈	—	4	16	no	—
182	C₂×Q₈	C₂²⋊C₄×C₂₃	1	4	yes	—
183	C₂×Q₈	D₈×C₂₃	1	4	yes	—
184	C₂×Q₈	SD₁₆×C₂₃	1	4	yes	—
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	2	6	no	—
187	C₂×D₄	—	2	6	no	—
188	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	12	no	—
191	C₂²×C₄	—	2	3	no	—
192	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
193	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
194	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
195	C₂×C₈	—	2	4	no	—
196	C₄⋊C₄	—	2	6	no	—
197	C₄⋊C₄	—	4	6	no	—
198	C₄⋊C₄	—	4	6	no	—
199	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	12	yes	—
200	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
201	C₂²⋊C₄	—	2	12	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₄²	—	4	6	no	—
204	C₄²	C₂²×C₄×C₂₃	1	6	yes	—
205	C₄²	C₂×D₄×C₂₃	1	6	yes	—
206	C₂³	—	2	4	no	—
207	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
208	Q₈	—	4	12	no	—
209	Q₈	C₂₃×#102	1	16	yes	—
210	D₄	—	4	4	no	—
211	D₄	—	4	4	no	—
212	D₄	—	4	4	no	—
213	D₄	—	4	12	no	—
214	D₄	—	8	6	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	3	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	24	no	—
222	C₂×C₄	—	4	3	no	—
223	C₂×C₄	—	4	3	no	—
224	C₂×C₄	—	4	3	no	—
225	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	6	yes	—
226	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	6	yes	—
227	C₂×C₄	C₂×D₈×C₂₃	1	6	yes	—
228	C₂×C₄	C₂×SD₁₆×C₂₃	1	6	yes	—
229	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
231	C₈	—	4	4	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	4	4	no	—
235	C₂²	—	4	4	no	—
236	C₂²	—	4	4	no	—
237	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
238	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
239	C₂²	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	6	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	4	no	—
245	C₄	—	8	3	no	—
246	C₄	C₂×C₂₃×#102	1	12	yes	—
247	C₂	—	2	1	no	—
248	C₂	—	2	1	no	—
249	C₂	—	8	4	no	—
250	C₂	C₂₃×#2884	1	1	yes	characteristic
251	C₂	C₂²×C₂₃×#102	1	1	yes	characteristic
252	C₂	C₂₃×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#3497	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2982	C₂	semidirect
3. C₂₃×#2351	C₂	semidirect
4. C₂₃×#1517	C₂	semidirect
5. C₂₃×#1502	C₂	semidirect
7. C₂₃×#550	C₄	semidirect
8. D₄×Q₈×C₂₃	C₄	semidirect
79. #17618	C₂₃	direct (N × H)
107. #2982	C₂×C₂₃	semidirect
108. #2351	C₂×C₂₃	semidirect
109. #1517	C₂×C₂₃	semidirect
110. #1502	C₂×C₂₃	semidirect
127. #550	C₄×C₂₃	semidirect
128. D₄×Q₈	C₄×C₂₃	semidirect
176. C₂₃	#17618	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×#3497	2
C₂₃×#2982	2
C₂₃×#2351	2
C₂₃×#1517	2
C₂₃×#1502	2
#17618	23

Subgroups — C23×#17618 #19165244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17618 #19165244