Subgroups — C₂₃×#11354 #19158980

Order 5888.

774 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#384	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#343	C₂	1	4	yes	maximal
3	C₂₃×#1728	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
6	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₂₃	C₄	1	4	yes	—
8	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
9	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#384	—	2	1	no	—
11	C₂₃×#384	—	2	2	no	—
12	C₂₃×#384	—	2	1	no	—
13	C₂₃×#343	—	2	8	no	—
14	C₂₃×#336	—	2	1	no	—
15	C₂₃×#336	—	2	2	no	—
16	C₂₃×#336	—	2	1	no	—
17	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₂₃	—	2	2	no	—
21	C₂×M₁₆×C₂₃	—	2	2	no	—
22	C₂×M₁₆×C₂₃	—	2	4	no	—
23	C₂×M₁₆×C₂₃	—	2	4	no	—
24	C₂×M₁₆×C₂₃	—	2	4	no	—
25	C₂×M₁₆×C₂₃	—	2	4	no	—
26	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
27	C₂×M₁₆×C₂₃	Q₈	1	4	yes	—
28	C₂²×C₈×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
36	C₂×C₄⋊C₄×C₂₃	—	4	8	no	—
37	C₂×C₄²×C₂₃	—	2	4	no	—
38	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	D₄	1	2	yes	—
40	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
41	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	—	2	4	no	—
43	C₂²×C₄×C₂₃	—	2	4	no	—
44	C₂²×C₄×C₂₃	—	2	4	no	—
45	C₂²×C₄×C₂₃	—	2	4	no	—
46	C₂²×C₄×C₂₃	—	2	8	no	—
47	C₂²×C₄×C₂₃	—	4	8	no	—
48	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
54	C₂²×C₄×C₂₃	#41	1	2	yes	—
55	C₂²×C₄×C₂₃	#41	1	4	yes	—
56	C₂²×C₄×C₂₃	#41	1	2	yes	—
57	M₁₆×C₂₃	—	2	8	no	—
58	M₁₆×C₂₃	—	2	8	no	—
59	M₁₆×C₂₃	—	4	4	no	—
60	M₁₆×C₂₃	—	4	4	no	—
61	C₂×C₈×C₂₃	—	2	4	no	—
62	C₂×C₈×C₂₃	—	2	4	no	—
63	C₂×C₈×C₂₃	—	4	2	no	—
64	C₂×C₈×C₂₃	—	4	2	no	—
65	C₂×C₈×C₂₃	—	4	2	no	—
66	C₂×C₈×C₂₃	—	4	2	no	—
67	C₄⋊C₄×C₂₃	—	4	8	no	—
68	C₄⋊C₄×C₂₃	—	4	8	no	—
69	C₄⋊C₄×C₂₃	—	4	4	no	—
70	C₄⋊C₄×C₂₃	—	4	2	no	—
71	C₄⋊C₄×C₂₃	—	4	2	no	—
72	C₄²×C₂₃	—	2	2	no	—
73	C₄²×C₂₃	—	2	2	no	—
74	C₄²×C₂₃	—	2	4	no	—
75	#11354	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₃	—	2	2	no	—
77	C₂³×C₂₃	—	2	2	no	—
78	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂³×C₂₃	#117	1	2	yes	—
80	C₂³×C₂₃	C₄×D₄	1	2	yes	—
81	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
82	C₂×C₄×C₂₃	—	2	16	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	4	4	no	—
88	C₂×C₄×C₂₃	—	4	8	no	—
89	C₂×C₄×C₂₃	—	8	8	no	—
90	C₂×C₄×C₂₃	#100	1	4	yes	—
91	C₂×C₄×C₂₃	#101	1	4	yes	—
92	C₂×C₄×C₂₃	#121	1	2	yes	—
93	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
94	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
95	C₂×C₄×C₂₃	#123	1	2	yes	—
96	C₂×C₄×C₂₃	#123	1	4	yes	—
97	C₂×C₄×C₂₃	#123	1	2	yes	—
98	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
99	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
100	C₂×C₄×C₂₃	#125	1	4	yes	—
101	C₂×C₄×C₂₃	#126	1	4	yes	—
102	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
104	C₈×C₂₃	—	4	4	no	—
105	C₈×C₂₃	—	4	4	no	—
106	C₂×#384	C₂×C₂₃	1	1	yes	characteristic
107	C₂×#343	C₂×C₂₃	1	4	yes	—
108	#1728	C₂×C₂₃	1	1	yes	characteristic
109	#1728	C₂×C₂₃	1	1	yes	characteristic
110	C₂²×C₂₃	—	2	4	no	—
111	C₂²×C₂₃	—	2	2	no	—
112	C₂²×C₂₃	—	2	2	no	—
113	C₂²×C₂₃	—	2	4	no	—
114	C₂²×C₂₃	#381	1	1	yes	characteristic
115	C₂²×C₂₃	#387	1	4	yes	—
116	C₂²×C₂₃	#390	1	1	yes	characteristic
117	C₂²×C₂₃	#391	1	1	yes	characteristic
118	C₂²×C₂₃	C₂×#100	1	2	yes	—
119	C₂²×C₂₃	C₂×#101	1	2	yes	—
120	C₄×C₂₃	—	2	4	no	—
121	C₄×C₂₃	—	2	4	no	—
122	C₄×C₂₃	—	8	4	no	—
123	C₄×C₂₃	#397	1	2	yes	—
124	C₄×C₂₃	#398	1	2	yes	—
125	C₄×C₂₃	#399	1	2	yes	—
126	C₄×C₂₃	#400	1	2	yes	—
127	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
128	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
129	C₂²×C₄⋊C₄	C₄×C₂₃	1	4	yes	—
130	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
131	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
132	#384	—	2	1	no	—
133	#384	—	2	2	no	—
134	#384	—	2	1	no	—
135	#343	—	2	8	no	—
136	#336	—	2	1	no	—
137	#336	—	2	2	no	—
138	#336	—	2	1	no	—
139	C₂×C₂₃	—	2	4	no	—
140	C₂×C₂₃	#1445	1	1	yes	characteristic
141	C₂×C₂₃	#1821	1	2	yes	—
142	C₂×C₂₃	#1965	1	1	yes	characteristic
143	C₂×C₂₃	#1966	1	1	yes	characteristic
144	C₂×C₂₃	#1997	1	2	yes	—
145	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
146	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
147	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	—	2	4	no	—
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
155	C₂×M₁₆	Q₈×C₂₃	1	4	yes	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	4	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	—	4	8	no	—
165	C₂×C₄²	—	2	4	no	—
166	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
167	C₂×C₄²	D₄×C₂₃	1	2	yes	—
168	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
169	C₂₃	#11354	1	1	yes	characteristic, direct product factor, Sylow
170	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	4	8	no	—
177	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
178	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
179	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
183	C₂²×C₄	#41×C₂₃	1	2	yes	—
184	C₂²×C₄	#41×C₂₃	1	4	yes	—
185	C₂²×C₄	#41×C₂₃	1	2	yes	—
186	M₁₆	—	2	8	no	—
187	M₁₆	—	2	8	no	—
188	M₁₆	—	4	4	no	—
189	M₁₆	—	4	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₄⋊C₄	—	4	8	no	—
197	C₄⋊C₄	—	4	8	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	4	no	—
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
207	C₂³	C₂₃×#117	1	2	yes	—
208	C₂³	C₄×D₄×C₂₃	1	2	yes	—
209	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
210	C₂×C₄	—	2	16	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	—	8	8	no	—
218	C₂×C₄	C₂₃×#100	1	4	yes	—
219	C₂×C₄	C₂₃×#101	1	4	yes	—
220	C₂×C₄	C₂₃×#121	1	2	yes	—
221	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
222	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
223	C₂×C₄	C₂₃×#123	1	2	yes	—
224	C₂×C₄	C₂₃×#123	1	4	yes	—
225	C₂×C₄	C₂₃×#123	1	2	yes	—
226	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
227	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
228	C₂×C₄	C₂₃×#125	1	4	yes	—
229	C₂×C₄	C₂₃×#126	1	4	yes	—
230	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
231	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
232	C₈	—	4	4	no	—
233	C₈	—	4	4	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	C₂₃×#381	1	1	yes	characteristic
239	C₂²	C₂₃×#387	1	4	yes	—
240	C₂²	C₂₃×#390	1	1	yes	characteristic
241	C₂²	C₂₃×#391	1	1	yes	characteristic
242	C₂²	C₂×C₂₃×#100	1	2	yes	—
243	C₂²	C₂×C₂₃×#101	1	2	yes	—
244	C₄	—	2	4	no	—
245	C₄	—	2	4	no	—
246	C₄	—	8	4	no	—
247	C₄	C₂₃×#397	1	2	yes	—
248	C₄	C₂₃×#398	1	2	yes	—
249	C₄	C₂₃×#399	1	2	yes	—
250	C₄	C₂₃×#400	1	2	yes	—
251	C₂	—	2	4	no	—
252	C₂	C₂₃×#1445	1	1	yes	characteristic
253	C₂	C₂₃×#1821	1	2	yes	—
254	C₂	C₂₃×#1965	1	1	yes	characteristic
255	C₂	C₂₃×#1966	1	1	yes	characteristic
256	C₂	C₂₃×#1997	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
75. #11354	C₂₃	direct (N × H)
169. C₂₃	#11354	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#384	2
C₂×C₂₃×#343	2
C₂₃×#1728	2
#11354	23

Subgroups — C23×#11354 #19158980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11354 #19158980