Subgroups — C₂₃×#29149 #19176775

Order 5888.

878 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2859	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2025	C₂	1	4	yes	maximal, complemented
4	C₂₃×#1961	C₂	1	2	yes	maximal
5	C₂₃×#1836	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1802	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#125	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₂₃	—	2	2	no	—
11	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂₃×#102	C₂²	1	8	yes	—
14	C₂×C₂₃×#97	C₂²	1	2	yes	—
15	C₂₃×#390	—	2	2	no	—
16	C₂₃×#390	C₂²	1	2	yes	—
17	C₂₃×#389	—	2	2	no	—
18	C₂₃×#389	—	2	2	no	—
19	C₂₃×#384	C₂²	1	1	yes	characteristic
20	C₂₃×#384	C₂²	1	2	yes	—
21	C₂₃×#382	C₂²	1	2	yes	—
22	C₄³×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#340	C₂²	1	8	yes	—
24	C₂₃×#336	C₂²	1	2	yes	—
25	C₂²×D₄×C₂₃	—	2	2	no	—
26	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
27	C₂³×C₄×C₂₃	—	2	2	no	—
28	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
29	C₂₃×#127	—	2	2	no	—
30	C₂₃×#127	Q₈	1	4	yes	—
31	C₂₃×#125	—	4	4	no	—
32	C₄×D₄×C₂₃	—	2	8	no	—
33	C₄×D₄×C₂₃	—	4	8	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
40	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
41	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
42	C₂×C₄²×C₂₃	D₄	1	2	yes	—
43	C₂×C₄²×C₂₃	D₄	1	2	yes	—
44	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×C₄⋊C₈	—	2	4	no	—
48	C₂₃×#102	—	4	8	no	—
49	C₂₃×#97	—	2	4	no	—
50	C₂₃×#95	—	2	4	no	—
51	C₂⁴×C₂₃	—	2	2	no	—
52	C₂×D₄×C₂₃	—	2	8	no	—
53	C₂×D₄×C₂₃	—	2	4	no	—
54	C₂×D₄×C₂₃	—	2	2	no	—
55	C₂×D₄×C₂₃	—	2	4	no	—
56	C₂×D₄×C₂₃	—	4	4	no	—
57	C₂×D₄×C₂₃	—	4	4	no	—
58	C₂×D₄×C₂₃	C₂×Q₈	1	4	yes	—
59	C₂²×C₄×C₂₃	—	2	2	no	—
60	C₂²×C₄×C₂₃	—	2	4	no	—
61	C₂²×C₄×C₂₃	—	4	4	no	—
62	C₂²×C₄×C₂₃	—	4	8	no	—
63	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
68	C₂×C₈×C₂₃	—	2	4	no	—
69	C₂×C₈×C₂₃	—	2	4	no	—
70	C₂×C₈×C₂₃	—	2	4	no	—
71	C₄⋊C₄×C₂₃	—	2	4	no	—
72	C₄⋊C₄×C₂₃	—	2	8	no	—
73	C₄⋊C₄×C₂₃	—	4	4	no	—
74	C₄⋊C₄×C₂₃	—	4	4	no	—
75	C₄⋊C₄×C₂₃	—	4	8	no	—
76	C₄⋊C₄×C₂₃	—	4	4	no	—
77	C₂²⋊C₄×C₂₃	—	4	8	no	—
78	C₂²⋊C₄×C₂₃	—	4	4	no	—
79	C₄²×C₂₃	—	2	4	no	—
80	C₄²×C₂₃	—	2	2	no	—
81	C₄²×C₂₃	—	4	2	no	—
82	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
83	C₄²×C₂₃	#41	1	2	yes	—
84	C₄²×C₂₃	#41	1	4	yes	—
85	#29149	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	—	2	4	no	—
87	C₂³×C₂₃	—	4	2	no	—
88	C₂³×C₂₃	—	4	2	no	—
89	C₂³×C₂₃	—	4	4	no	—
90	C₂³×C₂₃	—	4	4	no	—
91	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
92	D₄×C₂₃	—	2	8	no	—
93	D₄×C₂₃	—	4	16	no	—
94	D₄×C₂₃	—	4	4	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	—	4	4	no	—
104	C₂×C₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	—	8	8	no	—
107	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
108	C₂×C₄×C₂₃	#124	1	2	yes	—
109	C₂×C₄×C₂₃	#124	1	2	yes	—
110	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
111	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
113	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
114	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
115	C₈×C₂₃	—	4	4	no	—
116	#2859	C₂×C₂₃	1	1	yes	characteristic, complemented
117	C₄×#127	C₂×C₂₃	1	1	yes	characteristic
118	#2025	C₂×C₂₃	1	4	yes	complemented
119	#1961	C₂×C₂₃	1	2	yes	—
120	#1836	C₂×C₂₃	1	2	yes	complemented
121	#1802	C₂×C₂₃	1	4	yes	complemented
122	#1767	C₂×C₂₃	1	1	yes	characteristic, complemented
123	C₂²×C₂₃	—	4	4	no	—
124	C₂²×C₂₃	—	4	8	no	—
125	C₂²×C₂₃	—	8	2	no	—
126	C₂²×C₂₃	—	8	2	no	—
127	C₂²×C₂₃	—	8	4	no	—
128	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
129	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
130	C₂²×C₂₃	#535	1	1	yes	characteristic
131	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
132	C₂²×C₂₃	#575	1	2	yes	—
133	C₄×C₂₃	—	2	4	no	—
134	C₄×C₂₃	—	2	2	no	—
135	C₄×C₂₃	—	4	2	no	—
136	C₄×C₂₃	—	8	2	no	—
137	C₄×C₂₃	—	8	4	no	—
138	C₄×C₂₃	#490	1	4	yes	—
139	C₄×C₂₃	#554	1	2	yes	—
140	C₄×C₂₃	#555	1	2	yes	—
141	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×#125	C₂²×C₂₃	1	2	yes	—
143	C₂×C₄×D₄	—	2	2	no	—
144	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
145	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
146	C₂×#102	C₂²×C₂₃	1	8	yes	—
147	C₂×#97	C₂²×C₂₃	1	2	yes	—
148	#390	—	2	2	no	—
149	#390	C₂²×C₂₃	1	2	yes	—
150	#389	—	2	2	no	—
151	#389	—	2	2	no	—
152	#384	C₂²×C₂₃	1	1	yes	characteristic
153	#384	C₂²×C₂₃	1	2	yes	—
154	#382	C₂²×C₂₃	1	2	yes	—
155	C₄³	C₂²×C₂₃	1	1	yes	characteristic
156	#340	C₂²×C₂₃	1	8	yes	—
157	#336	C₂²×C₂₃	1	2	yes	—
158	C₂×C₂₃	—	8	4	no	—
159	C₂×C₂₃	#2563	1	1	yes	characteristic
160	C₂×C₂₃	#3077	1	2	yes	—
161	C₂×C₂₃	C₂×#490	1	1	yes	characteristic
162	C₂×C₂₃	#3142	1	1	yes	characteristic
163	C₂×C₂₃	#3214	1	2	yes	—
164	C₂²×D₄	—	2	2	no	—
165	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
166	C₂³×C₄	—	2	2	no	—
167	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
168	#127	—	2	2	no	—
169	#127	Q₈×C₂₃	1	4	yes	—
170	#125	—	4	4	no	—
171	C₄×D₄	—	2	8	no	—
172	C₄×D₄	—	4	8	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	4	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
178	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
179	C₂×C₂²⋊C₄	—	2	4	no	—
180	C₂×C₂²⋊C₄	—	2	2	no	—
181	C₂×C₄²	D₄×C₂₃	1	2	yes	—
182	C₂×C₄²	D₄×C₂₃	1	2	yes	—
183	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
186	C₄⋊C₈	—	2	4	no	—
187	#102	—	4	8	no	—
188	#97	—	2	4	no	—
189	#95	—	2	4	no	—
190	C₂₃	#29149	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁴	—	2	2	no	—
192	C₂×D₄	—	2	8	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	4	4	no	—
197	C₂×D₄	—	4	4	no	—
198	C₂×D₄	C₂×Q₈×C₂₃	1	4	yes	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	8	no	—
203	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
204	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	8	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	8	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₂²⋊C₄	—	4	8	no	—
218	C₂²⋊C₄	—	4	4	no	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	4	2	no	—
222	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
223	C₄²	#41×C₂₃	1	2	yes	—
224	C₄²	#41×C₂₃	1	4	yes	—
225	C₂³	—	2	4	no	—
226	C₂³	—	4	4	no	—
227	C₂³	—	4	4	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	2	no	—
230	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
231	D₄	—	2	8	no	—
232	D₄	—	4	16	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	8	8	no	—
246	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
247	C₂×C₄	C₂₃×#124	1	2	yes	—
248	C₂×C₄	C₂₃×#124	1	2	yes	—
249	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
250	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
252	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
254	C₈	—	4	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	4	8	no	—
257	C₂²	—	8	4	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
261	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
262	C₂²	C₂₃×#535	1	1	yes	characteristic
263	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
264	C₂²	C₂₃×#575	1	2	yes	—
265	C₄	—	2	4	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	8	4	no	—
269	C₄	—	8	2	no	—
270	C₄	C₂₃×#490	1	4	yes	—
271	C₄	C₂₃×#554	1	2	yes	—
272	C₄	C₂₃×#555	1	2	yes	—
273	C₂	—	8	4	no	—
274	C₂	C₂₃×#2563	1	1	yes	characteristic
275	C₂	C₂₃×#3077	1	2	yes	—
276	C₂	C₂×C₂₃×#490	1	1	yes	characteristic
277	C₂	C₂₃×#3142	1	1	yes	characteristic
278	C₂	C₂₃×#3214	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2859	C₂	semidirect
3. C₂₃×#2025	C₂	semidirect
5. C₂₃×#1836	C₂	semidirect
6. C₂₃×#1802	C₂	semidirect
7. C₂₃×#1767	C₂	semidirect
85. #29149	C₂₃	direct (N × H)
116. #2859	C₂×C₂₃	semidirect
118. #2025	C₂×C₂₃	semidirect
120. #1836	C₂×C₂₃	semidirect
121. #1802	C₂×C₂₃	semidirect
122. #1767	C₂×C₂₃	semidirect
190. C₂₃	#29149	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×#2859	2
C₄×C₂₃×#127	2
C₂₃×#2025	2
C₂₃×#1961	2
C₂₃×#1836	2
C₂₃×#1802	2
C₂₃×#1767	2
#29149	23

Subgroups — C23×#29149 #19176775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29149 #19176775