Subgroups — C₂₃×#29150 #19176776

Order 5888.

710 conjugacy classes of subgroups.

The subgroup lattice has 252 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
2	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2026	C₂	1	4	yes	maximal
4	C₂₃×#1960	C₂	1	2	yes	maximal
5	C₂₃×#1836	C₂	1	2	yes	maximal
6	C₂₃×#1803	C₂	1	4	yes	maximal
7	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#125	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₂₃×#103	C₂²	1	8	yes	—
13	C₂×C₂₃×#97	C₂²	1	2	yes	—
14	C₂₃×#391	C₂²	1	2	yes	—
15	C₂₃×#389	—	2	2	no	—
16	C₂₃×#389	—	2	2	no	—
17	C₂₃×#389	—	2	2	no	—
18	C₂₃×#384	C₂²	1	1	yes	characteristic
19	C₂₃×#384	C₂²	1	2	yes	—
20	C₂₃×#382	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
22	C₄³×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#340	C₂²	1	8	yes	—
24	C₂₃×#336	C₂²	1	2	yes	—
25	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
26	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
27	C₂₃×C₄⋊Q₈	—	2	2	no	—
28	C₂₃×C₄⋊Q₈	Q₈	1	4	yes	—
29	C₂₃×#125	—	4	4	no	—
30	C₄×Q₈×C₂₃	—	2	8	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
38	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
39	C₂×C₄²×C₂₃	—	2	2	no	—
40	C₂×C₄²×C₂₃	D₄	1	2	yes	—
41	C₂×C₄²×C₂₃	D₄	1	2	yes	—
42	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂₃×C₄⋊C₈	—	2	4	no	—
46	C₂₃×#103	—	4	8	no	—
47	C₂₃×#97	—	2	4	no	—
48	C₂₃×#95	—	2	4	no	—
49	C₂₃×#95	—	2	4	no	—
50	C₂×Q₈×C₂₃	—	2	2	no	—
51	C₂×Q₈×C₂₃	—	2	4	no	—
52	C₂×Q₈×C₂₃	—	2	4	no	—
53	C₂×Q₈×C₂₃	C₂×Q₈	1	4	yes	—
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	2	4	no	—
57	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
62	C₂×C₈×C₂₃	—	2	4	no	—
63	C₂×C₈×C₂₃	—	2	4	no	—
64	C₂×C₈×C₂₃	—	2	4	no	—
65	C₄⋊C₄×C₂₃	—	2	8	no	—
66	C₄⋊C₄×C₂₃	—	2	4	no	—
67	C₄⋊C₄×C₂₃	—	2	8	no	—
68	C₄⋊C₄×C₂₃	—	4	4	no	—
69	C₄⋊C₄×C₂₃	—	4	4	no	—
70	C₄⋊C₄×C₂₃	—	4	8	no	—
71	C₄⋊C₄×C₂₃	—	4	4	no	—
72	C₄⋊C₄×C₂₃	—	4	4	no	—
73	C₄²×C₂₃	—	2	4	no	—
74	C₄²×C₂₃	—	2	2	no	—
75	C₄²×C₂₃	—	4	2	no	—
76	C₄²×C₂₃	—	4	4	no	—
77	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
78	C₄²×C₂₃	#41	1	2	yes	—
79	C₄²×C₂₃	#41	1	4	yes	—
80	#29150	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
82	Q₈×C₂₃	—	2	8	no	—
83	Q₈×C₂₃	—	4	4	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	—	2	4	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	4	2	no	—
92	C₂×C₄×C₂₃	—	4	2	no	—
93	C₂×C₄×C₂₃	—	4	2	no	—
94	C₂×C₄×C₂₃	—	4	4	no	—
95	C₂×C₄×C₂₃	—	4	4	no	—
96	C₂×C₄×C₂₃	—	4	4	no	—
97	C₂×C₄×C₂₃	—	4	4	no	—
98	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
99	C₂×C₄×C₂₃	#124	1	2	yes	—
100	C₂×C₄×C₂₃	#124	1	2	yes	—
101	C₂×C₄×C₂₃	#137	1	4	yes	—
102	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
104	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
106	C₈×C₂₃	—	4	4	no	—
107	#2859	C₂×C₂₃	1	1	yes	characteristic
108	C₄×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
109	#2026	C₂×C₂₃	1	4	yes	—
110	#1960	C₂×C₂₃	1	2	yes	—
111	#1836	C₂×C₂₃	1	2	yes	—
112	#1803	C₂×C₂₃	1	4	yes	—
113	#1767	C₂×C₂₃	1	1	yes	characteristic
114	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
115	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
116	C₂²×C₂₃	#535	1	1	yes	characteristic
117	C₂²×C₂₃	C₂×#137	1	2	yes	—
118	C₂²×C₂₃	#575	1	2	yes	—
119	C₄×C₂₃	—	2	4	no	—
120	C₄×C₂₃	—	2	2	no	—
121	C₄×C₂₃	—	4	4	no	—
122	C₄×C₂₃	—	4	2	no	—
123	C₄×C₂₃	—	8	2	no	—
124	C₄×C₂₃	—	8	4	no	—
125	C₄×C₂₃	#491	1	4	yes	—
126	C₄×C₂₃	#554	1	2	yes	—
127	C₄×C₂₃	#555	1	2	yes	—
128	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
129	C₂×#125	C₂²×C₂₃	1	2	yes	—
130	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
131	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
132	C₂×#103	C₂²×C₂₃	1	8	yes	—
133	C₂×#97	C₂²×C₂₃	1	2	yes	—
134	#391	C₂²×C₂₃	1	2	yes	—
135	#389	—	2	2	no	—
136	#389	—	2	2	no	—
137	#389	—	2	2	no	—
138	#384	C₂²×C₂₃	1	1	yes	characteristic
139	#384	C₂²×C₂₃	1	2	yes	—
140	#382	C₂²×C₂₃	1	2	yes	—
141	C₄×C₄⋊C₄	—	2	2	no	—
142	C₄³	C₂²×C₂₃	1	1	yes	characteristic
143	#340	C₂²×C₂₃	1	8	yes	—
144	#336	C₂²×C₂₃	1	2	yes	—
145	C₂×C₂₃	#2563	1	1	yes	characteristic
146	C₂×C₂₃	#3078	1	2	yes	—
147	C₂×C₂₃	C₂×#491	1	1	yes	characteristic
148	C₂×C₂₃	#3142	1	1	yes	characteristic
149	C₂×C₂₃	#3216	1	2	yes	—
150	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
151	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	Q₈×C₂₃	1	4	yes	—
154	#125	—	4	4	no	—
155	C₄×Q₈	—	2	8	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
163	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	D₄×C₂₃	1	2	yes	—
166	C₂×C₄²	D₄×C₂₃	1	2	yes	—
167	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
168	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
170	C₄⋊C₈	—	2	4	no	—
171	#103	—	4	8	no	—
172	#97	—	2	4	no	—
173	#95	—	2	4	no	—
174	#95	—	2	4	no	—
175	C₂₃	#29150	1	1	yes	characteristic, direct product factor, Sylow
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	C₂×Q₈×C₂₃	1	4	yes	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
184	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
185	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
187	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	4	8	no	—
195	C₄⋊C₄	—	4	4	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	4	2	no	—
202	C₄²	—	4	4	no	—
203	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
204	C₄²	#41×C₂₃	1	2	yes	—
205	C₄²	#41×C₂₃	1	4	yes	—
206	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
207	Q₈	—	2	8	no	—
208	Q₈	—	4	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
224	C₂×C₄	C₂₃×#124	1	2	yes	—
225	C₂×C₄	C₂₃×#124	1	2	yes	—
226	C₂×C₄	C₂₃×#137	1	4	yes	—
227	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
228	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
229	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
230	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
231	C₈	—	4	4	no	—
232	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
233	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
234	C₂²	C₂₃×#535	1	1	yes	characteristic
235	C₂²	C₂×C₂₃×#137	1	2	yes	—
236	C₂²	C₂₃×#575	1	2	yes	—
237	C₄	—	2	4	no	—
238	C₄	—	2	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	4	no	—
241	C₄	—	8	4	no	—
242	C₄	—	8	2	no	—
243	C₄	C₂₃×#491	1	4	yes	—
244	C₄	C₂₃×#554	1	2	yes	—
245	C₄	C₂₃×#555	1	2	yes	—
246	C₂	C₂₃×#2563	1	1	yes	characteristic
247	C₂	C₂₃×#3078	1	2	yes	—
248	C₂	C₂×C₂₃×#491	1	1	yes	characteristic
249	C₂	C₂₃×#3142	1	1	yes	characteristic
250	C₂	C₂₃×#3216	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
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
80. #29150	C₂₃	direct (N × H)
175. C₂₃	#29150	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×C₄⋊Q₈	2
C₂₃×#2026	2
C₂₃×#1960	2
C₂₃×#1836	2
C₂₃×#1803	2
C₂₃×#1767	2
#29150	23

Subgroups — C23×#29150 #19176776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29150 #19176776