Subgroups — C₂₃×#7031 #19154657

Order 5888.

352 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1842	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
5	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
6	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
7	C₂×C₂₃×C₄⋊C₈	C₄	1	2	yes	—
8	C₂×C₂₃×C₄⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₂₃	—	2	1	no	—
10	C₂×C₄×C₈×C₂₃	—	2	1	no	—
11	C₄³×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#336	—	2	1	no	—
13	C₂₃×#336	C₄	1	2	yes	—
14	C₂²×C₈×C₂₃	—	2	2	no	—
15	C₂²×C₈×C₂₃	—	2	2	no	—
16	C₂²×C₈×C₂₃	—	2	2	no	—
17	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₂₃	Q₈	1	1	yes	characteristic
24	C₂₃×C₄⋊C₈	—	2	4	no	—
25	C₂₃×C₄⋊C₈	—	2	2	no	—
26	C₂₃×C₄⋊C₈	—	4	2	no	—
27	C₂₃×C₄⋊C₈	—	4	1	no	—
28	C₂₃×C₄⋊C₈	—	4	1	no	—
29	C₂₃×C₄⋊C₈	—	4	1	no	—
30	C₂₃×C₄⋊C₈	—	4	1	no	—
31	C₂₃×C₄⋊C₈	C₈	1	4	yes	complemented
32	C₄×C₈×C₂₃	—	4	1	no	—
33	C₄×C₈×C₂₃	—	4	1	no	—
34	C₄×C₈×C₂₃	—	4	1	no	—
35	C₄×C₈×C₂₃	—	4	1	no	—
36	C₂²×C₄×C₂₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₂₃	—	2	4	no	—
44	C₂×C₈×C₂₃	—	2	4	no	—
45	C₂×C₈×C₂₃	—	4	2	no	—
46	C₂×C₈×C₂₃	—	4	2	no	—
47	C₂×C₈×C₂₃	—	4	2	no	—
48	C₂×C₈×C₂₃	—	4	2	no	—
49	C₂×C₈×C₂₃	—	4	2	no	—
50	C₂×C₈×C₂₃	—	4	2	no	—
51	C₂×C₈×C₂₃	—	4	2	no	—
52	C₄²×C₂₃	—	2	1	no	—
53	C₄²×C₂₃	—	2	1	no	—
54	C₄²×C₂₃	—	2	2	no	—
55	C₄²×C₂₃	—	2	2	no	—
56	C₄²×C₂₃	—	4	1	no	—
57	C₄²×C₂₃	C₂×C₈	1	2	yes	—
58	C₄²×C₂₃	M₁₆	1	2	yes	—
59	C₄²×C₂₃	D₈	1	1	yes	characteristic
60	C₄²×C₂₃	D₈	1	1	yes	characteristic
61	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
62	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
63	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
64	C₄²×C₂₃	SD₁₆	1	1	yes	characteristic
65	C₄²×C₂₃	Q₁₆	1	1	yes	characteristic
66	C₄²×C₂₃	Q₁₆	1	1	yes	characteristic
67	#7031	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₂₃	#95	1	1	yes	characteristic
69	C₂×C₄×C₂₃	—	2	1	no	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
79	C₂×C₄×C₂₃	#97	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
81	C₂×C₄×C₂₃	#100	1	1	yes	characteristic
82	C₂×C₄×C₂₃	#101	1	1	yes	characteristic
83	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
84	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₂₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
88	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
89	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
90	C₂×C₄×C₂₃	#103	1	1	yes	characteristic
91	C₂×C₄×C₂₃	#104	1	2	yes	—
92	C₂×C₄×C₂₃	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
94	C₂×C₄×C₂₃	#106	1	1	yes	characteristic
95	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
96	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#108	1	2	yes	—
98	C₈×C₂₃	—	4	4	no	—
99	C₈×C₂₃	—	4	4	no	—
100	C₈×C₂₃	—	8	2	no	—
101	#1842	C₂×C₂₃	1	1	yes	characteristic
102	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
103	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
104	C₂²×C₂₃	#336	1	1	yes	characteristic
105	C₂²×C₂₃	#339	1	1	yes	characteristic
106	C₂²×C₂₃	#340	1	1	yes	characteristic
107	C₂²×C₂₃	#340	1	1	yes	characteristic
108	C₂²×C₂₃	#341	1	1	yes	characteristic
109	C₂²×C₂₃	#343	1	1	yes	characteristic
110	C₂²×C₂₃	#344	1	1	yes	characteristic
111	C₄×C₂₃	—	2	1	no	—
112	C₄×C₂₃	—	2	1	no	—
113	C₄×C₂₃	—	2	1	no	—
114	C₄×C₂₃	—	2	1	no	—
115	C₄×C₂₃	—	2	1	no	—
116	C₄×C₂₃	—	2	1	no	—
117	C₄×C₂₃	—	4	1	no	—
118	C₄×C₂₃	D₄⋊C₈	1	2	yes	—
119	C₄×C₂₃	Q₈⋊C₈	1	2	yes	—
120	C₄×C₂₃	#331	1	1	yes	characteristic
121	C₄×C₂₃	#332	1	1	yes	characteristic
122	C₄×C₂₃	#332	1	1	yes	characteristic
123	C₄×C₂₃	#333	1	1	yes	characteristic
124	C₄×C₂₃	#334	1	2	yes	—
125	C₄×C₂₃	#335	1	2	yes	—
126	C₂×C₄⋊C₈	—	2	1	no	—
127	C₂×C₄⋊C₈	—	2	1	no	—
128	C₂×C₄⋊C₈	—	2	1	no	—
129	C₂×C₄⋊C₈	C₄×C₂₃	1	2	yes	—
130	C₂×C₄⋊C₈	C₄×C₂₃	1	2	yes	—
131	C₂×C₄×C₈	—	2	1	no	—
132	C₂×C₄×C₈	—	2	1	no	—
133	C₄³	C₂²×C₂₃	1	1	yes	characteristic
134	#336	—	2	1	no	—
135	#336	C₄×C₂₃	1	2	yes	—
136	C₂×C₂₃	#1277	1	1	yes	characteristic
137	C₂×C₂₃	#1279	1	1	yes	characteristic
138	C₂×C₂₃	#1283	1	1	yes	characteristic
139	C₂×C₂₃	#1296	1	1	yes	characteristic
140	C₂×C₂₃	#1299	1	1	yes	characteristic
141	C₂×C₂₃	#1301	1	1	yes	characteristic
142	C₂×C₂₃	#1303	1	1	yes	characteristic
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
147	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
148	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
149	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
150	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
151	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
152	C₂×C₄²	Q₈×C₂₃	1	1	yes	characteristic
153	C₄⋊C₈	—	2	4	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	4	2	no	—
156	C₄⋊C₈	—	4	1	no	—
157	C₄⋊C₈	—	4	1	no	—
158	C₄⋊C₈	—	4	1	no	—
159	C₄⋊C₈	—	4	1	no	—
160	C₄⋊C₈	C₈×C₂₃	1	4	yes	complemented
161	C₄×C₈	—	4	1	no	—
162	C₄×C₈	—	4	1	no	—
163	C₄×C₈	—	4	1	no	—
164	C₄×C₈	—	4	1	no	—
165	C₂₃	#7031	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²×C₄	C₄²×C₂₃	1	1	yes	characteristic
167	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
170	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
172	C₂²×C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	4	1	no	—
187	C₄²	C₂×C₈×C₂₃	1	2	yes	—
188	C₄²	M₁₆×C₂₃	1	2	yes	—
189	C₄²	D₈×C₂₃	1	1	yes	characteristic
190	C₄²	D₈×C₂₃	1	1	yes	characteristic
191	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
192	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
193	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
194	C₄²	SD₁₆×C₂₃	1	1	yes	characteristic
195	C₄²	Q₁₆×C₂₃	1	1	yes	characteristic
196	C₄²	Q₁₆×C₂₃	1	1	yes	characteristic
197	C₂³	C₂₃×#95	1	1	yes	characteristic
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	C₄×C₈×C₂₃	1	1	yes	characteristic
208	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
209	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
210	C₂×C₄	C₂₃×#100	1	1	yes	characteristic
211	C₂×C₄	C₂₃×#101	1	1	yes	characteristic
212	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
213	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
214	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
215	C₂×C₄	C₂₃×#102	1	1	yes	characteristic
216	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
217	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
218	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
219	C₂×C₄	C₂₃×#103	1	1	yes	characteristic
220	C₂×C₄	C₂₃×#104	1	2	yes	—
221	C₂×C₄	C₂₃×C₄⋊C₈	1	2	yes	—
222	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
223	C₂×C₄	C₂₃×#106	1	1	yes	characteristic
224	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
225	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
226	C₂×C₄	C₂₃×#108	1	2	yes	—
227	C₈	—	4	4	no	—
228	C₈	—	4	4	no	—
229	C₈	—	8	2	no	—
230	C₂²	C₂₃×#336	1	1	yes	characteristic
231	C₂²	C₂₃×#339	1	1	yes	characteristic
232	C₂²	C₂₃×#340	1	1	yes	characteristic
233	C₂²	C₂₃×#340	1	1	yes	characteristic
234	C₂²	C₂₃×#341	1	1	yes	characteristic
235	C₂²	C₂₃×#343	1	1	yes	characteristic
236	C₂²	C₂₃×#344	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	1	no	—
244	C₄	C₂₃×D₄⋊C₈	1	2	yes	—
245	C₄	C₂₃×Q₈⋊C₈	1	2	yes	—
246	C₄	C₂₃×#331	1	1	yes	characteristic
247	C₄	C₂₃×#332	1	1	yes	characteristic
248	C₄	C₂₃×#332	1	1	yes	characteristic
249	C₄	C₂₃×#333	1	1	yes	characteristic
250	C₄	C₂₃×#334	1	2	yes	—
251	C₄	C₂₃×#335	1	2	yes	—
252	C₂	C₂₃×#1277	1	1	yes	characteristic
253	C₂	C₂₃×#1279	1	1	yes	characteristic
254	C₂	C₂₃×#1283	1	1	yes	characteristic
255	C₂	C₂₃×#1296	1	1	yes	characteristic
256	C₂	C₂₃×#1299	1	1	yes	characteristic
257	C₂	C₂₃×#1301	1	1	yes	characteristic
258	C₂	C₂₃×#1303	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₈×C₂₃
Frattini, cosocle	C₄³	C₂²×C₂₃
socle, prime radical	C₂³×C₂₃	#95
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
31. C₂₃×C₄⋊C₈	C₈	semidirect
67. #7031	C₂₃	direct (N × H)
160. C₄⋊C₈	C₈×C₂₃	semidirect
165. C₂₃	#7031	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×#1842	2
C₄×C₂₃×C₄⋊C₈	2
#7031	23

Subgroups — C23×#7031 #19154657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7031 #19154657