Subgroups — C₂₃×#2604 #15670223

Order 2944.

470 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#126	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#399	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#393	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#389	C₂	1	2	yes	maximal, complemented
8	C₂₃×#388	C₂	1	2	yes	maximal
9	C₂₃×#388	C₂	1	2	yes	maximal
10	C₂₃×#387	C₂	1	2	yes	maximal, complemented
11	C₂₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#126	—	2	4	no	—
14	C₄×Q₈×C₂₃	—	2	4	no	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
28	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
29	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
30	C₂₃×#95	C₂²	1	2	yes	—
31	C₂₃×#95	C₂²	1	1	yes	characteristic
32	C₂₃×#95	C₂²	1	1	yes	characteristic
33	C₂₃×#95	C₂²	1	2	yes	—
34	C₂₃×#95	C₂²	1	2	yes	—
35	C₂₃×#95	C₂²	1	2	yes	—
36	C₂₃×#95	C₂²	1	2	yes	—
37	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₂₃	—	2	2	no	—
39	C₂×Q₈×C₂₃	—	2	4	no	—
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
50	C₄⋊C₄×C₂₃	—	2	4	no	—
51	C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	—	2	2	no	—
55	C₄⋊C₄×C₂₃	—	2	4	no	—
56	C₄⋊C₄×C₂₃	—	2	4	no	—
57	C₂²⋊C₄×C₂₃	—	2	2	no	—
58	C₂²⋊C₄×C₂₃	—	2	2	no	—
59	C₂²⋊C₄×C₂₃	—	2	2	no	—
60	C₂²⋊C₄×C₂₃	—	2	4	no	—
61	C₂²⋊C₄×C₂₃	—	2	4	no	—
62	C₄²×C₂₃	—	2	4	no	—
63	C₄²×C₂₃	—	2	4	no	—
64	C₄²×C₂₃	D₄	1	4	yes	—
65	C₂³×C₂₃	—	2	1	no	—
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	2	1	no	—
69	C₂³×C₂₃	—	2	1	no	—
70	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
71	Q₈×C₂₃	—	2	8	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	—	2	1	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	1	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	2	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₂₃	#41	1	4	yes	—
97	C₂×C₄×C₂₃	#41	1	4	yes	—
98	#2604	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₂₃	—	4	1	no	—
100	C₂²×C₂₃	—	4	1	no	—
101	C₂²×C₂₃	—	4	1	no	—
102	C₂²×C₂₃	—	4	2	no	—
103	C₂²×C₂₃	—	4	2	no	—
104	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
105	C₂²×C₂₃	C₂×#41	1	2	yes	—
106	C₂²×C₂₃	C₂×#41	1	2	yes	—
107	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	—	2	4	no	—
111	C₄×C₂₃	—	2	2	no	—
112	C₄×C₂₃	—	2	2	no	—
113	C₄×C₂₃	—	2	4	no	—
114	C₄×C₂₃	—	4	1	no	—
115	C₄×C₂₃	—	4	2	no	—
116	C₄×C₂₃	—	4	1	no	—
117	C₄×C₂₃	—	4	1	no	—
118	C₄×C₂₃	—	4	2	no	—
119	C₂×#126	C₂×C₂₃	1	1	yes	characteristic
120	C₂×C₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
121	#400	C₂×C₂₃	1	1	yes	characteristic, complemented
122	#399	C₂×C₂₃	1	1	yes	characteristic
123	#395	C₂×C₂₃	1	1	yes	characteristic, complemented
124	#393	C₂×C₂₃	1	1	yes	characteristic
125	#389	C₂×C₂₃	1	2	yes	complemented
126	#388	C₂×C₂₃	1	2	yes	—
127	#388	C₂×C₂₃	1	2	yes	—
128	#387	C₂×C₂₃	1	2	yes	complemented
129	#376	C₂×C₂₃	1	1	yes	characteristic, complemented
130	C₂×C₂₃	—	8	1	no	—
131	C₂×C₂₃	#525	1	1	yes	characteristic
132	C₂×C₂₃	#532	1	1	yes	characteristic
133	C₂×C₂₃	#536	1	1	yes	characteristic
134	C₂×C₂₃	#552	1	2	yes	—
135	C₂×C₂₃	#556	1	2	yes	—
136	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
137	#126	—	2	4	no	—
138	C₄×Q₈	—	2	4	no	—
139	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
143	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
145	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
149	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
151	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
152	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
153	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
154	#95	C₂²×C₂₃	1	2	yes	—
155	#95	C₂²×C₂₃	1	1	yes	characteristic
156	#95	C₂²×C₂₃	1	1	yes	characteristic
157	#95	C₂²×C₂₃	1	2	yes	—
158	#95	C₂²×C₂₃	1	2	yes	—
159	#95	C₂²×C₂₃	1	2	yes	—
160	#95	C₂²×C₂₃	1	2	yes	—
161	C₂₃	#2604	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
166	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
168	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
172	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
174	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₄²	—	2	4	no	—
188	C₄²	—	2	4	no	—
189	C₄²	D₄×C₂₃	1	4	yes	—
190	C₂³	—	2	1	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	1	no	—
195	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
196	Q₈	—	2	8	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	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	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
219	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
220	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
221	C₂×C₄	#41×C₂₃	1	4	yes	—
222	C₂×C₄	#41×C₂₃	1	4	yes	—
223	C₂²	—	4	2	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
229	C₂²	C₂×#41×C₂₃	1	2	yes	—
230	C₂²	C₂×#41×C₂₃	1	2	yes	—
231	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
232	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	2	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	1	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	1	no	—
243	C₂	—	8	1	no	—
244	C₂	C₂₃×#525	1	1	yes	characteristic
245	C₂	C₂₃×#532	1	1	yes	characteristic
246	C₂	C₂₃×#536	1	1	yes	characteristic
247	C₂	C₂₃×#552	1	2	yes	—
248	C₂	C₂₃×#556	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₈×C₂₃	C₂	semidirect
3. C₂₃×#400	C₂	semidirect
5. C₂₃×#395	C₂	semidirect
7. C₂₃×#389	C₂	semidirect
10. C₂₃×#387	C₂	semidirect
11. C₂₃×#376	C₂	semidirect
98. #2604	C₂₃	direct (N × H)
120. C₂×C₄×Q₈	C₂×C₂₃	semidirect
121. #400	C₂×C₂₃	semidirect
123. #395	C₂×C₂₃	semidirect
125. #389	C₂×C₂₃	semidirect
128. #387	C₂×C₂₃	semidirect
129. #376	C₂×C₂₃	semidirect
161. C₂₃	#2604	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×#126	2
C₂×C₄×Q₈×C₂₃	2
C₂₃×#400	2
C₂₃×#399	2
C₂₃×#395	2
C₂₃×#393	2
C₂₃×#389	2
C₂₃×#388	2
C₂₃×#387	2
C₂₃×#376	2
#2604	23

Subgroups — C23×#2604 #15670223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2604 #15670223