Subgroups — C₂₃×#2550 #15670169

Order 2944.

468 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#398	C₂	1	2	yes	maximal
4	C₂₃×#395	C₂	1	2	yes	maximal
5	C₂₃×#391	C₂	1	2	yes	maximal
6	C₂₃×#389	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#389	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#389	C₂	1	2	yes	maximal
9	C₂₃×#382	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
13	C₂₃×C₄⋊Q₈	—	2	4	no	—
14	C₂₃×#125	—	2	4	no	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
29	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
30	C₂₃×#95	C₂²	1	1	yes	characteristic
31	C₂₃×#95	C₂²	1	1	yes	characteristic
32	C₂₃×#95	C₂²	1	2	yes	—
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	C₂×Q₈×C₂₃	—	2	4	no	—
38	C₂×Q₈×C₂₃	—	2	4	no	—
39	C₂×Q₈×C₂₃	—	4	2	no	—
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₂₃	—	2	4	no	—
52	C₄⋊C₄×C₂₃	—	2	4	no	—
53	C₄⋊C₄×C₂₃	—	2	2	no	—
54	C₄⋊C₄×C₂₃	—	2	4	no	—
55	C₄⋊C₄×C₂₃	—	2	2	no	—
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	2	4	no	—
58	C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₄⋊C₄×C₂₃	—	4	2	no	—
60	C₄⋊C₄×C₂₃	D₄	1	4	yes	—
61	C₄⋊C₄×C₂₃	Q₈	1	4	yes	complemented
62	C₄²×C₂₃	—	2	2	no	—
63	C₄²×C₂₃	—	2	2	no	—
64	C₄²×C₂₃	—	4	2	no	—
65	C₄²×C₂₃	Q₈	1	4	yes	complemented
66	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
67	Q₈×C₂₃	—	4	8	no	—
68	C₂×C₄×C₂₃	—	2	1	no	—
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	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	1	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	1	no	—
80	C₂×C₄×C₂₃	—	2	1	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
93	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
95	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
96	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
97	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₂₃	#41	1	2	yes	—
100	C₂×C₄×C₂₃	#41	1	2	yes	—
101	#2550	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
103	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
104	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
105	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
106	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
107	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
108	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	—	2	4	no	—
112	C₄×C₂₃	—	2	2	no	—
113	C₄×C₂₃	—	2	2	no	—
114	C₄×C₂₃	—	2	4	no	—
115	C₄×C₂₃	—	4	1	no	—
116	C₄×C₂₃	—	4	2	no	—
117	C₄×C₂₃	—	4	1	no	—
118	C₄×C₂₃	—	4	2	no	—
119	C₄×C₂₃	C₄⋊Q₈	1	4	yes	—
120	C₂×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
121	C₂×#125	C₂×C₂₃	1	1	yes	characteristic
122	#398	C₂×C₂₃	1	2	yes	—
123	#395	C₂×C₂₃	1	2	yes	—
124	#391	C₂×C₂₃	1	2	yes	—
125	#389	C₂×C₂₃	1	2	yes	—
126	#389	C₂×C₂₃	1	1	yes	characteristic
127	#389	C₂×C₂₃	1	1	yes	characteristic
128	#382	C₂×C₂₃	1	1	yes	characteristic
129	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
130	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
131	C₂×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₂₃	#533	1	1	yes	characteristic
133	C₂×C₂₃	#544	1	1	yes	characteristic
134	C₂×C₂₃	#546	1	1	yes	characteristic
135	C₂×C₂₃	#548	1	1	yes	characteristic
136	C₂×C₂₃	#554	1	1	yes	characteristic
137	C₂×C₂₃	#557	1	1	yes	characteristic
138	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
139	C₄⋊Q₈	—	2	4	no	—
140	#125	—	2	4	no	—
141	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
143	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
144	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
149	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
150	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
154	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
155	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
156	#95	C₂²×C₂₃	1	1	yes	characteristic
157	#95	C₂²×C₂₃	1	1	yes	characteristic
158	#95	C₂²×C₂₃	1	2	yes	—
159	#95	C₂²×C₂₃	1	2	yes	—
160	#95	C₂²×C₂₃	1	2	yes	—
161	#95	C₂²×C₂₃	1	2	yes	—
162	#95	C₂²×C₂₃	1	2	yes	—
163	C₂₃	#2550	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
169	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
170	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
172	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
175	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
176	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
177	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	D₄×C₂₃	1	4	yes	—
188	C₄⋊C₄	Q₈×C₂₃	1	4	yes	complemented
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	4	2	no	—
192	C₄²	Q₈×C₂₃	1	4	yes	complemented
193	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
194	Q₈	—	4	8	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	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	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
220	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
221	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
222	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
223	C₂×C₄	C₂×Q₈×C₂₃	1	4	yes	—
224	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
225	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
226	C₂×C₄	#41×C₂₃	1	2	yes	—
227	C₂×C₄	#41×C₂₃	1	2	yes	—
228	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
229	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
230	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
231	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
232	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
233	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
234	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	4	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	1	no	—
245	C₄	C₂₃×C₄⋊Q₈	1	4	yes	—
246	C₂	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
247	C₂	C₂₃×#533	1	1	yes	characteristic
248	C₂	C₂₃×#544	1	1	yes	characteristic
249	C₂	C₂₃×#546	1	1	yes	characteristic
250	C₂	C₂₃×#548	1	1	yes	characteristic
251	C₂	C₂₃×#554	1	1	yes	characteristic
252	C₂	C₂₃×#557	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×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
61. C₄⋊C₄×C₂₃	Q₈	semidirect
65. C₄²×C₂₃	Q₈	semidirect
101. #2550	C₂₃	direct (N × H)
163. C₂₃	#2550	direct (N × H)
188. C₄⋊C₄	Q₈×C₂₃	semidirect
192. C₄²	Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×C₄⋊Q₈	2
C₂×C₂₃×#125	2
C₂₃×#398	2
C₂₃×#395	2
C₂₃×#391	2
C₂₃×#389	2
C₂₃×#382	2
C₄×C₄⋊C₄×C₂₃	2
#2550	23

Subgroups — C23×#2550 #15670169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2550 #15670169