Subgroups — #15676281

Order 2944.

504 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625063	C₂	1	1	yes	maximal, characteristic
2	#13092316	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092309	C₂	1	2	yes	maximal
4	#13092307	C₂	1	2	yes	maximal
5	#13092282	C₂	1	2	yes	maximal, complemented
6	#13092274	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×(C₂×D₂₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
10	#13092262	C₂	1	2	yes	maximal, complemented
11	C₂₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
20	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
25	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
26	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
28	#10625063	C₄	1	8	yes	complemented
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃⋊#95	C₂²	1	1	yes	characteristic
31	C₂₃⋊#95	C₂²	1	2	yes	—
32	C₂₃⋊#95	C₂²	1	2	yes	—
33	C₂₃⋊#95	C₂²	1	1	yes	characteristic
34	C₂₃⋊#95	C₂²	1	2	yes	—
35	C₂₃⋊#95	C₂²	1	2	yes	—
36	C₂³×D₂₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₂₃	C₂³	1	2	yes	—
43	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₂₃	C₂³	1	2	yes	—
45	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
46	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₂₃	—	2	2	no	—
48	C₂×C₄×D₂₃	—	2	2	no	—
49	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
50	C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₄⋊C₄×C₂₃	—	4	2	no	—
52	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
53	C₄²×C₂₃	—	2	2	no	—
54	(C₂×D₂₃)⋊C₄	—	2	2	no	—
55	(C₂×D₂₃)⋊C₄	—	2	2	no	—
56	(C₂×D₂₃)⋊C₄	—	4	2	no	—
57	(C₂×D₂₃)⋊C₄	C₂×C₄	1	8	yes	—
58	C₄⋊Dic₂₃	—	2	2	no	—
59	C₄⋊Dic₂₃	—	4	2	no	—
60	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
61	C₄×Dic₂₃	—	2	4	no	—
62	C₄×Dic₂₃	—	2	2	no	—
63	C₄×Dic₂₃	—	2	4	no	—
64	C₄×Dic₂₃	C₂×C₄	1	8	yes	—
65	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
66	C₂²×D₂₃	—	2	1	no	—
67	C₂²×D₂₃	—	2	1	no	—
68	C₂²×D₂₃	—	2	1	no	—
69	C₂²×D₂₃	—	2	1	no	—
70	C₂²×D₂₃	—	2	1	no	—
71	C₂²×D₂₃	—	2	1	no	—
72	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	1	no	—
79	C₂×C₄×C₂₃	—	2	1	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
84	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₂₃	#41	1	2	yes	—
86	C₂×C₄×C₂₃	#41	1	2	yes	—
87	C₂×C₄×C₂₃	#41	1	2	yes	—
88	C₂×Dic₂₃	—	2	2	no	—
89	C₂×Dic₂₃	—	2	1	no	—
90	C₂×Dic₂₃	—	2	2	no	—
91	C₂×Dic₂₃	—	2	1	no	—
92	C₂×Dic₂₃	—	2	1	no	—
93	C₂×Dic₂₃	—	2	1	no	—
94	C₂×Dic₂₃	—	2	1	no	—
95	C₂×Dic₂₃	—	2	2	no	—
96	C₂×Dic₂₃	—	2	2	no	—
97	C₂×Dic₂₃	—	2	1	no	—
98	C₂×Dic₂₃	—	2	1	no	—
99	C₂×Dic₂₃	—	2	2	no	—
100	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
101	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
102	C₂×Dic₂₃	#41	1	2	yes	—
103	C₄×D₂₃	—	2	8	no	—
104	#2346	—	23	1	no	maximal, Sylow
105	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
106	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
107	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
108	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
109	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
110	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
112	C₂×D₂₃	—	2	2	no	—
113	C₂×D₂₃	—	2	2	no	—
114	C₂×D₂₃	—	2	2	no	—
115	C₂×D₂₃	—	4	1	no	—
116	C₂×D₂₃	—	4	1	no	—
117	C₂×D₂₃	—	4	1	no	—
118	C₂×D₂₃	—	4	1	no	—
119	C₄×C₂₃	—	2	4	no	—
120	C₄×C₂₃	—	2	2	no	—
121	C₄×C₂₃	—	4	2	no	—
122	C₄×C₂₃	—	4	1	no	—
123	C₄×C₂₃	#117	1	4	yes	—
124	Dic₂₃	—	2	2	no	—
125	Dic₂₃	—	2	4	no	—
126	Dic₂₃	—	2	2	no	—
127	Dic₂₃	—	4	2	no	—
128	Dic₂₃	—	4	1	no	—
129	C₂×#117	—	23	1	no	—
130	#389	—	23	1	no	—
131	#389	D₂₃	1	1	yes	characteristic, complemented
132	#388	—	23	2	no	—
133	#388	—	23	2	no	—
134	#382	—	23	2	no	—
135	#380	—	23	1	no	—
136	C₄×C₄⋊C₄	—	23	1	no	—
137	C₄×C₄⋊C₄	—	23	1	no	—
138	C₄×C₂²⋊C₄	—	23	1	no	—
139	#376	—	23	2	no	—
140	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
141	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
142	C₂×C₂₃	#520	1	1	yes	characteristic
143	C₂×C₂₃	#552	1	1	yes	characteristic
144	C₂×C₂₃	#552	1	1	yes	characteristic
145	C₂×C₂₃	#555	1	1	yes	characteristic
146	C₂×C₂₃	#555	1	1	yes	characteristic
147	D₂₃	—	4	2	no	—
148	C₂³×C₄	—	23	1	no	—
149	#117	—	23	8	no	—
150	C₂×C₄⋊C₄	—	23	1	no	—
151	C₂×C₄⋊C₄	—	23	1	no	—
152	C₂×C₄⋊C₄	—	23	2	no	—
153	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
154	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
155	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	—	23	2	no	—
157	C₂×C₂²⋊C₄	—	23	1	no	—
158	C₂×C₂²⋊C₄	—	23	1	no	—
159	C₂×C₂²⋊C₄	—	23	2	no	—
160	C₂×C₄²	—	23	2	no	—
161	C₂×C₄²	—	23	2	no	—
162	C₂×C₄²	—	23	2	no	—
163	C₂×C₄²	—	23	1	no	—
164	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
165	#95	—	23	1	no	—
166	#95	—	23	2	no	—
167	#95	—	23	2	no	—
168	#95	—	23	1	no	—
169	#95	—	23	2	no	—
170	#95	—	23	2	no	—
171	#95	C₂×D₂₃	1	2	yes	—
172	C₂₃	#2346	1	1	yes	characteristic, complemented, Sylow
173	C₂⁴	—	23	1	no	—
174	C₂²×C₄	—	23	1	no	—
175	C₂²×C₄	—	23	1	no	—
176	C₂²×C₄	—	23	2	no	—
177	C₂²×C₄	—	23	4	no	—
178	C₂²×C₄	—	23	2	no	—
179	C₂²×C₄	—	23	1	no	—
180	C₂²×C₄	—	46	2	no	—
181	C₂²×C₄	—	46	2	no	—
182	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
184	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
185	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	23	4	no	—
190	C₄⋊C₄	—	46	2	no	—
191	C₄⋊C₄	—	92	2	no	—
192	C₄⋊C₄	C₄×D₂₃	1	4	yes	—
193	C₂²⋊C₄	—	23	8	no	—
194	C₂²⋊C₄	—	46	2	no	—
195	C₂²⋊C₄	—	46	2	no	—
196	C₂²⋊C₄	—	92	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	23	8	no	—
199	C₄²	—	46	4	no	—
200	C₄²	—	46	4	no	—
201	C₄²	—	46	2	no	—
202	C₂³	—	23	2	no	—
203	C₂³	—	46	1	no	—
204	C₂³	—	46	1	no	—
205	C₂³	—	46	1	no	—
206	C₂³	—	46	1	no	—
207	C₂³	—	46	1	no	—
208	C₂³	—	46	1	no	—
209	C₂³	C₂³×D₂₃	1	1	yes	characteristic
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	23	2	no	—
221	C₂×C₄	—	23	4	no	—
222	C₂×C₄	—	23	2	no	—
223	C₂×C₄	—	46	1	no	—
224	C₂×C₄	—	46	2	no	—
225	C₂×C₄	—	46	1	no	—
226	C₂×C₄	—	46	1	no	—
227	C₂×C₄	—	46	2	no	—
228	C₂×C₄	—	46	1	no	—
229	C₂×C₄	—	46	8	no	—
230	C₂×C₄	—	46	2	no	—
231	C₂×C₄	—	46	1	no	—
232	C₂×C₄	—	46	1	no	—
233	C₂×C₄	—	46	2	no	—
234	C₂×C₄	—	46	1	no	—
235	C₂×C₄	—	46	2	no	—
236	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
237	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
238	C₂×C₄	#67290	1	2	yes	—
239	C₂×C₄	#67292	1	2	yes	—
240	C₂×C₄	#67292	1	2	yes	—
241	C₂²	—	46	2	no	—
242	C₂²	—	46	2	no	—
243	C₂²	—	46	2	no	—
244	C₂²	—	92	1	no	—
245	C₂²	—	92	1	no	—
246	C₂²	—	92	1	no	—
247	C₂²	—	92	1	no	—
248	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
249	C₂²	C₂×#67290	1	1	yes	characteristic
250	C₂²	#10625155	1	1	yes	characteristic
251	C₂²	C₂×#67292	1	1	yes	characteristic
252	C₂²	C₂×#67292	1	1	yes	characteristic
253	C₂²	#41×D₂₃	1	1	yes	characteristic
254	C₂²	#10625161	1	1	yes	characteristic
255	C₄	—	2	4	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	2	no	—
259	C₄	—	46	2	no	—
260	C₄	—	46	4	no	—
261	C₄	—	46	2	no	—
262	C₄	—	92	2	no	—
263	C₄	—	92	1	no	—
264	C₄	#10625063	1	4	yes	—
265	C₂	—	92	2	no	—
266	C₂	C₂×#10625063	1	1	yes	characteristic
267	C₂	C₄×#67292	1	1	yes	characteristic
268	C₂	#13092896	1	1	yes	characteristic
269	C₂	#13093105	1	1	yes	characteristic
270	C₂	#13093109	1	1	yes	characteristic
271	C₂	#13093131	1	1	yes	characteristic
272	C₂	#13093133	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#389	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
prime radical	C₂³×D₂₃	C₂³
hypercenter	#389	D₂₃
hypocenter	C₂₃	#2346
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
2. #13092316	C₂	semidirect
5. #13092282	C₂	semidirect
7. C₄⋊C₄×Dic₂₃	C₂	semidirect
8. C₄×C₄⋊Dic₂₃	C₂	semidirect
10. #13092262	C₂	semidirect
11. C₂₃×#389	C₂	semidirect
28. #10625063	C₄	semidirect
131. #389	D₂₃	semidirect
172. C₂₃	#2346	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625063	2
#13092316	2
#13092309	2
#13092307	2
#13092282	2
#13092274	2
C₄⋊C₄×Dic₂₃	2
C₄×C₄⋊Dic₂₃	2
C₄×(C₂×D₂₃)⋊C₄	2
#13092262	2
C₂₃×#389	2
#2346	23