Subgroups — #15671626

Order 2944.

656 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃⋊#95	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×Dic₂₃	C₂²	1	2	yes	—
8	C₂³×Dic₂₃	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
10	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
11	C₂×C₄⋊C₄×C₂₃	C₄	1	4	yes	complemented
12	C₂×C₄⋊Dic₂₃	—	2	2	no	—
13	C₂×C₄⋊Dic₂₃	—	2	2	no	—
14	C₂×C₄⋊Dic₂₃	C₄	1	4	yes	complemented
15	C₂×C₄×Dic₂₃	—	2	2	no	—
16	C₂×C₄×Dic₂₃	—	2	2	no	—
17	C₂×C₄×Dic₂₃	C₄	1	4	yes	complemented
18	C₂₃⋊#95	—	2	8	no	—
19	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	—	2	2	no	—
21	C₂²×C₄×C₂₃	—	2	2	no	—
22	C₂²×C₄×C₂₃	—	2	4	no	—
23	C₂²×C₄×C₂₃	—	2	4	no	—
24	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₂₃	D₄	1	2	yes	—
29	C₂²×C₄×C₂₃	D₄	1	2	yes	—
30	C₂²×C₄×C₂₃	D₄	1	2	yes	—
31	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
32	C₂²×Dic₂₃	—	2	2	no	—
33	C₂²×Dic₂₃	—	2	2	no	—
34	C₂²×Dic₂₃	—	2	2	no	—
35	C₂²×Dic₂₃	—	2	2	no	—
36	C₂²×Dic₂₃	—	2	2	no	—
37	C₂²×Dic₂₃	—	2	2	no	—
38	C₂²×Dic₂₃	—	2	4	no	—
39	C₂²×Dic₂₃	—	2	4	no	—
40	C₂²×Dic₂₃	C₂×C₄	1	4	yes	—
41	C₂²×Dic₂₃	C₂×C₄	1	4	yes	—
42	C₂²×Dic₂₃	D₄	1	4	yes	—
43	C₂²×Dic₂₃	Q₈	1	4	yes	—
44	C₄⋊C₄×C₂₃	—	2	8	no	—
45	C₄⋊Dic₂₃	—	2	8	no	—
46	C₄×Dic₂₃	—	2	8	no	—
47	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
48	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
50	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂³×C₂₃	C₂×D₄	1	2	yes	—
52	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
55	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
56	C₂³×C₂₃	#41	1	2	yes	—
57	C₂³×C₂₃	#41	1	2	yes	—
58	C₂×C₄×C₂₃	—	2	4	no	—
59	C₂×C₄×C₂₃	—	2	8	no	—
60	C₂×C₄×C₂₃	—	4	4	no	—
61	C₂×C₄×C₂₃	—	4	2	no	—
62	C₂×C₄×C₂₃	—	4	2	no	—
63	C₂×C₄×C₂₃	C₄²	1	4	yes	—
64	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
65	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
66	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
67	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
68	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
69	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
70	C₂×Dic₂₃	—	2	4	no	—
71	C₂×Dic₂₃	—	2	4	no	—
72	C₂×Dic₂₃	—	2	4	no	—
73	C₂×Dic₂₃	—	2	4	no	—
74	C₂×Dic₂₃	—	2	4	no	—
75	C₂×Dic₂₃	—	2	4	no	—
76	C₂×Dic₂₃	—	2	4	no	—
77	C₂×Dic₂₃	—	2	8	no	—
78	C₂×Dic₂₃	—	4	4	no	—
79	C₂×Dic₂₃	—	4	2	no	—
80	C₂×Dic₂₃	—	4	2	no	—
81	C₂×Dic₂₃	C₄⋊C₄	1	8	yes	complemented
82	C₄⋊#95	—	23	1	no	maximal, Sylow
83	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
84	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
88	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₂₃	#117	1	2	yes	—
92	C₂²×C₂₃	C₄×D₄	1	2	yes	—
93	C₂²×C₂₃	C₄×D₄	1	2	yes	—
94	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
95	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
96	C₂²×C₂₃	#121	1	2	yes	—
97	C₂²×C₂₃	#121	1	2	yes	—
98	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
99	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
100	C₂²×C₂₃	#124	1	1	yes	characteristic
101	C₂²×C₂₃	#124	1	1	yes	characteristic
102	C₂²×C₂₃	#125	1	2	yes	—
103	C₂²×C₂₃	#127	1	1	yes	characteristic
104	C₂²×C₂₃	C₄⋊Q₈	1	2	yes	—
105	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
106	C₄×C₂₃	—	4	4	no	—
107	C₄×C₂₃	#95	1	8	yes	complemented
108	Dic₂₃	—	2	8	no	—
109	Dic₂₃	—	4	4	no	—
110	C₂²×C₄⋊C₄	—	23	1	no	—
111	C₂²×C₄⋊C₄	D₂₃	1	1	yes	characteristic
112	C₂²×C₄²	—	23	1	no	—
113	C₂×#95	—	23	4	no	—
114	C₂×C₂₃	C₂×#95	1	1	yes	characteristic
115	C₂×C₂₃	C₄×C₄⋊C₄	1	2	yes	—
116	C₂×C₂₃	#380	1	2	yes	—
117	C₂×C₂₃	#382	1	1	yes	characteristic
118	C₂×C₂₃	#384	1	1	yes	characteristic
119	C₂×C₂₃	#389	1	2	yes	—
120	C₂×C₂₃	#389	1	2	yes	—
121	C₂×C₂₃	#390	1	1	yes	characteristic
122	C₂×C₂₃	#390	1	1	yes	characteristic
123	C₂×C₂₃	#391	1	1	yes	characteristic
124	C₂×C₂₃	#391	1	1	yes	characteristic
125	C₂³×C₄	—	23	2	no	—
126	C₂³×C₄	—	23	2	no	—
127	C₂³×C₄	C₂×D₂₃	1	2	yes	—
128	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	23	4	no	—
132	C₂×C₄⋊C₄	—	46	2	no	—
133	C₂×C₄⋊C₄	—	46	2	no	—
134	C₂×C₄⋊C₄	Dic₂₃	1	4	yes	complemented
135	C₂×C₄²	—	23	4	no	—
136	C₂×C₄²	—	46	2	no	—
137	C₂×C₄²	—	46	2	no	—
138	#95	—	46	8	no	—
139	C₂₃	C₄⋊#95	1	1	yes	characteristic, complemented, Sylow
140	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	23	4	no	—
146	C₂²×C₄	—	23	4	no	—
147	C₂²×C₄	—	23	4	no	—
148	C₂²×C₄	—	23	4	no	—
149	C₂²×C₄	—	46	2	no	—
150	C₂²×C₄	—	46	2	no	—
151	C₂²×C₄	—	46	2	no	—
152	C₂²×C₄	—	46	2	no	—
153	C₂²×C₄	—	46	2	no	—
154	C₂²×C₄	—	46	4	no	—
155	C₂²×C₄	—	46	4	no	—
156	C₂²×C₄	—	46	2	no	—
157	C₂²×C₄	Dic₄₆	1	2	yes	—
158	C₂²×C₄	C₄×D₂₃	1	2	yes	—
159	C₂²×C₄	C₄×D₂₃	1	2	yes	—
160	C₂²×C₄	D₉₂	1	2	yes	—
161	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
162	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
163	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
164	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	46	8	no	—
167	C₄²	—	46	8	no	—
168	C₂³	C₂×Dic₄₆	1	1	yes	characteristic
169	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
170	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
171	C₂³	C₂×D₉₂	1	1	yes	characteristic
172	C₂³	D₄×D₂₃	1	2	yes	—
173	C₂³	#67290	1	2	yes	—
174	C₂³	Q₈×D₂₃	1	2	yes	—
175	C₂³	#67292	1	2	yes	—
176	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
177	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
178	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	23	8	no	—
185	C₂×C₄	—	46	4	no	—
186	C₂×C₄	—	46	4	no	—
187	C₂×C₄	—	46	4	no	—
188	C₂×C₄	—	46	4	no	—
189	C₂×C₄	—	46	4	no	—
190	C₂×C₄	—	46	4	no	—
191	C₂×C₄	—	46	8	no	—
192	C₂×C₄	—	46	4	no	—
193	C₂×C₄	—	92	4	no	—
194	C₂×C₄	—	92	2	no	—
195	C₂×C₄	—	92	2	no	—
196	C₂×C₄	C₄×Dic₂₃	1	4	yes	—
197	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
198	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
199	C₂×C₄	C₄⋊Dic₂₃	1	4	yes	—
200	C₂×C₄	(C₂×D₂₃)⋊C₄	1	4	yes	—
201	C₂×C₄	(C₂×D₂₃)⋊C₄	1	4	yes	—
202	C₂×C₄	C₂²⋊Dic₂₃	1	4	yes	—
203	C₂²	#10625058	1	2	yes	—
204	C₂²	#10625059	1	2	yes	—
205	C₂²	#10625061	1	2	yes	—
206	C₂²	C₄⋊C₄×D₂₃	1	2	yes	—
207	C₂²	#10625063	1	2	yes	—
208	C₂²	#10625064	1	2	yes	—
209	C₂²	#10625066	1	2	yes	—
210	C₂²	#10625068	1	2	yes	—
211	C₂²	C₂×C₄×Dic₂₃	1	1	yes	characteristic
212	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
213	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
214	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
215	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
216	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
217	C₂²	D₄×Dic₂₃	1	2	yes	—
218	C₂²	#10625107	1	1	yes	characteristic
219	C₂²	#10625109	1	2	yes	—
220	C₂²	#10625111	1	1	yes	characteristic
221	C₂²	#10625115	1	1	yes	characteristic
222	C₂²	Q₈×Dic₂₃	1	2	yes	—
223	C₂²	#10625117	1	2	yes	—
224	C₂²	#10625118	1	1	yes	characteristic
225	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
226	C₄	—	4	4	no	—
227	C₄	—	46	8	no	—
228	C₄	—	92	4	no	—
229	C₄	C₂₃⋊#95	1	8	yes	complemented
230	C₂	C₂×C₂₃⋊#95	1	1	yes	characteristic
231	C₂	C₄⋊C₄×Dic₂₃	1	2	yes	—
232	C₂	#13092274	1	2	yes	—
233	C₂	#13092282	1	1	yes	characteristic
234	C₂	#13092286	1	1	yes	characteristic
235	C₂	#13092313	1	2	yes	—
236	C₂	#13092316	1	2	yes	—
237	C₂	#13092318	1	1	yes	characteristic
238	C₂	#13092320	1	1	yes	characteristic
239	C₂	#13092322	1	1	yes	characteristic
240	C₂	#13092324	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂×C₄²
Fitting	C₂²×C₄⋊C₄×C₂₃	C₂
socle, prime radical, cosocle	C₂⁴×C₂₃	C₂³
hypercenter	C₂²×C₄⋊C₄	D₂₃
hypocenter	C₂₃	C₄⋊#95
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
11. C₂×C₄⋊C₄×C₂₃	C₄	semidirect
14. C₂×C₄⋊Dic₂₃	C₄	semidirect
17. C₂×C₄×Dic₂₃	C₄	semidirect
81. C₂×Dic₂₃	C₄⋊C₄	semidirect
107. C₄×C₂₃	#95	semidirect
134. C₂×C₄⋊C₄	Dic₂₃	semidirect
139. C₂₃	C₄⋊#95	semidirect
229. C₄	C₂₃⋊#95	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₂₃	2
C₂²×C₄×Dic₂₃	2
C₂×C₂₃⋊#95	2
C₂²×C₄⋊C₄×C₂₃	2
C₄⋊#95	23