Subgroups — #15676042

Order 2944.

444 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic
2	#13092324	C₂	1	1	yes	maximal, characteristic
3	#13092315	C₂	1	1	yes	maximal, characteristic
4	#13092313	C₂	1	1	yes	maximal, characteristic
5	#13092302	C₂	1	2	yes	maximal
6	#13092300	C₂	1	2	yes	maximal
7	#13092284	C₂	1	2	yes	maximal
8	#13092281	C₂	1	2	yes	maximal
9	C₄⋊C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
10	#13092263	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#391	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	Q₈×Dic₂₃	C₄	1	8	yes	complemented
16	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₂₃⋊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	C₂₃×#95	C₂²	1	2	yes	—
26	C₂₃×#95	C₂²	1	2	yes	—
27	C₂₃⋊#95	C₂²	1	1	yes	characteristic
28	C₂₃⋊#95	C₂²	1	1	yes	characteristic
29	C₂₃⋊#95	C₂²	1	1	yes	characteristic
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	2	yes	—
34	C₂₃⋊#95	C₂²	1	2	yes	—
35	C₂₃⋊#95	C₂²	1	2	yes	—
36	C₂×Q₈×C₂₃	—	2	2	no	—
37	C₂×Q₈×C₂₃	—	2	2	no	—
38	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	—
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×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₂₃	C₂³	1	2	yes	—
46	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
47	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
48	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
49	C₂²×Dic₂₃	C₂³	1	2	yes	—
50	C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₄²×C₂₃	—	2	2	no	—
52	C₄⋊Dic₂₃	—	2	2	no	—
53	C₄⋊Dic₂₃	C₂×C₄	1	8	yes	—
54	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
55	Dic₂₃⋊C₄	—	4	2	no	—
56	Dic₂₃⋊C₄	—	4	2	no	—
57	C₄×Dic₂₃	—	2	2	no	—
58	C₄×Dic₂₃	—	4	2	no	—
59	C₄×Dic₂₃	C₂×C₄	1	8	yes	—
60	C₄×Dic₂₃	C₂×C₄	1	4	yes	—
61	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
62	Q₈×C₂₃	—	2	8	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	1	no	—
66	C₂×C₄×C₂₃	—	2	1	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	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	1	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₂₃	#41	1	2	yes	—
77	C₂×C₄×C₂₃	#41	1	2	yes	—
78	C₂×Dic₂₃	—	2	1	no	—
79	C₂×Dic₂₃	—	2	2	no	—
80	C₂×Dic₂₃	—	2	1	no	—
81	C₂×Dic₂₃	—	2	1	no	—
82	C₂×Dic₂₃	—	2	2	no	—
83	C₂×Dic₂₃	—	2	1	no	—
84	C₂×Dic₂₃	—	2	1	no	—
85	C₂×Dic₂₃	—	2	1	no	—
86	C₂×Dic₂₃	—	2	2	no	—
87	C₂×Dic₂₃	—	2	2	no	—
88	C₂×Dic₂₃	—	2	1	no	—
89	C₂×Dic₂₃	—	2	1	no	—
90	C₂×Dic₂₃	—	2	1	no	—
91	C₂×Dic₂₃	—	2	2	no	—
92	C₂×Dic₂₃	—	2	1	no	—
93	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
94	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
95	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
96	#2320	—	23	1	no	maximal, Sylow
97	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
98	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
99	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
100	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
103	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
104	C₄×C₂₃	—	2	4	no	—
105	C₄×C₂₃	—	4	1	no	—
106	C₄×C₂₃	—	4	2	no	—
107	C₄×C₂₃	—	4	1	no	—
108	C₄×C₂₃	#117	1	4	yes	—
109	Dic₂₃	—	2	2	no	—
110	Dic₂₃	—	2	2	no	—
111	Dic₂₃	—	2	4	no	—
112	Dic₂₃	—	4	1	no	—
113	Dic₂₃	—	4	1	no	—
114	Dic₂₃	—	4	2	no	—
115	C₂×C₄×Q₈	—	23	1	no	—
116	#391	—	23	1	no	—
117	#391	D₂₃	1	1	yes	characteristic
118	#389	—	23	1	no	—
119	#389	—	23	1	no	—
120	#387	—	23	2	no	—
121	#387	—	23	2	no	—
122	#383	—	23	2	no	—
123	#382	—	23	2	no	—
124	C₄×C₄⋊C₄	—	23	1	no	—
125	#376	—	23	1	no	—
126	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
127	C₂×C₂₃	#519	1	1	yes	characteristic
128	C₂×C₂₃	#520	1	1	yes	characteristic
129	C₂×C₂₃	#541	1	1	yes	characteristic
130	C₂×C₂₃	#541	1	1	yes	characteristic
131	C₂×C₂₃	#542	1	1	yes	characteristic
132	C₂×C₂₃	#542	1	1	yes	characteristic
133	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
134	C₄×Q₈	—	23	8	no	—
135	C₂×C₄⋊C₄	—	23	2	no	—
136	C₂×C₄⋊C₄	—	23	1	no	—
137	C₂×C₄⋊C₄	—	23	1	no	—
138	C₂×C₄⋊C₄	—	23	2	no	—
139	C₂×C₄⋊C₄	—	23	2	no	—
140	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
141	C₂×C₄²	—	23	1	no	—
142	C₂×C₄²	—	23	2	no	—
143	C₂×C₄²	—	23	1	no	—
144	C₂×C₄²	—	23	2	no	—
145	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
146	#95	—	23	1	no	—
147	#95	—	23	1	no	—
148	#95	—	23	1	no	—
149	#95	—	23	1	no	—
150	#95	—	23	2	no	—
151	#95	—	23	2	no	—
152	#95	—	23	2	no	—
153	#95	—	23	2	no	—
154	#95	—	23	2	no	—
155	#95	C₂×D₂₃	1	2	yes	—
156	#95	C₂×D₂₃	1	2	yes	—
157	C₂₃	#2320	1	1	yes	characteristic, complemented, Sylow
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	C₄×D₂₃	1	4	yes	—
161	C₂²×C₄	—	23	1	no	—
162	C₂²×C₄	—	23	1	no	—
163	C₂²×C₄	—	23	2	no	—
164	C₂²×C₄	—	23	1	no	—
165	C₂²×C₄	—	23	1	no	—
166	C₂²×C₄	—	23	2	no	—
167	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
169	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
171	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	23	8	no	—
174	C₄⋊C₄	—	23	4	no	—
175	C₄⋊C₄	—	46	2	no	—
176	C₄⋊C₄	—	92	2	no	—
177	C₄⋊C₄	—	92	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	23	8	no	—
180	C₄²	—	23	4	no	—
181	C₄²	—	46	2	no	—
182	C₄²	—	92	2	no	—
183	C₂³	C₂³×D₂₃	1	1	yes	characteristic
184	Q₈	—	2	8	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	23	2	no	—
197	C₂×C₄	—	23	2	no	—
198	C₂×C₄	—	23	4	no	—
199	C₂×C₄	—	46	1	no	—
200	C₂×C₄	—	46	1	no	—
201	C₂×C₄	—	46	2	no	—
202	C₂×C₄	—	46	1	no	—
203	C₂×C₄	—	46	2	no	—
204	C₂×C₄	—	46	1	no	—
205	C₂×C₄	—	46	1	no	—
206	C₂×C₄	—	46	1	no	—
207	C₂×C₄	—	46	2	no	—
208	C₂×C₄	—	46	1	no	—
209	C₂×C₄	—	46	1	no	—
210	C₂×C₄	—	46	2	no	—
211	C₂×C₄	—	46	2	no	—
212	C₂×C₄	—	46	1	no	—
213	C₂×C₄	—	46	1	no	—
214	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
215	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
216	C₂×C₄	#67290	1	2	yes	—
217	C₂×C₄	#67290	1	2	yes	—
218	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
219	C₂²	C₂×#67290	1	1	yes	characteristic
220	C₂²	C₂×#67290	1	1	yes	characteristic
221	C₂²	#10625158	1	1	yes	characteristic
222	C₂²	#10625158	1	1	yes	characteristic
223	C₂²	#10625160	1	1	yes	characteristic
224	C₂²	#10625161	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	1	no	—
229	C₄	—	46	2	no	—
230	C₄	—	46	2	no	—
231	C₄	—	46	4	no	—
232	C₄	—	92	1	no	—
233	C₄	—	92	1	no	—
234	C₄	—	92	2	no	—
235	C₄	#10625048	1	4	yes	—
236	C₂	C₂×#10625048	1	1	yes	characteristic
237	C₂	#13092893	1	1	yes	characteristic
238	C₂	#13092899	1	1	yes	characteristic
239	C₂	#13093027	1	1	yes	characteristic
240	C₂	#13093028	1	1	yes	characteristic
241	C₂	#13093036	1	1	yes	characteristic
242	C₂	#13093040	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#391	C₂
socle, prime radical, cosocle	C₂³×C₂₃	C₂⁴
hypercenter	#391	D₂₃
hypocenter	C₂₃	#2320
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
15. Q₈×Dic₂₃	C₄	semidirect
157. C₂₃	#2320	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₂₃	2
#13092324	2
#13092315	2
#13092313	2
#13092302	2
#13092300	2
#13092284	2
#13092281	2
C₄⋊C₄×Dic₂₃	2
#13092263	2
C₂₃×#391	2
#2320	23