Subgroups — #15676316

Order 2944.

552 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625058	C₂	1	1	yes	maximal, characteristic
3	C₂×#10625058	C₂	1	1	yes	maximal, characteristic
4	#13092316	C₂	1	1	yes	maximal, characteristic
5	#13092314	C₂	1	2	yes	maximal
6	#13092312	C₂	1	2	yes	maximal
7	#13092282	C₂	1	1	yes	maximal, characteristic
8	#13092281	C₂	1	2	yes	maximal
9	C₄⋊C₄×Dic₂₃	C₂	1	2	yes	maximal
10	C₄×Dic₂₃⋊C₄	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#389	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₄₆	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×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
19	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
24	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
26	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
27	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
29	#10625058	—	2	4	no	—
30	#10625058	C₄	1	8	yes	complemented
31	C₄×Dic₄₆	—	2	4	no	—
32	C₂₃×#95	C₂²	1	2	yes	—
33	C₂₃⋊#95	C₂²	1	1	yes	characteristic
34	C₂₃⋊#95	C₂²	1	1	yes	characteristic
35	C₂₃⋊#95	C₂²	1	2	yes	—
36	C₂₃⋊#95	C₂²	1	2	yes	—
37	C₂₃⋊#95	C₂²	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
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	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
48	C₂²×Dic₂₃	C₂³	1	2	yes	—
49	C₂×Dic₄₆	C₂×C₄	1	4	yes	—
50	C₂×Dic₄₆	D₄	1	4	yes	—
51	C₂×Dic₄₆	Q₈	1	4	yes	—
52	C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₄⋊C₄×C₂₃	—	2	4	no	—
54	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
55	C₄²×C₂₃	—	2	2	no	—
56	C₄⋊Dic₂₃	—	2	2	no	—
57	C₄⋊Dic₂₃	—	2	4	no	—
58	Dic₂₃⋊C₄	—	2	2	no	—
59	Dic₂₃⋊C₄	—	2	4	no	—
60	Dic₂₃⋊C₄	—	2	4	no	—
61	Dic₂₃⋊C₄	—	2	2	no	—
62	Dic₂₃⋊C₄	C₂×C₄	1	8	yes	—
63	C₄×Dic₂₃	—	2	2	no	—
64	C₄×Dic₂₃	—	2	4	no	—
65	C₄×Dic₂₃	—	2	4	no	—
66	C₄×Dic₂₃	C₂×C₄	1	4	yes	—
67	C₄×Dic₂₃	C₂×C₄	1	8	yes	—
68	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₄×C₂₃	—	2	1	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
80	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
83	C₂×C₄×C₂₃	#41	1	2	yes	—
84	C₂×Dic₂₃	—	2	1	no	—
85	C₂×Dic₂₃	—	2	1	no	—
86	C₂×Dic₂₃	—	2	1	no	—
87	C₂×Dic₂₃	—	2	1	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	—	2	2	no	—
94	C₂×Dic₂₃	—	2	2	no	—
95	C₂×Dic₂₃	—	2	1	no	—
96	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
97	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
98	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
99	C₂×Dic₂₃	C₂×D₄	1	4	yes	—
100	C₂×Dic₂₃	C₂×Q₈	1	4	yes	—
101	C₂×Dic₂₃	#41	1	2	yes	—
102	Dic₄₆	C₄⋊C₄	1	16	yes	complemented
103	#2351	—	23	1	no	maximal, Sylow
104	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
106	C₂²×C₂₃	C₂²×Q₈	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	2₋¹⁺⁴	1	1	yes	characteristic
110	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
111	C₄×C₂₃	—	2	4	no	—
112	C₄×C₂₃	—	2	2	no	—
113	C₄×C₂₃	—	4	2	no	—
114	C₄×C₂₃	—	4	1	no	—
115	C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
116	Dic₂₃	—	2	2	no	—
117	Dic₂₃	—	2	2	no	—
118	Dic₂₃	—	2	2	no	—
119	Dic₂₃	—	4	1	no	—
120	Dic₂₃	—	4	2	no	—
121	Dic₂₃	C₂×C₄⋊C₄	1	8	yes	—
122	C₂×C₄×Q₈	—	23	1	no	—
123	C₂×C₄×Q₈	—	23	1	no	—
124	C₂×C₄×Q₈	—	23	1	no	—
125	#389	—	23	2	no	—
126	#389	—	23	1	no	—
127	#389	—	23	2	no	—
128	#389	D₂₃	1	1	yes	characteristic
129	#382	—	23	1	no	—
130	#382	—	23	2	no	—
131	C₄×C₄⋊C₄	—	23	2	no	—
132	C₄×C₄⋊C₄	—	23	1	no	—
133	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
135	C₂×C₂₃	#519	1	1	yes	characteristic
136	C₂×C₂₃	#548	1	1	yes	characteristic
137	C₂×C₂₃	#548	1	1	yes	characteristic
138	C₂×C₂₃	#557	1	1	yes	characteristic
139	C₂×C₂₃	#557	1	1	yes	characteristic
140	C₂²×Q₈	—	23	1	no	—
141	C₄×Q₈	—	23	8	no	—
142	C₄×Q₈	—	46	4	no	—
143	C₄×Q₈	—	46	4	no	—
144	C₂×C₄⋊C₄	—	23	2	no	—
145	C₂×C₄⋊C₄	—	23	2	no	—
146	C₂×C₄⋊C₄	—	23	2	no	—
147	C₂×C₄⋊C₄	—	23	1	no	—
148	C₂×C₄⋊C₄	—	23	1	no	—
149	C₂×C₄⋊C₄	—	23	1	no	—
150	C₂×C₄⋊C₄	—	23	2	no	—
151	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
153	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
154	C₂×C₄²	—	23	2	no	—
155	C₂×C₄²	—	23	1	no	—
156	C₂×C₄²	—	23	2	no	—
157	C₂×C₄²	—	23	2	no	—
158	C₂×C₄²	—	23	1	no	—
159	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
160	#95	—	23	1	no	—
161	#95	—	23	2	no	—
162	#95	—	23	1	no	—
163	#95	—	23	2	no	—
164	#95	—	23	2	no	—
165	#95	C₂×D₂₃	1	2	yes	—
166	C₂₃	#2351	1	1	yes	characteristic, complemented, Sylow
167	C₂×Q₈	—	23	4	no	—
168	C₂×Q₈	—	23	4	no	—
169	C₂×Q₈	—	23	4	no	—
170	C₂²×C₄	—	23	2	no	—
171	C₂²×C₄	—	23	1	no	—
172	C₂²×C₄	—	23	1	no	—
173	C₂²×C₄	—	23	1	no	—
174	C₂²×C₄	—	23	1	no	—
175	C₂²×C₄	—	23	2	no	—
176	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
177	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
179	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	23	8	no	—
184	C₄⋊C₄	—	46	4	no	—
185	C₄⋊C₄	—	46	4	no	—
186	C₄⋊C₄	—	46	2	no	—
187	C₄⋊C₄	—	46	4	no	—
188	C₄⋊C₄	—	46	2	no	—
189	C₄⋊C₄	—	46	2	no	—
190	C₄⋊C₄	C₄×D₂₃	1	4	yes	—
191	C₄²	—	2	2	no	—
192	C₄²	—	23	8	no	—
193	C₄²	—	23	4	no	—
194	C₄²	—	46	4	no	—
195	C₄²	—	46	4	no	—
196	C₄²	—	46	2	no	—
197	C₂³	C₂³×D₂₃	1	1	yes	characteristic
198	Q₈	—	23	16	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	23	4	no	—
210	C₂×C₄	—	23	4	no	—
211	C₂×C₄	—	23	4	no	—
212	C₂×C₄	—	23	2	no	—
213	C₂×C₄	—	23	2	no	—
214	C₂×C₄	—	23	2	no	—
215	C₂×C₄	—	46	1	no	—
216	C₂×C₄	—	46	1	no	—
217	C₂×C₄	—	46	1	no	—
218	C₂×C₄	—	46	1	no	—
219	C₂×C₄	—	46	1	no	—
220	C₂×C₄	—	46	1	no	—
221	C₂×C₄	—	46	1	no	—
222	C₂×C₄	—	46	2	no	—
223	C₂×C₄	—	46	1	no	—
224	C₂×C₄	—	46	2	no	—
225	C₂×C₄	—	46	2	no	—
226	C₂×C₄	—	46	1	no	—
227	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
228	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
229	C₂×C₄	D₄×D₂₃	1	2	yes	—
230	C₂×C₄	#67290	1	2	yes	—
231	C₂×C₄	Q₈×D₂₃	1	2	yes	—
232	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
233	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
234	C₂²	C₂×#67290	1	1	yes	characteristic
235	C₂²	C₂×Q₈×D₂₃	1	1	yes	characteristic
236	C₂²	#10625158	1	1	yes	characteristic
237	C₂²	#41×D₂₃	1	1	yes	characteristic
238	C₂²	#10625161	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	23	8	no	—
244	C₄	—	46	2	no	—
245	C₄	—	46	2	no	—
246	C₄	—	46	2	no	—
247	C₄	—	92	1	no	—
248	C₄	—	92	2	no	—
249	C₄	C₄⋊C₄×D₂₃	1	4	yes	—
250	C₂	C₂×C₄⋊C₄×D₂₃	1	1	yes	characteristic
251	C₂	C₄×#67290	1	1	yes	characteristic
252	C₂	#13092893	1	1	yes	characteristic
253	C₂	#13093073	1	1	yes	characteristic
254	C₂	#13093077	1	1	yes	characteristic
255	C₂	#13093142	1	1	yes	characteristic
256	C₂	#13093144	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#389	C₂
socle, prime radical, cosocle	C₂³×C₂₃	C₂⁴
hypercenter	#389	D₂₃
hypocenter	C₂₃	#2351
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
30. #10625058	C₄	semidirect
102. Dic₄₆	C₄⋊C₄	semidirect
166. C₂₃	#2351	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₄₆	2
C₂×#10625058	2
#13092316	2
#13092314	2
#13092312	2
#13092282	2
#13092281	2
C₄⋊C₄×Dic₂₃	2
C₄×Dic₂₃⋊C₄	2
C₂₃×#389	2
#2351	23