Subgroups — #15682348

Order 2944.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 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	#13092546	C₂	1	4	yes	maximal, complemented
3	#13092544	C₂	1	2	yes	maximal, complemented
4	#13092541	C₂	1	2	yes	maximal, complemented
5	#13092440	C₂	1	2	yes	maximal
6	C₄×#67267	C₂	1	1	yes	maximal, characteristic
7	M₁₆×Dic₂₃	C₂	1	2	yes	maximal
8	C₂₃×#423	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₄₆	C₄	1	1	yes	characteristic
10	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₂₃	C₄	1	1	yes	characteristic
13	C₂×Dic₂₃⋊C₄	C₄	1	2	yes	—
14	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
15	C₂×#67267	C₂²	1	1	yes	characteristic
16	C₂×#67267	C₂²	1	1	yes	characteristic
17	C₄×Dic₄₆	C₄	1	1	yes	characteristic
18	C₄×Dic₄₆	C₄	1	2	yes	—
19	C₄×Dic₄₆	C₄	1	1	yes	characteristic
20	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
21	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
22	C₄×Dic₄₆	C₂²	1	2	yes	—
23	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
25	#10624997	C₂²	1	4	yes	—
26	Dic₂₃⋊C₈	C₂²	1	4	yes	—
27	Dic₂₃⋊C₈	C₂²	1	4	yes	—
28	C₈×Dic₂₃	C₂²	1	4	yes	—
29	#10624985	C₂²	1	2	yes	—
30	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
31	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
35	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
36	C₂×Dic₄₆	—	2	4	no	—
37	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
38	C₂×Dic₄₆	C₂×C₄	1	2	yes	—
39	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
40	M₁₆×C₂₃	—	2	4	no	—
41	C₂×C₈×C₂₃	C₂³	1	4	yes	—
42	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
48	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
49	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
50	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
51	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
52	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
53	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
54	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
55	C₄×Dic₂₃	C₂³	1	2	yes	—
56	C₄×Dic₂₃	C₂³	1	2	yes	—
57	#67267	—	2	2	no	—
58	#67267	Q₈	1	4	yes	complemented
59	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
60	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
61	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	4	1	no	—
64	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₂₃	#41	1	2	yes	—
71	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₂₃	—	2	4	no	—
73	C₂×Dic₂₃	—	4	2	no	—
74	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
75	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
76	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
77	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
78	Dic₄₆	—	2	2	no	—
79	Dic₄₆	—	2	4	no	—
80	Dic₄₆	M₁₆	1	4	yes	complemented
81	C₈×C₂₃	—	2	4	no	—
82	C₂₃⋊C₈	—	2	2	no	—
83	C₂₃⋊C₈	C₂×Q₈	1	4	yes	complemented
84	Q₈×M₁₆	—	23	1	no	maximal, Sylow
85	C₂²×C₂₃	—	2	1	no	—
86	C₂²×C₂₃	—	2	1	no	—
87	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
88	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
89	C₄×C₂₃	—	2	1	no	—
90	C₄×C₂₃	—	2	1	no	—
91	C₄×C₂₃	—	2	1	no	—
92	C₄×C₂₃	C₄×Q₈	1	2	yes	—
93	C₄×C₂₃	C₂×M₁₆	1	2	yes	—
94	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
95	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
96	Dic₂₃	—	2	2	no	—
97	Dic₂₃	—	2	2	no	—
98	Dic₂₃	—	2	2	no	—
99	Dic₂₃	C₂×M₁₆	1	4	yes	—
100	C₂×C₄×Q₈	—	23	1	no	—
101	#446	—	23	2	no	—
102	#446	—	23	4	no	—
103	C₈×Q₈	—	23	2	no	—
104	#423	—	23	2	no	—
105	#423	D₂₃	1	1	yes	characteristic
106	C₄×M₁₆	—	23	2	no	—
107	C₄×M₁₆	—	23	1	no	—
108	C₂×C₂₃	—	2	2	no	—
109	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
110	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
111	C₂×C₂₃	#568	1	1	yes	characteristic
112	C₂²×Q₈	—	23	1	no	—
113	C₂×M₁₆	—	23	1	no	—
114	C₂×M₁₆	—	23	1	no	—
115	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
116	C₄×Q₈	—	23	1	no	—
117	C₄×Q₈	—	23	2	no	—
118	C₄×Q₈	—	23	1	no	—
119	C₄×Q₈	—	23	1	no	—
120	C₄×Q₈	—	23	2	no	—
121	C₄×Q₈	—	23	1	no	—
122	C₂×C₄⋊C₄	—	23	2	no	—
123	C₂×C₄⋊C₄	—	23	1	no	—
124	C₂×C₄²	—	23	2	no	—
125	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
126	C₄⋊C₈	—	23	4	no	—
127	C₄⋊C₈	—	23	4	no	—
128	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
129	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
130	#97	—	23	4	no	—
131	#97	—	23	2	no	—
132	C₄×C₈	—	23	4	no	—
133	C₄×C₈	—	23	2	no	—
134	C₂₃	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
135	C₂×Q₈	—	23	1	no	—
136	C₂×Q₈	—	23	2	no	—
137	C₂×Q₈	—	23	1	no	—
138	C₂×Q₈	—	46	4	no	—
139	C₂²×C₄	—	23	2	no	—
140	C₂²×C₄	—	23	2	no	—
141	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
142	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
144	M₁₆	—	2	4	no	—
145	M₁₆	—	23	4	no	—
146	M₁₆	—	46	2	no	—
147	C₂×C₈	—	23	2	no	—
148	C₂×C₈	—	23	2	no	—
149	C₂×C₈	C₂²×D₂₃	1	4	yes	—
150	C₄⋊C₄	—	23	2	no	—
151	C₄⋊C₄	—	23	2	no	—
152	C₄⋊C₄	—	23	2	no	—
153	C₄⋊C₄	—	23	2	no	—
154	C₄⋊C₄	—	23	2	no	—
155	C₄⋊C₄	—	23	1	no	—
156	C₄⋊C₄	—	23	1	no	—
157	C₄²	—	23	2	no	—
158	C₄²	—	23	2	no	—
159	C₄²	—	23	2	no	—
160	C₄²	—	23	2	no	—
161	C₄²	C₄×D₂₃	1	1	yes	characteristic
162	C₄²	C₄×D₂₃	1	1	yes	characteristic
163	C₄²	C₂²×D₂₃	1	1	yes	characteristic
164	C₄²	C₂²×D₂₃	1	1	yes	characteristic
165	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
166	Q₈	—	23	4	no	—
167	Q₈	—	46	4	no	—
168	Q₈	—	46	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	23	2	no	—
172	C₂×C₄	—	23	2	no	—
173	C₂×C₄	—	23	2	no	—
174	C₂×C₄	—	23	2	no	—
175	C₂×C₄	—	46	4	no	—
176	C₂×C₄	—	92	2	no	—
177	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
182	C₂×C₄	#67290	1	2	yes	—
183	C₂×C₄	Q₈×D₂₃	1	2	yes	—
184	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	23	4	no	—
187	C₈	—	46	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	#10625058	1	2	yes	—
191	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	23	4	no	—
196	C₄	—	46	2	no	—
197	C₄	—	46	2	no	—
198	C₄	—	46	2	no	—
199	C₄	#10625058	1	2	yes	—
200	C₄	M₁₆×D₂₃	1	2	yes	—
201	C₄	C₂×#67290	1	1	yes	characteristic
202	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₂×#10625058	1	1	yes	characteristic
205	C₂	C₂×M₁₆×D₂₃	1	1	yes	characteristic
206	C₂	#13093183	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#423	C₂
prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#423	D₂₃
hypocenter	C₂₃	Q₈×M₁₆
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
2. #13092546	C₂	semidirect
3. #13092544	C₂	semidirect
4. #13092541	C₂	semidirect
58. #67267	Q₈	semidirect
80. Dic₄₆	M₁₆	semidirect
83. C₂₃⋊C₈	C₂×Q₈	semidirect
134. C₂₃	Q₈×M₁₆	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
#13092546	2
#13092544	2
#13092541	2
#13092440	2
C₄×#67267	2
M₁₆×Dic₂₃	2
C₂₃×#423	2
Q₈×M₁₆	23