Subgroups — #15681986

Order 2944.

492 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#67267	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625097	C₂	1	1	yes	maximal, characteristic
3	#13092460	C₂	1	4	yes	maximal, complemented
4	#13092457	C₂	1	4	yes	maximal, complemented
5	#13092453	C₂	1	4	yes	maximal, complemented
6	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#117	—	2	1	no	—
9	C₂₃×#117	—	2	1	no	—
10	C₂₃×#117	C₂²	1	2	yes	—
11	C₂₃×#117	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
14	C₂×C₄²×C₂₃	—	2	1	no	—
15	C₂×C₂²⋊Dic₂₃	—	2	1	no	—
16	#10625097	—	2	1	no	—
17	#10625097	—	2	1	no	—
18	#10625097	C₂²	1	2	yes	—
19	#10625097	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
21	C₂×C₄×Dic₂₃	—	2	1	no	—
22	C₂×#67267	C₄	1	8	yes	complemented
23	C₂×#67267	C₂²	1	2	yes	—
24	C₂×#67267	C₂²	1	2	yes	—
25	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
26	#10624990	C₂²	1	8	yes	complemented
27	#10624989	C₂²	1	8	yes	complemented
28	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	—	2	2	no	—
30	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	Q₈	1	4	yes	—
35	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×Dic₂₃	—	2	2	no	—
39	C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
41	C₂²⋊C₄×C₂₃	—	2	2	no	—
42	C₂²⋊C₄×C₂₃	—	2	2	no	—
43	C₄²×C₂₃	—	2	2	no	—
44	C₄²×C₂₃	—	2	2	no	—
45	C₂²⋊Dic₂₃	—	2	2	no	—
46	C₂²⋊Dic₂₃	—	2	2	no	—
47	C₄⋊Dic₂₃	—	2	2	no	—
48	C₄⋊Dic₂₃	C₂³	1	4	yes	—
49	C₄×Dic₂₃	—	2	2	no	—
50	C₄×Dic₂₃	—	2	2	no	—
51	#67267	C₂×C₄	1	8	yes	complemented
52	#67267	C₂×C₄	1	8	yes	complemented
53	C₂×C₂₃⋊C₈	C₂×C₄	1	8	yes	complemented
54	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
55	C₂³×C₂₃	—	4	2	no	—
56	C₂³×C₂₃	C₄⋊C₄	1	4	yes	—
57	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
59	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₂₃	—	2	4	no	—
61	C₂×C₄×C₂₃	—	4	4	no	—
62	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
63	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
64	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
65	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
72	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
75	C₂×Dic₂₃	—	2	4	no	—
76	C₂×Dic₂₃	—	4	4	no	—
77	C₂₃⋊C₈	C₂²×C₄	1	8	yes	—
78	#2911	—	23	1	no	maximal, Sylow
79	C₂²×C₂₃	—	2	2	no	—
80	C₂²×C₂₃	—	2	2	no	—
81	C₂²×C₂₃	—	2	2	no	—
82	C₂²×C₂₃	—	4	4	no	—
83	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
86	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
87	C₄×C₂₃	—	4	4	no	—
88	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
89	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
92	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
93	Dic₂₃	—	4	4	no	—
94	C₂²×M₁₆	—	23	1	no	—
95	C₂×#117	—	23	1	no	—
96	C₂×#117	D₂₃	1	1	yes	characteristic
97	#428	—	23	4	no	—
98	#428	—	23	4	no	—
99	#427	—	23	4	no	—
100	C₂×C₂₃	—	2	2	no	—
101	C₂×C₂₃	—	2	2	no	—
102	C₂×C₂₃	—	2	2	no	—
103	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₂₃	#575	1	2	yes	—
105	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
106	C₂×M₁₆	—	23	8	no	—
107	C₂×M₁₆	—	23	2	no	—
108	C₂×M₁₆	—	23	2	no	—
109	C₂²×C₈	—	23	2	no	—
110	#117	—	2	1	no	—
111	#117	—	2	1	no	—
112	#117	—	23	2	no	—
113	#117	—	23	2	no	—
114	#117	—	46	1	no	—
115	#117	—	46	1	no	—
116	#117	C₂×D₂₃	1	2	yes	—
117	#117	C₂×D₂₃	1	2	yes	—
118	C₂×C₄⋊C₄	—	23	2	no	—
119	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
120	C₂×C₂²⋊C₄	—	2	1	no	—
121	C₂×C₂²⋊C₄	—	46	1	no	—
122	C₂×C₄²	—	2	1	no	—
123	C₂×C₄²	—	46	1	no	—
124	#107	—	23	8	no	—
125	#106	—	23	8	no	—
126	C₂₃	#2911	1	1	yes	characteristic, complemented, Sylow
127	C₂⁴	C₂²⋊D₂₃	1	1	yes	characteristic
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	46	2	no	—
130	C₂²×C₄	Dic₄₆	1	4	yes	—
131	C₂²×C₄	C₄×D₂₃	1	4	yes	—
132	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
133	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
134	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
138	M₁₆	—	23	8	no	—
139	M₁₆	—	23	8	no	—
140	C₂×C₈	—	23	8	no	—
141	C₂×C₈	—	23	4	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	23	4	no	—
144	C₄⋊C₄	—	46	2	no	—
145	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₂²⋊C₄	—	46	2	no	—
149	C₂²⋊C₄	—	46	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	46	2	no	—
153	C₄²	—	46	2	no	—
154	C₂³	—	4	2	no	—
155	C₂³	Dic₂₃⋊C₄	1	4	yes	—
156	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
157	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
158	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	46	4	no	—
162	C₂×C₄	—	92	4	no	—
163	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
164	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
165	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
166	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
167	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
168	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
169	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
170	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
171	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
172	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
175	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
176	C₈	—	23	8	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	C₂×Dic₂₃⋊C₄	1	2	yes	—
182	C₂²	C₂×Dic₂₃⋊C₄	1	2	yes	—
183	C₂²	C₂×Dic₂₃⋊C₄	1	2	yes	—
184	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
185	C₄	—	4	4	no	—
186	C₄	—	92	4	no	—
187	C₄	C₂×Dic₂₃⋊C₄	1	2	yes	—
188	C₄	C₂×Dic₂₃⋊C₄	1	2	yes	—
189	C₄	C₂×Dic₂₃⋊C₄	1	2	yes	—
190	C₄	C₂²×Dic₄₆	1	1	yes	characteristic
191	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
192	C₂	—	2	2	no	—
193	C₂	—	2	2	no	—
194	C₂	—	2	2	no	—
195	C₂	C₂²×Dic₂₃⋊C₄	1	1	yes	characteristic
196	C₂	#13093215	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#117	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂⁴×C₂₃	D₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×#117	D₂₃
hypocenter	C₂₃	#2911
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #13092460	C₂	semidirect
4. #13092457	C₂	semidirect
5. #13092453	C₂	semidirect
22. C₂×#67267	C₄	semidirect
26. #10624990	C₂²	semidirect
27. #10624989	C₂²	semidirect
51. #67267	C₂×C₄	semidirect
52. #67267	C₂×C₄	semidirect
53. C₂×C₂₃⋊C₈	C₂×C₄	semidirect
126. C₂₃	#2911	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#67267	2
C₂×#10625097	2
#13092460	2
#13092457	2
#13092453	2
C₂×C₂₃×#117	2
#2911	23