Subgroups — #15673106

Order 2944.

404 conjugacy classes of subgroups.

The subgroup lattice has 210 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	#13092176	C₂	1	4	yes	maximal, complemented
4	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#117	—	2	1	no	—
7	C₂₃×#117	—	2	1	no	—
8	C₂₃×#117	C₄	1	2	yes	complemented
9	C₂₃×#117	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₂₃	C₄	1	2	yes	complemented
11	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
12	C₂×C₄²×C₂₃	—	2	1	no	—
13	C₂×C₂²⋊Dic₂₃	—	2	1	no	—
14	#10625097	—	2	2	no	—
15	#10625097	C₄	1	4	yes	complemented
16	C₂×C₄⋊Dic₂₃	—	2	1	no	—
17	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
18	C₂×#67267	—	2	2	no	—
19	C₂×#67267	—	2	1	no	—
20	C₂×#67267	C₄	1	4	yes	complemented
21	C₂×#67267	C₂²	1	2	yes	—
22	C₂²×C₂₃⋊C₈	—	2	1	no	—
23	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₃	—	2	2	no	—
25	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	D₄	1	2	yes	—
30	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
34	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₂₃	—	2	2	no	—
36	C₄⋊C₄×C₂₃	—	2	2	no	—
37	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
38	C₂²⋊C₄×C₂₃	—	2	2	no	—
39	C₂²⋊C₄×C₂₃	—	2	2	no	—
40	C₄²×C₂₃	—	2	2	no	—
41	C₄²×C₂₃	—	2	2	no	—
42	C₂²⋊Dic₂₃	—	2	4	no	—
43	C₄⋊Dic₂₃	—	2	4	no	—
44	C₄×Dic₂₃	—	2	2	no	—
45	C₄×Dic₂₃	C₂×C₄	1	4	yes	complemented
46	#67267	—	2	4	no	—
47	#67267	—	2	2	no	—
48	#67267	C₂×C₄	1	4	yes	complemented
49	C₂×C₂₃⋊C₈	—	2	4	no	—
50	C₂×C₂₃⋊C₈	—	2	2	no	—
51	C₂³×C₂₃	—	4	2	no	—
52	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
53	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂³×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
56	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₃	—	2	4	no	—
58	C₂×C₄×C₂₃	—	4	4	no	—
59	C₂×C₄×C₂₃	C₄²	1	2	yes	—
60	C₂×C₄×C₂₃	C₄²	1	2	yes	—
61	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
68	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
77	C₂×Dic₂₃	—	2	4	no	—
78	C₂×Dic₂₃	—	4	4	no	—
79	C₂₃⋊C₈	—	2	4	no	—
80	#1733	—	23	1	no	maximal, Sylow
81	C₂²×C₂₃	—	2	2	no	—
82	C₂²×C₂₃	—	2	2	no	—
83	C₂²×C₂₃	—	2	2	no	—
84	C₂²×C₂₃	—	4	2	no	—
85	C₂²×C₂₃	—	4	2	no	—
86	C₂²×C₂₃	#95	1	2	yes	—
87	C₂²×C₂₃	#95	1	2	yes	—
88	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₄×C₂₃	—	4	4	no	—
92	C₄×C₂₃	#95	1	2	yes	—
93	C₄×C₂₃	#95	1	2	yes	—
94	C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
95	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	Dic₂₃	—	4	4	no	—
99	C₂²×M₁₆	—	23	1	no	—
100	C₂×#117	—	23	1	no	—
101	C₂×#117	D₂₃	1	1	yes	characteristic
102	#339	—	23	4	no	—
103	C₂×C₂₃	—	2	1	no	—
104	C₂×C₂₃	—	2	2	no	—
105	C₂×C₂₃	—	2	2	no	—
106	C₂×C₂₃	—	2	1	no	—
107	C₂×C₂₃	C₂×#95	1	1	yes	characteristic
108	C₂×C₂₃	#421	1	2	yes	—
109	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
110	C₂×M₁₆	—	23	4	no	—
111	C₂×M₁₆	—	23	2	no	—
112	C₂×M₁₆	—	46	2	no	—
113	C₂×M₁₆	—	46	1	no	—
114	C₂²×C₈	—	46	1	no	—
115	#117	—	2	1	no	—
116	#117	—	2	1	no	—
117	#117	—	23	4	no	—
118	#117	—	46	2	no	—
119	#117	Dic₂₃	1	2	yes	complemented
120	#117	C₂×D₂₃	1	2	yes	—
121	C₂×C₄⋊C₄	—	46	1	no	—
122	C₂×C₄⋊C₄	Dic₂₃	1	2	yes	complemented
123	C₂×C₂²⋊C₄	—	2	1	no	—
124	C₂×C₂²⋊C₄	—	46	1	no	—
125	C₂×C₄²	—	2	1	no	—
126	C₂×C₄²	—	23	2	no	—
127	C₂₃	#1733	1	1	yes	characteristic, complemented, Sylow
128	C₂⁴	C₂²⋊D₂₃	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	46	2	no	—
131	C₂²×C₄	Dic₄₆	1	2	yes	—
132	C₂²×C₄	C₄×D₂₃	1	2	yes	—
133	C₂²×C₄	C₄×D₂₃	1	2	yes	—
134	C₂²×C₄	D₉₂	1	2	yes	—
135	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
137	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
138	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
141	M₁₆	—	23	4	no	—
142	M₁₆	—	46	4	no	—
143	M₁₆	—	46	2	no	—
144	C₂×C₈	—	46	4	no	—
145	C₂×C₈	—	46	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	46	4	no	—
148	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
149	C₂²⋊C₄	—	2	2	no	—
150	C₂²⋊C₄	—	2	2	no	—
151	C₂²⋊C₄	—	46	4	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	2	no	—
154	C₄²	—	23	4	no	—
155	C₄²	—	46	2	no	—
156	C₂³	—	4	2	no	—
157	C₂³	Dic₂₃⋊C₄	1	2	yes	—
158	C₂³	(C₂×D₂₃)⋊C₄	1	2	yes	—
159	C₂³	C₂²⋊Dic₂₃	1	1	yes	characteristic
160	C₂³	C₂²⋊Dic₂₃	1	1	yes	characteristic
161	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	46	4	no	—
165	C₂×C₄	—	92	4	no	—
166	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
167	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
168	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
169	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
170	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
171	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
172	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
173	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
174	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
175	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
176	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
177	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
178	C₂×C₄	C₂×Dic₄₆	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×D₉₂	1	1	yes	characteristic
182	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
184	C₈	—	46	4	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	C₂₃⋊#95	1	2	yes	—
191	C₂²	C₂₃⋊#95	1	2	yes	—
192	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
193	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
194	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
195	C₄	—	4	4	no	—
196	C₄	—	92	4	no	—
197	C₄	C₂₃⋊#95	1	2	yes	—
198	C₄	C₂₃⋊#95	1	2	yes	—
199	C₄	C₂×C₄×Dic₂₃	1	1	yes	characteristic
200	C₄	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
201	C₄	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
202	C₄	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
203	C₂	—	2	1	no	—
204	C₂	—	2	1	no	—
205	C₂	—	2	2	no	—
206	C₂	—	2	2	no	—
207	C₂	C₂×C₂₃⋊#95	1	1	yes	characteristic
208	C₂	#13092434	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#117	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#117	D₂₃
hypocenter	C₂₃	#1733
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. #13092176	C₂	semidirect
8. C₂₃×#117	C₄	semidirect
10. C₂×C₄⋊C₄×C₂₃	C₄	semidirect
15. #10625097	C₄	semidirect
20. C₂×#67267	C₄	semidirect
45. C₄×Dic₂₃	C₂×C₄	semidirect
48. #67267	C₂×C₄	semidirect
119. #117	Dic₂₃	semidirect
122. C₂×C₄⋊C₄	Dic₂₃	semidirect
127. C₂₃	#1733	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
#13092176	2
C₂×C₂₃×#117	2
#1733	23