Subgroups — C₄×#10625064 #15675742

Order 2944.

924 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₉₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625064	C₂	1	2	yes	maximal
3	C₂×C₄²×D₂₃	C₂	1	2	yes	maximal, complemented
4	C₄⋊C₄×Dic₂₃	C₂	1	2	yes	maximal, complemented
5	C₄×(C₂×D₂₃)⋊C₄	C₂	1	2	yes	maximal, complemented
6	#13092264	C₂	1	4	yes	maximal, complemented
7	C₄²×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₉₂	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₂₃	C₂²	1	4	yes	—
11	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
16	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
19	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
21	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
22	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
23	#10625064	C₄	1	16	yes	direct product factor
24	C₄×D₉₂	C₄	1	8	yes	complemented
25	C₄²×D₂₃	—	2	8	no	—
26	C₂₃×#95	C₂²	1	2	yes	—
27	C₂₃⋊#95	C₂²	1	4	yes	—
28	C₂³×D₂₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₂₃	C₂³	1	2	yes	—
34	C₂²×Dic₂₃	C₂³	1	2	yes	—
35	C₂²×Dic₂₃	C₂³	1	2	yes	—
36	C₂×D₉₂	C₂×C₄	1	4	yes	—
37	C₂×D₉₂	C₂×C₄	1	8	yes	—
38	C₂×C₄×D₂₃	—	2	4	no	—
39	C₂×C₄×D₂₃	—	2	4	no	—
40	C₂×C₄×D₂₃	—	2	8	no	—
41	C₂×C₄×D₂₃	—	2	8	no	—
42	C₂×C₄×D₂₃	C₂×C₄	1	8	yes	complemented
43	C₂×C₄×D₂₃	C₂×C₄	1	16	yes	complemented
44	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	complemented
45	C₄²×C₂₃	—	2	4	no	—
46	C₄²×C₂₃	C₂×C₄	1	4	yes	complemented
47	(C₂×D₂₃)⋊C₄	C₂×C₄	1	8	yes	complemented
48	(C₂×D₂₃)⋊C₄	C₂×C₄	1	16	yes	complemented
49	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	complemented
50	C₄×Dic₂₃	—	2	2	no	—
51	C₄×Dic₂₃	—	2	8	no	—
52	C₄×Dic₂₃	—	2	4	no	—
53	C₄×Dic₂₃	C₂×C₄	1	8	yes	complemented
54	C₄×Dic₂₃	D₄	1	4	yes	complemented
55	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
56	C₂²×D₂₃	—	2	2	no	—
57	C₂²×D₂₃	—	2	2	no	—
58	C₂²×D₂₃	—	2	4	no	—
59	C₂²×D₂₃	C₂²×C₄	1	4	yes	—
60	C₂²×D₂₃	C₂²×C₄	1	8	yes	—
61	C₂×C₄×C₂₃	—	2	4	no	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	4	no	—
64	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
66	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
67	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₂₃	#41	1	2	yes	—
71	C₂×Dic₂₃	—	2	2	no	—
72	C₂×Dic₂₃	—	2	2	no	—
73	C₂×Dic₂₃	—	2	2	no	—
74	C₂×Dic₂₃	—	2	2	no	—
75	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
76	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
77	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
78	C₂×Dic₂₃	C₂×D₄	1	4	yes	—
79	C₂×Dic₂₃	#41	1	4	yes	—
80	D₉₂	C₄²	1	16	yes	complemented
81	C₄×D₂₃	—	2	16	no	—
82	C₄×D₂₃	—	2	32	no	—
83	C₄²×D₄	—	23	1	no	maximal, Sylow
84	C₂²×C₂₃	C₂³×C₄	1	2	yes	—
85	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
86	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
87	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
88	C₂²×C₂₃	C₂×#41	1	2	yes	—
89	C₂×D₂₃	—	2	8	no	—
90	C₂×D₂₃	—	2	4	no	—
91	C₂×D₂₃	—	2	4	no	—
92	C₂×D₂₃	—	2	8	no	—
93	C₂×D₂₃	C₂×C₄²	1	8	yes	complemented
94	C₄×C₂₃	—	2	8	no	—
95	C₄×C₂₃	—	2	2	no	—
96	C₄×C₂₃	C₂×C₄²	1	4	yes	complemented
97	C₄×C₂₃	C₄×D₄	1	4	yes	complemented
98	Dic₂₃	—	2	4	no	—
99	Dic₂₃	—	2	4	no	—
100	Dic₂₃	C₄×D₄	1	8	yes	complemented
101	C₂×C₄×D₄	—	23	2	no	—
102	C₂×C₄×D₄	—	23	1	no	—
103	C₂²×C₄²	—	23	2	no	—
104	C₄×C₄⋊C₄	—	23	2	no	—
105	C₄×C₄⋊C₄	D₂₃	1	1	yes	characteristic, complemented
106	C₄×C₂²⋊C₄	—	23	2	no	—
107	C₄×C₂²⋊C₄	—	23	4	no	—
108	C₄³	—	23	1	no	—
109	C₂×C₂₃	C₂²×C₄²	1	1	yes	characteristic
110	C₂×C₂₃	C₂×C₄×D₄	1	2	yes	—
111	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
113	C₂×C₂₃	C₄×#41	1	2	yes	—
114	D₂₃	—	2	8	no	—
115	C₂²×D₄	—	23	1	no	—
116	C₂³×C₄	—	23	4	no	—
117	C₂³×C₄	—	23	2	no	—
118	C₄×D₄	—	23	16	no	—
119	C₄×D₄	—	23	8	no	—
120	C₂×C₄⋊C₄	—	23	1	no	—
121	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
122	C₂×C₂²⋊C₄	—	23	4	no	—
123	C₂×C₂²⋊C₄	—	23	2	no	—
124	C₂×C₄²	—	23	2	no	—
125	C₂×C₄²	—	23	1	no	—
126	C₂×C₄²	—	23	2	no	—
127	C₂×C₄²	—	23	4	no	—
128	C₂×C₄²	—	23	1	no	—
129	C₂×C₄²	—	46	8	no	—
130	C₂×C₄²	C₂×D₂₃	1	2	yes	—
131	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
132	#95	—	23	4	no	—
133	#95	C₂×D₂₃	1	2	yes	—
134	C₂₃	C₄²×D₄	1	1	yes	characteristic, complemented, Sylow
135	C₂⁴	—	23	2	no	—
136	C₂×D₄	—	23	8	no	—
137	C₂×D₄	—	23	4	no	—
138	C₂²×C₄	—	23	2	no	—
139	C₂²×C₄	—	23	2	no	—
140	C₂²×C₄	—	23	2	no	—
141	C₂²×C₄	—	23	16	no	—
142	C₂²×C₄	—	23	8	no	—
143	C₂²×C₄	—	46	8	no	—
144	C₂²×C₄	—	46	8	no	—
145	C₂²×C₄	—	46	4	no	—
146	C₂²×C₄	—	46	4	no	—
147	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
148	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
151	C₄⋊C₄	—	23	4	no	—
152	C₄⋊C₄	C₄×D₂₃	1	8	yes	complemented
153	C₂²⋊C₄	—	23	16	no	—
154	C₂²⋊C₄	—	23	8	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	23	8	no	—
157	C₄²	—	23	4	no	—
158	C₄²	—	46	8	no	—
159	C₄²	—	46	4	no	—
160	C₄²	—	46	2	no	—
161	C₄²	C₄×D₂₃	1	4	yes	complemented
162	C₂³	—	23	8	no	—
163	C₂³	—	23	4	no	—
164	C₂³	—	46	2	no	—
165	C₂³	—	46	2	no	—
166	C₂³	—	46	4	no	—
167	C₂³	C₂³×D₂₃	1	1	yes	characteristic
168	D₄	—	23	16	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	23	4	no	—
173	C₂×C₄	—	23	4	no	—
174	C₂×C₄	—	23	4	no	—
175	C₂×C₄	—	23	4	no	—
176	C₂×C₄	—	23	4	no	—
177	C₂×C₄	—	46	32	no	—
178	C₂×C₄	—	46	2	no	—
179	C₂×C₄	—	46	2	no	—
180	C₂×C₄	—	46	2	no	—
181	C₂×C₄	—	46	2	no	—
182	C₂×C₄	—	46	16	no	—
183	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
184	C₂×C₄	C₂×C₄×D₂₃	1	8	yes	—
185	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
186	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
187	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
188	C₂×C₄	D₄×D₂₃	1	2	yes	—
189	C₂×C₄	#67292	1	2	yes	—
190	C₂²	—	23	8	no	—
191	C₂²	—	46	8	no	—
192	C₂²	—	46	4	no	—
193	C₂²	—	46	4	no	—
194	C₂²	—	46	8	no	—
195	C₂²	C₂²×C₄×D₂₃	1	2	yes	—
196	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
197	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
198	C₂²	C₂×#67292	1	1	yes	characteristic
199	C₂²	#41×D₂₃	1	2	yes	—
200	C₄	—	2	8	no	—
201	C₄	—	2	2	no	—
202	C₄	—	23	8	no	—
203	C₄	—	46	4	no	—
204	C₄	—	46	4	no	—
205	C₄	C₄²×D₂₃	1	4	yes	complemented
206	C₄	#10625064	1	4	yes	direct product factor
207	C₂	—	46	8	no	—
208	C₂	C₂×C₄²×D₂₃	1	1	yes	characteristic
209	C₂	C₄×D₄×D₂₃	1	2	yes	—
210	C₂	C₂×#10625064	1	1	yes	characteristic
211	C₂	#13092885	1	1	yes	characteristic
212	C₂	C₄×#67292	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₃
commutator	C₂×C₂₃	C₂²×C₄²
Frattini	C₂³	C₂³×D₂₃
Fitting	C₄×C₄⋊C₄×C₂₃	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
prime radical	C₂²×D₉₂	C₂²
hypercenter	C₄×C₄⋊C₄	D₂₃
hypocenter	C₂₃	C₄²×D₄
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄²×D₂₃	C₂	semidirect
4. C₄⋊C₄×Dic₂₃	C₂	semidirect
5. C₄×(C₂×D₂₃)⋊C₄	C₂	semidirect
6. #13092264	C₂	semidirect
7. C₄²×Dic₂₃	C₂	semidirect
8. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
23. #10625064	C₄	direct (N × H)
24. C₄×D₉₂	C₄	semidirect
42. C₂×C₄×D₂₃	C₂×C₄	semidirect
43. C₂×C₄×D₂₃	C₂×C₄	semidirect
44. C₄⋊C₄×C₂₃	C₂×C₄	semidirect
46. C₄²×C₂₃	C₂×C₄	semidirect
47. (C₂×D₂₃)⋊C₄	C₂×C₄	semidirect
48. (C₂×D₂₃)⋊C₄	C₂×C₄	semidirect
49. C₄⋊Dic₂₃	C₂×C₄	semidirect
53. C₄×Dic₂₃	C₂×C₄	semidirect
54. C₄×Dic₂₃	D₄	semidirect
80. D₉₂	C₄²	semidirect
93. C₂×D₂₃	C₂×C₄²	semidirect
96. C₄×C₂₃	C₂×C₄²	semidirect
97. C₄×C₂₃	C₄×D₄	semidirect
100. Dic₂₃	C₄×D₄	semidirect
105. C₄×C₄⋊C₄	D₂₃	semidirect
134. C₂₃	C₄²×D₄	semidirect
152. C₄⋊C₄	C₄×D₂₃	semidirect
161. C₄²	C₄×D₂₃	semidirect
205. C₄	C₄²×D₂₃	semidirect
206. C₄	#10625064	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₉₂	2
C₂×#10625064	2
C₂×C₄²×D₂₃	2
C₄⋊C₄×Dic₂₃	2
C₄×(C₂×D₂₃)⋊C₄	2
#13092264	2
C₄²×Dic₂₃	2
C₄×C₄⋊C₄×C₂₃	2
C₄²×D₄	23

Subgroups — C4×#10625064 #15675742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#10625064 #15675742