Subgroups — C₄×#10625063 #15675737

Order 2944.

752 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625063	C₂	1	2	yes	maximal
2	C₂×C₄²×D₂₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₂₃	C₂	1	2	yes	maximal, complemented
4	C₄×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×(C₂×D₂₃)⋊C₄	C₂	1	2	yes	maximal
6	#13092264	C₂	1	2	yes	maximal
7	#13092262	C₂	1	2	yes	maximal, complemented
8	C₄²×Dic₂₃	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
17	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
18	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
19	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
20	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₂₃	C₂²	1	4	yes	—
24	#10625063	C₄	1	16	yes	direct product factor
25	C₄²×D₂₃	C₄	1	8	yes	complemented
26	C₂₃×#95	C₂²	1	2	yes	—
27	C₂₃⋊#95	C₂²	1	2	yes	—
28	C₂₃⋊#95	C₂²	1	4	yes	—
29	C₂³×D₂₃	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×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₂₃	C₂³	1	4	yes	—
35	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
36	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
37	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×D₂₃	—	2	2	no	—
39	C₂×C₄×D₂₃	—	2	2	no	—
40	C₂×C₄×D₂₃	—	2	4	no	—
41	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
42	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
43	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
44	C₂×C₄×D₂₃	C₂×C₄	1	8	yes	—
45	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	complemented
46	C₄²×C₂₃	—	2	4	no	—
47	C₄²×C₂₃	C₂×C₄	1	4	yes	—
48	(C₂×D₂₃)⋊C₄	C₂×C₄	1	16	yes	—
49	C₄⋊Dic₂₃	C₂×C₄	1	8	yes	complemented
50	C₄×Dic₂₃	—	2	4	no	—
51	C₄×Dic₂₃	—	2	8	no	—
52	C₄×Dic₂₃	C₂×C₄	1	4	yes	—
53	C₄×Dic₂₃	C₂×C₄	1	4	yes	—
54	C₄×Dic₂₃	C₂×C₄	1	4	yes	—
55	C₄×Dic₂₃	C₂×C₄	1	16	yes	complemented
56	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
57	C₂²×D₂₃	—	2	1	no	—
58	C₂²×D₂₃	—	2	1	no	—
59	C₂²×D₂₃	—	2	2	no	—
60	C₂²×D₂₃	C₂²×C₄	1	4	yes	—
61	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₂₃	—	2	4	no	—
63	C₂×C₄×C₂₃	—	2	4	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
67	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
69	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₂₃	#41	1	2	yes	—
71	C₂×C₄×C₂₃	#41	1	2	yes	—
72	C₂×Dic₂₃	—	2	4	no	—
73	C₂×Dic₂₃	—	2	4	no	—
74	C₂×Dic₂₃	—	2	1	no	—
75	C₂×Dic₂₃	—	2	1	no	—
76	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
77	C₂×Dic₂₃	C₂²×C₄	1	8	yes	—
78	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
79	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
80	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
81	C₂×Dic₂₃	#41	1	4	yes	—
82	C₄×D₂₃	—	2	8	no	—
83	C₄×D₂₃	—	2	8	no	—
84	C₄×D₂₃	C₄²	1	16	yes	complemented
85	C₄×#117	—	23	1	no	maximal, Sylow
86	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
87	C₂²×C₂₃	C₂³×C₄	1	2	yes	—
88	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
89	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
90	C₂²×C₂₃	C₂×#41	1	2	yes	—
91	C₂×D₂₃	—	2	2	no	—
92	C₂×D₂₃	—	2	2	no	—
93	C₂×D₂₃	—	2	4	no	—
94	C₂×D₂₃	—	2	4	no	—
95	C₂×D₂₃	C₂×C₄²	1	4	yes	—
96	C₄×C₂₃	—	2	2	no	—
97	C₄×C₂₃	—	2	8	no	—
98	C₄×C₂₃	C₂×C₄²	1	4	yes	—
99	C₄×C₂₃	#117	1	4	yes	complemented
100	Dic₂₃	—	2	2	no	—
101	Dic₂₃	—	2	8	no	—
102	Dic₂₃	C₂×C₄²	1	4	yes	—
103	Dic₂₃	#117	1	4	yes	complemented
104	C₂×#117	—	23	2	no	—
105	C₂²×C₄²	—	23	1	no	—
106	C₄×C₄⋊C₄	—	23	2	no	—
107	C₄×C₄⋊C₄	—	23	1	no	—
108	C₄×C₄⋊C₄	D₂₃	1	1	yes	characteristic, complemented
109	C₄×C₂²⋊C₄	—	23	2	no	—
110	C₄×C₂²⋊C₄	—	23	2	no	—
111	#376	—	23	2	no	—
112	C₄³	—	23	2	no	—
113	C₂×C₂₃	C₂²×C₄²	1	1	yes	characteristic
114	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
115	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
116	C₂×C₂₃	C₄×#41	1	2	yes	—
117	C₂×C₂₃	C₄×#41	1	2	yes	—
118	D₂₃	—	2	4	no	—
119	C₂³×C₄	—	23	1	no	—
120	C₂³×C₄	—	23	1	no	—
121	C₂³×C₄	—	23	1	no	—
122	#117	—	23	16	no	—
123	C₂×C₄⋊C₄	—	23	2	no	—
124	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
125	C₂×C₂²⋊C₄	—	23	4	no	—
126	C₂×C₄²	—	23	4	no	—
127	C₂×C₄²	—	23	1	no	—
128	C₂×C₄²	—	23	1	no	—
129	C₂×C₄²	—	23	1	no	—
130	C₂×C₄²	—	23	8	no	—
131	C₂×C₄²	—	23	4	no	—
132	C₂×C₄²	—	23	2	no	—
133	C₂×C₄²	C₂×D₂₃	1	2	yes	—
134	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
135	#95	—	23	4	no	—
136	#95	—	23	2	no	—
137	#95	C₂×D₂₃	1	2	yes	—
138	C₂₃	C₄×#117	1	1	yes	characteristic, complemented, Sylow
139	C₂⁴	—	23	1	no	—
140	C₂²×C₄	—	23	1	no	—
141	C₂²×C₄	—	23	1	no	—
142	C₂²×C₄	—	23	4	no	—
143	C₂²×C₄	—	23	8	no	—
144	C₂²×C₄	—	23	1	no	—
145	C₂²×C₄	—	23	4	no	—
146	C₂²×C₄	—	23	4	no	—
147	C₂²×C₄	—	23	4	no	—
148	C₂²×C₄	—	46	4	no	—
149	C₂²×C₄	—	46	2	no	—
150	C₂²×C₄	—	46	2	no	—
151	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
152	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
153	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
155	C₄⋊C₄	—	23	8	no	—
156	C₄⋊C₄	C₄×D₂₃	1	8	yes	complemented
157	C₂²⋊C₄	—	23	16	no	—
158	C₄²	—	2	4	no	—
159	C₄²	—	23	4	no	—
160	C₄²	—	23	4	no	—
161	C₄²	—	23	4	no	—
162	C₄²	—	23	16	no	—
163	C₄²	—	46	8	no	—
164	C₄²	—	46	4	no	—
165	C₄²	C₄×D₂₃	1	4	yes	—
166	C₂³	—	23	4	no	—
167	C₂³	—	23	2	no	—
168	C₂³	—	46	1	no	—
169	C₂³	—	46	1	no	—
170	C₂³	—	46	2	no	—
171	C₂³	C₂³×D₂₃	1	1	yes	characteristic
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	23	2	no	—
176	C₂×C₄	—	23	2	no	—
177	C₂×C₄	—	23	4	no	—
178	C₂×C₄	—	23	4	no	—
179	C₂×C₄	—	23	16	no	—
180	C₂×C₄	—	23	2	no	—
181	C₂×C₄	—	23	8	no	—
182	C₂×C₄	—	46	8	no	—
183	C₂×C₄	—	46	1	no	—
184	C₂×C₄	—	46	4	no	—
185	C₂×C₄	—	46	1	no	—
186	C₂×C₄	—	46	4	no	—
187	C₂×C₄	—	46	8	no	—
188	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
189	C₂×C₄	C₂×C₄×D₂₃	1	8	yes	—
190	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
191	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
192	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
193	C₂×C₄	#67290	1	2	yes	—
194	C₂×C₄	#67292	1	2	yes	—
195	C₂²	—	23	4	no	—
196	C₂²	—	46	2	no	—
197	C₂²	—	46	2	no	—
198	C₂²	—	46	4	no	—
199	C₂²	—	46	4	no	—
200	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
201	C₂²	C₂²×C₄×D₂₃	1	2	yes	—
202	C₂²	C₂×#67290	1	1	yes	characteristic
203	C₂²	C₂×#67292	1	1	yes	characteristic
204	C₂²	#41×D₂₃	1	2	yes	—
205	C₄	—	2	2	no	—
206	C₄	—	2	8	no	—
207	C₄	—	23	4	no	—
208	C₄	—	23	4	no	—
209	C₄	—	46	2	no	—
210	C₄	—	46	8	no	—
211	C₄	C₄²×D₂₃	1	4	yes	—
212	C₄	#10625063	1	4	yes	direct product factor
213	C₂	—	46	4	no	—
214	C₂	C₂×C₄²×D₂₃	1	1	yes	characteristic
215	C₂	#117×D₂₃	1	1	yes	characteristic
216	C₂	C₂×#10625063	1	1	yes	characteristic
217	C₂	C₄×#67290	1	2	yes	—
218	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	C₂³×D₂₃	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₃
hypocenter	C₂₃	C₄×#117
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. C₄⋊C₄×Dic₂₃	C₂	semidirect
4. C₄×C₄⋊Dic₂₃	C₂	semidirect
7. #13092262	C₂	semidirect
8. C₄²×Dic₂₃	C₂	semidirect
9. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
24. #10625063	C₄	direct (N × H)
25. C₄²×D₂₃	C₄	semidirect
45. C₄⋊C₄×C₂₃	C₂×C₄	semidirect
49. C₄⋊Dic₂₃	C₂×C₄	semidirect
55. C₄×Dic₂₃	C₂×C₄	semidirect
84. C₄×D₂₃	C₄²	semidirect
99. C₄×C₂₃	#117	semidirect
103. Dic₂₃	#117	semidirect
108. C₄×C₄⋊C₄	D₂₃	semidirect
138. C₂₃	C₄×#117	semidirect
156. C₄⋊C₄	C₄×D₂₃	semidirect
212. C₄	#10625063	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×#10625063	2
C₂×C₄²×D₂₃	2
C₄⋊C₄×Dic₂₃	2
C₄×C₄⋊Dic₂₃	2
C₄×(C₂×D₂₃)⋊C₄	2
#13092264	2
#13092262	2
C₄²×Dic₂₃	2
C₄×C₄⋊C₄×C₂₃	2
C₄×#117	23

Subgroups — C4×#10625063 #15675737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#10625063 #15675737