Subgroups — C₄×#10625083 #15682164

Order 2944.

500 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625083	C₂	1	1	yes	maximal, characteristic
2	C₄×#67290	C₂	1	2	yes	maximal, complemented
3	C₄×Dic₉₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092511	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092508	C₂	1	2	yes	maximal, complemented
6	C₄×#67273	C₂	1	2	yes	maximal, complemented
7	D₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092454	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092410	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67290	C₂²	1	2	yes	—
13	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₂₃	C₂²	1	2	yes	—
15	D₄×Dic₂₃	C₂²	1	2	yes	—
16	C₂×#67273	C₂²	1	2	yes	—
17	C₄×C₂²⋊D₂₃	—	2	2	no	—
18	C₂×C₄×Dic₂₃	—	2	2	no	—
19	#10625083	C₄	1	8	yes	direct product factor
20	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
23	#10625063	C₂²	1	2	yes	—
24	#10625058	C₂²	1	2	yes	—
25	(C₂²⋊D₂₃)⋊C₄	—	2	2	no	—
26	#10625048	—	2	2	no	—
27	C₄²×D₂₃	C₂²	1	1	yes	characteristic
28	C₄×Dic₄₆	C₂²	1	2	yes	—
29	C₂₃×#107	C₂²	1	1	yes	characteristic
30	C₂₃×#102	C₂²	1	2	yes	—
31	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
32	#10625014	C₂²	1	2	yes	—
33	#10625000	C₂²	1	1	yes	characteristic
34	#10624998	C₂²	1	2	yes	—
35	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
36	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
37	#10624992	C₂²	1	2	yes	—
38	#10624990	C₂²	1	2	yes	—
39	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×D₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	—	2	2	no	—
42	C₂×C₂²⋊D₂₃	—	2	2	no	—
43	C₂²×Dic₂₃	—	2	2	no	—
44	C₂²×Dic₂₃	—	2	2	no	—
45	#67290	—	2	4	no	—
46	#67290	C₂×C₄	1	8	yes	complemented
47	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
48	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
49	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
50	C₂×Dic₄₆	C₂³	1	2	yes	—
51	D₈×C₂₃	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₂₃	—	2	2	no	—
56	C₄²×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₂₃	—	2	2	no	—
58	#67273	C₂×C₄	1	8	yes	complemented
59	(C₂×D₂₃)⋊C₄	—	2	2	no	—
60	C₄⋊Dic₂₃	C₂³	1	2	yes	—
61	Dic₂₃⋊C₄	—	2	2	no	—
62	Dic₂₃⋊C₄	—	2	2	no	—
63	C₄×Dic₂₃	—	2	2	no	—
64	C₄×Dic₂₃	—	2	2	no	—
65	C₄×Dic₂₃	—	2	2	no	—
66	C₄×Dic₂₃	D₄	1	1	yes	characteristic
67	C₄×Dic₂₃	D₄	1	1	yes	characteristic
68	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
69	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
71	Dic₉₂	C₂×C₄	1	4	yes	complemented
72	C₈×D₂₃	—	2	2	no	—
73	C₈×D₂₃	C₂×C₄	1	4	yes	complemented
74	C₂³×C₂₃	—	2	2	no	—
75	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
76	D₄×C₂₃	—	2	2	no	—
77	D₄×C₂₃	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	4	4	no	—
80	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₂₃	—	2	8	no	—
84	C₂×Dic₂₃	—	2	2	no	—
85	C₂×Dic₂₃	—	2	8	no	—
86	C₂×Dic₂₃	—	2	2	no	—
87	C₂×Dic₂₃	—	4	4	no	—
88	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
89	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
91	C₄×D₂₃	—	2	2	no	—
92	C₄×D₂₃	—	2	2	no	—
93	C₄×D₂₃	C₂²×C₄	1	2	yes	—
94	C₄×D₂₃	#41	1	2	yes	—
95	Dic₄₆	—	2	2	no	—
96	Dic₄₆	C₂²×C₄	1	4	yes	—
97	C₈×C₂₃	—	2	1	no	—
98	C₈×C₂₃	C₂²×C₄	1	2	yes	—
99	C₂₃⋊C₈	—	2	1	no	—
100	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
101	C₄×#135	—	23	1	no	maximal, Sylow
102	C₂²×C₂₃	—	2	4	no	—
103	C₂²×C₂₃	—	4	2	no	—
104	C₂²×C₂₃	—	4	2	no	—
105	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×D₂₃	—	2	1	no	—
107	C₂×D₂₃	—	2	1	no	—
108	C₂×D₂₃	C₄×D₄	1	2	yes	—
109	C₄×C₂₃	—	2	1	no	—
110	C₄×C₂₃	—	4	2	no	—
111	C₄×C₂₃	#135	1	2	yes	complemented
112	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
113	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
114	Dic₂₃	—	2	4	no	—
115	Dic₂₃	—	2	1	no	—
116	Dic₂₃	—	4	2	no	—
117	Dic₂₃	C₄×D₄	1	2	yes	—
118	Dic₂₃	#135	1	2	yes	complemented
119	C₂×#135	—	23	1	no	—
120	C₄×#41	—	23	2	no	—
121	C₄×Q₁₆	—	23	1	no	—
122	C₄×Q₁₆	—	23	1	no	—
123	C₄×SD₁₆	—	23	2	no	—
124	C₄×SD₁₆	—	23	2	no	—
125	C₄×D₈	—	23	1	no	—
126	C₄×D₈	D₂₃	1	1	yes	characteristic, complemented
127	#427	—	23	1	no	—
128	#416	—	23	2	no	—
129	C₂×C₄×C₈	—	23	1	no	—
130	C₂×C₂₃	—	4	4	no	—
131	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
133	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
134	D₂₃	—	2	2	no	—
135	C₂×#41	—	23	2	no	—
136	#135	—	23	8	no	—
137	C₂×Q₁₆	—	23	1	no	—
138	C₂×SD₁₆	—	23	2	no	—
139	C₂×D₈	C₂×D₂₃	1	1	yes	characteristic
140	C₂²×C₈	—	23	1	no	—
141	C₂²×C₈	—	23	1	no	—
142	C₄×Q₈	—	23	2	no	—
143	C₄×Q₈	—	23	2	no	—
144	C₄×D₄	—	23	2	no	—
145	C₄×D₄	—	46	2	no	—
146	C₄×D₄	—	46	2	no	—
147	C₄×D₄	C₂×D₂₃	1	2	yes	—
148	#117	—	23	2	no	—
149	#117	—	46	2	no	—
150	C₂×C₄²	—	23	1	no	—
151	C₂×C₄²	—	46	2	no	—
152	#107	—	23	1	no	—
153	#107	C₂×D₂₃	1	1	yes	characteristic
154	#106	—	23	2	no	—
155	#103	—	23	2	no	—
156	#103	—	23	2	no	—
157	#102	—	23	2	no	—
158	#102	C₂×D₂₃	1	2	yes	—
159	C₄×C₈	—	23	1	no	—
160	C₄×C₈	—	23	1	no	—
161	C₄×C₈	—	23	1	no	—
162	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
163	C₂₃	C₄×#135	1	1	yes	characteristic, complemented, Sylow
164	#41	—	23	8	no	—
165	#41	—	46	4	no	—
166	C₂×Q₈	—	23	2	no	—
167	C₂×D₄	—	46	2	no	—
168	C₂×D₄	C₂²×D₂₃	1	2	yes	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	23	1	no	—
171	C₂²×C₄	—	23	1	no	—
172	C₂²×C₄	—	23	1	no	—
173	C₂²×C₄	—	46	2	no	—
174	C₂²×C₄	—	46	2	no	—
175	Q₁₆	—	23	4	no	—
176	SD₁₆	—	23	8	no	—
177	D₈	C₄×D₂₃	1	4	yes	complemented
178	C₂×C₈	—	23	1	no	—
179	C₂×C₈	—	23	4	no	—
180	C₂×C₈	—	23	1	no	—
181	C₂×C₈	—	46	2	no	—
182	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
184	C₄⋊C₄	—	23	2	no	—
185	C₄⋊C₄	—	46	2	no	—
186	C₄⋊C₄	—	46	2	no	—
187	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	46	2	no	—
190	C₂²⋊C₄	—	46	2	no	—
191	C₄²	—	23	1	no	—
192	C₄²	—	23	1	no	—
193	C₄²	—	23	1	no	—
194	C₄²	—	46	2	no	—
195	C₄²	—	46	2	no	—
196	C₄²	—	46	2	no	—
197	C₄²	C₂²×D₂₃	1	1	yes	characteristic
198	C₂³	—	2	2	no	—
199	C₂³	—	23	1	no	—
200	Q₈	—	23	4	no	—
201	Q₈	—	46	2	no	—
202	D₄	—	2	2	no	—
203	D₄	—	46	8	no	—
204	D₄	C₂×C₄×D₂₃	1	4	yes	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	23	1	no	—
208	C₂×C₄	—	23	1	no	—
209	C₂×C₄	—	23	2	no	—
210	C₂×C₄	—	23	2	no	—
211	C₂×C₄	—	23	1	no	—
212	C₂×C₄	—	46	2	no	—
213	C₂×C₄	—	46	8	no	—
214	C₂×C₄	—	46	2	no	—
215	C₂×C₄	—	46	2	no	—
216	C₂×C₄	—	46	2	no	—
217	C₂×C₄	—	92	4	no	—
218	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
219	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	23	2	no	—
223	C₈	—	46	1	no	—
224	C₈	C₂×C₄×D₂₃	1	2	yes	—
225	C₂²	—	2	4	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	23	2	no	—
229	C₂²	—	46	1	no	—
230	C₂²	—	46	1	no	—
231	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	23	2	no	—
235	C₄	—	23	2	no	—
236	C₄	—	46	4	no	—
237	C₄	—	46	1	no	—
238	C₄	—	92	2	no	—
239	C₄	#10625083	1	2	yes	direct product factor
240	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
241	C₄	#41×D₂₃	1	1	yes	characteristic
242	C₂	—	4	4	no	—
243	C₂	—	46	2	no	—
244	C₂	C₄×D₄×D₂₃	1	1	yes	characteristic
245	C₂	#135×D₂₃	1	1	yes	characteristic
246	C₂	C₂×#10625083	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₄×D₈×C₂₃	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#10625083	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₄×D₈	D₂₃
hypocenter	C₂₃	C₄×#135
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₄×#67290	C₂	semidirect
3. C₄×Dic₉₂	C₂	semidirect
4. #13092511	C₂	semidirect
5. #13092508	C₂	semidirect
6. C₄×#67273	C₂	semidirect
7. D₈×Dic₂₃	C₂	semidirect
8. #13092454	C₂	semidirect
9. #13092410	C₂	semidirect
10. C₄×C₈×D₂₃	C₂	semidirect
11. C₄×D₈×C₂₃	C₂	semidirect
19. #10625083	C₄	direct (N × H)
46. #67290	C₂×C₄	semidirect
51. D₈×C₂₃	C₂×C₄	semidirect
58. #67273	C₂×C₄	semidirect
71. Dic₉₂	C₂×C₄	semidirect
73. C₈×D₂₃	C₂×C₄	semidirect
111. C₄×C₂₃	#135	semidirect
118. Dic₂₃	#135	semidirect
126. C₄×D₈	D₂₃	semidirect
163. C₂₃	C₄×#135	semidirect
177. D₈	C₄×D₂₃	semidirect
239. C₄	#10625083	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×#10625083	2
C₄×#67290	2
C₄×Dic₉₂	2
#13092511	2
#13092508	2
C₄×#67273	2
D₈×Dic₂₃	2
#13092454	2
#13092410	2
C₄×C₈×D₂₃	2
C₄×D₈×C₂₃	2
C₄×#135	23

Subgroups — C4×#10625083 #15682164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#10625083 #15682164