Subgroups — C₄²×#10625325 #26214751

Order 12032.

1230 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625325	C₂	1	3	yes	maximal
2	C₄²×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₁₈₈	C₂	1	1	yes	maximal, characteristic
4	C₄×#13093956	C₂	1	3	yes	maximal, complemented
5	C₄×#13093949	C₂	1	3	yes	maximal, complemented
6	C₄×#13093946	C₂	1	3	yes	maximal
7	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625325	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₉₄	C₂²	1	3	yes	—
10	C₂×C₄×D₁₈₈	C₂²	1	3	yes	—
11	C₂×C₄²×D₄₇	—	2	1	no	—
12	C₄×#10625325	C₄	1	24	yes	direct product factor
13	C₂×#13093956	C₂²	1	3	yes	—
14	C₂×#13093949	C₂²	1	3	yes	—
15	C₂×#13093946	C₂²	1	3	yes	—
16	C₄×Dic₄₇⋊C₄	—	2	3	no	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	3	yes	—
18	C₄×C₄⋊Dic₄₇	C₂²	1	3	yes	—
19	C₄×(C₂×D₄₇)⋊C₄	—	2	3	no	—
20	C₄²×Dic₄₇	—	2	1	no	—
21	#15688886	C₂²	1	6	yes	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
24	C₄³×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×Dic₉₄	C₂³	1	1	yes	characteristic
26	C₂²×D₁₈₈	C₂³	1	1	yes	characteristic
27	C₂²×C₄×D₄₇	—	2	3	no	—
28	C₂×#10625325	C₂×C₄	1	12	yes	—
29	C₄×Dic₉₄	—	2	6	no	—
30	C₄×Dic₉₄	C₂×C₄	1	12	yes	complemented
31	C₄×D₁₈₈	—	2	6	no	—
32	C₄×D₁₈₈	C₂×C₄	1	12	yes	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	3	yes	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	3	yes	—
35	C₂×Dic₄₇⋊C₄	—	2	3	no	—
36	C₂×(C₂×D₄₇)⋊C₄	—	2	3	no	—
37	C₂×C₄×Dic₄₇	—	2	3	no	—
38	C₂×C₄×Dic₄₇	—	2	3	no	—
39	C₂×C₄×Dic₄₇	—	2	3	no	—
40	C₄²×D₄₇	—	4	4	no	—
41	#13093956	C₂×C₄	1	12	yes	complemented
42	#13093949	C₂×C₄	1	12	yes	complemented
43	#13093946	C₂×C₄	1	12	yes	—
44	C₄₇⋊#95	—	2	3	no	—
45	C₄₇⋊#95	—	2	3	no	—
46	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	3	yes	—
49	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
51	C₄×C₈×C₄₇	—	2	6	no	—
52	C₄×C₈×C₄₇	C₂×C₄	1	12	yes	complemented
53	C₂³×D₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₄₇	—	2	1	no	—
58	C₂²×Dic₄₇	—	2	3	no	—
59	C₂²×Dic₄₇	—	2	3	no	—
60	C₂×D₁₈₈	—	2	3	no	—
61	C₂×D₁₈₈	C₂²×C₄	1	6	yes	—
62	C₂×C₄×D₄₇	—	2	12	no	—
63	C₂×C₄×D₄₇	—	4	6	no	—
64	C₂×C₄×D₄₇	—	4	6	no	—
65	C₂×Dic₉₄	—	2	3	no	—
66	C₂×Dic₉₄	C₂²×C₄	1	6	yes	—
67	C₂×C₈×C₄₇	—	2	6	no	—
68	C₂×C₈×C₄₇	C₂²×C₄	1	6	yes	—
69	C₂×C₈×C₄₇	C₂²×C₄	1	6	yes	—
70	C₄²×C₄₇	—	2	6	no	—
71	C₄²×C₄₇	SD₁₆	1	4	yes	complemented
72	C₄²×C₄₇	C₂²×C₄	1	6	yes	—
73	C₄²×C₄₇	#41	1	6	yes	—
74	(C₂×D₄₇)⋊C₄	—	2	12	no	—
75	C₄⋊Dic₄₇	—	2	3	no	—
76	C₄⋊Dic₄₇	—	2	3	no	—
77	C₄⋊Dic₄₇	C₂²×C₄	1	6	yes	—
78	C₄⋊Dic₄₇	C₂²×C₄	1	6	yes	—
79	Dic₄₇⋊C₄	—	2	12	no	—
80	C₄×Dic₄₇	—	2	12	no	—
81	C₄×Dic₄₇	—	4	6	no	—
82	C₄×Dic₄₇	—	4	6	no	—
83	#10625325	C₄²	1	16	yes	direct product factor
84	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
85	C₂²×D₄₇	—	2	6	no	—
86	C₂²×D₄₇	—	4	1	no	—
87	C₂²×D₄₇	—	4	3	no	—
88	C₂×C₄×C₄₇	—	2	6	no	—
89	C₂×C₄×C₄₇	C₄×D₄	1	6	yes	—
90	C₂×C₄×C₄₇	C₄×D₄	1	6	yes	—
91	C₂×C₄×C₄₇	C₂×SD₁₆	1	6	yes	—
92	C₂×C₄×C₄₇	#135	1	6	yes	—
93	C₂×C₄×C₄₇	C₂³×C₄	1	3	yes	—
94	C₂×C₄×C₄₇	C₂×#41	1	3	yes	—
95	C₂×Dic₄₇	—	2	6	no	—
96	C₂×Dic₄₇	—	2	6	no	—
97	C₂×Dic₄₇	—	2	6	no	—
98	C₂×Dic₄₇	—	4	3	no	—
99	C₂×Dic₄₇	—	4	6	no	—
100	C₂×Dic₄₇	—	4	3	no	—
101	D₁₈₈	—	2	6	no	—
102	D₁₈₈	C₂×C₄²	1	4	yes	—
103	C₄×D₄₇	—	4	24	no	—
104	Dic₉₄	—	2	6	no	—
105	Dic₉₄	C₂×C₄²	1	4	yes	complemented
106	C₈×C₄₇	—	2	6	no	—
107	C₈×C₄₇	C₂×C₄²	1	4	yes	complemented
108	C₄²×SD₁₆	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	C₂×C₄×D₄	1	3	yes	—
110	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
111	C₂²×C₄₇	C₂×#135	1	3	yes	—
112	C₂×D₄₇	—	2	4	no	—
113	C₂×D₄₇	—	4	6	no	—
114	C₂×D₄₇	—	4	6	no	—
115	C₄×C₄₇	—	2	6	no	—
116	C₄×C₄₇	C₄×SD₁₆	1	12	yes	complemented
117	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
118	C₄×C₄₇	C₄×#41	1	3	yes	—
119	Dic₄₇	—	2	4	no	—
120	Dic₄₇	—	4	6	no	—
121	Dic₄₇	—	4	6	no	—
122	C₂×C₄×SD₁₆	—	47	3	no	—
123	C₄²×Q₈	—	47	1	no	—
124	C₄²×D₄	—	47	1	no	—
125	C₄×#106	—	47	3	no	—
126	C₄×#103	—	47	3	no	—
127	C₄×#102	—	47	3	no	—
128	C₄²×C₈	D₄₇	1	1	yes	characteristic, complemented
129	C₂×C₄₇	C₄²×D₄	1	1	yes	characteristic
130	C₂×C₄₇	C₂×C₄×SD₁₆	1	3	yes	—
131	C₂×C₄₇	C₄×#135	1	3	yes	—
132	D₄₇	—	4	4	no	—
133	C₂²×SD₁₆	—	47	1	no	—
134	C₂×C₄×Q₈	—	47	3	no	—
135	C₂×C₄×D₄	—	47	3	no	—
136	C₂²×C₄²	—	94	1	no	—
137	C₄×SD₁₆	—	47	24	no	—
138	C₂×#106	—	47	3	no	—
139	C₂×#103	—	47	3	no	—
140	C₂×#102	—	47	3	no	—
141	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
142	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
143	C₄×C₄⋊C₄	—	47	3	no	—
144	C₄×C₄⋊C₄	—	47	3	no	—
145	C₄×C₄⋊C₄	—	94	3	no	—
146	C₄×C₂²⋊C₄	—	94	3	no	—
147	C₄³	—	94	1	no	—
148	C₄³	C₂×D₄₇	1	1	yes	characteristic
149	#340	—	47	6	no	—
150	C₄₇	C₄²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₈	—	47	1	no	—
152	C₂²×D₄	—	47	1	no	—
153	C₂³×C₄	—	94	3	no	—
154	C₂×SD₁₆	—	47	12	no	—
155	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
156	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
157	C₄×Q₈	—	47	12	no	—
158	C₄×Q₈	—	94	6	no	—
159	C₄×D₄	—	47	12	no	—
160	C₄×D₄	—	94	6	no	—
161	C₂×C₄⋊C₄	—	47	3	no	—
162	C₂×C₄⋊C₄	—	47	3	no	—
163	C₂×C₄⋊C₄	—	94	3	no	—
164	C₂×C₂²⋊C₄	—	94	3	no	—
165	C₂×C₄²	—	94	3	no	—
166	C₂×C₄²	—	94	3	no	—
167	C₂×C₄²	—	94	3	no	—
168	C₂×C₄²	—	188	4	no	—
169	C₂×C₄²	D₁₈₈	1	3	yes	—
170	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
172	#106	—	47	12	no	—
173	#103	—	47	12	no	—
174	#102	—	47	12	no	—
175	C₄×C₈	—	2	6	no	—
176	C₄×C₈	C₄×D₄₇	1	12	yes	complemented
177	#95	—	94	3	no	—
178	#95	—	94	3	no	—
179	C₂⁴	—	94	1	no	—
180	C₂×Q₈	—	47	6	no	—
181	C₂×Q₈	—	94	3	no	—
182	C₂×D₄	—	47	6	no	—
183	C₂×D₄	—	94	3	no	—
184	C₂²×C₄	—	94	1	no	—
185	C₂²×C₄	—	94	3	no	—
186	C₂²×C₄	—	94	3	no	—
187	C₂²×C₄	—	94	12	no	—
188	C₂²×C₄	—	188	6	no	—
189	C₂²×C₄	—	188	6	no	—
190	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
191	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
192	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
193	SD₁₆	—	47	16	no	—
194	C₂×C₈	—	2	6	no	—
195	C₂×C₈	C₂×C₄×D₄₇	1	6	yes	—
196	C₂×C₈	C₂×C₄×D₄₇	1	6	yes	—
197	C₄⋊C₄	—	47	6	no	—
198	C₄⋊C₄	—	47	6	no	—
199	C₄⋊C₄	—	94	12	no	—
200	C₄⋊C₄	—	94	3	no	—
201	C₄⋊C₄	—	94	3	no	—
202	C₂²⋊C₄	—	94	12	no	—
203	C₄²	—	2	6	no	—
204	C₄²	—	94	12	no	—
205	C₄²	—	188	6	no	—
206	C₄²	—	188	6	no	—
207	C₄²	#10625325	1	4	yes	direct product factor
208	C₄²	C₂×C₄×D₄₇	1	6	yes	—
209	C₄²	#10625350	1	6	yes	—
210	C₂³	—	94	6	no	—
211	C₂³	—	188	1	no	—
212	C₂³	—	188	3	no	—
213	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
214	Q₈	—	47	4	no	—
215	Q₈	—	94	6	no	—
216	D₄	—	47	4	no	—
217	D₄	—	94	6	no	—
218	C₂×C₄	—	2	6	no	—
219	C₂×C₄	—	94	6	no	—
220	C₂×C₄	—	94	6	no	—
221	C₂×C₄	—	94	6	no	—
222	C₂×C₄	—	188	3	no	—
223	C₂×C₄	—	188	6	no	—
224	C₂×C₄	—	188	3	no	—
225	C₂×C₄	—	188	24	no	—
226	C₂×C₄	C₄×D₁₈₈	1	6	yes	—
227	C₂×C₄	C₄×D₁₈₈	1	6	yes	—
228	C₂×C₄	C₂×#10625325	1	6	yes	—
229	C₂×C₄	#13094046	1	6	yes	—
230	C₂×C₄	C₂²×C₄×D₄₇	1	3	yes	—
231	C₂×C₄	C₂×#10625350	1	3	yes	—
232	C₈	—	2	6	no	—
233	C₈	C₄²×D₄₇	1	4	yes	complemented
234	C₂²	—	94	4	no	—
235	C₂²	—	188	6	no	—
236	C₂²	—	188	6	no	—
237	C₂²	C₂×C₄×D₁₈₈	1	3	yes	—
238	C₂²	C₂²×#10625325	1	1	yes	characteristic
239	C₂²	C₂×#13094046	1	3	yes	—
240	C₄	—	2	6	no	—
241	C₄	—	94	4	no	—
242	C₄	—	188	6	no	—
243	C₄	—	188	6	no	—
244	C₄	C₄×#10625325	1	12	yes	direct product factor
245	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
246	C₄	C₄×#10625350	1	3	yes	—
247	C₂	—	188	4	no	—
248	C₂	C₄²×D₁₈₈	1	1	yes	characteristic
249	C₂	C₂×C₄×#10625325	1	3	yes	—
250	C₂	C₄×#13094046	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²	D₁₈₈
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄²×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₁₈₈	C₂³
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	C₄²×SD₁₆
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
2. C₄²×Dic₉₄	C₂	semidirect
4. C₄×#13093956	C₂	semidirect
5. C₄×#13093949	C₂	semidirect
7. C₄²×C₈×C₄₇	C₂	semidirect
12. C₄×#10625325	C₄	direct (N × H)
30. C₄×Dic₉₄	C₂×C₄	semidirect
41. #13093956	C₂×C₄	semidirect
42. #13093949	C₂×C₄	semidirect
52. C₄×C₈×C₄₇	C₂×C₄	semidirect
71. C₄²×C₄₇	SD₁₆	semidirect
83. #10625325	C₄²	direct (N × H)
105. Dic₉₄	C₂×C₄²	semidirect
107. C₈×C₄₇	C₂×C₄²	semidirect
116. C₄×C₄₇	C₄×SD₁₆	semidirect
128. C₄²×C₈	D₄₇	semidirect
150. C₄₇	C₄²×SD₁₆	semidirect
176. C₄×C₈	C₄×D₄₇	semidirect
207. C₄²	#10625325	direct (N × H)
233. C₈	C₄²×D₄₇	semidirect
244. C₄	C₄×#10625325	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×#10625325	2
C₄²×Dic₉₄	2
C₄²×D₁₈₈	2
C₄×#13093956	2
C₄×#13093949	2
C₄×#13093946	2
C₄²×C₈×C₄₇	2
C₄²×SD₁₆	47

Subgroups — C42×#10625325 #26214751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#10625325 #26214751