Subgroups — C₄²×#10621873 #25095070

Order 11008.

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×#10621873	C₂	1	3	yes	maximal
2	C₄²×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₁₇₂	C₂	1	1	yes	maximal, characteristic
4	C₄×#13063257	C₂	1	3	yes	maximal, complemented
5	C₄×#13063250	C₂	1	3	yes	maximal, complemented
6	C₄×#13063247	C₂	1	3	yes	maximal
7	C₄²×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621873	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₈₆	C₂²	1	3	yes	—
10	C₂×C₄×D₁₇₂	C₂²	1	3	yes	—
11	C₂×C₄²×D₄₃	—	2	1	no	—
12	C₄×#10621873	C₄	1	24	yes	direct product factor
13	C₂×#13063257	C₂²	1	3	yes	—
14	C₂×#13063250	C₂²	1	3	yes	—
15	C₂×#13063247	C₂²	1	3	yes	—
16	C₄×Dic₄₃⋊C₄	—	2	3	no	—
17	C₄×C₄⋊Dic₄₃	C₂²	1	3	yes	—
18	C₄×C₄⋊Dic₄₃	C₂²	1	3	yes	—
19	C₄×(C₂×D₄₃)⋊C₄	—	2	3	no	—
20	C₄²×Dic₄₃	—	2	1	no	—
21	#14582662	C₂²	1	6	yes	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₄₃	C₂²	1	3	yes	—
24	C₄³×C₄₃	C₂²	1	1	yes	characteristic
25	C₂²×Dic₈₆	C₂³	1	1	yes	characteristic
26	C₂²×D₁₇₂	C₂³	1	1	yes	characteristic
27	C₂²×C₄×D₄₃	—	2	3	no	—
28	C₂×#10621873	C₂×C₄	1	12	yes	—
29	C₄×Dic₈₆	—	2	6	no	—
30	C₄×Dic₈₆	C₂×C₄	1	12	yes	complemented
31	C₄×D₁₇₂	—	2	6	no	—
32	C₄×D₁₇₂	C₂×C₄	1	12	yes	—
33	C₂×C₄⋊Dic₄₃	C₂³	1	3	yes	—
34	C₂×C₄⋊Dic₄₃	C₂³	1	3	yes	—
35	C₂×Dic₄₃⋊C₄	—	2	3	no	—
36	C₂×(C₂×D₄₃)⋊C₄	—	2	3	no	—
37	C₂×C₄×Dic₄₃	—	2	3	no	—
38	C₂×C₄×Dic₄₃	—	2	3	no	—
39	C₂×C₄×Dic₄₃	—	2	3	no	—
40	C₄²×D₄₃	—	4	4	no	—
41	#13063257	C₂×C₄	1	12	yes	complemented
42	#13063250	C₂×C₄	1	12	yes	complemented
43	#13063247	C₂×C₄	1	12	yes	—
44	C₄₃⋊#95	—	2	3	no	—
45	C₄₃⋊#95	—	2	3	no	—
46	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
47	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	D₄	1	3	yes	—
49	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
51	C₄×C₈×C₄₃	—	2	6	no	—
52	C₄×C₈×C₄₃	C₂×C₄	1	12	yes	complemented
53	C₂³×D₄₃	—	2	1	no	—
54	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₄₃	—	2	1	no	—
58	C₂²×Dic₄₃	—	2	3	no	—
59	C₂²×Dic₄₃	—	2	3	no	—
60	C₂×D₁₇₂	—	2	3	no	—
61	C₂×D₁₇₂	C₂²×C₄	1	6	yes	—
62	C₂×C₄×D₄₃	—	2	12	no	—
63	C₂×C₄×D₄₃	—	4	6	no	—
64	C₂×C₄×D₄₃	—	4	6	no	—
65	C₂×Dic₈₆	—	2	3	no	—
66	C₂×Dic₈₆	C₂²×C₄	1	6	yes	—
67	C₂×C₈×C₄₃	—	2	6	no	—
68	C₂×C₈×C₄₃	C₂²×C₄	1	6	yes	—
69	C₂×C₈×C₄₃	C₂²×C₄	1	6	yes	—
70	C₄²×C₄₃	—	2	6	no	—
71	C₄²×C₄₃	SD₁₆	1	4	yes	complemented
72	C₄²×C₄₃	C₂²×C₄	1	6	yes	—
73	C₄²×C₄₃	#41	1	6	yes	—
74	(C₂×D₄₃)⋊C₄	—	2	12	no	—
75	C₄⋊Dic₄₃	—	2	3	no	—
76	C₄⋊Dic₄₃	—	2	3	no	—
77	C₄⋊Dic₄₃	C₂²×C₄	1	6	yes	—
78	C₄⋊Dic₄₃	C₂²×C₄	1	6	yes	—
79	Dic₄₃⋊C₄	—	2	12	no	—
80	C₄×Dic₄₃	—	2	12	no	—
81	C₄×Dic₄₃	—	4	6	no	—
82	C₄×Dic₄₃	—	4	6	no	—
83	#10621873	C₄²	1	16	yes	direct product factor
84	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
85	C₂²×D₄₃	—	2	6	no	—
86	C₂²×D₄₃	—	4	1	no	—
87	C₂²×D₄₃	—	4	3	no	—
88	C₂×C₄×C₄₃	—	2	6	no	—
89	C₂×C₄×C₄₃	C₄×D₄	1	6	yes	—
90	C₂×C₄×C₄₃	C₄×D₄	1	6	yes	—
91	C₂×C₄×C₄₃	C₂×SD₁₆	1	6	yes	—
92	C₂×C₄×C₄₃	#135	1	6	yes	—
93	C₂×C₄×C₄₃	C₂³×C₄	1	3	yes	—
94	C₂×C₄×C₄₃	C₂×#41	1	3	yes	—
95	C₂×Dic₄₃	—	2	6	no	—
96	C₂×Dic₄₃	—	2	6	no	—
97	C₂×Dic₄₃	—	2	6	no	—
98	C₂×Dic₄₃	—	4	3	no	—
99	C₂×Dic₄₃	—	4	6	no	—
100	C₂×Dic₄₃	—	4	3	no	—
101	D₁₇₂	—	2	6	no	—
102	D₁₇₂	C₂×C₄²	1	4	yes	—
103	C₄×D₄₃	—	4	24	no	—
104	Dic₈₆	—	2	6	no	—
105	Dic₈₆	C₂×C₄²	1	4	yes	complemented
106	C₈×C₄₃	—	2	6	no	—
107	C₈×C₄₃	C₂×C₄²	1	4	yes	complemented
108	C₄²×SD₁₆	—	43	1	no	maximal, Sylow
109	C₂²×C₄₃	C₂×C₄×D₄	1	3	yes	—
110	C₂²×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
111	C₂²×C₄₃	C₂×#135	1	3	yes	—
112	C₂×D₄₃	—	2	4	no	—
113	C₂×D₄₃	—	4	6	no	—
114	C₂×D₄₃	—	4	6	no	—
115	C₄×C₄₃	—	2	6	no	—
116	C₄×C₄₃	C₄×SD₁₆	1	12	yes	complemented
117	C₄×C₄₃	C₂²×C₄²	1	1	yes	characteristic
118	C₄×C₄₃	C₄×#41	1	3	yes	—
119	Dic₄₃	—	2	4	no	—
120	Dic₄₃	—	4	6	no	—
121	Dic₄₃	—	4	6	no	—
122	C₂×C₄×SD₁₆	—	43	3	no	—
123	C₄²×Q₈	—	43	1	no	—
124	C₄²×D₄	—	43	1	no	—
125	C₄×#106	—	43	3	no	—
126	C₄×#103	—	43	3	no	—
127	C₄×#102	—	43	3	no	—
128	C₄²×C₈	D₄₃	1	1	yes	characteristic, complemented
129	C₂×C₄₃	C₄²×D₄	1	1	yes	characteristic
130	C₂×C₄₃	C₂×C₄×SD₁₆	1	3	yes	—
131	C₂×C₄₃	C₄×#135	1	3	yes	—
132	D₄₃	—	4	4	no	—
133	C₂²×SD₁₆	—	43	1	no	—
134	C₂×C₄×Q₈	—	43	3	no	—
135	C₂×C₄×D₄	—	43	3	no	—
136	C₂²×C₄²	—	86	1	no	—
137	C₄×SD₁₆	—	43	24	no	—
138	C₂×#106	—	43	3	no	—
139	C₂×#103	—	43	3	no	—
140	C₂×#102	—	43	3	no	—
141	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
142	C₂×C₄×C₈	C₂×D₄₃	1	3	yes	—
143	C₄×C₄⋊C₄	—	43	3	no	—
144	C₄×C₄⋊C₄	—	43	3	no	—
145	C₄×C₄⋊C₄	—	86	3	no	—
146	C₄×C₂²⋊C₄	—	86	3	no	—
147	C₄³	—	86	1	no	—
148	C₄³	C₂×D₄₃	1	1	yes	characteristic
149	#340	—	43	6	no	—
150	C₄₃	C₄²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₈	—	43	1	no	—
152	C₂²×D₄	—	43	1	no	—
153	C₂³×C₄	—	86	3	no	—
154	C₂×SD₁₆	—	43	12	no	—
155	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
156	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
157	C₄×Q₈	—	43	12	no	—
158	C₄×Q₈	—	86	6	no	—
159	C₄×D₄	—	43	12	no	—
160	C₄×D₄	—	86	6	no	—
161	C₂×C₄⋊C₄	—	43	3	no	—
162	C₂×C₄⋊C₄	—	43	3	no	—
163	C₂×C₄⋊C₄	—	86	3	no	—
164	C₂×C₂²⋊C₄	—	86	3	no	—
165	C₂×C₄²	—	86	3	no	—
166	C₂×C₄²	—	86	3	no	—
167	C₂×C₄²	—	86	3	no	—
168	C₂×C₄²	—	172	4	no	—
169	C₂×C₄²	D₁₇₂	1	3	yes	—
170	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
172	#106	—	43	12	no	—
173	#103	—	43	12	no	—
174	#102	—	43	12	no	—
175	C₄×C₈	—	2	6	no	—
176	C₄×C₈	C₄×D₄₃	1	12	yes	complemented
177	#95	—	86	3	no	—
178	#95	—	86	3	no	—
179	C₂⁴	—	86	1	no	—
180	C₂×Q₈	—	43	6	no	—
181	C₂×Q₈	—	86	3	no	—
182	C₂×D₄	—	43	6	no	—
183	C₂×D₄	—	86	3	no	—
184	C₂²×C₄	—	86	1	no	—
185	C₂²×C₄	—	86	3	no	—
186	C₂²×C₄	—	86	3	no	—
187	C₂²×C₄	—	86	12	no	—
188	C₂²×C₄	—	172	6	no	—
189	C₂²×C₄	—	172	6	no	—
190	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
191	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
192	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
193	SD₁₆	—	43	16	no	—
194	C₂×C₈	—	2	6	no	—
195	C₂×C₈	C₂×C₄×D₄₃	1	6	yes	—
196	C₂×C₈	C₂×C₄×D₄₃	1	6	yes	—
197	C₄⋊C₄	—	43	6	no	—
198	C₄⋊C₄	—	43	6	no	—
199	C₄⋊C₄	—	86	12	no	—
200	C₄⋊C₄	—	86	3	no	—
201	C₄⋊C₄	—	86	3	no	—
202	C₂²⋊C₄	—	86	12	no	—
203	C₄²	—	2	6	no	—
204	C₄²	—	86	12	no	—
205	C₄²	—	172	6	no	—
206	C₄²	—	172	6	no	—
207	C₄²	#10621873	1	4	yes	direct product factor
208	C₄²	C₂×C₄×D₄₃	1	6	yes	—
209	C₄²	#10621898	1	6	yes	—
210	C₂³	—	86	6	no	—
211	C₂³	—	172	1	no	—
212	C₂³	—	172	3	no	—
213	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
214	Q₈	—	43	4	no	—
215	Q₈	—	86	6	no	—
216	D₄	—	43	4	no	—
217	D₄	—	86	6	no	—
218	C₂×C₄	—	2	6	no	—
219	C₂×C₄	—	86	6	no	—
220	C₂×C₄	—	86	6	no	—
221	C₂×C₄	—	86	6	no	—
222	C₂×C₄	—	172	3	no	—
223	C₂×C₄	—	172	6	no	—
224	C₂×C₄	—	172	3	no	—
225	C₂×C₄	—	172	24	no	—
226	C₂×C₄	C₄×D₁₇₂	1	6	yes	—
227	C₂×C₄	C₄×D₁₇₂	1	6	yes	—
228	C₂×C₄	C₂×#10621873	1	6	yes	—
229	C₂×C₄	#13063347	1	6	yes	—
230	C₂×C₄	C₂²×C₄×D₄₃	1	3	yes	—
231	C₂×C₄	C₂×#10621898	1	3	yes	—
232	C₈	—	2	6	no	—
233	C₈	C₄²×D₄₃	1	4	yes	complemented
234	C₂²	—	86	4	no	—
235	C₂²	—	172	6	no	—
236	C₂²	—	172	6	no	—
237	C₂²	C₂×C₄×D₁₇₂	1	3	yes	—
238	C₂²	C₂²×#10621873	1	1	yes	characteristic
239	C₂²	C₂×#13063347	1	3	yes	—
240	C₄	—	2	6	no	—
241	C₄	—	86	4	no	—
242	C₄	—	172	6	no	—
243	C₄	—	172	6	no	—
244	C₄	C₄×#10621873	1	12	yes	direct product factor
245	C₄	C₂×C₄²×D₄₃	1	1	yes	characteristic
246	C₄	C₄×#10621898	1	3	yes	—
247	C₂	—	172	4	no	—
248	C₂	C₄²×D₁₇₂	1	1	yes	characteristic
249	C₂	C₂×C₄×#10621873	1	3	yes	—
250	C₂	C₄×#13063347	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²	D₁₇₂
commutator	C₄×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄²×C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₁₇₂	C₂³
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄²×C₈	D₄₃
hypocenter	C₄₃	C₄²×SD₁₆
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
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×Dic₈₆	C₂	semidirect
4. C₄×#13063257	C₂	semidirect
5. C₄×#13063250	C₂	semidirect
7. C₄²×C₈×C₄₃	C₂	semidirect
12. C₄×#10621873	C₄	direct (N × H)
30. C₄×Dic₈₆	C₂×C₄	semidirect
41. #13063257	C₂×C₄	semidirect
42. #13063250	C₂×C₄	semidirect
52. C₄×C₈×C₄₃	C₂×C₄	semidirect
71. C₄²×C₄₃	SD₁₆	semidirect
83. #10621873	C₄²	direct (N × H)
105. Dic₈₆	C₂×C₄²	semidirect
107. C₈×C₄₃	C₂×C₄²	semidirect
116. C₄×C₄₃	C₄×SD₁₆	semidirect
128. C₄²×C₈	D₄₃	semidirect
150. C₄₃	C₄²×SD₁₆	semidirect
176. C₄×C₈	C₄×D₄₃	semidirect
207. C₄²	#10621873	direct (N × H)
233. C₈	C₄²×D₄₃	semidirect
244. C₄	C₄×#10621873	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×#10621873	2
C₄²×Dic₈₆	2
C₄²×D₁₇₂	2
C₄×#13063257	2
C₄×#13063250	2
C₄×#13063247	2
C₄²×C₈×C₄₃	2
C₄²×SD₁₆	43

Subgroups — C42×#10621873 #25095070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#10621873 #25095070