Subgroups — C₂²×#14582934 #25960244

Order 11008.

3006 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×D₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13063283	C₂	1	2	yes	maximal
3	C₂×#14582934	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13063258	C₂	1	2	yes	maximal, complemented
5	C₂²×#13063257	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#106×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₄₃	C₂²	1	1	yes	characteristic
8	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₈×D₄₃	C₄	1	16	yes	complemented
10	C₂²×C₈×D₄₃	C₂²	1	12	yes	—
11	C₂×#13063283	—	2	12	no	—
12	C₂×#13063283	C₂²	1	24	yes	—
13	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
14	C₂²×(C₂×D₄₃)⋊C₄	—	2	2	no	—
15	C₂²×C₄×Dic₄₃	—	2	2	no	—
16	#14582934	C₂²	1	64	yes	direct product factor
17	C₂×#13063258	C₂²	1	24	yes	complemented
18	C₂×#13063257	C₂²	1	12	yes	complemented
19	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
21	C₂×#106×C₄₃	C₂²	1	12	yes	complemented
22	C₂⁴×D₄₃	D₄	1	1	yes	characteristic
23	C₂³×Dic₄₃	—	2	2	no	—
24	C₂³×Dic₄₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×D₄₃	C₂×C₄	1	8	yes	—
26	C₂²×C₄×D₄₃	D₄	1	6	yes	—
27	C₂²×C₄×D₄₃	Q₈	1	8	yes	—
28	C₂²×C₄×D₄₃	C₂³	1	6	yes	—
29	C₂²×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
30	C₂²×C₄₃⋊C₈	C₂³	1	6	yes	—
31	C₂×C₈×D₄₃	C₂×C₄	1	96	yes	complemented
32	C₂×C₈×D₄₃	C₂³	1	16	yes	—
33	#13063283	—	2	48	no	—
34	#13063283	C₂³	1	32	yes	—
35	C₂×C₄⋊Dic₄₃	—	2	6	no	—
36	C₂×C₄⋊Dic₄₃	C₂³	1	12	yes	—
37	C₂×(C₂×D₄₃)⋊C₄	—	2	24	no	—
38	C₂×C₄×Dic₄₃	—	2	24	no	—
39	#13063258	C₂³	1	32	yes	complemented
40	#13063257	C₂³	1	16	yes	complemented
41	C₂³×C₄×C₄₃	—	2	2	no	—
42	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₄₃	C₂×C₄	1	8	yes	—
44	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
45	C₂×C₄⋊C₄×C₄₃	—	2	6	no	—
46	C₂×C₄⋊C₄×C₄₃	C₂³	1	12	yes	—
47	#106×C₄₃	C₂³	1	16	yes	complemented
48	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
49	C₂³×D₄₃	—	2	8	no	—
50	C₂³×D₄₃	C₄⋊C₄	1	8	yes	—
51	C₂³×D₄₃	C₂×D₄	1	6	yes	—
52	C₂²×C₄×C₄₃	—	2	12	no	—
53	C₂²×C₄×C₄₃	—	4	8	no	—
54	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
57	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
58	C₂²×Dic₄₃	—	2	12	no	—
59	C₂²×Dic₄₃	—	4	8	no	—
60	C₂²×Dic₄₃	C₄⋊C₄	1	8	yes	—
61	C₂²×Dic₄₃	C₂×D₄	1	6	yes	—
62	C₂×C₄×D₄₃	C₄⋊C₄	1	48	yes	—
63	C₂×C₄×D₄₃	C₂²×C₄	1	24	yes	—
64	C₂×C₄×D₄₃	C₂×D₄	1	12	yes	—
65	C₂×C₄×D₄₃	C₂×Q₈	1	24	yes	—
66	C₂×C₄×D₄₃	C₂⁴	1	4	yes	—
67	C₂×C₈×C₄₃	C₂²×C₄	1	24	yes	—
68	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
69	C₄⋊C₄×C₄₃	—	2	12	no	—
70	C₄⋊C₄×C₄₃	C₂⁴	1	8	yes	—
71	(C₂×D₄₃)⋊C₄	—	2	32	no	—
72	C₄⋊Dic₄₃	—	2	12	no	—
73	C₄⋊Dic₄₃	C₂⁴	1	8	yes	—
74	C₄×Dic₄₃	—	2	32	no	—
75	C₂×C₄₃⋊C₈	C₂²×C₄	1	24	yes	—
76	C₂×C₄₃⋊C₈	C₂⁴	1	4	yes	—
77	C₈×D₄₃	C₂²×C₄	1	64	yes	complemented
78	C₂³×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
79	C₂³×C₄₃	#135	1	8	yes	—
80	C₂³×C₄₃	C₂²×D₄	1	3	yes	—
81	C₂²×D₄₃	—	2	24	no	—
82	C₂²×D₄₃	—	2	32	no	—
83	C₂²×D₄₃	C₂×C₄⋊C₄	1	24	yes	—
84	C₂²×D₄₃	C₂²×D₄	1	4	yes	—
85	C₂×C₄×C₄₃	—	2	8	no	—
86	C₂×C₄×C₄₃	—	4	24	no	—
87	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	12	yes	—
88	C₂×C₄×C₄₃	C₂³×C₄	1	6	yes	—
89	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
90	C₂×C₄×C₄₃	C₂²×Q₈	1	6	yes	—
91	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
92	C₂×Dic₄₃	—	2	8	no	—
93	C₂×Dic₄₃	—	4	24	no	—
94	C₂×Dic₄₃	C₂×C₄⋊C₄	1	24	yes	—
95	C₂×Dic₄₃	C₂²×D₄	1	4	yes	—
96	C₄×D₄₃	C₂×C₄⋊C₄	1	48	yes	—
97	C₄×D₄₃	C₂³×C₄	1	8	yes	—
98	C₄×D₄₃	C₂²×Q₈	1	8	yes	—
99	C₈×C₄₃	C₂³×C₄	1	8	yes	—
100	C₄₃⋊C₈	C₂³×C₄	1	8	yes	—
101	C₂²×#427	—	43	1	no	maximal, Sylow
102	C₂²×C₄₃	#427	1	16	yes	complemented
103	C₂²×C₄₃	C₂²×C₄⋊C₄	1	6	yes	—
104	C₂²×C₄₃	C₂×#135	1	12	yes	—
105	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
106	C₂×D₄₃	—	2	48	no	—
107	C₂×D₄₃	—	2	8	no	—
108	C₂×D₄₃	C₂²×C₄⋊C₄	1	8	yes	—
109	C₄×C₄₃	—	4	8	no	—
110	C₄×C₄₃	C₂²×C₄⋊C₄	1	6	yes	—
111	C₄×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
112	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
113	Dic₄₃	—	4	8	no	—
114	Dic₄₃	C₂²×C₄⋊C₄	1	8	yes	—
115	C₂⁴×C₈	—	43	1	no	—
116	C₂²×#117	—	43	2	no	—
117	C₂×#427	—	43	24	no	—
118	C₂²×#107	—	43	2	no	—
119	C₂²×#106	—	43	1	no	—
120	C₂²×#106	D₄₃	1	1	yes	characteristic, complemented
121	C₂×C₄₃	C₂×#427	1	12	yes	complemented
122	C₂×C₄₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄₃	C₂²×#135	1	2	yes	—
124	D₄₃	—	2	8	no	—
125	C₂⁴×C₄	—	43	1	no	—
126	C₂³×C₈	—	43	1	no	—
127	C₂³×C₈	—	43	12	no	—
128	C₂³×C₈	—	43	16	no	—
129	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
130	C₂×#117	—	43	24	no	—
131	C₂×#117	—	86	12	no	—
132	C₂²×C₄⋊C₄	—	43	2	no	—
133	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
134	C₂²×C₂²⋊C₄	—	86	2	no	—
135	C₂²×C₄²	—	86	2	no	—
136	#427	—	43	64	no	—
137	C₂×#107	—	43	24	no	—
138	C₂×#106	—	43	12	no	—
139	C₂×#106	C₂×D₄₃	1	12	yes	complemented
140	C₄₃	C₂²×#427	1	1	yes	characteristic, complemented, Sylow
141	C₂⁵	—	43	1	no	—
142	C₂³×C₄	—	2	2	no	—
143	C₂³×C₄	—	43	1	no	—
144	C₂³×C₄	—	43	8	no	—
145	C₂³×C₄	—	43	6	no	—
146	C₂³×C₄	—	43	8	no	—
147	C₂³×C₄	—	43	6	no	—
148	C₂³×C₄	—	86	2	no	—
149	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
150	C₂²×C₈	—	43	8	no	—
151	C₂²×C₈	—	43	6	no	—
152	C₂²×C₈	—	43	96	no	—
153	C₂²×C₈	—	43	16	no	—
154	C₂²×C₈	C₄×D₄₃	1	8	yes	—
155	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
156	#117	—	43	32	no	—
157	#117	—	86	48	no	—
158	C₂×C₄⋊C₄	—	2	6	no	—
159	C₂×C₄⋊C₄	—	43	12	no	—
160	C₂×C₄⋊C₄	—	86	6	no	—
161	C₂×C₄⋊C₄	C₂²×D₄₃	1	12	yes	—
162	C₂×C₂²⋊C₄	—	86	24	no	—
163	C₂×C₄²	—	86	24	no	—
164	#107	—	43	32	no	—
165	#106	—	43	16	no	—
166	#106	C₂²×D₄₃	1	16	yes	complemented
167	C₂⁴	—	43	8	no	—
168	C₂⁴	—	43	6	no	—
169	C₂⁴	—	86	8	no	—
170	C₂⁴	D₄×D₄₃	1	1	yes	characteristic
171	C₂²×C₄	—	2	12	no	—
172	C₂²×C₄	—	4	8	no	—
173	C₂²×C₄	—	43	8	no	—
174	C₂²×C₄	—	43	6	no	—
175	C₂²×C₄	—	43	48	no	—
176	C₂²×C₄	—	43	24	no	—
177	C₂²×C₄	—	43	12	no	—
178	C₂²×C₄	—	43	24	no	—
179	C₂²×C₄	—	43	4	no	—
180	C₂²×C₄	—	86	12	no	—
181	C₂²×C₄	—	172	8	no	—
182	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
183	C₂²×C₄	D₄×D₄₃	1	3	yes	—
184	C₂²×C₄	Q₈×D₄₃	1	4	yes	—
185	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
186	C₂×C₈	—	43	24	no	—
187	C₂×C₈	—	43	4	no	—
188	C₂×C₈	—	43	64	no	—
189	C₂×C₈	C₂×C₄×D₄₃	1	24	yes	—
190	C₂×C₈	C₂³×D₄₃	1	4	yes	—
191	C₄⋊C₄	—	2	12	no	—
192	C₄⋊C₄	—	43	8	no	—
193	C₄⋊C₄	—	86	12	no	—
194	C₄⋊C₄	C₂³×D₄₃	1	8	yes	—
195	C₂²⋊C₄	—	86	32	no	—
196	C₄²	—	86	32	no	—
197	C₂³	—	43	24	no	—
198	C₂³	—	43	4	no	—
199	C₂³	—	86	24	no	—
200	C₂³	—	86	32	no	—
201	C₂³	C₄⋊C₄×D₄₃	1	4	yes	—
202	C₂³	#13063346	1	8	yes	—
203	C₂³	C₂×D₄×D₄₃	1	3	yes	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	4	24	no	—
206	C₂×C₄	—	43	24	no	—
207	C₂×C₄	—	43	4	no	—
208	C₂×C₄	—	43	48	no	—
209	C₂×C₄	—	43	8	no	—
210	C₂×C₄	—	43	8	no	—
211	C₂×C₄	—	86	8	no	—
212	C₂×C₄	—	172	24	no	—
213	C₂×C₄	C₄⋊C₄×D₄₃	1	12	yes	—
214	C₂×C₄	C₂²×C₄×D₄₃	1	6	yes	—
215	C₂×C₄	C₂×D₄×D₄₃	1	3	yes	—
216	C₂×C₄	C₂×Q₈×D₄₃	1	6	yes	—
217	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
218	C₈	—	43	8	no	—
219	C₈	C₂²×C₄×D₄₃	1	8	yes	—
220	C₂²	—	43	8	no	—
221	C₂²	—	86	48	no	—
222	C₂²	—	86	8	no	—
223	C₂²	#14582934	1	16	yes	direct product factor
224	C₂²	C₂×C₄⋊C₄×D₄₃	1	6	yes	—
225	C₂²	C₂×#13063346	1	12	yes	—
226	C₂²	C₂²×D₄×D₄₃	1	1	yes	characteristic
227	C₄	—	4	8	no	—
228	C₄	—	43	8	no	—
229	C₄	—	172	8	no	—
230	C₄	C₂×C₄⋊C₄×D₄₃	1	6	yes	—
231	C₄	C₂³×C₄×D₄₃	1	1	yes	characteristic
232	C₄	C₂²×Q₈×D₄₃	1	1	yes	characteristic
233	C₂	—	86	8	no	—
234	C₂	C₂×#14582934	1	12	yes	direct product factor
235	C₂	C₂²×C₄⋊C₄×D₄₃	1	1	yes	characteristic
236	C₂	C₂²×#13063346	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₄₃
commutator	C₄×C₄₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂²×#106×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂×D₄
prime radical	C₂⁴×D₄₃	D₄
cosocle	C₂×C₄×C₄₃	C₂⁵
hypercenter	C₂²×#106	D₄₃
hypocenter	C₄₃	C₂²×#427
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
3. C₂×#14582934	C₂	direct (N × H)
4. C₂²×#13063258	C₂	semidirect
5. C₂²×#13063257	C₂	semidirect
6. C₂²×#106×C₄₃	C₂	semidirect
9. C₂²×C₈×D₄₃	C₄	semidirect
16. #14582934	C₂²	direct (N × H)
17. C₂×#13063258	C₂²	semidirect
18. C₂×#13063257	C₂²	semidirect
21. C₂×#106×C₄₃	C₂²	semidirect
31. C₂×C₈×D₄₃	C₂×C₄	semidirect
39. #13063258	C₂³	semidirect
40. #13063257	C₂³	semidirect
47. #106×C₄₃	C₂³	semidirect
77. C₈×D₄₃	C₂²×C₄	semidirect
102. C₂²×C₄₃	#427	semidirect
120. C₂²×#106	D₄₃	semidirect
121. C₂×C₄₃	C₂×#427	semidirect
139. C₂×#106	C₂×D₄₃	semidirect
140. C₄₃	C₂²×#427	semidirect
166. #106	C₂²×D₄₃	semidirect
223. C₂²	#14582934	direct (N × H)
234. C₂	C₂×#14582934	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×D₄₃	2
C₂²×#13063283	2
C₂×#14582934	2
C₂²×#13063258	2
C₂²×#13063257	2
C₂²×#106×C₄₃	2
C₂²×#427	43

Subgroups — C22×#14582934 #25960244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14582934 #25960244