Subgroups — C₂²×#14583511 #26003415

Order 11008.

7110 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583511	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13063325	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13063318	C₂	1	4	yes	maximal
4	C₂²×#13063317	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13063315	C₂	1	4	yes	maximal, complemented
6	C₂²×#13063304	C₂	1	2	yes	maximal
7	C₂²×C₄×Dic₈₆	C₂	1	2	yes	maximal, complemented
8	C₂²×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₈₆	C₂²	1	2	yes	—
10	#14583511	C₂²	1	256	yes	direct product factor
11	C₂×#13063325	C₂²	1	24	yes	complemented
12	C₂×#13063318	C₂²	1	96	yes	—
13	C₂×#13063317	C₂²	1	24	yes	complemented
14	C₂×#13063315	C₂²	1	96	yes	complemented
15	C₂×#13063304	C₂²	1	48	yes	—
16	C₂×C₄×Dic₈₆	C₂²	1	48	yes	complemented
17	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
18	C₂²×C₄⋊Dic₄₃	C₂²	1	4	yes	—
19	C₂²×Dic₄₃⋊C₄	C₂²	1	4	yes	—
20	C₂²×Dic₄₃⋊C₄	C₂²	1	4	yes	—
21	C₂²×Dic₄₃⋊C₄	C₂²	1	4	yes	—
22	C₂²×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
23	C₂²×C₄×Dic₄₃	C₂²	1	4	yes	—
24	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#117×C₄₃	C₂²	1	24	yes	—
26	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
28	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
30	C₂²×Dic₈₆	—	2	16	no	—
31	C₂²×Dic₈₆	C₂³	1	24	yes	—
32	C₂³×Dic₄₃	C₂³	1	4	yes	—
33	C₂³×Dic₄₃	C₂³	1	4	yes	—
34	#13063325	C₂³	1	64	yes	complemented
35	#13063318	C₂³	1	256	yes	—
36	#13063317	C₂³	1	64	yes	complemented
37	#13063315	C₂³	1	256	yes	complemented
38	#13063304	C₂³	1	128	yes	—
39	C₄×Dic₈₆	C₂³	1	128	yes	complemented
40	C₂×C₄⋊Dic₄₃	C₂³	1	24	yes	—
41	C₂×C₄⋊Dic₄₃	C₂³	1	48	yes	—
42	C₂×Dic₄₃⋊C₄	C₂³	1	48	yes	—
43	C₂×Dic₄₃⋊C₄	C₂³	1	48	yes	—
44	C₂×Dic₄₃⋊C₄	C₂³	1	48	yes	—
45	C₂×C₂²⋊Dic₄₃	C₂³	1	48	yes	—
46	C₂×C₄×Dic₄₃	C₂³	1	48	yes	—
47	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₄₃	—	2	8	no	—
49	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
50	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
51	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₄₃	C₂³	1	12	yes	—
53	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
54	#117×C₄₃	C₂³	1	64	yes	—
55	C₂×C₄⋊C₄×C₄₃	C₂³	1	24	yes	—
56	C₂×C₂²⋊C₄×C₄₃	C₂³	1	24	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	12	yes	—
58	C₂×C₄²×C₄₃	C₂³	1	12	yes	—
59	C₂⁴×C₄₃	—	2	4	no	—
60	C₂⁴×C₄₃	—	2	8	no	—
61	C₂⁴×C₄₃	C₂⁴	1	6	yes	—
62	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	—	2	48	no	—
64	C₂²×C₄×C₄₃	—	2	4	no	—
65	C₂²×C₄×C₄₃	—	2	8	no	—
66	C₂²×C₄×C₄₃	—	2	8	no	—
67	C₂²×C₄×C₄₃	#41	1	8	yes	—
68	C₂²×C₄×C₄₃	C₂⁴	1	12	yes	—
69	C₂²×C₄×C₄₃	C₂⁴	1	12	yes	—
70	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
71	C₂²×C₄×C₄₃	C₂⁴	1	16	yes	—
72	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
73	C₂²×Dic₄₃	—	2	16	no	—
74	C₂²×Dic₄₃	—	2	16	no	—
75	C₂²×Dic₄₃	C₂⁴	1	24	yes	—
76	C₂²×Dic₄₃	C₂⁴	1	24	yes	—
77	C₂×Dic₈₆	—	2	96	no	—
78	C₂×Dic₈₆	C₂⁴	1	32	yes	—
79	C₄⋊C₄×C₄₃	C₂⁴	1	32	yes	—
80	C₂²⋊C₄×C₄₃	C₂⁴	1	32	yes	—
81	C₄²×C₄₃	C₂⁴	1	16	yes	—
82	C₄²×C₄₃	C₂⁴	1	16	yes	—
83	C₂²⋊Dic₄₃	C₂⁴	1	64	yes	—
84	C₄⋊Dic₄₃	C₂⁴	1	32	yes	—
85	C₄⋊Dic₄₃	C₂⁴	1	64	yes	—
86	Dic₄₃⋊C₄	C₂⁴	1	64	yes	—
87	Dic₄₃⋊C₄	C₂⁴	1	64	yes	—
88	Dic₄₃⋊C₄	C₂⁴	1	64	yes	—
89	C₄×Dic₄₃	C₂⁴	1	64	yes	—
90	C₂³×C₄₃	—	2	12	no	—
91	C₂³×C₄₃	—	2	24	no	—
92	C₂³×C₄₃	—	4	16	no	—
93	C₂³×C₄₃	C₂×#41	1	4	yes	—
94	C₂³×C₄₃	2₋¹⁺⁴	1	8	yes	—
95	C₂³×C₄₃	C₂⁵	1	4	yes	—
96	C₂³×C₄₃	C₂⁵	1	3	yes	—
97	C₂×C₄×C₄₃	—	2	12	no	—
98	C₂×C₄×C₄₃	—	2	24	no	—
99	C₂×C₄×C₄₃	—	2	24	no	—
100	C₂×C₄×C₄₃	—	2	32	no	—
101	C₂×C₄×C₄₃	C₂×#41	1	24	yes	—
102	C₂×C₄×C₄₃	C₂⁵	1	8	yes	—
103	C₂×C₄×C₄₃	C₂⁵	1	4	yes	—
104	C₂×C₄×C₄₃	C₂⁵	1	8	yes	—
105	C₂×C₄×C₄₃	C₂⁵	1	4	yes	—
106	C₂×Dic₄₃	—	2	48	no	—
107	C₂×Dic₄₃	—	2	48	no	—
108	C₂×Dic₄₃	C₂⁵	1	16	yes	—
109	C₂×Dic₄₃	C₂⁵	1	16	yes	—
110	Dic₈₆	—	2	64	no	—
111	C₂²×#541	—	43	1	no	maximal, Sylow
112	C₂²×C₄₃	—	2	4	no	—
113	C₂²×C₄₃	—	2	8	no	—
114	C₂²×C₄₃	—	4	24	no	—
115	C₂²×C₄₃	#541	1	16	yes	complemented
116	C₂²×C₄₃	C₂²×#41	1	6	yes	—
117	C₂²×C₄₃	C₂×2₋¹⁺⁴	1	12	yes	—
118	C₂²×C₄₃	C₂⁶	1	1	yes	characteristic
119	C₄×C₄₃	—	2	4	no	—
120	C₄×C₄₃	—	2	8	no	—
121	C₄×C₄₃	—	2	8	no	—
122	C₄×C₄₃	C₂²×#41	1	8	yes	—
123	Dic₄₃	—	2	16	no	—
124	Dic₄₃	—	2	16	no	—
125	C₂×#541	—	43	48	no	—
126	C₂²×C₄⋊Q₈	—	43	1	no	—
127	C₂²×#126	—	43	4	no	—
128	C₂²×#125	—	43	1	no	—
129	C₂²×#125	—	43	4	no	—
130	C₂²×C₂²⋊Q₈	—	43	2	no	—
131	C₂²×C₄×Q₈	—	43	2	no	—
132	C₂²×#117	D₄₃	1	1	yes	characteristic
133	C₂×C₄₃	—	4	4	no	—
134	C₂×C₄₃	C₂×#541	1	12	yes	complemented
135	C₂×C₄₃	C₂³×#41	1	1	yes	characteristic
136	C₂×C₄₃	C₂²×2₋¹⁺⁴	1	2	yes	—
137	C₂³×Q₈	—	43	2	no	—
138	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
139	#541	—	43	256	no	—
140	C₂×C₄⋊Q₈	—	43	24	no	—
141	C₂×#126	—	43	96	no	—
142	C₂×#125	—	43	96	no	—
143	C₂×#125	—	43	24	no	—
144	C₂×C₂²⋊Q₈	—	43	48	no	—
145	C₂×C₄×Q₈	—	43	48	no	—
146	C₂×#117	C₂×D₄₃	1	24	yes	—
147	C₂²×C₄⋊C₄	—	43	4	no	—
148	C₂²×C₄⋊C₄	—	43	4	no	—
149	C₂²×C₄⋊C₄	—	43	4	no	—
150	C₂²×C₄⋊C₄	—	43	2	no	—
151	C₂²×C₄⋊C₄	—	43	4	no	—
152	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
153	C₂²×C₂²⋊C₄	—	43	4	no	—
154	C₂²×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
155	C₂²×C₄²	—	43	4	no	—
156	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
157	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
158	C₄₃	C₂²×#541	1	1	yes	characteristic, complemented, Sylow
159	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
160	C₂²×Q₈	—	43	24	no	—
161	C₂²×Q₈	—	86	16	no	—
162	C₂³×C₄	—	2	8	no	—
163	C₂³×C₄	—	43	4	no	—
164	C₂³×C₄	—	43	4	no	—
165	C₂³×C₄	C₂²×D₄₃	1	12	yes	—
166	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
167	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
168	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
169	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
170	C₄⋊Q₈	—	43	64	no	—
171	#126	—	43	256	no	—
172	#125	—	43	256	no	—
173	#125	—	43	64	no	—
174	C₂²⋊Q₈	—	43	128	no	—
175	C₄×Q₈	—	43	128	no	—
176	#117	C₂²×D₄₃	1	64	yes	—
177	C₂×C₄⋊C₄	—	43	24	no	—
178	C₂×C₄⋊C₄	—	43	48	no	—
179	C₂×C₄⋊C₄	—	43	48	no	—
180	C₂×C₄⋊C₄	—	43	48	no	—
181	C₂×C₄⋊C₄	—	43	48	no	—
182	C₂×C₄⋊C₄	C₂²×D₄₃	1	24	yes	—
183	C₂×C₂²⋊C₄	—	43	48	no	—
184	C₂×C₂²⋊C₄	C₂²×D₄₃	1	24	yes	—
185	C₂×C₄²	—	43	48	no	—
186	C₂×C₄²	C₂²×D₄₃	1	12	yes	—
187	C₂×C₄²	C₂²×D₄₃	1	12	yes	—
188	C₂⁴	—	2	4	no	—
189	C₂⁴	—	2	8	no	—
190	C₂⁴	C₂³×D₄₃	1	6	yes	—
191	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
192	C₂×Q₈	—	43	32	no	—
193	C₂×Q₈	—	86	96	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	48	no	—
198	C₂²×C₄	—	43	24	no	—
199	C₂²×C₄	—	43	24	no	—
200	C₂²×C₄	—	86	16	no	—
201	C₂²×C₄	—	86	16	no	—
202	C₂²×C₄	#10621898	1	8	yes	—
203	C₂²×C₄	C₂³×D₄₃	1	12	yes	—
204	C₂²×C₄	C₂³×D₄₃	1	12	yes	—
205	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
206	C₂²×C₄	C₂³×D₄₃	1	16	yes	—
207	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
208	C₄⋊C₄	—	43	32	no	—
209	C₄⋊C₄	—	43	64	no	—
210	C₄⋊C₄	—	43	64	no	—
211	C₄⋊C₄	—	43	64	no	—
212	C₄⋊C₄	—	43	64	no	—
213	C₄⋊C₄	C₂³×D₄₃	1	32	yes	—
214	C₂²⋊C₄	—	43	64	no	—
215	C₂²⋊C₄	C₂³×D₄₃	1	32	yes	—
216	C₄²	—	43	64	no	—
217	C₄²	C₂³×D₄₃	1	16	yes	—
218	C₄²	C₂³×D₄₃	1	16	yes	—
219	C₂³	—	2	12	no	—
220	C₂³	—	2	24	no	—
221	C₂³	—	4	16	no	—
222	C₂³	C₂×#10621898	1	4	yes	—
223	C₂³	#13063372	1	8	yes	—
224	C₂³	C₂⁴×D₄₃	1	4	yes	—
225	C₂³	C₂⁴×D₄₃	1	3	yes	—
226	Q₈	—	86	64	no	—
227	C₂×C₄	—	2	12	no	—
228	C₂×C₄	—	2	24	no	—
229	C₂×C₄	—	2	24	no	—
230	C₂×C₄	—	2	32	no	—
231	C₂×C₄	—	43	16	no	—
232	C₂×C₄	—	43	16	no	—
233	C₂×C₄	—	86	48	no	—
234	C₂×C₄	—	86	48	no	—
235	C₂×C₄	C₂×#10621898	1	24	yes	—
236	C₂×C₄	C₂⁴×D₄₃	1	8	yes	—
237	C₂×C₄	C₂⁴×D₄₃	1	8	yes	—
238	C₂×C₄	C₂⁴×D₄₃	1	4	yes	—
239	C₂×C₄	C₂⁴×D₄₃	1	4	yes	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	4	24	no	—
243	C₂²	#14583511	1	16	yes	direct product factor
244	C₂²	C₂²×#10621898	1	6	yes	—
245	C₂²	C₂×#13063372	1	12	yes	—
246	C₂²	C₂⁵×D₄₃	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	—	2	8	no	—
249	C₄	—	2	8	no	—
250	C₄	—	86	16	no	—
251	C₄	—	86	16	no	—
252	C₄	C₂²×#10621898	1	8	yes	—
253	C₂	—	4	4	no	—
254	C₂	C₂×#14583511	1	12	yes	direct product factor
255	C₂	C₂³×#10621898	1	1	yes	characteristic
256	C₂	C₂²×#13063372	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₃
commutator, cosocle	C₂²×C₄₃	C₂⁶
Frattini	C₂²	C₂⁵×D₄₃
Fitting	C₂²×#117×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂⁴
prime radical	C₂⁵×C₄₃	C₂³
hypercenter	C₂²×#117	D₄₃
hypocenter	C₄₃	C₂²×#541
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
1. C₂×#14583511	C₂	direct (N × H)
2. C₂²×#13063325	C₂	semidirect
4. C₂²×#13063317	C₂	semidirect
5. C₂²×#13063315	C₂	semidirect
7. C₂²×C₄×Dic₈₆	C₂	semidirect
10. #14583511	C₂²	direct (N × H)
11. C₂×#13063325	C₂²	semidirect
13. C₂×#13063317	C₂²	semidirect
14. C₂×#13063315	C₂²	semidirect
16. C₂×C₄×Dic₈₆	C₂²	semidirect
34. #13063325	C₂³	semidirect
36. #13063317	C₂³	semidirect
37. #13063315	C₂³	semidirect
39. C₄×Dic₈₆	C₂³	semidirect
115. C₂²×C₄₃	#541	semidirect
134. C₂×C₄₃	C₂×#541	semidirect
158. C₄₃	C₂²×#541	semidirect
243. C₂²	#14583511	direct (N × H)
254. C₂	C₂×#14583511	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×#14583511	2
C₂²×#13063325	2
C₂²×#13063318	2
C₂²×#13063317	2
C₂²×#13063315	2
C₂²×#13063304	2
C₂²×C₄×Dic₈₆	2
C₂²×#117×C₄₃	2
C₂²×#541	43

Subgroups — C22×#14583511 #26003415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583511 #26003415