Subgroups — C₂²×#14583376 #26003181

Order 11008.

11230 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	C₂⁴×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583376	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₄×Dic₈₆	C₂	1	4	yes	maximal, complemented
4	C₂²×#13063290	C₂	1	4	yes	maximal, complemented
5	C₂²×#13063284	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13063281	C₂	1	4	yes	maximal
7	C₂²×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₈₆	C₄	1	32	yes	complemented
9	C₂³×Dic₈₆	C₂²	1	2	yes	—
10	C₂³×Dic₈₆	C₂²	1	2	yes	—
11	C₂³×Dic₈₆	C₂²	1	24	yes	—
12	C₂⁴×Dic₄₃	C₂²	1	2	yes	—
13	#14583376	C₂²	1	256	yes	direct product factor
14	C₂×C₄×Dic₈₆	C₂²	1	96	yes	complemented
15	C₂×#13063290	C₂²	1	96	yes	complemented
16	C₂×#13063284	C₂²	1	24	yes	—
17	C₂×#13063281	C₂²	1	96	yes	—
18	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
19	C₂²×Dic₄₃⋊C₄	C₂²	1	8	yes	—
20	C₂²×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
21	C₂²×C₄×Dic₄₃	C₂²	1	8	yes	—
22	C₂²×C₄×Dic₄₃	C₂²	1	2	yes	—
23	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×#117×C₄₃	C₂²	1	24	yes	—
25	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
27	C₂²×C₄²×C₄₃	C₂²	1	2	yes	—
28	C₂²×Dic₈₆	C₂×C₄	1	32	yes	—
29	C₂²×Dic₈₆	C₂×C₄	1	32	yes	—
30	C₂²×Dic₈₆	C₂×C₄	1	384	yes	complemented
31	C₂²×Dic₈₆	C₂³	1	24	yes	—
32	C₂²×Dic₈₆	C₂³	1	24	yes	—
33	C₂²×Dic₈₆	C₂³	1	64	yes	—
34	C₂³×Dic₄₃	C₂×C₄	1	32	yes	—
35	C₂³×Dic₄₃	C₂³	1	4	yes	—
36	C₂³×Dic₄₃	C₂³	1	4	yes	—
37	C₂³×Dic₄₃	C₂³	1	24	yes	—
38	C₄×Dic₈₆	C₂³	1	256	yes	complemented
39	#13063290	C₂³	1	256	yes	complemented
40	#13063284	C₂³	1	64	yes	—
41	#13063281	C₂³	1	256	yes	—
42	C₂×C₄⋊Dic₄₃	C₂³	1	24	yes	—
43	C₂×Dic₄₃⋊C₄	C₂³	1	96	yes	—
44	C₂×C₂²⋊Dic₄₃	C₂³	1	24	yes	—
45	C₂×C₄×Dic₄₃	C₂³	1	96	yes	—
46	C₂×C₄×Dic₄₃	C₂³	1	24	yes	—
47	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₄₃	C₂×C₄	1	16	yes	—
49	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
50	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₄₃	C₂³	1	12	yes	—
53	#117×C₄₃	C₂³	1	64	yes	—
54	C₂×C₄⋊C₄×C₄₃	C₂³	1	24	yes	—
55	C₂×C₂²⋊C₄×C₄₃	C₂³	1	24	yes	—
56	C₂×C₄²×C₄₃	C₂³	1	24	yes	—
57	C₂⁴×C₄₃	—	2	8	no	—
58	C₂⁴×C₄₃	C₂²×C₄	1	8	yes	—
59	C₂⁴×C₄₃	C₂⁴	1	6	yes	—
60	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	2	16	no	—
62	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
64	C₂²×C₄×C₄₃	C₂²×C₄	1	96	yes	—
65	C₂²×C₄×C₄₃	C₂⁴	1	24	yes	—
66	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
67	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
68	C₂²×C₄×C₄₃	C₂⁴	1	16	yes	—
69	C₂²×Dic₄₃	—	2	16	no	—
70	C₂²×Dic₄₃	C₂²×C₄	1	32	yes	—
71	C₂²×Dic₄₃	C₂²×C₄	1	192	yes	—
72	C₂²×Dic₄₃	C₂⁴	1	24	yes	—
73	C₂²×Dic₄₃	C₂⁴	1	24	yes	—
74	C₂²×Dic₄₃	C₂⁴	1	32	yes	—
75	C₂×Dic₈₆	C₂²×C₄	1	192	yes	—
76	C₂×Dic₈₆	C₂²×C₄	1	192	yes	—
77	C₂×Dic₈₆	C₂²×C₄	1	512	yes	complemented
78	C₂×Dic₈₆	C₂⁴	1	32	yes	—
79	C₂×Dic₈₆	C₂⁴	1	32	yes	—
80	C₄⋊C₄×C₄₃	C₂⁴	1	32	yes	—
81	C₂²⋊C₄×C₄₃	C₂⁴	1	32	yes	—
82	C₄²×C₄₃	C₂⁴	1	32	yes	—
83	C₂²⋊Dic₄₃	C₂⁴	1	32	yes	—
84	C₄⋊Dic₄₃	C₂⁴	1	32	yes	—
85	Dic₄₃⋊C₄	C₂⁴	1	128	yes	—
86	C₄×Dic₄₃	C₂⁴	1	128	yes	—
87	C₄×Dic₄₃	C₂⁴	1	32	yes	—
88	C₂³×C₄₃	—	2	32	no	—
89	C₂³×C₄₃	—	2	24	no	—
90	C₂³×C₄₃	C₂³×C₄	1	24	yes	—
91	C₂³×C₄₃	C₂³×C₄	1	4	yes	—
92	C₂³×C₄₃	2₋¹⁺⁴	1	8	yes	—
93	C₂³×C₄₃	C₂⁵	1	4	yes	—
94	C₂³×C₄₃	C₂⁵	1	3	yes	—
95	C₂×C₄×C₄₃	—	2	48	no	—
96	C₂×C₄×C₄₃	C₂³×C₄	1	24	yes	—
97	C₂×C₄×C₄₃	C₂³×C₄	1	24	yes	—
98	C₂×C₄×C₄₃	C₂³×C₄	1	64	yes	—
99	C₂×C₄×C₄₃	C₂⁵	1	16	yes	—
100	C₂×C₄×C₄₃	C₂⁵	1	4	yes	—
101	C₂×C₄×C₄₃	C₂⁵	1	4	yes	—
102	C₂×Dic₄₃	—	2	48	no	—
103	C₂×Dic₄₃	C₂³×C₄	1	96	yes	—
104	C₂×Dic₄₃	C₂³×C₄	1	128	yes	—
105	C₂×Dic₄₃	C₂⁵	1	16	yes	—
106	C₂×Dic₄₃	C₂⁵	1	16	yes	—
107	Dic₈₆	C₂³×C₄	1	128	yes	—
108	Dic₈₆	C₂³×C₄	1	128	yes	—
109	C₂²×#519	—	43	1	no	maximal, Sylow
110	C₂²×C₄₃	—	2	8	no	—
111	C₂²×C₄₃	—	2	48	no	—
112	C₂²×C₄₃	#519	1	16	yes	complemented
113	C₂²×C₄₃	C₂⁴×C₄	1	8	yes	—
114	C₂²×C₄₃	C₂⁴×C₄	1	6	yes	—
115	C₂²×C₄₃	C₂×2₋¹⁺⁴	1	12	yes	—
116	C₂²×C₄₃	C₂⁶	1	1	yes	characteristic
117	C₄×C₄₃	—	2	16	no	—
118	C₄×C₄₃	C₂⁴×C₄	1	8	yes	—
119	C₄×C₄₃	C₂⁴×C₄	1	8	yes	—
120	Dic₄₃	—	2	16	no	—
121	Dic₄₃	C₂⁴×C₄	1	32	yes	—
122	C₂⁴×Q₈	—	43	1	no	—
123	C₂×#519	—	43	48	no	—
124	C₂²×C₄×Q₈	—	43	4	no	—
125	C₂²×C₄×Q₈	—	43	4	no	—
126	C₂²×#117	—	43	1	no	—
127	C₂²×#117	—	43	4	no	—
128	C₂²×#117	D₄₃	1	1	yes	characteristic
129	C₂×C₄₃	—	2	8	no	—
130	C₂×C₄₃	C₂×#519	1	12	yes	complemented
131	C₂×C₄₃	C₂⁵×C₄	1	1	yes	characteristic
132	C₂×C₄₃	C₂²×2₋¹⁺⁴	1	2	yes	—
133	C₂³×Q₈	—	43	2	no	—
134	C₂³×Q₈	—	43	2	no	—
135	C₂³×Q₈	—	43	24	no	—
136	C₂³×Q₈	—	43	32	no	—
137	C₂⁴×C₄	—	43	2	no	—
138	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
139	#519	—	43	256	no	—
140	C₂×C₄×Q₈	—	43	96	no	—
141	C₂×C₄×Q₈	—	43	96	no	—
142	C₂×#117	—	43	24	no	—
143	C₂×#117	—	43	96	no	—
144	C₂×#117	C₂×D₄₃	1	24	yes	—
145	C₂²×C₄⋊C₄	—	43	8	no	—
146	C₂²×C₄⋊C₄	—	43	2	no	—
147	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
148	C₂²×C₂²⋊C₄	—	43	2	no	—
149	C₂²×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
150	C₂²×C₄²	—	43	8	no	—
151	C₂²×C₄²	—	43	2	no	—
152	C₂²×C₄²	C₂×D₄₃	1	2	yes	—
153	C₄₃	C₂²×#519	1	1	yes	characteristic, complemented, Sylow
154	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
155	C₂²×Q₈	—	43	32	no	—
156	C₂²×Q₈	—	43	32	no	—
157	C₂²×Q₈	—	43	384	no	—
158	C₂²×Q₈	—	43	24	no	—
159	C₂²×Q₈	—	43	24	no	—
160	C₂²×Q₈	—	43	64	no	—
161	C₂³×C₄	—	43	32	no	—
162	C₂³×C₄	—	43	24	no	—
163	C₂³×C₄	—	43	4	no	—
164	C₂³×C₄	—	43	4	no	—
165	C₂³×C₄	C₄×D₄₃	1	16	yes	—
166	C₂³×C₄	C₂²×D₄₃	1	4	yes	—
167	C₂³×C₄	C₂²×D₄₃	1	12	yes	—
168	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
169	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
170	C₄×Q₈	—	43	256	no	—
171	C₄×Q₈	—	43	256	no	—
172	#117	—	43	64	no	—
173	#117	—	43	256	no	—
174	#117	C₂²×D₄₃	1	64	yes	—
175	C₂×C₄⋊C₄	—	43	96	no	—
176	C₂×C₄⋊C₄	—	43	24	no	—
177	C₂×C₄⋊C₄	C₂²×D₄₃	1	24	yes	—
178	C₂×C₂²⋊C₄	—	43	24	no	—
179	C₂×C₂²⋊C₄	C₂²×D₄₃	1	24	yes	—
180	C₂×C₄²	—	43	96	no	—
181	C₂×C₄²	—	43	24	no	—
182	C₂×C₄²	C₂²×D₄₃	1	24	yes	—
183	C₂⁴	—	2	8	no	—
184	C₂⁴	C₂×C₄×D₄₃	1	8	yes	—
185	C₂⁴	C₂³×D₄₃	1	6	yes	—
186	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
187	C₂×Q₈	—	43	192	no	—
188	C₂×Q₈	—	43	192	no	—
189	C₂×Q₈	—	43	512	no	—
190	C₂×Q₈	—	43	32	no	—
191	C₂×Q₈	—	43	32	no	—
192	C₂²×C₄	—	2	16	no	—
193	C₂²×C₄	—	43	192	no	—
194	C₂²×C₄	—	43	32	no	—
195	C₂²×C₄	—	43	24	no	—
196	C₂²×C₄	—	43	32	no	—
197	C₂²×C₄	—	43	24	no	—
198	C₂²×C₄	—	86	16	no	—
199	C₂²×C₄	C₂×C₄×D₄₃	1	96	yes	—
200	C₂²×C₄	C₂×C₄×D₄₃	1	8	yes	—
201	C₂²×C₄	C₂×C₄×D₄₃	1	8	yes	—
202	C₂²×C₄	C₂³×D₄₃	1	24	yes	—
203	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
204	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
205	C₂²×C₄	C₂³×D₄₃	1	16	yes	—
206	C₄⋊C₄	—	43	128	no	—
207	C₄⋊C₄	—	43	32	no	—
208	C₄⋊C₄	C₂³×D₄₃	1	32	yes	—
209	C₂²⋊C₄	—	43	32	no	—
210	C₂²⋊C₄	C₂³×D₄₃	1	32	yes	—
211	C₄²	—	43	128	no	—
212	C₄²	—	43	32	no	—
213	C₄²	C₂³×D₄₃	1	32	yes	—
214	C₂³	—	2	32	no	—
215	C₂³	—	2	24	no	—
216	C₂³	C₂²×C₄×D₄₃	1	24	yes	—
217	C₂³	C₂²×C₄×D₄₃	1	4	yes	—
218	C₂³	#13063372	1	8	yes	—
219	C₂³	C₂⁴×D₄₃	1	4	yes	—
220	C₂³	C₂⁴×D₄₃	1	3	yes	—
221	Q₈	—	43	128	no	—
222	Q₈	—	43	128	no	—
223	C₂×C₄	—	2	48	no	—
224	C₂×C₄	—	43	128	no	—
225	C₂×C₄	—	43	96	no	—
226	C₂×C₄	—	43	16	no	—
227	C₂×C₄	—	43	16	no	—
228	C₂×C₄	—	86	48	no	—
229	C₂×C₄	C₂²×C₄×D₄₃	1	64	yes	—
230	C₂×C₄	C₂²×C₄×D₄₃	1	24	yes	—
231	C₂×C₄	C₂²×C₄×D₄₃	1	24	yes	—
232	C₂×C₄	C₂⁴×D₄₃	1	16	yes	—
233	C₂×C₄	C₂⁴×D₄₃	1	4	yes	—
234	C₂×C₄	C₂⁴×D₄₃	1	4	yes	—
235	C₂²	—	2	48	no	—
236	C₂²	—	2	8	no	—
237	C₂²	#14583376	1	16	yes	direct product factor
238	C₂²	C₂³×C₄×D₄₃	1	8	yes	—
239	C₂²	C₂³×C₄×D₄₃	1	6	yes	—
240	C₂²	C₂×#13063372	1	12	yes	—
241	C₂²	C₂⁵×D₄₃	1	1	yes	characteristic
242	C₄	—	2	16	no	—
243	C₄	—	43	32	no	—
244	C₄	—	86	16	no	—
245	C₄	C₂³×C₄×D₄₃	1	8	yes	—
246	C₄	C₂³×C₄×D₄₃	1	8	yes	—
247	C₂	—	2	8	no	—
248	C₂	C₂×#14583376	1	12	yes	direct product factor
249	C₂	C₂⁴×C₄×D₄₃	1	1	yes	characteristic
250	C₂	C₂²×#13063372	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₄₃
commutator	C₂×C₄₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₄₃
Fitting	C₂²×#117×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂⁴
prime radical	C₂⁵×C₄₃	C₂³
cosocle	C₂²×C₄₃	C₂⁶
hypercenter	C₂²×#117	D₄₃
hypocenter	C₄₃	C₂²×#519
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
2. C₂×#14583376	C₂	direct (N × H)
3. C₂²×C₄×Dic₈₆	C₂	semidirect
4. C₂²×#13063290	C₂	semidirect
8. C₂³×Dic₈₆	C₄	semidirect
13. #14583376	C₂²	direct (N × H)
14. C₂×C₄×Dic₈₆	C₂²	semidirect
15. C₂×#13063290	C₂²	semidirect
30. C₂²×Dic₈₆	C₂×C₄	semidirect
38. C₄×Dic₈₆	C₂³	semidirect
39. #13063290	C₂³	semidirect
77. C₂×Dic₈₆	C₂²×C₄	semidirect
112. C₂²×C₄₃	#519	semidirect
130. C₂×C₄₃	C₂×#519	semidirect
153. C₄₃	C₂²×#519	semidirect
237. C₂²	#14583376	direct (N × H)
248. C₂	C₂×#14583376	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₈₆	2
C₂×#14583376	2
C₂²×C₄×Dic₈₆	2
C₂²×#13063290	2
C₂²×#13063284	2
C₂²×#13063281	2
C₂²×#117×C₄₃	2
C₂²×#519	43

Subgroups — C22×#14583376 #26003181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583376 #26003181