Subgroups — C₂²×#14582942 #25960253

Order 11008.

3182 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#10621877	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582942	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13063258	C₂	1	2	yes	maximal, complemented
5	C₂²×#13063256	C₂	1	2	yes	maximal, complemented
6	C₂²×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₄₃	—	2	1	no	—
8	C₂²×#10621877	C₄	1	16	yes	complemented
9	C₂²×#10621877	C₂²	1	12	yes	—
10	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₄⋊Dic₄₃	—	2	1	no	—
12	C₂²×C₄⋊Dic₄₃	—	2	6	no	—
13	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
14	C₂²×C₄⋊Dic₄₃	C₂²	1	12	yes	—
15	#14582942	C₂²	1	64	yes	direct product factor
16	C₂×#13063258	C₂²	1	24	yes	complemented
17	C₂×#13063256	C₂²	1	24	yes	complemented
18	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×#117×C₄₃	—	2	6	no	—
20	C₂×#117×C₄₃	C₂²	1	12	yes	—
21	C₂²×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₄₃	—	2	1	no	—
24	C₂²×C₄²×C₄₃	—	2	1	no	—
25	C₂³×Dic₄₃	—	2	12	no	—
26	C₂³×Dic₄₃	—	2	2	no	—
27	C₂³×Dic₄₃	—	4	8	no	—
28	C₂×#10621877	C₂×C₄	1	96	yes	complemented
29	C₂×#10621877	C₂³	1	16	yes	—
30	C₂²×C₄₃⋊C₈	C₂×C₄	1	16	yes	complemented
31	C₂²×C₄₃⋊C₈	C₂³	1	12	yes	—
32	C₂×C₄⋊Dic₄₃	—	2	12	no	—
33	C₂×C₄⋊Dic₄₃	—	2	6	no	—
34	C₂×C₄⋊Dic₄₃	—	2	24	no	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	12	yes	—
36	C₂×C₄⋊Dic₄₃	C₂³	1	16	yes	—
37	#13063258	C₂³	1	32	yes	complemented
38	#13063256	C₂³	1	32	yes	complemented
39	C₂⁵×C₄₃	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₃	—	2	1	no	—
41	C₂³×C₄×C₄₃	—	2	1	no	—
42	C₂³×C₄×C₄₃	C₂×C₄	1	8	yes	—
43	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₃	D₄	1	6	yes	—
45	C₂³×C₄×C₄₃	Q₈	1	8	yes	—
46	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₄₃	C₂³	1	6	yes	—
48	#117×C₄₃	—	2	24	no	—
49	#117×C₄₃	C₂³	1	16	yes	—
50	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
51	C₂×C₄⋊C₄×C₄₃	—	2	3	no	—
52	C₂×C₄⋊C₄×C₄₃	C₂³	1	6	yes	—
53	C₂×C₄⋊C₄×C₄₃	C₂³	1	6	yes	—
54	C₂×C₂²⋊C₄×C₄₃	—	2	12	no	—
55	C₂×C₄²×C₄₃	—	2	12	no	—
56	C₂⁴×C₄₃	—	2	4	no	—
57	C₂⁴×C₄₃	—	2	4	no	—
58	C₂⁴×C₄₃	C₄⋊C₄	1	8	yes	—
59	C₂⁴×C₄₃	C₂×D₄	1	6	yes	—
60	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	2	6	no	—
62	C₂²×C₄×C₄₃	—	2	6	no	—
63	C₂²×C₄×C₄₃	—	4	4	no	—
64	C₂²×C₄×C₄₃	—	4	4	no	—
65	C₂²×C₄×C₄₃	C₄⋊C₄	1	48	yes	—
66	C₂²×C₄×C₄₃	C₄⋊C₄	1	8	yes	—
67	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₃	C₂²×C₄	1	24	yes	—
69	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
70	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
71	C₂²×C₄×C₄₃	C₂×D₄	1	12	yes	—
72	C₂²×C₄×C₄₃	C₂×Q₈	1	24	yes	—
73	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
74	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
75	C₂²×C₄×C₄₃	C₂⁴	1	4	yes	—
76	C₂²×Dic₄₃	—	2	16	no	—
77	C₂²×Dic₄₃	—	2	12	no	—
78	C₂²×Dic₄₃	—	4	48	no	—
79	C₂²×Dic₄₃	—	4	8	no	—
80	C₄⋊C₄×C₄₃	—	2	6	no	—
81	C₄⋊C₄×C₄₃	—	2	6	no	—
82	C₄⋊C₄×C₄₃	C₂⁴	1	4	yes	—
83	C₄⋊C₄×C₄₃	C₂⁴	1	4	yes	—
84	C₂²⋊C₄×C₄₃	—	2	16	no	—
85	C₄²×C₄₃	—	2	16	no	—
86	C₄⋊Dic₄₃	—	2	16	no	—
87	C₄⋊Dic₄₃	—	2	12	no	—
88	C₄⋊Dic₄₃	C₂⁴	1	8	yes	—
89	#10621877	C₂²×C₄	1	64	yes	complemented
90	C₂×C₄₃⋊C₈	C₂²×C₄	1	48	yes	complemented
91	C₂×C₄₃⋊C₈	C₂⁴	1	8	yes	—
92	C₂³×C₄₃	—	2	12	no	—
93	C₂³×C₄₃	—	2	12	no	—
94	C₂³×C₄₃	—	2	32	no	—
95	C₂³×C₄₃	C₂×C₄⋊C₄	1	24	yes	—
96	C₂³×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
97	C₂³×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
98	C₂³×C₄₃	#137	1	4	yes	—
99	C₂³×C₄₃	C₂²×D₄	1	4	yes	—
100	C₂³×C₄₃	C₂²×D₄	1	3	yes	—
101	C₂×C₄×C₄₃	—	2	4	no	—
102	C₂×C₄×C₄₃	—	2	4	no	—
103	C₂×C₄×C₄₃	—	4	12	no	—
104	C₂×C₄×C₄₃	—	4	12	no	—
105	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	48	yes	—
106	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	12	yes	—
107	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	24	yes	—
108	C₂×C₄×C₄₃	C₂³×C₄	1	6	yes	—
109	C₂×C₄×C₄₃	C₂³×C₄	1	8	yes	—
110	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
111	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
112	C₂×C₄×C₄₃	C₂²×Q₈	1	8	yes	—
113	C₂×C₄×C₄₃	C₂²×Q₈	1	6	yes	—
114	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
115	C₂×Dic₄₃	—	2	8	no	—
116	C₂×Dic₄₃	—	4	32	no	—
117	C₂×Dic₄₃	—	4	24	no	—
118	C₄₃⋊C₈	C₂³×C₄	1	16	yes	complemented
119	C₂²×#428	—	43	1	no	maximal, Sylow
120	C₂²×C₄₃	—	2	48	no	—
121	C₂²×C₄₃	—	2	4	no	—
122	C₂²×C₄₃	—	2	4	no	—
123	C₂²×C₄₃	#428	1	16	yes	complemented
124	C₂²×C₄₃	C₂²×C₄⋊C₄	1	8	yes	—
125	C₂²×C₄₃	C₂²×C₄⋊C₄	1	6	yes	—
126	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
127	C₂²×C₄₃	C₂×#137	1	6	yes	—
128	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
129	C₄×C₄₃	—	4	4	no	—
130	C₄×C₄₃	—	4	4	no	—
131	C₄×C₄₃	C₂²×C₄⋊C₄	1	6	yes	—
132	C₄×C₄₃	C₂²×C₄⋊C₄	1	8	yes	—
133	C₄×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
134	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
135	Dic₄₃	—	4	8	no	—
136	C₂³×M₁₆	—	43	1	no	—
137	C₂²×#117	D₄₃	1	1	yes	characteristic
138	C₂³×C₄⋊C₄	—	43	1	no	—
139	C₂×#428	—	43	24	no	—
140	C₂²×#107	—	43	2	no	—
141	C₂²×#106	—	43	2	no	—
142	C₂×C₄₃	—	2	8	no	—
143	C₂×C₄₃	C₂×#428	1	12	yes	complemented
144	C₂×C₄₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄₃	C₂²×#137	1	1	yes	characteristic
147	C₂⁴×C₄	—	86	1	no	—
148	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
149	C₂²×M₁₆	—	43	12	no	—
150	C₂²×M₁₆	—	43	16	no	—
151	C₂³×C₈	—	43	2	no	—
152	C₂×#117	—	2	6	no	—
153	C₂×#117	C₂×D₄₃	1	12	yes	—
154	C₂²×C₄⋊C₄	—	43	2	no	—
155	C₂²×C₄⋊C₄	—	43	12	no	—
156	C₂²×C₄⋊C₄	—	86	6	no	—
157	C₂²×C₄⋊C₄	—	86	1	no	—
158	C₂²×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
159	C₂²×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
160	C₂²×C₂²⋊C₄	—	2	1	no	—
161	C₂²×C₄²	—	2	1	no	—
162	#428	—	43	64	no	—
163	C₂×#107	—	43	24	no	—
164	C₂×#106	—	43	24	no	—
165	C₄₃	C₂²×#428	1	1	yes	characteristic, complemented, Sylow
166	C₂⁵	C₂²⋊D₄₃	1	1	yes	characteristic
167	C₂³×C₄	—	2	1	no	—
168	C₂³×C₄	—	2	1	no	—
169	C₂³×C₄	—	86	12	no	—
170	C₂³×C₄	—	86	2	no	—
171	C₂³×C₄	—	172	8	no	—
172	C₂³×C₄	Dic₈₆	1	8	yes	—
173	C₂³×C₄	C₄×D₄₃	1	8	yes	—
174	C₂³×C₄	C₂²⋊D₄₃	1	6	yes	—
175	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
176	C₂³×C₄	C₂²×D₄₃	1	6	yes	—
177	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
178	C₂×M₁₆	—	43	96	no	—
179	C₂×M₁₆	—	43	16	no	—
180	C₂²×C₈	—	43	16	no	—
181	C₂²×C₈	—	43	12	no	—
182	#117	—	2	24	no	—
183	#117	C₂²×D₄₃	1	16	yes	—
184	C₂×C₄⋊C₄	—	2	3	no	—
185	C₂×C₄⋊C₄	—	2	3	no	—
186	C₂×C₄⋊C₄	—	43	12	no	—
187	C₂×C₄⋊C₄	—	43	16	no	—
188	C₂×C₄⋊C₄	—	86	12	no	—
189	C₂×C₄⋊C₄	—	86	6	no	—
190	C₂×C₄⋊C₄	—	86	24	no	—
191	C₂×C₄⋊C₄	C₂²×D₄₃	1	6	yes	—
192	C₂×C₄⋊C₄	C₂²×D₄₃	1	6	yes	—
193	C₂×C₂²⋊C₄	—	2	12	no	—
194	C₂×C₄²	—	2	12	no	—
195	#107	—	43	32	no	—
196	#106	—	43	32	no	—
197	C₂⁴	—	2	4	no	—
198	C₂⁴	—	2	4	no	—
199	C₂⁴	Dic₄₃⋊C₄	1	8	yes	—
200	C₂⁴	C₂×C₂²⋊D₄₃	1	6	yes	—
201	C₂⁴	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
202	C₂²×C₄	—	2	6	no	—
203	C₂²×C₄	—	2	6	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	—	86	16	no	—
207	C₂²×C₄	—	86	12	no	—
208	C₂²×C₄	—	172	48	no	—
209	C₂²×C₄	—	172	8	no	—
210	C₂²×C₄	Dic₄₃⋊C₄	1	48	yes	—
211	C₂²×C₄	Dic₄₃⋊C₄	1	8	yes	—
212	C₂²×C₄	C₂×Dic₈₆	1	24	yes	—
213	C₂²×C₄	C₂×Dic₈₆	1	4	yes	—
214	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
215	C₂²×C₄	C₂×C₄×D₄₃	1	24	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₄₃	1	12	yes	—
219	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
220	C₂²×C₄	C₂³×D₄₃	1	4	yes	—
221	M₁₆	—	43	64	no	—
222	C₂×C₈	—	43	48	no	—
223	C₂×C₈	—	43	8	no	—
224	C₄⋊C₄	—	2	6	no	—
225	C₄⋊C₄	—	2	6	no	—
226	C₄⋊C₄	—	43	8	no	—
227	C₄⋊C₄	—	86	16	no	—
228	C₄⋊C₄	—	86	12	no	—
229	C₄⋊C₄	C₂³×D₄₃	1	4	yes	—
230	C₄⋊C₄	C₂³×D₄₃	1	4	yes	—
231	C₂²⋊C₄	—	2	16	no	—
232	C₄²	—	2	16	no	—
233	C₂³	—	2	12	no	—
234	C₂³	—	2	12	no	—
235	C₂³	—	2	32	no	—
236	C₂³	C₂×Dic₄₃⋊C₄	1	24	yes	—
237	C₂³	C₂×Dic₄₃⋊C₄	1	4	yes	—
238	C₂³	#13063350	1	4	yes	—
239	C₂³	#13063355	1	4	yes	—
240	C₂³	C₂²×C₂²⋊D₄₃	1	4	yes	—
241	C₂³	C₂²×C₂²⋊D₄₃	1	3	yes	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	12	no	—
245	C₂×C₄	—	4	12	no	—
246	C₂×C₄	—	86	8	no	—
247	C₂×C₄	—	172	32	no	—
248	C₂×C₄	—	172	24	no	—
249	C₂×C₄	C₂×Dic₄₃⋊C₄	1	48	yes	—
250	C₂×C₄	C₂×Dic₄₃⋊C₄	1	12	yes	—
251	C₂×C₄	C₂×Dic₄₃⋊C₄	1	24	yes	—
252	C₂×C₄	C₂²×C₄×D₄₃	1	6	yes	—
253	C₂×C₄	C₂²×C₄×D₄₃	1	8	yes	—
254	C₂×C₄	C₂²×C₂²⋊D₄₃	1	4	yes	—
255	C₂×C₄	C₂²×C₂²⋊D₄₃	1	3	yes	—
256	C₂×C₄	C₂²×Dic₈₆	1	8	yes	—
257	C₂×C₄	C₂²×Dic₈₆	1	6	yes	—
258	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
259	C₈	—	43	16	no	—
260	C₂²	—	2	48	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	4	no	—
263	C₂²	#14582942	1	16	yes	direct product factor
264	C₂²	C₂²×Dic₄₃⋊C₄	1	8	yes	—
265	C₂²	C₂²×Dic₄₃⋊C₄	1	6	yes	—
266	C₂²	C₂×#13063350	1	6	yes	—
267	C₂²	C₂×#13063355	1	6	yes	—
268	C₂²	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
269	C₄	—	4	4	no	—
270	C₄	—	4	4	no	—
271	C₄	—	172	8	no	—
272	C₄	C₂²×Dic₄₃⋊C₄	1	6	yes	—
273	C₄	C₂²×Dic₄₃⋊C₄	1	8	yes	—
274	C₄	C₂³×C₄×D₄₃	1	1	yes	characteristic
275	C₄	C₂³×Dic₈₆	1	1	yes	characteristic
276	C₂	—	2	8	no	—
277	C₂	C₂×#14582942	1	12	yes	direct product factor
278	C₂	C₂³×Dic₄₃⋊C₄	1	1	yes	characteristic
279	C₂	C₂²×#13063350	1	1	yes	characteristic
280	C₂	C₂²×#13063355	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₃
commutator	C₄×C₄₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂²×#117×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂×D₄
prime radical	C₂⁵×C₄₃	D₄
cosocle	C₂×C₄×C₄₃	C₂⁵
hypercenter	C₂²×#117	D₄₃
hypocenter	C₄₃	C₂²×#428
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
3. C₂×#14582942	C₂	direct (N × H)
4. C₂²×#13063258	C₂	semidirect
5. C₂²×#13063256	C₂	semidirect
8. C₂²×#10621877	C₄	semidirect
15. #14582942	C₂²	direct (N × H)
16. C₂×#13063258	C₂²	semidirect
17. C₂×#13063256	C₂²	semidirect
28. C₂×#10621877	C₂×C₄	semidirect
30. C₂²×C₄₃⋊C₈	C₂×C₄	semidirect
37. #13063258	C₂³	semidirect
38. #13063256	C₂³	semidirect
89. #10621877	C₂²×C₄	semidirect
90. C₂×C₄₃⋊C₈	C₂²×C₄	semidirect
118. C₄₃⋊C₈	C₂³×C₄	semidirect
123. C₂²×C₄₃	#428	semidirect
143. C₂×C₄₃	C₂×#428	semidirect
165. C₄₃	C₂²×#428	semidirect
263. C₂²	#14582942	direct (N × H)
277. C₂	C₂×#14582942	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×#10621877	2
C₂³×C₄⋊Dic₄₃	2
C₂×#14582942	2
C₂²×#13063258	2
C₂²×#13063256	2
C₂²×#117×C₄₃	2
C₂²×#428	43

Subgroups — C22×#14582942 #25960253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14582942 #25960253