Subgroups — C₂²×#14583015 #25960887

Order 11008.

2750 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063347	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13063333	C₂	1	2	yes	maximal, complemented
3	C₂×#14583015	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13063251	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#108×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#10621898	C₂²	1	2	yes	—
8	C₂×#13063347	C₄	1	12	yes	complemented
9	C₂×#13063347	C₂²	1	12	yes	—
10	C₂²×Dic₁₇₂	C₄	1	1	yes	characteristic, complemented
11	C₂²×#10621873	C₄	1	2	yes	complemented
12	C₂²×D₃₄₄	C₄	1	1	yes	characteristic, complemented
13	C₂×#13063333	—	2	12	no	—
14	C₂×#13063333	C₂²	1	24	yes	complemented
15	C₂²×#10621872	—	2	2	no	—
16	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂²×C₈×D₄₃	—	2	2	no	—
18	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
19	#14583015	C₂²	1	64	yes	direct product factor
20	C₂×#13063251	C₂²	1	24	yes	complemented
21	C₂×C₈×Dic₄₃	C₂²	1	12	yes	complemented
22	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
23	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×#108×C₄₃	C₂²	1	12	yes	complemented
25	C₂×#10621898	—	2	6	no	—
26	C₂×#10621898	—	2	6	no	—
27	C₂×#10621898	C₂×C₄	1	12	yes	—
28	C₂×#10621898	C₂³	1	12	yes	—
29	C₂²×Dic₈₆	C₂×C₄	1	2	yes	—
30	C₂²×D₁₇₂	C₂×C₄	1	2	yes	—
31	C₂²×C₂²⋊D₄₃	—	2	2	no	—
32	C₂³×Dic₄₃	—	2	1	no	—
33	C₂²×C₄×D₄₃	—	2	2	no	—
34	#13063347	C₂×C₄	1	48	yes	complemented
35	#13063347	C₂³	1	16	yes	—
36	C₂×Dic₁₇₂	C₂×C₄	1	12	yes	complemented
37	C₂×#10621873	C₂×C₄	1	24	yes	complemented
38	C₂×D₃₄₄	C₂×C₄	1	12	yes	complemented
39	#13063333	—	2	48	no	—
40	#13063333	C₂³	1	32	yes	complemented
41	C₂×#10621872	—	2	24	no	—
42	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
43	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
44	C₂²×C₄₃⋊C₈	D₄	1	6	yes	—
45	C₂²×C₄₃⋊C₈	C₂³	1	6	yes	—
46	C₂×C₈×D₄₃	—	2	24	no	—
47	C₂×C₄×Dic₄₃	—	2	3	no	—
48	C₂×C₄×Dic₄₃	C₂³	1	6	yes	—
49	#13063251	C₂³	1	32	yes	complemented
50	C₈×Dic₄₃	C₂³	1	16	yes	complemented
51	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₄₃	—	2	6	no	—
53	C₂×M₁₆×C₄₃	C₂³	1	12	yes	—
54	C₂²×C₈×C₄₃	—	2	2	no	—
55	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₃	C₂×C₄	1	6	yes	—
58	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
59	#108×C₄₃	C₂³	1	16	yes	complemented
60	C₂⁴×C₄₃	#41	1	1	yes	characteristic
61	C₂³×D₄₃	—	2	2	no	—
62	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	#41	1	3	yes	—
67	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
68	C₂×C₂²⋊D₄₃	—	2	12	no	—
69	C₂×C₂²⋊D₄₃	—	2	12	no	—
70	C₂²×Dic₄₃	—	2	8	no	—
71	C₂²×Dic₄₃	—	2	2	no	—
72	C₂²×Dic₄₃	—	2	6	no	—
73	#10621898	—	2	24	no	—
74	#10621898	—	2	12	no	—
75	#10621898	—	2	12	no	—
76	#10621898	C₂²×C₄	1	24	yes	—
77	#10621898	C₂⁴	1	8	yes	—
78	C₂×D₁₇₂	—	2	6	no	—
79	C₂×D₁₇₂	C₂²×C₄	1	12	yes	—
80	C₂×C₄×D₄₃	—	2	12	no	—
81	C₂×C₄×D₄₃	—	2	12	no	—
82	C₂×Dic₈₆	—	2	6	no	—
83	C₂×Dic₈₆	C₂²×C₄	1	12	yes	—
84	M₁₆×C₄₃	—	2	12	no	—
85	M₁₆×C₄₃	C₂⁴	1	8	yes	—
86	C₂×C₈×C₄₃	—	2	12	no	—
87	C₂×C₈×C₄₃	C₂²×C₄	1	6	yes	—
88	C₂×C₈×C₄₃	C₂²×C₄	1	6	yes	—
89	C₂×C₈×C₄₃	C₂²×C₄	1	12	yes	—
90	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
91	C₄×Dic₄₃	—	2	6	no	—
92	C₄×Dic₄₃	C₂⁴	1	4	yes	—
93	C₂×C₄₃⋊C₈	C₂×D₄	1	6	yes	—
94	C₂×C₄₃⋊C₈	C₂×D₄	1	6	yes	—
95	C₂×C₄₃⋊C₈	C₂×D₄	1	12	yes	—
96	C₂×C₄₃⋊C₈	C₂⁴	1	4	yes	—
97	Dic₁₇₂	C₂²×C₄	1	16	yes	complemented
98	D₃₄₄	C₂²×C₄	1	16	yes	complemented
99	#10621873	C₂²×C₄	1	32	yes	complemented
100	#10621872	—	2	32	no	—
101	C₈×D₄₃	—	2	32	no	—
102	C₂³×C₄₃	—	2	4	no	—
103	C₂³×C₄₃	C₄×D₄	1	3	yes	—
104	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
105	C₂³×C₄₃	C₂×#41	1	3	yes	—
106	C₂²×D₄₃	—	2	12	no	—
107	C₂²×D₄₃	—	4	8	no	—
108	C₂×C₄×C₄₃	C₄×D₄	1	3	yes	—
109	C₂×C₄×C₄₃	C₄×D₄	1	3	yes	—
110	C₂×C₄×C₄₃	C₄×D₄	1	6	yes	—
111	C₂×C₄×C₄₃	C₂³×C₄	1	3	yes	—
112	C₂×C₄×C₄₃	C₂³×C₄	1	3	yes	—
113	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
114	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
115	C₂×C₄×C₄₃	C₂×#41	1	3	yes	—
116	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
117	C₂²⋊D₄₃	—	2	24	no	—
118	C₂²⋊D₄₃	—	2	8	no	—
119	C₂×Dic₄₃	—	2	24	no	—
120	C₂×Dic₄₃	—	2	12	no	—
121	C₂×Dic₄₃	—	2	4	no	—
122	D₁₇₂	—	2	12	no	—
123	D₁₇₂	C₂³×C₄	1	8	yes	—
124	C₄×D₄₃	—	2	24	no	—
125	C₄×D₄₃	—	2	8	no	—
126	Dic₈₆	—	2	12	no	—
127	Dic₈₆	C₂³×C₄	1	8	yes	—
128	C₈×C₄₃	—	2	8	no	—
129	C₈×C₄₃	C₂³×C₄	1	4	yes	—
130	C₈×C₄₃	C₂³×C₄	1	4	yes	—
131	C₄₃⋊C₈	C₂²×D₄	1	4	yes	—
132	C₄₃⋊C₈	C₂²×D₄	1	4	yes	—
133	C₂²×#443	—	43	1	no	maximal, Sylow
134	C₂²×C₄₃	—	2	6	no	—
135	C₂²×C₄₃	—	2	6	no	—
136	C₂²×C₄₃	#443	1	4	yes	complemented
137	C₂²×C₄₃	C₂×C₄×D₄	1	3	yes	—
138	C₂²×C₄₃	C₂×C₄×D₄	1	3	yes	—
139	C₂²×C₄₃	C₂²×#41	1	1	yes	characteristic
140	C₂×D₄₃	—	2	8	no	—
141	C₂×D₄₃	—	4	24	no	—
142	C₄×C₄₃	C₂×C₄×D₄	1	3	yes	—
143	C₄×C₄₃	C₂×C₄×D₄	1	3	yes	—
144	C₄×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
145	C₄×C₄₃	C₂³×D₄	1	1	yes	characteristic
146	Dic₄₃	—	2	8	no	—
147	Dic₄₃	—	2	8	no	—
148	C₂²×#135	—	43	1	no	—
149	C₂²×#131	—	43	2	no	—
150	C₂×#443	—	43	24	no	—
151	C₂²×#108	D₄₃	1	1	yes	characteristic, complemented
152	C₂²×#104	—	43	2	no	—
153	C₂²×C₄×C₈	—	43	1	no	—
154	C₂×C₄₃	—	2	1	no	—
155	C₂×C₄₃	—	2	3	no	—
156	C₂×C₄₃	C₂×#443	1	6	yes	complemented
157	C₂×C₄₃	C₂²×C₄×D₄	1	1	yes	characteristic
158	D₄₃	—	4	8	no	—
159	C₂²×#41	—	43	2	no	—
160	C₂×#135	—	43	12	no	—
161	C₂×#135	—	43	12	no	—
162	C₂²×Q₁₆	—	43	1	no	—
163	C₂²×SD₁₆	—	43	2	no	—
164	C₂²×D₈	—	43	1	no	—
165	C₂×#131	—	43	24	no	—
166	C₂×#131	—	86	12	no	—
167	C₂²×M₁₆	—	86	2	no	—
168	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
169	C₂³×C₈	—	43	1	no	—
170	C₂³×C₈	—	86	2	no	—
171	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
172	C₂²×C₄²	—	43	1	no	—
173	#443	—	43	64	no	—
174	C₂×#108	C₂×D₄₃	1	12	yes	complemented
175	C₂×#104	—	43	24	no	—
176	C₂×C₄×C₈	—	43	12	no	—
177	C₄₃	C₂²×#443	1	1	yes	characteristic, complemented, Sylow
178	C₂×#41	—	43	12	no	—
179	C₂×#41	—	43	12	no	—
180	C₂×#41	—	86	6	no	—
181	C₂×#41	—	86	6	no	—
182	C₂²×Q₈	—	43	2	no	—
183	C₂²×D₄	—	43	2	no	—
184	C₂²×D₄	—	86	2	no	—
185	C₂³×C₄	—	86	1	no	—
186	C₂³×C₄	—	86	2	no	—
187	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
188	#135	—	43	48	no	—
189	#135	—	43	16	no	—
190	C₂×Q₁₆	—	43	12	no	—
191	C₂×SD₁₆	—	43	24	no	—
192	C₂×D₈	—	43	12	no	—
193	#131	—	43	32	no	—
194	#131	—	86	48	no	—
195	C₂×M₁₆	—	2	6	no	—
196	C₂×M₁₆	—	86	24	no	—
197	C₂×M₁₆	C₂²×D₄₃	1	12	yes	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	43	6	no	—
200	C₂²×C₈	—	43	1	no	—
201	C₂²×C₈	—	43	1	no	—
202	C₂²×C₈	—	43	6	no	—
203	C₂²×C₈	—	86	24	no	—
204	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
205	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
206	C₂²×C₈	C₄×D₄₃	1	6	yes	—
207	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
208	C₂×C₄²	—	43	6	no	—
209	C₂×C₄²	—	86	3	no	—
210	#108	C₂²×D₄₃	1	16	yes	complemented
211	#104	—	43	32	no	—
212	C₄×C₈	—	43	16	no	—
213	C₂⁴	—	86	2	no	—
214	C₂⁴	#10621902	1	1	yes	characteristic
215	#41	—	43	24	no	—
216	#41	—	43	8	no	—
217	#41	—	86	24	no	—
218	#41	—	86	12	no	—
219	#41	—	86	12	no	—
220	C₂×Q₈	—	43	12	no	—
221	C₂×Q₈	—	86	6	no	—
222	C₂×D₄	—	43	12	no	—
223	C₂×D₄	—	86	12	no	—
224	C₂×D₄	—	86	12	no	—
225	C₂×D₄	—	86	6	no	—
226	C₂²×C₄	—	86	8	no	—
227	C₂²×C₄	—	86	2	no	—
228	C₂²×C₄	—	86	6	no	—
229	C₂²×C₄	—	86	12	no	—
230	C₂²×C₄	—	86	12	no	—
231	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×D₄₃	1	3	yes	—
233	C₂²×C₄	D₄×D₄₃	1	3	yes	—
234	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
235	C₂²×C₄	#10621902	1	3	yes	—
236	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
237	Q₁₆	—	43	16	no	—
238	SD₁₆	—	43	32	no	—
239	D₈	—	43	16	no	—
240	M₁₆	—	2	12	no	—
241	M₁₆	—	86	32	no	—
242	M₁₆	C₂³×D₄₃	1	8	yes	—
243	C₂×C₈	—	2	12	no	—
244	C₂×C₈	—	43	6	no	—
245	C₂×C₈	—	43	6	no	—
246	C₂×C₈	—	43	12	no	—
247	C₂×C₈	—	43	4	no	—
248	C₂×C₈	—	86	32	no	—
249	C₂×C₈	C₂×C₄×D₄₃	1	6	yes	—
250	C₂×C₈	C₂×C₄×D₄₃	1	6	yes	—
251	C₂×C₈	C₂×C₄×D₄₃	1	12	yes	—
252	C₂×C₈	C₂³×D₄₃	1	4	yes	—
253	C₄²	—	43	4	no	—
254	C₄²	—	86	6	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	86	12	no	—
257	C₂³	—	172	8	no	—
258	C₂³	#13063287	1	3	yes	—
259	C₂³	#13063287	1	1	yes	characteristic
260	C₂³	C₂×#10621902	1	3	yes	—
261	Q₈	—	43	8	no	—
262	Q₈	—	86	12	no	—
263	D₄	—	43	8	no	—
264	D₄	—	86	24	no	—
265	D₄	—	86	8	no	—
266	D₄	—	86	12	no	—
267	C₂×C₄	—	86	24	no	—
268	C₂×C₄	—	86	12	no	—
269	C₂×C₄	—	86	4	no	—
270	C₂×C₄	—	86	24	no	—
271	C₂×C₄	—	86	8	no	—
272	C₂×C₄	#13063287	1	3	yes	—
273	C₂×C₄	#13063287	1	3	yes	—
274	C₂×C₄	#13063287	1	6	yes	—
275	C₂×C₄	C₂²×C₄×D₄₃	1	3	yes	—
276	C₂×C₄	C₂²×C₄×D₄₃	1	3	yes	—
277	C₂×C₄	C₂×D₄×D₄₃	1	3	yes	—
278	C₂×C₄	C₂×D₄×D₄₃	1	3	yes	—
279	C₂×C₄	C₂×#10621902	1	3	yes	—
280	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
281	C₈	—	2	8	no	—
282	C₈	—	43	4	no	—
283	C₈	—	43	4	no	—
284	C₈	C₂²×C₄×D₄₃	1	4	yes	—
285	C₈	C₂²×C₄×D₄₃	1	4	yes	—
286	C₂²	—	2	6	no	—
287	C₂²	—	2	6	no	—
288	C₂²	—	86	8	no	—
289	C₂²	—	172	24	no	—
290	C₂²	#14583015	1	4	yes	direct product factor
291	C₂²	C₂×#13063287	1	3	yes	—
292	C₂²	C₂×#13063287	1	3	yes	—
293	C₂²	C₂²×#10621902	1	1	yes	characteristic
294	C₄	—	86	8	no	—
295	C₄	—	86	8	no	—
296	C₄	C₂×#13063287	1	3	yes	—
297	C₄	C₂×#13063287	1	3	yes	—
298	C₄	C₂³×C₄×D₄₃	1	1	yes	characteristic
299	C₄	C₂²×D₄×D₄₃	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	3	no	—
302	C₂	—	172	8	no	—
303	C₂	C₂×#14583015	1	6	yes	direct product factor
304	C₂	C₂²×#13063287	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₄×C₄₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂²×#108×C₄₃	C₂
socle	C₂³×C₄₃	C₄×D₄
prime radical	C₂²×#13063347	C₂
cosocle	C₂×C₄×C₄₃	C₂⁵
hypercenter	C₂²×#108	D₄₃
hypocenter	C₄₃	C₂²×#443
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
2. C₂²×#13063333	C₂	semidirect
3. C₂×#14583015	C₂	direct (N × H)
4. C₂²×#13063251	C₂	semidirect
5. C₂²×C₈×Dic₄₃	C₂	semidirect
6. C₂²×#108×C₄₃	C₂	semidirect
8. C₂×#13063347	C₄	semidirect
10. C₂²×Dic₁₇₂	C₄	semidirect
11. C₂²×#10621873	C₄	semidirect
12. C₂²×D₃₄₄	C₄	semidirect
14. C₂×#13063333	C₂²	semidirect
19. #14583015	C₂²	direct (N × H)
20. C₂×#13063251	C₂²	semidirect
21. C₂×C₈×Dic₄₃	C₂²	semidirect
24. C₂×#108×C₄₃	C₂²	semidirect
34. #13063347	C₂×C₄	semidirect
36. C₂×Dic₁₇₂	C₂×C₄	semidirect
37. C₂×#10621873	C₂×C₄	semidirect
38. C₂×D₃₄₄	C₂×C₄	semidirect
40. #13063333	C₂³	semidirect
49. #13063251	C₂³	semidirect
50. C₈×Dic₄₃	C₂³	semidirect
59. #108×C₄₃	C₂³	semidirect
97. Dic₁₇₂	C₂²×C₄	semidirect
98. D₃₄₄	C₂²×C₄	semidirect
99. #10621873	C₂²×C₄	semidirect
136. C₂²×C₄₃	#443	semidirect
151. C₂²×#108	D₄₃	semidirect
156. C₂×C₄₃	C₂×#443	semidirect
174. C₂×#108	C₂×D₄₃	semidirect
177. C₄₃	C₂²×#443	semidirect
210. #108	C₂²×D₄₃	semidirect
290. C₂²	#14583015	direct (N × H)
303. C₂	C₂×#14583015	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13063347	2
C₂²×#13063333	2
C₂×#14583015	2
C₂²×#13063251	2
C₂²×C₈×Dic₄₃	2
C₂²×#108×C₄₃	2
C₂²×#443	43

Subgroups — C22×#14583015 #25960887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583015 #25960887