Subgroups — C₂²×#14583135 #25963689

Order 11008.

3694 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃₄₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13063299	C₂	1	2	yes	maximal, complemented
3	C₂×#14583135	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13063259	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13063247	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₇₂	C₂²	1	2	yes	—
8	C₂³×C₂²⋊D₄₃	—	2	2	no	—
9	C₂²×D₃₄₄	—	2	8	no	—
10	C₂²×D₃₄₄	C₂²	1	12	yes	complemented
11	C₂×#13063299	—	2	12	no	—
12	C₂×#13063299	C₂²	1	24	yes	complemented
13	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
14	C₂²×(C₂×D₄₃)⋊C₄	—	2	2	no	—
15	#14583135	C₂²	1	64	yes	direct product factor
16	C₂×#13063259	C₂²	1	12	yes	complemented
17	C₂×#13063247	C₂²	1	24	yes	complemented
18	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₄₃	—	2	8	no	—
20	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₂³×C₈×C₄₃	C₂²	1	12	yes	complemented
23	C₂⁴×D₄₃	—	2	2	no	—
24	C₂²×D₁₇₂	—	2	8	no	—
25	C₂²×D₁₇₂	—	2	8	no	—
26	C₂²×D₁₇₂	—	2	6	no	—
27	C₂²×D₁₇₂	C₂³	1	12	yes	—
28	C₂²×C₂²⋊D₄₃	—	2	24	no	—
29	C₂²×C₂²⋊D₄₃	—	4	16	no	—
30	C₂³×Dic₄₃	—	2	2	no	—
31	C₂×D₃₄₄	—	2	48	no	—
32	C₂×D₃₄₄	C₂³	1	16	yes	complemented
33	#13063299	—	2	48	no	—
34	#13063299	C₂³	1	32	yes	complemented
35	C₂×C₄⋊Dic₄₃	—	2	6	no	—
36	C₂×C₄⋊Dic₄₃	C₂³	1	12	yes	—
37	C₂×(C₂×D₄₃)⋊C₄	—	2	24	no	—
38	#13063259	C₂³	1	16	yes	complemented
39	#13063247	C₂³	1	32	yes	complemented
40	C₂⁵×C₄₃	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₄₃	—	2	4	no	—
42	C₂³×C₄×C₄₃	—	2	4	no	—
43	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₃	D₄	1	6	yes	—
45	C₂³×C₄×C₄₃	C₂³	1	6	yes	—
46	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₃	—	2	4	no	—
48	C₂²×C₈×C₄₃	—	2	48	no	—
49	C₂²×C₈×C₄₃	D₄	1	8	yes	complemented
50	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
51	C₂²×C₈×C₄₃	C₂³	1	6	yes	—
52	C₂²×C₈×C₄₃	C₂³	1	16	yes	complemented
53	C₂⁴×C₄₃	—	2	4	no	—
54	C₂⁴×C₄₃	—	2	4	no	—
55	C₂⁴×C₄₃	D₈	1	8	yes	complemented
56	C₂⁴×C₄₃	C₂×D₄	1	6	yes	—
57	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂³×D₄₃	—	2	12	no	—
59	C₂³×D₄₃	—	4	8	no	—
60	C₂³×D₄₃	—	4	8	no	—
61	C₂³×D₄₃	—	4	8	no	—
62	C₂²×C₄×C₄₃	—	2	24	no	—
63	C₂²×C₄×C₄₃	—	2	12	no	—
64	C₂²×C₄×C₄₃	—	2	12	no	—
65	C₂²×C₄×C₄₃	—	2	4	no	—
66	C₂²×C₄×C₄₃	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
68	C₂²×C₄×C₄₃	C₂×D₄	1	12	yes	—
69	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
70	C₂²×C₄×C₄₃	#41	1	4	yes	—
71	C₂²×C₄×C₄₃	C₂⁴	1	4	yes	—
72	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
73	C₂×C₂²⋊D₄₃	—	2	32	no	—
74	C₂×C₂²⋊D₄₃	—	4	96	no	—
75	C₂²×Dic₄₃	—	2	12	no	—
76	C₂²×Dic₄₃	—	4	8	no	—
77	C₂×D₁₇₂	—	2	12	no	—
78	C₂×D₁₇₂	—	2	24	no	—
79	C₂×D₁₇₂	—	2	24	no	—
80	C₂×D₁₇₂	—	4	24	no	—
81	C₂×D₁₇₂	C₂⁴	1	8	yes	—
82	C₂×C₈×C₄₃	—	2	12	no	—
83	C₂×C₈×C₄₃	—	2	32	no	—
84	C₂×C₈×C₄₃	C₂×D₄	1	24	yes	complemented
85	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
86	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
87	(C₂×D₄₃)⋊C₄	—	2	32	no	—
88	C₄⋊Dic₄₃	—	2	12	no	—
89	C₄⋊Dic₄₃	C₂⁴	1	8	yes	—
90	D₃₄₄	—	2	32	no	—
91	C₂³×C₄₃	—	2	12	no	—
92	C₂³×C₄₃	—	2	12	no	—
93	C₂³×C₄₃	—	2	32	no	—
94	C₂³×C₄₃	#121	1	4	yes	—
95	C₂³×C₄₃	C₂×D₈	1	4	yes	—
96	C₂³×C₄₃	C₂×D₈	1	24	yes	complemented
97	C₂³×C₄₃	#135	1	4	yes	—
98	C₂³×C₄₃	C₂²×D₄	1	4	yes	—
99	C₂³×C₄₃	C₂²×D₄	1	3	yes	—
100	C₂²×D₄₃	—	2	8	no	—
101	C₂²×D₄₃	—	4	24	no	—
102	C₂²×D₄₃	—	4	24	no	—
103	C₂²×D₄₃	—	4	24	no	—
104	C₂²×D₄₃	—	8	32	no	—
105	C₂×C₄×C₄₃	—	2	12	no	—
106	C₂×C₄×C₄₃	—	2	24	no	—
107	C₂×C₄×C₄₃	—	2	4	no	—
108	C₂×C₄×C₄₃	—	2	4	no	—
109	C₂×C₄×C₄₃	#121	1	12	yes	—
110	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
111	C₂×C₄×C₄₃	C₂²×D₄	1	4	yes	—
112	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
113	C₂×C₄×C₄₃	C₂×#41	1	6	yes	—
114	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
115	C₂²⋊D₄₃	—	4	64	no	—
116	C₂×Dic₄₃	—	2	8	no	—
117	C₂×Dic₄₃	—	4	24	no	—
118	D₁₇₂	—	2	8	no	—
119	D₁₇₂	—	2	8	no	—
120	D₁₇₂	—	4	24	no	—
121	C₈×C₄₃	—	2	4	no	—
122	C₈×C₄₃	C₂²×D₄	1	8	yes	complemented
123	C₂²×#466	—	43	1	no	maximal, Sylow
124	C₂²×C₄₃	—	2	4	no	—
125	C₂²×C₄₃	—	2	48	no	—
126	C₂²×C₄₃	—	2	4	no	—
127	C₂²×C₄₃	#466	1	16	yes	complemented
128	C₂²×C₄₃	C₂×#121	1	6	yes	—
129	C₂²×C₄₃	C₂²×D₈	1	6	yes	—
130	C₂²×C₄₃	C₂²×D₈	1	8	yes	complemented
131	C₂²×C₄₃	C₂×#135	1	6	yes	—
132	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
133	C₂×D₄₃	—	4	8	no	—
134	C₂×D₄₃	—	4	8	no	—
135	C₂×D₄₃	—	4	8	no	—
136	C₂×D₄₃	—	8	48	no	—
137	C₄×C₄₃	—	2	4	no	—
138	C₄×C₄₃	C₂×#121	1	6	yes	—
139	C₄×C₄₃	C₂³×D₄	1	1	yes	characteristic
140	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
141	Dic₄₃	—	4	8	no	—
142	C₂³×D₈	—	43	1	no	—
143	C₂⁴×C₈	D₄₃	1	1	yes	characteristic, complemented
144	C₂²×#121	—	43	2	no	—
145	C₂×#466	—	43	24	no	—
146	C₂²×#107	—	43	1	no	—
147	C₂²×#102	—	43	2	no	—
148	C₂×C₄₃	—	2	8	no	—
149	C₂×C₄₃	C₂×#466	1	12	yes	complemented
150	C₂×C₄₃	C₂²×#121	1	1	yes	characteristic
151	C₂×C₄₃	C₂³×D₈	1	1	yes	characteristic
152	C₂×C₄₃	C₂²×#135	1	1	yes	characteristic
153	D₄₃	—	8	8	no	—
154	C₂³×D₄	—	43	2	no	—
155	C₂³×D₄	—	86	2	no	—
156	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
157	C₂²×D₈	—	43	12	no	—
158	C₂²×D₈	—	86	8	no	—
159	C₂³×C₈	—	2	8	no	—
160	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
161	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
162	C₂³×C₈	C₂×D₄₃	1	12	yes	complemented
163	C₂×#121	—	43	24	no	—
164	C₂×#121	—	86	12	no	—
165	C₂²×C₄⋊C₄	—	43	2	no	—
166	C₂²×C₂²⋊C₄	—	86	2	no	—
167	#466	—	43	64	no	—
168	C₂×#107	—	43	12	no	—
169	C₂×#102	—	43	24	no	—
170	C₄₃	C₂²×#466	1	1	yes	characteristic, complemented, Sylow
171	C₂⁵	—	86	2	no	—
172	C₂⁵	D₁₇₂	1	1	yes	characteristic
173	C₂²×D₄	—	43	12	no	—
174	C₂²×D₄	—	86	24	no	—
175	C₂²×D₄	—	86	6	no	—
176	C₂²×D₄	—	86	8	no	—
177	C₂²×D₄	—	86	8	no	—
178	C₂²×D₄	—	172	16	no	—
179	C₂³×C₄	—	2	4	no	—
180	C₂³×C₄	—	2	4	no	—
181	C₂³×C₄	—	86	2	no	—
182	C₂³×C₄	D₁₇₂	1	6	yes	—
183	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
184	C₂³×C₄	C₂²×D₄₃	1	6	yes	—
185	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
186	C₂×D₈	—	43	16	no	—
187	C₂×D₈	—	86	48	no	—
188	C₂²×C₈	—	2	4	no	—
189	C₂²×C₈	—	2	48	no	—
190	C₂²×C₈	C₂²⋊D₄₃	1	8	yes	complemented
191	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
192	C₂²×C₈	C₂²×D₄₃	1	6	yes	—
193	C₂²×C₈	C₂²×D₄₃	1	16	yes	complemented
194	#121	—	43	32	no	—
195	#121	—	86	48	no	—
196	C₂×C₄⋊C₄	—	43	12	no	—
197	C₂×C₄⋊C₄	—	86	6	no	—
198	C₂×C₂²⋊C₄	—	86	24	no	—
199	#107	—	43	16	no	—
200	#102	—	43	32	no	—
201	C₂⁴	—	2	4	no	—
202	C₂⁴	—	2	4	no	—
203	C₂⁴	—	86	12	no	—
204	C₂⁴	—	172	8	no	—
205	C₂⁴	—	172	8	no	—
206	C₂⁴	—	172	8	no	—
207	C₂⁴	D₃₄₄	1	8	yes	complemented
208	C₂⁴	C₂×D₁₇₂	1	6	yes	—
209	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
210	C₂×D₄	—	43	8	no	—
211	C₂×D₄	—	86	32	no	—
212	C₂×D₄	—	86	24	no	—
213	C₂×D₄	—	86	12	no	—
214	C₂×D₄	—	86	24	no	—
215	C₂×D₄	—	172	96	no	—
216	C₂×D₄	—	172	24	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	24	no	—
219	C₂²×C₄	—	2	12	no	—
220	C₂²×C₄	—	2	12	no	—
221	C₂²×C₄	—	86	12	no	—
222	C₂²×C₄	—	172	8	no	—
223	C₂²×C₄	C₂×D₁₇₂	1	6	yes	—
224	C₂²×C₄	C₂×D₁₇₂	1	12	yes	—
225	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
226	C₂²×C₄	#10621898	1	4	yes	—
227	C₂²×C₄	C₂×C₂²⋊D₄₃	1	4	yes	—
228	C₂²×C₄	C₂³×D₄₃	1	4	yes	—
229	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
230	D₈	—	86	32	no	—
231	C₂×C₈	—	2	12	no	—
232	C₂×C₈	—	2	32	no	—
233	C₂×C₈	C₂×C₂²⋊D₄₃	1	24	yes	complemented
234	C₂×C₈	C₂³×D₄₃	1	4	yes	—
235	C₂×C₈	C₂³×D₄₃	1	4	yes	—
236	C₄⋊C₄	—	43	8	no	—
237	C₄⋊C₄	—	86	12	no	—
238	C₂²⋊C₄	—	86	32	no	—
239	C₂³	—	2	32	no	—
240	C₂³	—	2	12	no	—
241	C₂³	—	2	12	no	—
242	C₂³	—	86	8	no	—
243	C₂³	—	344	32	no	—
244	C₂³	—	172	24	no	—
245	C₂³	—	172	24	no	—
246	C₂³	—	172	24	no	—
247	C₂³	#13063299	1	4	yes	—
248	C₂³	C₂×D₃₄₄	1	4	yes	—
249	C₂³	C₂×D₃₄₄	1	24	yes	complemented
250	C₂³	#13063347	1	4	yes	—
251	C₂³	C₂²×D₁₇₂	1	4	yes	—
252	C₂³	C₂²×D₁₇₂	1	3	yes	—
253	D₄	—	86	8	no	—
254	D₄	—	86	8	no	—
255	D₄	—	172	64	no	—
256	D₄	—	172	24	no	—
257	C₂×C₄	—	2	12	no	—
258	C₂×C₄	—	2	24	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	86	8	no	—
262	C₂×C₄	—	172	24	no	—
263	C₂×C₄	#13063299	1	12	yes	—
264	C₂×C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
265	C₂×C₄	C₂²×D₁₇₂	1	4	yes	—
266	C₂×C₄	C₂²×D₁₇₂	1	3	yes	—
267	C₂×C₄	C₂×#10621898	1	6	yes	—
268	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
269	C₈	—	2	4	no	—
270	C₈	C₂²×C₂²⋊D₄₃	1	8	yes	complemented
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	48	no	—
274	C₂²	—	344	48	no	—
275	C₂²	—	172	8	no	—
276	C₂²	—	172	8	no	—
277	C₂²	—	172	8	no	—
278	C₂²	#14583135	1	16	yes	direct product factor
279	C₂²	C₂×#13063299	1	6	yes	—
280	C₂²	C₂²×D₃₄₄	1	6	yes	—
281	C₂²	C₂²×D₃₄₄	1	8	yes	complemented
282	C₂²	C₂×#13063347	1	6	yes	—
283	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	172	8	no	—
286	C₄	C₂×#13063299	1	6	yes	—
287	C₄	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
288	C₄	C₂²×#10621898	1	1	yes	characteristic
289	C₂	—	2	8	no	—
290	C₂	—	344	8	no	—
291	C₂	C₂×#14583135	1	12	yes	direct product factor
292	C₂	C₂²×#13063299	1	1	yes	characteristic
293	C₂	C₂³×D₃₄₄	1	1	yes	characteristic
294	C₂	C₂²×#13063347	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₁₇₂
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂⁴×C₈×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂×D₄
hypercenter	C₂⁴×C₈	D₄₃
hypocenter	C₄₃	C₂²×#466
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×D₃₄₄	C₂	semidirect
2. C₂²×#13063299	C₂	semidirect
3. C₂×#14583135	C₂	direct (N × H)
4. C₂²×#13063259	C₂	semidirect
5. C₂²×#13063247	C₂	semidirect
6. C₂⁴×C₈×C₄₃	C₂	semidirect
10. C₂²×D₃₄₄	C₂²	semidirect
12. C₂×#13063299	C₂²	semidirect
15. #14583135	C₂²	direct (N × H)
16. C₂×#13063259	C₂²	semidirect
17. C₂×#13063247	C₂²	semidirect
22. C₂³×C₈×C₄₃	C₂²	semidirect
32. C₂×D₃₄₄	C₂³	semidirect
34. #13063299	C₂³	semidirect
38. #13063259	C₂³	semidirect
39. #13063247	C₂³	semidirect
49. C₂²×C₈×C₄₃	D₄	semidirect
52. C₂²×C₈×C₄₃	C₂³	semidirect
55. C₂⁴×C₄₃	D₈	semidirect
84. C₂×C₈×C₄₃	C₂×D₄	semidirect
96. C₂³×C₄₃	C₂×D₈	semidirect
122. C₈×C₄₃	C₂²×D₄	semidirect
127. C₂²×C₄₃	#466	semidirect
130. C₂²×C₄₃	C₂²×D₈	semidirect
143. C₂⁴×C₈	D₄₃	semidirect
149. C₂×C₄₃	C₂×#466	semidirect
162. C₂³×C₈	C₂×D₄₃	semidirect
170. C₄₃	C₂²×#466	semidirect
190. C₂²×C₈	C₂²⋊D₄₃	semidirect
193. C₂²×C₈	C₂²×D₄₃	semidirect
207. C₂⁴	D₃₄₄	semidirect
233. C₂×C₈	C₂×C₂²⋊D₄₃	semidirect
249. C₂³	C₂×D₃₄₄	semidirect
270. C₈	C₂²×C₂²⋊D₄₃	semidirect
278. C₂²	#14583135	direct (N × H)
281. C₂²	C₂²×D₃₄₄	semidirect
291. C₂	C₂×#14583135	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×D₃₄₄	2
C₂²×#13063299	2
C₂×#14583135	2
C₂²×#13063259	2
C₂²×#13063247	2
C₂⁴×C₈×C₄₃	2
C₂²×#466	43

Subgroups — C22×#14583135 #25963689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#14583135 #25963689