Subgroups — #19144385

Order 5504.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583456	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#10621898	C₂	1	1	yes	maximal, characteristic
3	#14583207	C₂	1	2	yes	maximal, complemented
4	#14583200	C₂	1	2	yes	maximal, complemented
5	#14583195	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583190	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583185	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583180	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582937	C₂	1	2	yes	maximal, complemented
10	#14582896	C₂	1	2	yes	maximal
11	C₂×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621898	C₂²	1	1	yes	characteristic
13	#13063325	C₂²	1	1	yes	characteristic
14	#13063325	C₂²	1	1	yes	characteristic
15	#13063317	C₂²	1	2	yes	—
16	#13063304	—	2	1	no	—
17	#13063304	—	2	1	no	—
18	C₄×Dic₈₆	—	2	1	no	—
19	C₄×Dic₈₆	—	2	1	no	—
20	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
21	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
22	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
23	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊D₄₃	—	2	1	no	—
25	C₄×C₂²⋊D₄₃	—	2	1	no	—
26	#13063284	C₂²	1	1	yes	characteristic
27	#13063284	C₂²	1	2	yes	—
28	#13063282	—	2	1	no	—
29	C₄²×D₄₃	—	2	1	no	—
30	#13063259	C₂²	1	2	yes	—
31	#13063259	C₂²	1	2	yes	—
32	#13063257	C₂²	1	2	yes	—
33	#13063257	C₂²	1	2	yes	—
34	#13063250	C₂²	1	2	yes	—
35	#13063250	C₂²	1	2	yes	—
36	#13063247	C₂²	1	2	yes	—
37	#13063247	C₂²	1	2	yes	—
38	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
39	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
40	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
41	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
42	C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
43	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₄₃	—	2	1	no	—
47	#10621898	—	2	2	no	—
48	#10621898	Q₈	1	4	yes	complemented
49	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₄₃	—	2	1	no	—
51	C₂×C₄×D₄₃	—	2	1	no	—
52	C₂×C₄×D₄₃	—	2	1	no	—
53	C₂×Dic₈₆	—	2	1	no	—
54	C₂×Dic₈₆	—	2	1	no	—
55	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₄₃	—	2	4	no	—
57	C₂×C₈×C₄₃	C₂³	1	2	yes	—
58	C₂×C₈×C₄₃	C₂³	1	2	yes	—
59	C₄²×C₄₃	D₄	1	1	yes	characteristic
60	C₄²×C₄₃	D₄	1	1	yes	characteristic
61	C₄²×C₄₃	C₂³	1	1	yes	characteristic
62	C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₄₃	—	2	1	no	—
64	C₂²⋊Dic₄₃	—	2	2	no	—
65	(C₂×D₄₃)⋊C₄	—	2	1	no	—
66	(C₂×D₄₃)⋊C₄	—	2	1	no	—
67	C₄⋊Dic₄₃	—	2	1	no	—
68	C₄⋊Dic₄₃	—	2	1	no	—
69	C₄⋊Dic₄₃	C₂³	1	2	yes	—
70	C₄⋊Dic₄₃	C₂³	1	2	yes	—
71	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
73	Dic₄₃⋊C₄	—	2	1	no	—
74	Dic₄₃⋊C₄	—	2	1	no	—
75	Dic₄₃⋊C₄	—	2	2	no	—
76	Dic₄₃⋊C₄	—	2	2	no	—
77	C₄×Dic₄₃	—	2	1	no	—
78	C₄×Dic₄₃	—	2	1	no	—
79	C₄×Dic₄₃	—	2	1	no	—
80	C₄×Dic₄₃	—	2	2	no	—
81	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₄₃	—	2	1	no	—
83	C₂×C₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	—	4	1	no	—
85	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₄₃	#41	1	2	yes	—
92	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₄₃	—	2	4	no	—
94	C₂×Dic₄₃	—	2	1	no	—
95	C₂×Dic₄₃	—	2	1	no	—
96	C₂×Dic₄₃	—	2	1	no	—
97	C₂×Dic₄₃	—	2	2	no	—
98	C₂×Dic₄₃	—	2	2	no	—
99	D₁₇₂	—	2	1	no	—
100	D₁₇₂	C₂×Q₈	1	2	yes	—
101	C₄×D₄₃	—	2	4	no	—
102	C₄×D₄₃	—	4	2	no	—
103	C₄×D₄₃	—	4	2	no	—
104	Dic₈₆	—	2	1	no	—
105	Dic₈₆	—	4	2	no	—
106	Dic₈₆	—	4	2	no	—
107	Dic₈₆	C₂×Q₈	1	2	yes	—
108	C₈×C₄₃	—	2	2	no	—
109	C₈×C₄₃	—	2	2	no	—
110	C₂²×C₄₃	—	2	1	no	—
111	C₂²×C₄₃	—	2	1	no	—
112	C₂²×C₄₃	C₂²⋊Q₈	1	2	yes	—
113	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×D₄₃	—	2	2	no	—
115	C₂×D₄₃	—	4	1	no	—
116	C₂×D₄₃	—	4	1	no	—
117	C₄×C₄₃	—	2	1	no	—
118	C₄×C₄₃	—	2	1	no	—
119	C₄×C₄₃	—	2	1	no	—
120	C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
121	C₄×C₄₃	#135	1	2	yes	—
122	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
123	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
124	Dic₄₃	—	2	2	no	—
125	Dic₄₃	—	4	2	no	—
126	Dic₄₃	—	4	1	no	—
127	Dic₄₃	—	4	2	no	—
128	Dic₄₃	—	4	1	no	—
129	#3076	—	43	1	no	maximal, Sylow
130	C₂×C₄₃	—	2	2	no	—
131	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
133	C₂×C₄₃	#575	1	1	yes	characteristic
134	D₄₃	—	4	2	no	—
135	#533	—	43	1	no	—
136	C₄×#41	—	43	1	no	—
137	#479	—	43	2	no	—
138	#478	—	43	2	no	—
139	#477	—	43	1	no	—
140	#476	—	43	1	no	—
141	#475	—	43	1	no	—
142	#474	—	43	1	no	—
143	#427	—	43	2	no	—
144	C₂×C₄⋊C₈	D₄₃	1	1	yes	characteristic, complemented
145	#416	—	43	2	no	—
146	C₄₃	#3076	1	1	yes	characteristic, complemented, Sylow
147	C₂×#41	—	43	1	no	—
148	C₂²×C₈	C₂×D₄₃	1	2	yes	—
149	C₄⋊Q₈	—	43	1	no	—
150	C₄⋊Q₈	—	43	1	no	—
151	#125	—	43	2	no	—
152	C₂²⋊Q₈	—	86	1	no	—
153	C₂²⋊Q₈	—	86	1	no	—
154	C₄×Q₈	—	43	1	no	—
155	C₄×Q₈	—	43	1	no	—
156	C₄×Q₈	—	86	1	no	—
157	C₄×Q₈	—	86	1	no	—
158	C₄×D₄	—	43	1	no	—
159	C₄×D₄	—	43	1	no	—
160	C₄×D₄	—	86	1	no	—
161	C₄×D₄	—	86	1	no	—
162	#117	—	43	1	no	—
163	#117	—	43	2	no	—
164	#117	—	86	1	no	—
165	C₂×C₄²	—	86	1	no	—
166	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
167	#107	—	43	2	no	—
168	#107	—	43	2	no	—
169	#106	—	43	2	no	—
170	#106	—	43	2	no	—
171	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
172	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
173	C₄⋊C₈	C₂×D₄₃	1	2	yes	—
174	#103	—	43	2	no	—
175	#103	—	43	2	no	—
176	#102	—	43	2	no	—
177	#102	—	43	2	no	—
178	#41	—	43	4	no	—
179	#41	—	86	2	no	—
180	C₂×Q₈	—	43	1	no	—
181	C₂×Q₈	—	86	1	no	—
182	C₂×Q₈	—	86	1	no	—
183	C₂×D₄	—	43	1	no	—
184	C₂×D₄	—	86	1	no	—
185	C₂²×C₄	—	86	1	no	—
186	C₂²×C₄	—	86	1	no	—
187	C₂²×C₄	—	86	1	no	—
188	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
189	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	C₂²×D₄₃	1	2	yes	—
193	C₂×C₈	C₂²×D₄₃	1	2	yes	—
194	C₄⋊C₄	—	43	2	no	—
195	C₄⋊C₄	—	43	2	no	—
196	C₄⋊C₄	—	43	1	no	—
197	C₄⋊C₄	—	43	1	no	—
198	C₄⋊C₄	—	86	1	no	—
199	C₄⋊C₄	—	86	1	no	—
200	C₄⋊C₄	—	86	2	no	—
201	C₄⋊C₄	—	86	2	no	—
202	C₄⋊C₄	—	86	1	no	—
203	C₄⋊C₄	—	86	1	no	—
204	C₂²⋊C₄	—	86	1	no	—
205	C₂²⋊C₄	—	86	2	no	—
206	C₂²⋊C₄	—	86	1	no	—
207	C₂²⋊C₄	—	86	1	no	—
208	C₄²	—	86	1	no	—
209	C₄²	—	86	1	no	—
210	C₄²	—	86	1	no	—
211	C₄²	—	86	2	no	—
212	C₄²	D₁₇₂	1	1	yes	characteristic
213	C₄²	D₁₇₂	1	1	yes	characteristic
214	C₄²	C₂²×D₄₃	1	1	yes	characteristic
215	C₄²	C₂²×D₄₃	1	1	yes	characteristic
216	C₂³	—	86	1	no	—
217	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
218	Q₈	—	43	2	no	—
219	Q₈	—	86	1	no	—
220	Q₈	—	172	2	no	—
221	Q₈	—	172	2	no	—
222	D₄	—	43	2	no	—
223	D₄	—	86	4	no	—
224	D₄	—	86	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	86	1	no	—
228	C₂×C₄	—	86	4	no	—
229	C₂×C₄	—	86	1	no	—
230	C₂×C₄	—	86	1	no	—
231	C₂×C₄	—	86	2	no	—
232	C₂×C₄	—	86	2	no	—
233	C₂×C₄	—	172	2	no	—
234	C₂×C₄	—	172	2	no	—
235	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
236	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
237	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
238	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
239	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
240	C₂×C₄	#10621900	1	2	yes	—
241	C₂×C₄	Q₈×D₄₃	1	2	yes	—
242	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	86	2	no	—
248	C₂²	—	172	1	no	—
249	C₂²	—	172	1	no	—
250	C₂²	#13063305	1	2	yes	—
251	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	86	2	no	—
256	C₄	—	172	1	no	—
257	C₄	—	172	2	no	—
258	C₄	—	172	1	no	—
259	C₄	—	172	2	no	—
260	C₄	#13063305	1	2	yes	—
261	C₄	#13063347	1	2	yes	—
262	C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
263	C₄	C₂×#10621900	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	172	2	no	—
266	C₂	C₂×#13063305	1	1	yes	characteristic
267	C₂	C₂×#13063347	1	1	yes	characteristic
268	C₂	#14583699	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₂×C₄⋊C₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₂×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#3076
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #14583456	C₂	semidirect
3. #14583207	C₂	semidirect
4. #14583200	C₂	semidirect
5. #14583195	C₂	semidirect
6. #14583190	C₂	semidirect
7. #14583185	C₂	semidirect
8. #14583180	C₂	semidirect
9. #14582937	C₂	semidirect
11. C₂×C₄⋊C₈×C₄₃	C₂	semidirect
48. #10621898	Q₈	semidirect
144. C₂×C₄⋊C₈	D₄₃	semidirect
146. C₄₃	#3076	semidirect