Subgroups — #19132824

Order 5504.

656 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₃⋊#95	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₄₃	C₂²	1	2	yes	—
6	C₂³×Dic₄₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₄₃	—	2	2	no	—
8	C₂×C₄⋊Dic₄₃	—	2	2	no	—
9	C₂×C₄⋊Dic₄₃	C₄	1	4	yes	complemented
10	C₂×C₄×Dic₄₃	—	2	2	no	—
11	C₂×C₄×Dic₄₃	—	2	2	no	—
12	C₂×C₄×Dic₄₃	C₄	1	4	yes	complemented
13	C₄₃⋊#95	—	2	8	no	—
14	C₂³×C₄×C₄₃	C₂²	1	2	yes	—
15	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
17	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
18	C₂×C₄⋊C₄×C₄₃	C₄	1	4	yes	complemented
19	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₄₃	—	2	2	no	—
21	C₂²×C₄×C₄₃	—	2	2	no	—
22	C₂²×C₄×C₄₃	—	2	4	no	—
23	C₂²×C₄×C₄₃	—	2	4	no	—
24	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₄₃	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₄₃	D₄	1	2	yes	—
29	C₂²×C₄×C₄₃	D₄	1	2	yes	—
30	C₂²×C₄×C₄₃	D₄	1	2	yes	—
31	C₂²×C₄×C₄₃	Q₈	1	2	yes	—
32	C₂²×Dic₄₃	—	2	2	no	—
33	C₂²×Dic₄₃	—	2	2	no	—
34	C₂²×Dic₄₃	—	2	4	no	—
35	C₂²×Dic₄₃	—	2	2	no	—
36	C₂²×Dic₄₃	—	2	2	no	—
37	C₂²×Dic₄₃	—	2	2	no	—
38	C₂²×Dic₄₃	—	2	2	no	—
39	C₂²×Dic₄₃	—	2	4	no	—
40	C₂²×Dic₄₃	C₂×C₄	1	4	yes	—
41	C₂²×Dic₄₃	C₂×C₄	1	4	yes	—
42	C₂²×Dic₄₃	D₄	1	4	yes	—
43	C₂²×Dic₄₃	Q₈	1	4	yes	—
44	C₄⋊C₄×C₄₃	—	2	8	no	—
45	C₄⋊Dic₄₃	—	2	8	no	—
46	C₄×Dic₄₃	—	2	8	no	—
47	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
48	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
49	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
50	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂³×C₄₃	C₂×D₄	1	2	yes	—
52	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂³×C₄₃	C₂×Q₈	1	1	yes	characteristic
55	C₂³×C₄₃	C₂×Q₈	1	2	yes	—
56	C₂³×C₄₃	#41	1	2	yes	—
57	C₂³×C₄₃	#41	1	2	yes	—
58	C₂×C₄×C₄₃	—	2	4	no	—
59	C₂×C₄×C₄₃	—	2	8	no	—
60	C₂×C₄×C₄₃	—	4	4	no	—
61	C₂×C₄×C₄₃	—	4	2	no	—
62	C₂×C₄×C₄₃	—	4	2	no	—
63	C₂×C₄×C₄₃	C₄²	1	4	yes	—
64	C₂×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
65	C₂×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
66	C₂×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
67	C₂×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
68	C₂×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
69	C₂×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
70	C₂×Dic₄₃	—	2	4	no	—
71	C₂×Dic₄₃	—	2	4	no	—
72	C₂×Dic₄₃	—	2	4	no	—
73	C₂×Dic₄₃	—	2	4	no	—
74	C₂×Dic₄₃	—	2	4	no	—
75	C₂×Dic₄₃	—	2	4	no	—
76	C₂×Dic₄₃	—	2	4	no	—
77	C₂×Dic₄₃	—	2	8	no	—
78	C₂×Dic₄₃	—	4	4	no	—
79	C₂×Dic₄₃	—	4	2	no	—
80	C₂×Dic₄₃	—	4	2	no	—
81	C₂×Dic₄₃	C₄⋊C₄	1	8	yes	complemented
82	C₂²×C₄₃	C₂×C₄²	1	1	yes	characteristic
83	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄₃	#117	1	2	yes	—
91	C₂²×C₄₃	C₄×D₄	1	2	yes	—
92	C₂²×C₄₃	C₄×D₄	1	2	yes	—
93	C₂²×C₄₃	C₄×Q₈	1	2	yes	—
94	C₂²×C₄₃	C₄×Q₈	1	2	yes	—
95	C₂²×C₄₃	#121	1	2	yes	—
96	C₂²×C₄₃	#121	1	2	yes	—
97	C₂²×C₄₃	C₂²⋊Q₈	1	2	yes	—
98	C₂²×C₄₃	C₂²⋊Q₈	1	2	yes	—
99	C₂²×C₄₃	#124	1	1	yes	characteristic
100	C₂²×C₄₃	#124	1	1	yes	characteristic
101	C₂²×C₄₃	#125	1	2	yes	—
102	C₂²×C₄₃	#127	1	1	yes	characteristic
103	C₂²×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₄₃	C₄⋊Q₈	1	2	yes	—
105	C₄×C₄₃	—	4	4	no	—
106	C₄×C₄₃	#95	1	8	yes	complemented
107	Dic₄₃	—	2	8	no	—
108	Dic₄₃	—	4	4	no	—
109	C₄⋊#95	—	43	1	no	maximal, Sylow
110	C₂×C₄₃	C₂×#95	1	1	yes	characteristic
111	C₂×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
112	C₂×C₄₃	#380	1	2	yes	—
113	C₂×C₄₃	#382	1	1	yes	characteristic
114	C₂×C₄₃	#384	1	1	yes	characteristic
115	C₂×C₄₃	#389	1	2	yes	—
116	C₂×C₄₃	#389	1	2	yes	—
117	C₂×C₄₃	#390	1	1	yes	characteristic
118	C₂×C₄₃	#390	1	1	yes	characteristic
119	C₂×C₄₃	#391	1	1	yes	characteristic
120	C₂×C₄₃	#391	1	1	yes	characteristic
121	C₂²×C₄⋊C₄	—	43	1	no	—
122	C₂²×C₄⋊C₄	D₄₃	1	1	yes	characteristic
123	C₂²×C₄²	—	43	1	no	—
124	C₂×#95	—	43	4	no	—
125	C₄₃	C₄⋊#95	1	1	yes	characteristic, complemented, Sylow
126	C₂³×C₄	—	43	2	no	—
127	C₂³×C₄	—	43	2	no	—
128	C₂³×C₄	C₂×D₄₃	1	2	yes	—
129	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	43	4	no	—
133	C₂×C₄⋊C₄	—	86	2	no	—
134	C₂×C₄⋊C₄	—	86	2	no	—
135	C₂×C₄⋊C₄	Dic₄₃	1	4	yes	complemented
136	C₂×C₄²	—	43	4	no	—
137	C₂×C₄²	—	86	2	no	—
138	C₂×C₄²	—	86	2	no	—
139	#95	—	86	8	no	—
140	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	43	4	no	—
146	C₂²×C₄	—	43	4	no	—
147	C₂²×C₄	—	43	4	no	—
148	C₂²×C₄	—	43	4	no	—
149	C₂²×C₄	—	86	2	no	—
150	C₂²×C₄	—	86	2	no	—
151	C₂²×C₄	—	86	2	no	—
152	C₂²×C₄	—	86	2	no	—
153	C₂²×C₄	—	86	2	no	—
154	C₂²×C₄	—	86	4	no	—
155	C₂²×C₄	—	86	4	no	—
156	C₂²×C₄	—	86	2	no	—
157	C₂²×C₄	Dic₈₆	1	2	yes	—
158	C₂²×C₄	C₄×D₄₃	1	2	yes	—
159	C₂²×C₄	C₄×D₄₃	1	2	yes	—
160	C₂²×C₄	D₁₇₂	1	2	yes	—
161	C₂²×C₄	C₂×Dic₄₃	1	4	yes	—
162	C₂²×C₄	C₂×Dic₄₃	1	2	yes	—
163	C₂²×C₄	C₂²⋊D₄₃	1	2	yes	—
164	C₂²×C₄	C₂²⋊D₄₃	1	2	yes	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	86	8	no	—
167	C₄²	—	86	8	no	—
168	C₂³	C₂×Dic₈₆	1	1	yes	characteristic
169	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
170	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
171	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
172	C₂³	D₄×D₄₃	1	2	yes	—
173	C₂³	#10621900	1	2	yes	—
174	C₂³	Q₈×D₄₃	1	2	yes	—
175	C₂³	#10621902	1	2	yes	—
176	C₂³	C₂²×Dic₄₃	1	1	yes	characteristic
177	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
178	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	43	8	no	—
185	C₂×C₄	—	86	4	no	—
186	C₂×C₄	—	86	4	no	—
187	C₂×C₄	—	86	4	no	—
188	C₂×C₄	—	86	4	no	—
189	C₂×C₄	—	86	4	no	—
190	C₂×C₄	—	86	4	no	—
191	C₂×C₄	—	86	4	no	—
192	C₂×C₄	—	86	8	no	—
193	C₂×C₄	—	172	4	no	—
194	C₂×C₄	—	172	2	no	—
195	C₂×C₄	—	172	2	no	—
196	C₂×C₄	C₄×Dic₄₃	1	4	yes	—
197	C₂×C₄	Dic₄₃⋊C₄	1	4	yes	—
198	C₂×C₄	Dic₄₃⋊C₄	1	4	yes	—
199	C₂×C₄	C₄⋊Dic₄₃	1	4	yes	—
200	C₂×C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
201	C₂×C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
202	C₂×C₄	C₂²⋊Dic₄₃	1	4	yes	—
203	C₂²	C₂×C₄×Dic₄₃	1	1	yes	characteristic
204	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
205	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
206	C₂²	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
207	C₂²	C₄⋊C₄×D₄₃	1	2	yes	—
208	C₂²	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
209	C₂²	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
210	C₂²	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
211	C₂²	#13063283	1	2	yes	—
212	C₂²	#13063287	1	2	yes	—
213	C₂²	D₄×Dic₄₃	1	2	yes	—
214	C₂²	#13063290	1	2	yes	—
215	C₂²	Q₈×Dic₄₃	1	2	yes	—
216	C₂²	#13063296	1	2	yes	—
217	C₂²	#13063300	1	2	yes	—
218	C₂²	#13063301	1	2	yes	—
219	C₂²	#13063305	1	2	yes	—
220	C₂²	#13063312	1	1	yes	characteristic
221	C₂²	#13063314	1	1	yes	characteristic
222	C₂²	#13063316	1	2	yes	—
223	C₂²	#13063321	1	1	yes	characteristic
224	C₂²	#13063323	1	1	yes	characteristic
225	C₂²	#13063324	1	2	yes	—
226	C₄	—	4	4	no	—
227	C₄	—	86	8	no	—
228	C₄	—	172	4	no	—
229	C₄	C₄₃⋊#95	1	8	yes	complemented
230	C₂	C₂×C₄₃⋊#95	1	1	yes	characteristic
231	C₂	C₄⋊C₄×Dic₄₃	1	2	yes	—
232	C₂	#14582756	1	2	yes	—
233	C₂	#14582764	1	1	yes	characteristic
234	C₂	#14582768	1	1	yes	characteristic
235	C₂	#14582795	1	2	yes	—
236	C₂	#14582798	1	2	yes	—
237	C₂	#14582800	1	1	yes	characteristic
238	C₂	#14582802	1	1	yes	characteristic
239	C₂	#14582804	1	1	yes	characteristic
240	C₂	#14582806	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂×C₄²
Fitting	C₂²×C₄⋊C₄×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂³
hypercenter	C₂²×C₄⋊C₄	D₄₃
hypocenter	C₄₃	C₄⋊#95
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊Dic₄₃	C₄	semidirect
12. C₂×C₄×Dic₄₃	C₄	semidirect
18. C₂×C₄⋊C₄×C₄₃	C₄	semidirect
81. C₂×Dic₄₃	C₄⋊C₄	semidirect
106. C₄×C₄₃	#95	semidirect
125. C₄₃	C₄⋊#95	semidirect
135. C₂×C₄⋊C₄	Dic₄₃	semidirect
229. C₄	C₄₃⋊#95	semidirect