Subgroups — #19138456

Order 5504.

452 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063317	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic
3	#14582840	C₂	1	2	yes	maximal
4	#14582830	C₂	1	2	yes	maximal
5	#14582805	C₂	1	1	yes	maximal, characteristic
6	#14582805	C₂	1	1	yes	maximal, characteristic
7	#14582797	C₂	1	1	yes	maximal, characteristic
8	#14582797	C₂	1	1	yes	maximal, characteristic
9	#14582794	C₂	1	2	yes	maximal
10	#14582781	C₂	1	2	yes	maximal
11	C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
13	#13063317	—	2	4	no	—
14	C₄×Dic₈₆	—	2	4	no	—
15	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
22	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
24	C₄₃⋊#95	C₂²	1	2	yes	—
25	C₄₃⋊#95	C₂²	1	2	yes	—
26	C₄₃⋊#95	C₂²	1	2	yes	—
27	C₄₃⋊#95	C₂²	1	2	yes	—
28	C₄₃⋊#95	C₂²	1	2	yes	—
29	C₄₃⋊#95	C₂²	1	2	yes	—
30	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
35	#95×C₄₃	C₂²	1	1	yes	characteristic
36	#95×C₄₃	C₂²	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₄₃	C₂³	1	2	yes	—
45	C₂²×Dic₄₃	C₂³	1	2	yes	—
46	C₂²×Dic₄₃	C₂³	1	2	yes	—
47	C₂²×Dic₄₃	C₂³	1	2	yes	—
48	C₂×Dic₈₆	—	2	2	no	—
49	C₂×Dic₈₆	—	2	2	no	—
50	C₂×Dic₈₆	Q₈	1	4	yes	—
51	C₄⋊C₄×C₄₃	—	2	2	no	—
52	C₄⋊C₄×C₄₃	—	2	2	no	—
53	C₄²×C₄₃	—	2	2	no	—
54	C₄²×C₄₃	—	2	2	no	—
55	C₄²×C₄₃	—	2	2	no	—
56	C₄⋊Dic₄₃	—	2	2	no	—
57	C₄⋊Dic₄₃	—	2	2	no	—
58	C₄⋊Dic₄₃	D₄	1	4	yes	—
59	Dic₄₃⋊C₄	—	2	4	no	—
60	Dic₄₃⋊C₄	—	2	4	no	—
61	Dic₄₃⋊C₄	—	2	4	no	—
62	Dic₄₃⋊C₄	—	2	4	no	—
63	Dic₄₃⋊C₄	—	2	4	no	—
64	C₄×Dic₄₃	—	2	4	no	—
65	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
66	C₂×C₄×C₄₃	—	2	1	no	—
67	C₂×C₄×C₄₃	—	2	1	no	—
68	C₂×C₄×C₄₃	—	2	1	no	—
69	C₂×C₄×C₄₃	—	2	1	no	—
70	C₂×C₄×C₄₃	—	2	1	no	—
71	C₂×C₄×C₄₃	—	2	1	no	—
72	C₂×C₄×C₄₃	—	2	1	no	—
73	C₂×C₄×C₄₃	—	2	1	no	—
74	C₂×C₄×C₄₃	—	2	1	no	—
75	C₂×C₄×C₄₃	—	2	1	no	—
76	C₂×C₄×C₄₃	—	2	1	no	—
77	C₂×C₄×C₄₃	—	2	1	no	—
78	C₂×C₄×C₄₃	—	2	1	no	—
79	C₂×C₄×C₄₃	—	2	1	no	—
80	C₂×C₄×C₄₃	—	2	1	no	—
81	C₂×C₄×C₄₃	—	2	1	no	—
82	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
83	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
84	C₂×C₄×C₄₃	#41	1	2	yes	—
85	C₂×C₄×C₄₃	#41	1	2	yes	—
86	C₂×C₄×C₄₃	#41	1	2	yes	—
87	C₂×Dic₄₃	—	2	2	no	—
88	C₂×Dic₄₃	—	2	2	no	—
89	C₂×Dic₄₃	—	2	2	no	—
90	C₂×Dic₄₃	—	2	2	no	—
91	C₂×Dic₄₃	—	2	2	no	—
92	C₂×Dic₄₃	—	2	2	no	—
93	C₂×Dic₄₃	—	2	2	no	—
94	C₂×Dic₄₃	—	2	2	no	—
95	C₂×Dic₄₃	—	2	2	no	—
96	C₂×Dic₄₃	—	2	2	no	—
97	C₂×Dic₄₃	C₂×D₄	1	4	yes	—
98	C₂×Dic₄₃	C₂×Q₈	1	4	yes	—
99	Dic₈₆	—	2	8	no	—
100	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
103	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
104	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
105	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
106	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
107	C₄×C₄₃	—	2	2	no	—
108	C₄×C₄₃	—	2	2	no	—
109	C₄×C₄₃	—	4	1	no	—
110	C₄×C₄₃	—	4	1	no	—
111	C₄×C₄₃	—	4	1	no	—
112	C₄×C₄₃	—	4	1	no	—
113	C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
114	Dic₄₃	—	2	4	no	—
115	Dic₄₃	—	2	4	no	—
116	Dic₄₃	—	4	2	no	—
117	Dic₄₃	—	4	2	no	—
118	#2453	—	43	1	no	maximal, Sylow
119	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄₃	#529	1	1	yes	characteristic
121	C₂×C₄₃	#541	1	1	yes	characteristic
122	C₂×C₄₃	#547	1	1	yes	characteristic
123	C₂×C₄₃	#548	1	1	yes	characteristic
124	C₂×C₄₃	#557	1	1	yes	characteristic
125	C₂×C₄₃	#557	1	1	yes	characteristic
126	C₂×#125	—	43	1	no	—
127	C₂×C₄×Q₈	—	43	1	no	—
128	#400	—	43	2	no	—
129	#398	—	43	2	no	—
130	#391	—	43	1	no	—
131	#391	—	43	1	no	—
132	#389	—	43	2	no	—
133	#389	—	43	1	no	—
134	#389	—	43	1	no	—
135	#387	—	43	2	no	—
136	C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic
137	C₄₃	#2453	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	—	43	1	no	—
139	#125	—	86	4	no	—
140	C₄×Q₈	—	86	4	no	—
141	C₂×C₄⋊C₄	—	43	1	no	—
142	C₂×C₄⋊C₄	—	43	2	no	—
143	C₂×C₄⋊C₄	—	43	2	no	—
144	C₂×C₄⋊C₄	—	43	2	no	—
145	C₂×C₄⋊C₄	—	43	1	no	—
146	C₂×C₄⋊C₄	—	43	2	no	—
147	C₂×C₄⋊C₄	—	43	2	no	—
148	C₂×C₄⋊C₄	—	43	1	no	—
149	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
151	C₂×C₄²	—	43	2	no	—
152	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
153	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
154	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
155	#95	—	43	2	no	—
156	#95	—	43	2	no	—
157	#95	—	43	2	no	—
158	#95	—	43	2	no	—
159	#95	—	43	2	no	—
160	#95	—	43	2	no	—
161	#95	C₂×D₄₃	1	1	yes	characteristic
162	#95	C₂×D₄₃	1	1	yes	characteristic
163	C₂×Q₈	—	43	4	no	—
164	C₂×Q₈	—	86	2	no	—
165	C₂×Q₈	—	86	2	no	—
166	C₂²×C₄	—	43	2	no	—
167	C₂²×C₄	—	43	2	no	—
168	C₂²×C₄	—	43	2	no	—
169	C₂²×C₄	—	43	2	no	—
170	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
176	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	43	4	no	—
180	C₄⋊C₄	—	86	4	no	—
181	C₄⋊C₄	—	86	4	no	—
182	C₄⋊C₄	—	86	4	no	—
183	C₄⋊C₄	—	86	4	no	—
184	C₄⋊C₄	—	86	4	no	—
185	C₄⋊C₄	—	86	2	no	—
186	C₄⋊C₄	—	86	2	no	—
187	C₄²	—	2	2	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	86	4	no	—
191	C₂³	C₂³×D₄₃	1	1	yes	characteristic
192	Q₈	—	86	8	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	43	4	no	—
210	C₂×C₄	—	43	4	no	—
211	C₂×C₄	—	86	2	no	—
212	C₂×C₄	—	86	2	no	—
213	C₂×C₄	—	86	2	no	—
214	C₂×C₄	—	86	2	no	—
215	C₂×C₄	—	86	2	no	—
216	C₂×C₄	—	86	2	no	—
217	C₂×C₄	—	86	2	no	—
218	C₂×C₄	—	86	2	no	—
219	C₂×C₄	—	86	2	no	—
220	C₂×C₄	—	86	2	no	—
221	C₂×C₄	#10621898	1	2	yes	—
222	C₂×C₄	#10621898	1	2	yes	—
223	C₂×C₄	#10621898	1	2	yes	—
224	C₂×C₄	D₄×D₄₃	1	2	yes	—
225	C₂×C₄	Q₈×D₄₃	1	2	yes	—
226	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
227	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
228	C₂²	C₂×#10621898	1	1	yes	characteristic
229	C₂²	C₂×#10621898	1	1	yes	characteristic
230	C₂²	C₂×#10621898	1	1	yes	characteristic
231	C₂²	#13063372	1	1	yes	characteristic
232	C₂²	#13063372	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	1	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	1	no	—
239	C₄	—	86	4	no	—
240	C₄	—	86	4	no	—
241	C₄	—	172	2	no	—
242	C₄	—	172	2	no	—
243	C₄	#13063303	1	4	yes	—
244	C₂	C₂×#13063303	1	1	yes	characteristic
245	C₂	#14583433	1	1	yes	characteristic
246	C₂	#14583511	1	1	yes	characteristic
247	C₂	#14583554	1	1	yes	characteristic
248	C₂	#14583558	1	1	yes	characteristic
249	C₂	#14583627	1	1	yes	characteristic
250	C₂	#14583627	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₄×C₄⋊C₄×C₄₃	C₂
hypercenter	C₄×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#2453
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
137. C₄₃	#2453	semidirect