Subgroups — #19146338

Order 5504.

482 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063347	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583558	C₂	1	2	yes	maximal, complemented
4	#14583283	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583140	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583135	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583122	C₂	1	2	yes	maximal, complemented
8	#14583063	C₂	1	2	yes	maximal, complemented
9	#14583040	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621898	C₂²	1	2	yes	—
13	C₂²×Dic₈₆	C₂²	1	2	yes	—
14	#13063347	—	2	4	no	—
15	C₂×Dic₁₇₂	—	2	4	no	—
16	C₂×Dic₁₇₂	C₂²	1	2	yes	—
17	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
19	C₂×#10621873	C₂²	1	2	yes	—
20	C₂×#10621873	C₂²	1	2	yes	—
21	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
22	#13063313	C₂²	1	2	yes	—
23	#13063311	—	2	2	no	—
24	#13063304	C₂²	1	2	yes	—
25	#13063303	—	2	2	no	—
26	#13063299	C₂²	1	2	yes	—
27	#13063297	—	2	2	no	—
28	C₄×Dic₈₆	C₂²	1	2	yes	—
29	(C₂²⋊D₄₃)⋊C₄	—	2	2	no	—
30	#13063259	C₂²	1	1	yes	characteristic
31	#13063250	C₂²	1	2	yes	—
32	#13063250	C₂²	1	2	yes	—
33	#13063247	C₂²	1	2	yes	—
34	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
35	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₄₃	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₄₃	—	2	2	no	—
45	C₂×C₂²⋊D₄₃	—	2	2	no	—
46	C₂²×Dic₄₃	—	2	2	no	—
47	#10621898	—	2	4	no	—
48	#10621898	—	4	2	no	—
49	C₂×D₁₇₂	C₂³	1	2	yes	—
50	C₂×C₄×D₄₃	—	2	2	no	—
51	C₂×Dic₈₆	—	2	4	no	—
52	C₂×Dic₈₆	—	2	2	no	—
53	C₂×Dic₈₆	—	4	2	no	—
54	C₂×Dic₈₆	C₂³	1	2	yes	—
55	C₂×Dic₈₆	C₂³	1	2	yes	—
56	C₂×C₈×C₄₃	—	2	2	no	—
57	C₂×C₈×C₄₃	—	2	2	no	—
58	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
63	C₂²⋊C₄×C₄₃	D₄	1	1	yes	characteristic
64	C₂²⋊C₄×C₄₃	D₄	1	1	yes	characteristic
65	C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₂²⋊Dic₄₃	—	2	2	no	—
67	(C₂×D₄₃)⋊C₄	—	2	2	no	—
68	(C₂×D₄₃)⋊C₄	—	2	2	no	—
69	(C₂×D₄₃)⋊C₄	—	2	2	no	—
70	C₄⋊Dic₄₃	C₂³	1	2	yes	—
71	Dic₄₃⋊C₄	—	2	2	no	—
72	Dic₄₃⋊C₄	—	2	2	no	—
73	C₄×Dic₄₃	—	2	2	no	—
74	Dic₁₇₂	—	2	4	no	—
75	Dic₁₇₂	—	2	2	no	—
76	Dic₁₇₂	D₄	1	4	yes	complemented
77	D₃₄₄	—	2	2	no	—
78	#10621873	—	2	4	no	—
79	#10621873	—	4	2	no	—
80	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×D₄₃	—	2	2	no	—
83	D₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	—	2	1	no	—
85	C₂×C₄×C₄₃	—	2	1	no	—
86	C₂×C₄×C₄₃	—	2	1	no	—
87	C₂×C₄×C₄₃	—	2	1	no	—
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×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₄₃	—	4	4	no	—
94	C₂²⋊D₄₃	—	4	4	no	—
95	C₂×Dic₄₃	—	2	2	no	—
96	C₂×Dic₄₃	—	2	2	no	—
97	C₂×Dic₄₃	—	2	2	no	—
98	C₂×Dic₄₃	—	4	4	no	—
99	D₁₇₂	—	2	2	no	—
100	D₁₇₂	—	2	2	no	—
101	C₄×D₄₃	—	4	4	no	—
102	Dic₈₆	—	2	2	no	—
103	Dic₈₆	—	2	2	no	—
104	Dic₈₆	—	2	2	no	—
105	Dic₈₆	—	4	4	no	—
106	Dic₈₆	C₂×D₄	1	4	yes	—
107	C₈×C₄₃	—	2	1	no	—
108	C₈×C₄₃	—	2	1	no	—
109	C₈×C₄₃	—	2	1	no	—
110	C₈×C₄₃	C₂×D₄	1	2	yes	—
111	C₂²×C₄₃	—	2	1	no	—
112	C₂²×C₄₃	—	2	1	no	—
113	C₂²×C₄₃	—	2	1	no	—
114	C₂²×C₄₃	—	2	1	no	—
115	C₂²×C₄₃	—	2	1	no	—
116	C₂²×C₄₃	#135	1	2	yes	complemented
117	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₄₃	—	4	2	no	—
119	C₂×D₄₃	—	4	2	no	—
120	C₂×D₄₃	—	4	2	no	—
121	C₄×C₄₃	—	2	1	no	—
122	C₄×C₄₃	—	2	1	no	—
123	C₄×C₄₃	—	2	1	no	—
124	C₄×C₄₃	—	2	1	no	—
125	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
127	Dic₄₃	—	2	4	no	—
128	Dic₄₃	—	4	2	no	—
129	Dic₄₃	—	4	2	no	—
130	#3285	—	43	1	no	maximal, Sylow
131	C₂×C₄₃	—	2	2	no	—
132	C₂×C₄₃	—	4	1	no	—
133	C₂×C₄₃	D₄²	1	1	yes	characteristic
134	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
135	C₂×C₄₃	#578	1	1	yes	characteristic
136	D₄₃	—	8	2	no	—
137	C₂×#135	—	43	1	no	—
138	C₂²×Q₁₆	—	43	1	no	—
139	#548	—	43	2	no	—
140	#495	—	43	1	no	—
141	#467	—	43	1	no	—
142	#466	—	43	1	no	—
143	#464	—	43	2	no	—
144	#452	—	43	2	no	—
145	#448	—	43	2	no	—
146	C₄×Q₁₆	—	43	1	no	—
147	C₈×D₄	D₄₃	1	1	yes	characteristic, complemented
148	C₄₃	#3285	1	1	yes	characteristic, complemented, Sylow
149	C₂×#41	—	43	2	no	—
150	C₂²×Q₈	—	43	2	no	—
151	#135	—	86	4	no	—
152	C₂×Q₁₆	—	43	1	no	—
153	C₂×Q₁₆	—	43	2	no	—
154	C₂×Q₁₆	—	43	1	no	—
155	C₂×Q₁₆	—	86	4	no	—
156	C₂×SD₁₆	—	43	2	no	—
157	C₂×SD₁₆	—	43	2	no	—
158	C₂×D₈	—	43	1	no	—
159	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
160	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
161	#124	—	43	2	no	—
162	#124	—	86	2	no	—
163	C₂²⋊Q₈	—	43	2	no	—
164	C₂²⋊Q₈	—	86	2	no	—
165	#121	—	43	2	no	—
166	#121	—	86	2	no	—
167	C₄×Q₈	—	43	2	no	—
168	C₄×D₄	—	86	2	no	—
169	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
170	#107	—	43	1	no	—
171	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
172	#103	—	43	2	no	—
173	#103	—	43	2	no	—
174	#102	—	43	2	no	—
175	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
176	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
177	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
178	#41	—	86	4	no	—
179	#41	—	172	2	no	—
180	C₂×Q₈	—	43	2	no	—
181	C₂×Q₈	—	43	2	no	—
182	C₂×Q₈	—	86	4	no	—
183	C₂×Q₈	—	86	2	no	—
184	C₂×Q₈	—	172	2	no	—
185	C₂×D₄	—	43	2	no	—
186	C₂×D₄	—	86	2	no	—
187	C₂×D₄	—	86	2	no	—
188	C₂×D₄	D₁₇₂	1	1	yes	characteristic
189	C₂²×C₄	—	86	2	no	—
190	C₂²×C₄	—	86	2	no	—
191	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
193	Q₁₆	—	43	4	no	—
194	Q₁₆	—	86	4	no	—
195	Q₁₆	—	86	2	no	—
196	SD₁₆	—	86	4	no	—
197	SD₁₆	—	172	2	no	—
198	D₈	—	86	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
204	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
205	C₄⋊C₄	—	43	2	no	—
206	C₄⋊C₄	—	86	2	no	—
207	C₄⋊C₄	—	86	2	no	—
208	C₄⋊C₄	D₁₇₂	1	1	yes	characteristic
209	C₂²⋊C₄	—	86	2	no	—
210	C₂²⋊C₄	—	86	2	no	—
211	C₂²⋊C₄	—	86	2	no	—
212	C₂²⋊C₄	—	86	2	no	—
213	C₂²⋊C₄	D₁₇₂	1	1	yes	characteristic
214	C₂²⋊C₄	D₁₇₂	1	1	yes	characteristic
215	C₄²	—	86	2	no	—
216	C₄²	C₂²×D₄₃	1	1	yes	characteristic
217	C₂³	—	86	2	no	—
218	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
219	C₂³	C₂×D₁₇₂	1	1	yes	characteristic
220	Q₈	—	43	4	no	—
221	Q₈	—	86	2	no	—
222	Q₈	—	86	2	no	—
223	Q₈	—	86	2	no	—
224	Q₈	—	172	4	no	—
225	D₄	—	2	2	no	—
226	D₄	—	86	2	no	—
227	D₄	—	86	2	no	—
228	D₄	—	172	4	no	—
229	D₄	—	172	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	86	2	no	—
235	C₂×C₄	—	86	2	no	—
236	C₂×C₄	—	86	2	no	—
237	C₂×C₄	—	172	4	no	—
238	C₂×C₄	—	172	4	no	—
239	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
240	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
241	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
242	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	D₄×D₄₃	1	2	yes	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	172	2	no	—
254	C₂²	—	172	2	no	—
255	C₂²	—	172	2	no	—
256	C₂²	#13063347	1	2	yes	complemented
257	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	86	4	no	—
263	C₄	—	172	2	no	—
264	C₄	—	172	2	no	—
265	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
266	C₄	#13063371	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	4	1	no	—
269	C₂	—	344	2	no	—
270	C₂	D₄×D₁₇₂	1	1	yes	characteristic
271	C₂	C₂×#13063347	1	1	yes	characteristic
272	C₂	#14583716	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₈×D₄×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₈×D₄	D₄₃
hypocenter	C₄₃	#3285
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#13063347	C₂	semidirect
2. C₂²×Dic₁₇₂	C₂	semidirect
3. #14583558	C₂	semidirect
4. #14583283	C₂	semidirect
5. #14583140	C₂	semidirect
6. #14583135	C₂	semidirect
7. #14583122	C₂	semidirect
8. #14583063	C₂	semidirect
9. #14583040	C₂	semidirect
10. C₄×Dic₁₇₂	C₂	semidirect
11. C₈×D₄×C₄₃	C₂	semidirect
76. Dic₁₇₂	D₄	semidirect
116. C₂²×C₄₃	#135	semidirect
147. C₈×D₄	D₄₃	semidirect
148. C₄₃	#3285	semidirect
256. C₂²	#13063347	semidirect