Subgroups — #19137542

Order 5504.

632 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063283	C₂	1	2	yes	maximal, complemented
3	#14582801	C₂	1	2	yes	maximal
4	#14582791	C₂	1	4	yes	maximal, complemented
5	#14582769	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582756	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊Dic₄₃	C₂	1	2	yes	maximal, complemented
8	C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₄₃	C₂²	1	2	yes	—
11	C₄×D₁₇₂	C₄	1	8	yes	complemented
12	#13063283	—	2	8	no	—
13	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
17	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	—
18	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	—
19	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
20	C₂×C₄×Dic₄₃	C₂²	1	4	yes	—
21	C₄₃⋊#95	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
25	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
26	#95×C₄₃	C₂²	1	2	yes	—
27	C₂³×D₄₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₄₃	C₂³	1	4	yes	—
34	C₂²×Dic₄₃	C₂³	1	2	yes	—
35	C₂×D₁₇₂	—	2	2	no	—
36	C₂×D₁₇₂	—	2	2	no	—
37	C₂×D₁₇₂	C₂×C₄	1	4	yes	—
38	C₂×C₄×D₄₃	—	2	4	no	—
39	C₂×C₄×D₄₃	—	2	4	no	—
40	C₂×C₄×D₄₃	C₂×C₄	1	8	yes	—
41	C₄⋊C₄×C₄₃	—	2	2	no	—
42	C₄⋊C₄×C₄₃	D₄	1	4	yes	complemented
43	C₄²×C₄₃	—	2	4	no	—
44	C₄²×C₄₃	C₂×C₄	1	4	yes	—
45	(C₂×D₄₃)⋊C₄	—	2	8	no	—
46	(C₂×D₄₃)⋊C₄	—	4	4	no	—
47	(C₂×D₄₃)⋊C₄	C₂×C₄	1	8	yes	—
48	C₄⋊Dic₄₃	—	2	4	no	—
49	C₄⋊Dic₄₃	—	2	4	no	—
50	C₄⋊Dic₄₃	—	4	2	no	—
51	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
52	C₄×Dic₄₃	—	2	8	no	—
53	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
54	C₂²×D₄₃	—	2	2	no	—
55	C₂²×D₄₃	—	2	2	no	—
56	C₂²×D₄₃	—	2	2	no	—
57	C₂²×D₄₃	—	2	2	no	—
58	C₂²×D₄₃	—	2	2	no	—
59	C₂²×D₄₃	—	2	2	no	—
60	C₂²×D₄₃	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₄₃	—	2	2	no	—
62	C₂×C₄×C₄₃	—	2	1	no	—
63	C₂×C₄×C₄₃	—	2	1	no	—
64	C₂×C₄×C₄₃	—	2	2	no	—
65	C₂×C₄×C₄₃	—	2	2	no	—
66	C₂×C₄×C₄₃	—	2	2	no	—
67	C₂×C₄×C₄₃	—	2	2	no	—
68	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
72	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₄₃	#41	1	2	yes	—
74	C₂×C₄×C₄₃	#41	1	2	yes	—
75	C₂×C₄×C₄₃	#41	1	2	yes	—
76	C₂×Dic₄₃	—	2	2	no	—
77	C₂×Dic₄₃	—	2	2	no	—
78	C₂×Dic₄₃	—	2	4	no	—
79	C₂×Dic₄₃	—	2	4	no	—
80	C₂×Dic₄₃	—	2	4	no	—
81	C₂×Dic₄₃	C₂²×C₄	1	4	yes	—
82	D₁₇₂	—	2	8	no	—
83	C₄×D₄₃	—	2	16	no	—
84	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
85	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
87	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
88	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
89	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
90	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
91	C₂×D₄₃	—	2	4	no	—
92	C₂×D₄₃	—	2	4	no	—
93	C₂×D₄₃	—	2	4	no	—
94	C₂×D₄₃	—	4	2	no	—
95	C₂×D₄₃	—	4	2	no	—
96	C₂×D₄₃	—	4	2	no	—
97	C₂×D₄₃	—	4	2	no	—
98	C₄×C₄₃	—	2	2	no	—
99	C₄×C₄₃	—	2	4	no	—
100	C₄×C₄₃	—	4	2	no	—
101	C₄×C₄₃	#117	1	4	yes	complemented
102	C₄×C₄₃	C₄×D₄	1	4	yes	—
103	Dic₄₃	—	2	4	no	—
104	Dic₄₃	—	4	4	no	—
105	#2354	—	43	1	no	maximal, Sylow
106	C₂×C₄₃	C₂×#117	1	1	yes	characteristic
107	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
108	C₂×C₄₃	#520	1	1	yes	characteristic
109	C₂×C₄₃	#547	1	1	yes	characteristic
110	C₂×C₄₃	#550	1	1	yes	characteristic
111	C₂×C₄₃	#555	1	1	yes	characteristic
112	C₂×C₄₃	#556	1	1	yes	characteristic
113	D₄₃	—	4	4	no	—
114	C₂×C₄×D₄	—	43	1	no	—
115	C₂×#117	—	43	2	no	—
116	#390	—	43	2	no	—
117	#388	—	43	4	no	—
118	#384	—	43	1	no	—
119	#380	—	43	2	no	—
120	C₄×C₄⋊C₄	—	43	2	no	—
121	C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic, complemented
122	C₄₃	#2354	1	1	yes	characteristic, complemented, Sylow
123	C₂²×D₄	—	43	1	no	—
124	C₂³×C₄	—	43	2	no	—
125	C₄×D₄	—	43	8	no	—
126	#117	—	86	8	no	—
127	C₂×C₄⋊C₄	—	43	1	no	—
128	C₂×C₄⋊C₄	—	43	2	no	—
129	C₂×C₄⋊C₄	—	43	2	no	—
130	C₂×C₄⋊C₄	—	43	2	no	—
131	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
132	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	—	43	2	no	—
134	C₂×C₂²⋊C₄	—	43	4	no	—
135	C₂×C₂²⋊C₄	—	43	4	no	—
136	C₂×C₄²	—	43	4	no	—
137	C₂×C₄²	C₂×D₄₃	1	2	yes	—
138	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
139	#95	—	43	4	no	—
140	#95	C₂×D₄₃	1	2	yes	—
141	C₂⁴	—	43	2	no	—
142	C₂×D₄	—	43	4	no	—
143	C₂×D₄	—	86	2	no	—
144	C₂×D₄	—	86	2	no	—
145	C₂²×C₄	—	43	2	no	—
146	C₂²×C₄	—	43	8	no	—
147	C₂²×C₄	—	43	4	no	—
148	C₂²×C₄	—	86	4	no	—
149	C₂²×C₄	—	86	4	no	—
150	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
151	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
152	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
153	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	43	4	no	—
157	C₄⋊C₄	—	86	4	no	—
158	C₄⋊C₄	—	86	4	no	—
159	C₄⋊C₄	—	172	2	no	—
160	C₄⋊C₄	D₁₇₂	1	4	yes	complemented
161	C₂²⋊C₄	—	43	8	no	—
162	C₂²⋊C₄	—	86	8	no	—
163	C₂²⋊C₄	—	172	4	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	86	8	no	—
166	C₄²	C₄×D₄₃	1	4	yes	—
167	C₂³	—	43	4	no	—
168	C₂³	—	86	2	no	—
169	C₂³	—	86	2	no	—
170	C₂³	—	86	2	no	—
171	C₂³	—	86	2	no	—
172	C₂³	—	86	2	no	—
173	C₂³	—	86	2	no	—
174	C₂³	C₂³×D₄₃	1	1	yes	characteristic
175	D₄	—	86	8	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	43	4	no	—
184	C₂×C₄	—	86	2	no	—
185	C₂×C₄	—	86	2	no	—
186	C₂×C₄	—	86	16	no	—
187	C₂×C₄	—	86	4	no	—
188	C₂×C₄	—	86	4	no	—
189	C₂×C₄	—	86	4	no	—
190	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
191	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
192	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
193	C₂×C₄	C₂×D₁₇₂	1	4	yes	—
194	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
195	C₂×C₄	#10621898	1	2	yes	—
196	C₂×C₄	#10621900	1	2	yes	—
197	C₂×C₄	#10621902	1	2	yes	—
198	C₂²	—	86	4	no	—
199	C₂²	—	86	4	no	—
200	C₂²	—	86	4	no	—
201	C₂²	—	172	2	no	—
202	C₂²	—	172	2	no	—
203	C₂²	—	172	2	no	—
204	C₂²	—	172	2	no	—
205	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
206	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
207	C₂²	C₂×#10621900	1	1	yes	characteristic
208	C₂²	C₂×#10621898	1	1	yes	characteristic
209	C₂²	C₂×#10621902	1	1	yes	characteristic
210	C₂²	#13063371	1	1	yes	characteristic
211	C₂²	#13063372	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	2	2	no	—
214	C₄	—	4	2	no	—
215	C₄	—	86	4	no	—
216	C₄	—	172	4	no	—
217	C₄	#13063283	1	4	yes	complemented
218	C₄	C₄×D₁₇₂	1	4	yes	—
219	C₂	—	172	4	no	—
220	C₂	C₂×#13063283	1	1	yes	characteristic
221	C₂	C₂×C₄×D₁₇₂	1	1	yes	characteristic
222	C₂	#14583379	1	1	yes	characteristic
223	C₂	#14583553	1	1	yes	characteristic
224	C₂	#14583571	1	1	yes	characteristic
225	C₂	#14583614	1	1	yes	characteristic
226	C₂	#14583622	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#2354
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#13063283	C₂	semidirect
4. #14582791	C₂	semidirect
5. #14582769	C₂	semidirect
6. #14582756	C₂	semidirect
7. C₄×C₄⋊Dic₄₃	C₂	semidirect
8. C₄×C₄⋊C₄×C₄₃	C₂	semidirect
11. C₄×D₁₇₂	C₄	semidirect
42. C₄⋊C₄×C₄₃	D₄	semidirect
101. C₄×C₄₃	#117	semidirect
121. C₄×C₄⋊C₄	D₄₃	semidirect
122. C₄₃	#2354	semidirect
160. C₄⋊C₄	D₁₇₂	semidirect
217. C₄	#13063283	semidirect