Subgroups — #19139621

Order 5504.

556 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063325	C₂	1	2	yes	maximal
2	C₂×#13063324	C₂	1	2	yes	maximal
3	#14582820	C₂	1	4	yes	maximal
4	#14582798	C₂	1	2	yes	maximal
5	#14582797	C₂	1	2	yes	maximal
6	#14582769	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₈₆	C₂²	1	4	yes	—
10	#13063325	—	2	8	no	—
11	#13063324	—	2	8	no	—
12	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₄₃	C₂²	1	4	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×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
19	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
20	C₄₃⋊#95	C₂²	1	4	yes	—
21	C₄₃⋊#95	C₂²	1	2	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	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₄₃	C₂³	1	4	yes	—
33	C₂²×Dic₄₃	C₂³	1	2	yes	—
34	C₂²×Dic₄₃	C₂³	1	2	yes	—
35	C₂×Dic₈₆	—	2	8	no	—
36	C₂×Dic₈₆	—	2	8	no	—
37	C₂×Dic₈₆	—	4	4	no	—
38	C₄⋊C₄×C₄₃	D₄	1	4	yes	—
39	C₄⋊C₄×C₄₃	Q₈	1	4	yes	complemented
40	C₄²×C₄₃	—	2	4	no	—
41	C₄²×C₄₃	Q₈	1	4	yes	complemented
42	C₄⋊Dic₄₃	—	2	4	no	—
43	C₄⋊Dic₄₃	—	2	4	no	—
44	C₄⋊Dic₄₃	—	2	8	no	—
45	C₄⋊Dic₄₃	—	4	2	no	—
46	C₄⋊Dic₄₃	D₄	1	4	yes	—
47	C₄⋊Dic₄₃	Q₈	1	4	yes	complemented
48	Dic₄₃⋊C₄	—	2	8	no	—
49	C₄×Dic₄₃	—	2	4	no	—
50	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₄₃	—	2	2	no	—
52	C₂×C₄×C₄₃	—	2	2	no	—
53	C₂×C₄×C₄₃	—	2	1	no	—
54	C₂×C₄×C₄₃	—	2	2	no	—
55	C₂×C₄×C₄₃	—	2	1	no	—
56	C₂×C₄×C₄₃	—	2	2	no	—
57	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
58	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₄₃	C₂×Q₈	1	4	yes	—
62	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
66	C₂×Dic₄₃	—	2	2	no	—
67	C₂×Dic₄₃	—	2	2	no	—
68	C₂×Dic₄₃	—	2	2	no	—
69	C₂×Dic₄₃	—	2	2	no	—
70	C₂×Dic₄₃	—	2	4	no	—
71	C₂×Dic₄₃	—	2	4	no	—
72	C₂×Dic₄₃	—	2	4	no	—
73	C₂×Dic₄₃	C₂×D₄	1	4	yes	—
74	C₂×Dic₄₃	C₂×Q₈	1	4	yes	—
75	Dic₈₆	—	4	16	no	—
76	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
78	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
79	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
82	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
83	C₄×C₄₃	—	2	2	no	—
84	C₄×C₄₃	—	2	4	no	—
85	C₄×C₄₃	—	2	4	no	—
86	C₄×C₄₃	C₄⋊Q₈	1	4	yes	complemented
87	C₄×C₄₃	C₄⋊Q₈	1	4	yes	complemented
88	Dic₄₃	—	2	4	no	—
89	Dic₄₃	—	2	4	no	—
90	Dic₄₃	—	4	4	no	—
91	#2551	—	43	1	no	maximal, Sylow
92	C₂×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄₃	#544	1	1	yes	characteristic
95	C₂×C₄₃	D₄²	1	1	yes	characteristic
96	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
97	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
98	C₂×C₄₃	Q₈²	1	1	yes	characteristic
99	C₂×C₄⋊Q₈	—	43	2	no	—
100	C₂×C₄⋊Q₈	—	43	2	no	—
101	#395	—	43	4	no	—
102	#389	—	43	2	no	—
103	#389	—	43	2	no	—
104	#384	—	43	1	no	—
105	C₄×C₄⋊C₄	—	43	1	no	—
106	C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic
107	C₄₃	#2551	1	1	yes	characteristic, complemented, Sylow
108	C₂²×Q₈	—	43	4	no	—
109	C₄⋊Q₈	—	86	8	no	—
110	C₄⋊Q₈	—	86	8	no	—
111	C₂×C₄⋊C₄	—	43	1	no	—
112	C₂×C₄⋊C₄	—	43	1	no	—
113	C₂×C₄⋊C₄	—	43	4	no	—
114	C₂×C₄⋊C₄	—	43	2	no	—
115	C₂×C₄⋊C₄	—	43	2	no	—
116	C₂×C₄⋊C₄	—	43	4	no	—
117	C₂×C₄⋊C₄	—	43	2	no	—
118	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
120	C₂×C₄²	—	43	2	no	—
121	C₂×C₄²	C₂×D₄₃	1	2	yes	—
122	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
123	#95	—	43	2	no	—
124	#95	—	43	4	no	—
125	#95	C₂×D₄₃	1	2	yes	—
126	C₂×Q₈	—	86	8	no	—
127	C₂×Q₈	—	86	8	no	—
128	C₂×Q₈	—	172	4	no	—
129	C₂²×C₄	—	43	2	no	—
130	C₂²×C₄	—	43	2	no	—
131	C₂²×C₄	—	43	4	no	—
132	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
134	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
135	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
137	C₄⋊C₄	—	43	4	no	—
138	C₄⋊C₄	—	43	4	no	—
139	C₄⋊C₄	—	86	8	no	—
140	C₄⋊C₄	—	86	8	no	—
141	C₄⋊C₄	—	86	4	no	—
142	C₄⋊C₄	—	86	4	no	—
143	C₄⋊C₄	—	172	2	no	—
144	C₄⋊C₄	Dic₈₆	1	4	yes	complemented
145	C₄⋊C₄	D₁₇₂	1	4	yes	—
146	C₄²	—	2	4	no	—
147	C₄²	—	86	4	no	—
148	C₄²	Dic₈₆	1	4	yes	complemented
149	C₂³	C₂³×D₄₃	1	1	yes	characteristic
150	Q₈	—	172	16	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	43	4	no	—
158	C₂×C₄	—	43	4	no	—
159	C₂×C₄	—	86	2	no	—
160	C₂×C₄	—	86	2	no	—
161	C₂×C₄	—	86	2	no	—
162	C₂×C₄	—	86	2	no	—
163	C₂×C₄	—	86	4	no	—
164	C₂×C₄	—	86	4	no	—
165	C₂×C₄	—	86	4	no	—
166	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
167	C₂×C₄	C₂×Dic₈₆	1	4	yes	—
168	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
169	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
170	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
171	C₂×C₄	C₂×D₁₇₂	1	4	yes	—
172	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
173	C₂×C₄	D₄×D₄₃	1	2	yes	—
174	C₂×C₄	Q₈×D₄₃	1	2	yes	—
175	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
176	C₂²	C₂²×D₁₇₂	1	1	yes	characteristic
177	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
178	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
179	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
180	C₂²	#13063371	1	1	yes	characteristic
181	C₂²	#13063372	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	2	2	no	—
184	C₄	—	2	4	no	—
185	C₄	—	86	4	no	—
186	C₄	—	86	4	no	—
187	C₄	—	172	4	no	—
188	C₄	#13063324	1	4	yes	complemented
189	C₄	#13063325	1	4	yes	complemented
190	C₂	C₂×#13063324	1	1	yes	characteristic
191	C₂	C₂×#13063325	1	1	yes	characteristic
192	C₂	#14583527	1	1	yes	characteristic
193	C₂	D₄×D₁₇₂	1	1	yes	characteristic
194	C₂	D₄×Dic₈₆	1	1	yes	characteristic
195	C₂	Q₈×D₁₇₂	1	1	yes	characteristic
196	C₂	Q₈×Dic₈₆	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	#2551
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
39. C₄⋊C₄×C₄₃	Q₈	semidirect
41. C₄²×C₄₃	Q₈	semidirect
47. C₄⋊Dic₄₃	Q₈	semidirect
86. C₄×C₄₃	C₄⋊Q₈	semidirect
87. C₄×C₄₃	C₄⋊Q₈	semidirect
107. C₄₃	#2551	semidirect
144. C₄⋊C₄	Dic₈₆	semidirect
148. C₄²	Dic₈₆	semidirect
188. C₄	#13063324	semidirect
189. C₄	#13063325	semidirect