Subgroups — #19139517

Order 5504.

620 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063304	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582822	C₂	1	4	yes	maximal, complemented
4	#14582802	C₂	1	2	yes	maximal
5	#14582795	C₂	1	2	yes	maximal, complemented
6	#14582769	C₂	1	1	yes	maximal, characteristic, complemented
7	#14582759	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
10	#13063304	—	2	8	no	—
11	C₄×Dic₈₆	—	2	4	no	—
12	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
16	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
17	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
18	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
19	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
20	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
21	C₄₃⋊#95	C₂²	1	4	yes	—
22	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
23	C₂³×C₄×C₄₃	C₂²	1	2	yes	—
24	C₄×D₄×C₄₃	—	2	4	no	—
25	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
28	C₂⁴×C₄₃	C₂³	1	2	yes	—
29	C₂×D₄×C₄₃	—	2	2	no	—
30	C₂×D₄×C₄₃	—	2	2	no	—
31	C₂×D₄×C₄₃	Q₈	1	4	yes	complemented
32	C₂²×C₄×C₄₃	—	2	4	no	—
33	C₂²×C₄×C₄₃	—	2	4	no	—
34	C₂²×C₄×C₄₃	—	2	4	no	—
35	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
39	C₂²×Dic₄₃	C₂³	1	2	yes	—
40	C₂²×Dic₄₃	C₂³	1	4	yes	—
41	C₂²×Dic₄₃	C₂³	1	2	yes	—
42	C₂×Dic₈₆	—	2	2	no	—
43	C₂×Dic₈₆	—	2	2	no	—
44	C₂×Dic₈₆	D₄	1	4	yes	complemented
45	C₄⋊C₄×C₄₃	—	2	2	no	—
46	C₂²⋊C₄×C₄₃	—	2	4	no	—
47	C₄²×C₄₃	D₄	1	4	yes	—
48	C₂²⋊Dic₄₃	—	2	8	no	—
49	C₂²⋊Dic₄₃	—	4	4	no	—
50	C₄⋊Dic₄₃	—	2	2	no	—
51	C₄⋊Dic₄₃	—	2	4	no	—
52	C₄⋊Dic₄₃	—	2	4	no	—
53	C₄⋊Dic₄₃	—	4	2	no	—
54	Dic₄₃⋊C₄	—	2	4	no	—
55	Dic₄₃⋊C₄	—	2	8	no	—
56	C₄×Dic₄₃	—	2	4	no	—
57	C₂³×C₄₃	—	2	2	no	—
58	C₂³×C₄₃	—	2	2	no	—
59	C₂³×C₄₃	—	2	2	no	—
60	C₂³×C₄₃	—	2	2	no	—
61	C₂³×C₄₃	—	2	2	no	—
62	C₂³×C₄₃	—	2	2	no	—
63	C₂³×C₄₃	C₂×Q₈	1	4	yes	—
64	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
65	D₄×C₄₃	—	2	8	no	—
66	C₂×C₄×C₄₃	—	2	1	no	—
67	C₂×C₄×C₄₃	—	2	1	no	—
68	C₂×C₄×C₄₃	—	2	2	no	—
69	C₂×C₄×C₄₃	—	2	2	no	—
70	C₂×C₄×C₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	—	4	8	no	—
72	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
77	C₂×C₄×C₄₃	#41	1	2	yes	—
78	C₂×C₄×C₄₃	#41	1	2	yes	—
79	C₂×Dic₄₃	—	2	2	no	—
80	C₂×Dic₄₃	—	2	2	no	—
81	C₂×Dic₄₃	—	2	2	no	—
82	C₂×Dic₄₃	—	2	4	no	—
83	C₂×Dic₄₃	—	2	4	no	—
84	C₂×Dic₄₃	—	2	4	no	—
85	C₂×Dic₄₃	—	2	2	no	—
86	C₂×Dic₄₃	—	2	2	no	—
87	C₂×Dic₄₃	C₂×D₄	1	4	yes	—
88	Dic₈₆	—	2	8	no	—
89	C₂²×C₄₃	—	2	4	no	—
90	C₂²×C₄₃	—	2	4	no	—
91	C₂²×C₄₃	—	2	4	no	—
92	C₂²×C₄₃	—	4	2	no	—
93	C₂²×C₄₃	—	4	2	no	—
94	C₂²×C₄₃	—	4	2	no	—
95	C₂²×C₄₃	—	4	2	no	—
96	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
97	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
98	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
100	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
101	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₄₃	—	2	2	no	—
104	C₄×C₄₃	—	4	2	no	—
105	C₄×C₄₃	#121	1	4	yes	—
106	C₄×C₄₃	C₂²⋊Q₈	1	4	yes	complemented
107	Dic₄₃	—	2	4	no	—
108	Dic₄₃	—	4	2	no	—
109	Dic₄₃	—	4	4	no	—
110	#2538	—	43	1	no	maximal, Sylow
111	C₂×C₄₃	—	4	4	no	—
112	C₂×C₄₃	C₂×#121	1	1	yes	characteristic
113	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄₃	#537	1	1	yes	characteristic
115	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
116	C₂×C₄₃	#550	1	1	yes	characteristic
117	C₂×C₄₃	#554	1	1	yes	characteristic
118	C₂×C₄₃	#556	1	1	yes	characteristic
119	C₂×C₂²⋊Q₈	—	43	2	no	—
120	C₂×C₄×Q₈	—	43	1	no	—
121	C₂×C₄×D₄	D₄₃	1	1	yes	characteristic
122	#396	—	43	4	no	—
123	#390	—	43	2	no	—
124	#389	—	43	2	no	—
125	#384	—	43	1	no	—
126	#380	—	43	2	no	—
127	C₄₃	#2538	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	—	43	1	no	—
129	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
130	C₂³×C₄	C₂×D₄₃	1	2	yes	—
131	C₂²⋊Q₈	—	86	8	no	—
132	C₄×Q₈	—	86	4	no	—
133	C₄×D₄	—	2	4	no	—
134	C₂×C₄⋊C₄	—	43	2	no	—
135	C₂×C₄⋊C₄	—	43	1	no	—
136	C₂×C₄⋊C₄	—	43	4	no	—
137	C₂×C₄⋊C₄	—	43	2	no	—
138	C₂×C₄⋊C₄	—	43	2	no	—
139	C₂×C₄⋊C₄	—	43	2	no	—
140	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
141	C₂×C₂²⋊C₄	—	43	4	no	—
142	C₂×C₂²⋊C₄	—	43	4	no	—
143	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
144	C₂×C₄²	—	43	2	no	—
145	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
146	#95	—	43	4	no	—
147	C₂⁴	C₂²×D₄₃	1	2	yes	—
148	C₂×Q₈	—	43	4	no	—
149	C₂×Q₈	—	86	2	no	—
150	C₂×Q₈	—	86	2	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	Dic₈₆	1	4	yes	complemented
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	43	2	no	—
158	C₂²×C₄	—	43	2	no	—
159	C₂²×C₄	—	43	4	no	—
160	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
161	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	86	8	no	—
166	C₄⋊C₄	—	86	4	no	—
167	C₄⋊C₄	—	86	4	no	—
168	C₄⋊C₄	—	86	4	no	—
169	C₄⋊C₄	—	86	2	no	—
170	C₄⋊C₄	—	172	2	no	—
171	C₂²⋊C₄	—	2	4	no	—
172	C₂²⋊C₄	—	86	8	no	—
173	C₂²⋊C₄	—	172	4	no	—
174	C₄²	—	86	4	no	—
175	C₄²	C₂²⋊D₄₃	1	4	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	C₂×Dic₈₆	1	4	yes	—
183	C₂³	C₂³×D₄₃	1	1	yes	characteristic
184	Q₈	—	86	8	no	—
185	D₄	—	2	8	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	8	no	—
192	C₂×C₄	—	43	4	no	—
193	C₂×C₄	—	86	2	no	—
194	C₂×C₄	—	86	2	no	—
195	C₂×C₄	—	86	4	no	—
196	C₂×C₄	—	86	2	no	—
197	C₂×C₄	—	86	4	no	—
198	C₂×C₄	—	86	2	no	—
199	C₂×C₄	—	86	4	no	—
200	C₂×C₄	—	86	2	no	—
201	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
202	C₂×C₄	#10621898	1	2	yes	—
203	C₂×C₄	D₄×D₄₃	1	2	yes	—
204	C₂×C₄	#10621900	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
206	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
216	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
217	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
218	C₂²	C₂×#10621900	1	1	yes	characteristic
219	C₂²	C₂×#10621898	1	1	yes	characteristic
220	C₂²	#13063371	1	1	yes	characteristic
221	C₂²	#13063372	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	4	2	no	—
224	C₄	—	86	4	no	—
225	C₄	—	172	4	no	—
226	C₄	—	172	2	no	—
227	C₄	#13063296	1	4	yes	—
228	C₄	#13063304	1	4	yes	complemented
229	C₂	—	4	4	no	—
230	C₂	C₂×#13063296	1	1	yes	characteristic
231	C₂	C₂×#13063304	1	1	yes	characteristic
232	C₂	#14583477	1	1	yes	characteristic
233	C₂	D₄×Dic₈₆	1	1	yes	characteristic
234	C₂	#14583570	1	1	yes	characteristic
235	C₂	#14583606	1	1	yes	characteristic
236	C₂	#14583619	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₂×C₄×D₄×C₄₃	C₂
hypercenter	C₂×C₄×D₄	D₄₃
hypocenter	C₄₃	#2538
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#13063304	C₂	semidirect
2. C₂×C₄×Dic₈₆	C₂	semidirect
3. #14582822	C₂	semidirect
5. #14582795	C₂	semidirect
6. #14582769	C₂	semidirect
7. #14582759	C₂	semidirect
31. C₂×D₄×C₄₃	Q₈	semidirect
44. C₂×Dic₈₆	D₄	semidirect
106. C₄×C₄₃	C₂²⋊Q₈	semidirect
127. C₄₃	#2538	semidirect
153. C₂×D₄	Dic₈₆	semidirect
228. C₄	#13063304	semidirect