Subgroups — #41×Dic₈₆ #19148175

Order 5504.

1068 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583628	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583625	C₂	1	2	yes	maximal, complemented
4	#14583608	C₂	1	6	yes	maximal, complemented
5	#14583606	C₂	1	3	yes	maximal, complemented
6	D₄×Dic₈₆	C₂	1	3	yes	maximal, complemented
7	#14583456	C₂	1	3	yes	maximal, complemented
8	#14583452	C₂	1	6	yes	maximal, complemented
9	#41×Dic₄₃	C₂	1	2	yes	maximal
10	C₂×C₄×Dic₈₆	C₂	1	3	yes	maximal, complemented
11	C₄×#41×C₄₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₈₆	C₂²	1	3	yes	—
13	#13063325	C₂²	1	3	yes	—
14	#13063325	C₂²	1	3	yes	—
15	#13063324	C₂²	1	6	yes	—
16	#13063324	C₂²	1	6	yes	—
17	#13063317	C₂²	1	6	yes	—
18	#13063316	C₂²	1	6	yes	—
19	#13063316	C₂²	1	6	yes	—
20	#13063304	C₂²	1	6	yes	—
21	#13063304	C₂²	1	6	yes	—
22	#13063302	C₂²	1	12	yes	—
23	#13063302	C₂²	1	12	yes	—
24	Q₈×Dic₄₃	C₂²	1	2	yes	—
25	Q₈×Dic₄₃	C₂²	1	2	yes	—
26	C₄×Dic₈₆	C₂²	1	3	yes	—
27	C₄×Dic₈₆	C₂²	1	3	yes	—
28	C₄×Dic₈₆	C₂²	1	6	yes	—
29	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
30	C₄×Dic₈₆	C₂²	1	6	yes	—
31	C₄×Dic₈₆	C₂²	1	3	yes	—
32	C₄×Dic₈₆	C₂²	1	6	yes	—
33	D₄×Dic₄₃	C₂²	1	6	yes	—
34	D₄×Dic₄₃	C₂²	1	6	yes	—
35	#13063284	C₂²	1	6	yes	—
36	C₂×C₄⋊Dic₄₃	C₂²	1	3	yes	—
37	C₂×Dic₄₃⋊C₄	C₂²	1	6	yes	—
38	C₂×C₄×Dic₄₃	C₂²	1	6	yes	—
39	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
40	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
41	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₄₃	C₂²	1	3	yes	—
43	C₄×D₄×C₄₃	C₂²	1	3	yes	—
44	#117×C₄₃	C₂²	1	3	yes	—
45	C₂×C₄²×C₄₃	C₂²	1	3	yes	—
46	#41×C₄₃	Q₈	1	8	yes	complemented
47	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₄₃	C₂³	1	3	yes	—
49	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
50	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
51	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
52	C₂²×Dic₄₃	C₂³	1	6	yes	—
53	C₂²×Dic₄₃	C₂³	1	6	yes	—
54	C₂×Dic₈₆	—	2	12	no	—
55	C₂×Dic₈₆	C₂³	1	6	yes	—
56	C₂×Dic₈₆	C₂³	1	3	yes	—
57	C₂×Dic₈₆	C₂³	1	6	yes	—
58	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₄₃	C₂³	1	3	yes	—
60	C₄⋊C₄×C₄₃	C₂³	1	3	yes	—
61	C₂²⋊C₄×C₄₃	C₂³	1	6	yes	—
62	C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	C₄²×C₄₃	C₂³	1	3	yes	—
64	C₄²×C₄₃	C₂³	1	3	yes	—
65	C₄²×C₄₃	C₂³	1	3	yes	—
66	C₂²⋊Dic₄₃	C₂³	1	12	yes	—
67	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₄₃	C₂³	1	6	yes	—
69	C₄⋊Dic₄₃	C₂³	1	3	yes	—
70	C₄⋊Dic₄₃	C₂³	1	6	yes	—
71	C₄⋊Dic₄₃	C₂³	1	6	yes	—
72	C₄⋊Dic₄₃	C₂³	1	6	yes	—
73	Dic₄₃⋊C₄	C₂³	1	2	yes	—
74	Dic₄₃⋊C₄	C₂³	1	6	yes	—
75	Dic₄₃⋊C₄	C₂³	1	6	yes	—
76	Dic₄₃⋊C₄	C₂³	1	6	yes	—
77	Dic₄₃⋊C₄	C₂³	1	12	yes	—
78	C₄×Dic₄₃	C₂³	1	2	yes	—
79	C₄×Dic₄₃	C₂³	1	6	yes	—
80	C₄×Dic₄₃	C₂³	1	6	yes	—
81	C₄×Dic₄₃	C₂³	1	6	yes	—
82	C₂³×C₄₃	C₂⁴	1	3	yes	—
83	Q₈×C₄₃	C₂×Q₈	1	4	yes	complemented
84	D₄×C₄₃	C₂×Q₈	1	12	yes	complemented
85	C₂×C₄×C₄₃	—	2	6	no	—
86	C₂×C₄×C₄₃	—	4	3	no	—
87	C₂×C₄×C₄₃	C₂×Q₈	1	12	yes	complemented
88	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
89	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
90	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
91	C₂×C₄×C₄₃	C₂⁴	1	3	yes	—
92	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
94	C₂×Dic₄₃	—	2	12	no	—
95	C₂×Dic₄₃	—	4	6	no	—
96	C₂×Dic₄₃	C₂⁴	1	2	yes	—
97	C₂×Dic₄₃	C₂⁴	1	6	yes	—
98	C₂×Dic₄₃	C₂⁴	1	6	yes	—
99	C₂×Dic₄₃	C₂⁴	1	2	yes	—
100	Dic₈₆	—	2	6	no	—
101	Dic₈₆	—	2	12	no	—
102	Dic₈₆	—	4	6	no	—
103	Dic₈₆	#41	1	4	yes	direct product factor
104	C₂²×C₄₃	—	2	3	no	—
105	C₂²×C₄₃	—	2	3	no	—
106	C₂²×C₄₃	C₂²×Q₈	1	6	yes	—
107	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
108	C₄×C₄₃	—	2	3	no	—
109	C₄×C₄₃	—	2	3	no	—
110	C₄×C₄₃	—	2	1	no	—
111	C₄×C₄₃	C₂²×Q₈	1	2	yes	—
112	C₄×C₄₃	C₂²×Q₈	1	6	yes	—
113	C₄×C₄₃	C₂×#41	1	2	yes	—
114	Dic₄₃	—	2	6	no	—
115	Dic₄₃	—	2	6	no	—
116	Dic₄₃	—	2	2	no	—
117	Dic₄₃	C₂×#41	1	4	yes	—
118	Q₈×#41	—	43	1	no	maximal, Sylow
119	C₂×C₄₃	—	2	6	no	—
120	C₂×C₄₃	C₂³×Q₈	1	1	yes	characteristic
121	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
122	C₂×C₄₃	#585	1	1	yes	characteristic
123	Q₈²	—	43	1	no	—
124	#557	—	43	2	no	—
125	#557	—	43	1	no	—
126	#554	—	43	3	no	—
127	#554	—	43	6	no	—
128	D₄×Q₈	—	43	3	no	—
129	#533	—	43	6	no	—
130	#533	—	43	3	no	—
131	C₄×#41	—	43	2	no	—
132	C₄×#41	D₄₃	1	1	yes	characteristic
133	C₂×C₄×Q₈	—	43	3	no	—
134	C₄₃	Q₈×#41	1	1	yes	characteristic, complemented, Sylow
135	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
136	C₂²×Q₈	—	43	3	no	—
137	C₄⋊Q₈	—	43	6	no	—
138	C₄⋊Q₈	—	43	6	no	—
139	C₄⋊Q₈	—	43	3	no	—
140	C₄⋊Q₈	—	43	3	no	—
141	#125	—	43	6	no	—
142	#125	—	43	6	no	—
143	#125	—	43	6	no	—
144	C₂²⋊Q₈	—	43	6	no	—
145	C₂²⋊Q₈	—	43	6	no	—
146	C₂²⋊Q₈	—	43	12	no	—
147	C₂²⋊Q₈	—	43	12	no	—
148	C₄×Q₈	—	43	2	no	—
149	C₄×Q₈	—	43	2	no	—
150	C₄×Q₈	—	43	1	no	—
151	C₄×Q₈	—	43	6	no	—
152	C₄×Q₈	—	43	3	no	—
153	C₄×Q₈	—	43	3	no	—
154	C₄×Q₈	—	43	3	no	—
155	C₄×Q₈	—	43	6	no	—
156	C₄×Q₈	—	43	6	no	—
157	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
158	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
159	C₄×D₄	—	43	6	no	—
160	C₄×D₄	—	43	6	no	—
161	C₄×D₄	C₂×D₄₃	1	3	yes	—
162	C₄×D₄	C₂×D₄₃	1	3	yes	—
163	#117	—	43	6	no	—
164	#117	C₂×D₄₃	1	3	yes	—
165	C₂×C₄⋊C₄	—	43	6	no	—
166	C₂×C₄⋊C₄	—	43	3	no	—
167	C₂×C₄²	—	43	6	no	—
168	C₂×C₄²	C₂×D₄₃	1	3	yes	—
169	#41	Dic₈₆	1	8	yes	direct product factor
170	C₂×Q₈	—	43	1	no	—
171	C₂×Q₈	—	43	6	no	—
172	C₂×Q₈	—	43	3	no	—
173	C₂×Q₈	—	43	6	no	—
174	C₂×Q₈	—	86	12	no	—
175	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
176	C₂×D₄	C₂²×D₄₃	1	3	yes	—
177	C₂²×C₄	—	43	6	no	—
178	C₂²×C₄	—	43	6	no	—
179	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
180	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
181	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
182	C₄⋊C₄	—	43	2	no	—
183	C₄⋊C₄	—	43	1	no	—
184	C₄⋊C₄	—	43	6	no	—
185	C₄⋊C₄	—	43	6	no	—
186	C₄⋊C₄	—	43	6	no	—
187	C₄⋊C₄	—	43	6	no	—
188	C₄⋊C₄	—	43	3	no	—
189	C₄⋊C₄	—	43	6	no	—
190	C₄⋊C₄	—	43	6	no	—
191	C₄⋊C₄	—	43	12	no	—
192	C₄⋊C₄	—	43	6	no	—
193	C₄⋊C₄	C₂²×D₄₃	1	3	yes	—
194	C₄⋊C₄	C₂²×D₄₃	1	3	yes	—
195	C₂²⋊C₄	—	43	12	no	—
196	C₂²⋊C₄	C₂²×D₄₃	1	6	yes	—
197	C₄²	—	43	6	no	—
198	C₄²	—	43	6	no	—
199	C₄²	—	43	2	no	—
200	C₄²	—	43	6	no	—
201	C₄²	C₂²×D₄₃	1	3	yes	—
202	C₄²	C₂²×D₄₃	1	3	yes	—
203	C₄²	C₂²×D₄₃	1	1	yes	characteristic
204	C₄²	C₂²×D₄₃	1	3	yes	—
205	C₂³	C₂³×D₄₃	1	3	yes	—
206	Q₈	—	43	4	no	—
207	Q₈	—	86	12	no	—
208	Q₈	—	86	6	no	—
209	Q₈	—	172	6	no	—
210	Q₈	C₂×Dic₈₆	1	4	yes	complemented
211	D₄	C₂×Dic₈₆	1	12	yes	complemented
212	C₂×C₄	—	2	6	no	—
213	C₂×C₄	—	4	3	no	—
214	C₂×C₄	—	43	2	no	—
215	C₂×C₄	—	43	2	no	—
216	C₂×C₄	—	43	6	no	—
217	C₂×C₄	—	43	6	no	—
218	C₂×C₄	—	86	12	no	—
219	C₂×C₄	—	172	6	no	—
220	C₂×C₄	C₂×Dic₈₆	1	12	yes	complemented
221	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
222	C₂×C₄	C₂³×D₄₃	1	3	yes	—
223	C₂×C₄	C₂³×D₄₃	1	3	yes	—
224	C₂×C₄	C₂³×D₄₃	1	3	yes	—
225	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
227	C₂²	—	2	3	no	—
228	C₂²	—	2	3	no	—
229	C₂²	C₂²×Dic₈₆	1	6	yes	—
230	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
231	C₄	—	2	3	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	3	no	—
234	C₄	—	43	4	no	—
235	C₄	—	86	6	no	—
236	C₄	—	86	2	no	—
237	C₄	—	86	6	no	—
238	C₄	C₂²×Dic₈₆	1	2	yes	—
239	C₄	C₂²×Dic₈₆	1	6	yes	—
240	C₄	#41×D₄₃	1	2	yes	—
241	C₂	—	2	6	no	—
242	C₂	C₂³×Dic₈₆	1	1	yes	characteristic
243	C₂	C₂×#41×D₄₃	1	1	yes	characteristic
244	C₂	#14583735	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Frattini	C₂²	C₂⁴×D₄₃
Fitting	C₄×#41×C₄₃	C₂
hypercenter	C₄×#41	D₄₃
hypocenter	C₄₃	Q₈×#41
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₈₆	C₂	semidirect
2. #14583628	C₂	semidirect
3. #14583625	C₂	semidirect
4. #14583608	C₂	semidirect
5. #14583606	C₂	semidirect
6. D₄×Dic₈₆	C₂	semidirect
7. #14583456	C₂	semidirect
8. #14583452	C₂	semidirect
10. C₂×C₄×Dic₈₆	C₂	semidirect
46. #41×C₄₃	Q₈	semidirect
83. Q₈×C₄₃	C₂×Q₈	semidirect
84. D₄×C₄₃	C₂×Q₈	semidirect
87. C₂×C₄×C₄₃	C₂×Q₈	semidirect
103. Dic₈₆	#41	direct (N × H)
134. C₄₃	Q₈×#41	semidirect
169. #41	Dic₈₆	direct (N × H)
210. Q₈	C₂×Dic₈₆	semidirect
211. D₄	C₂×Dic₈₆	semidirect
220. C₂×C₄	C₂×Dic₈₆	semidirect

Subgroups — #41×Dic86 #19148175

Conjugacy classes of subgroups

Complements

Subgroups — #41×Dic₈₆ #19148175