Subgroups — #106×Dic₈₆ #25115423

Order 11008.

838 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×Dic₈₆	C₂	1	2	yes	maximal
3	#19135358	C₂	1	2	yes	maximal
4	#19135357	C₂	1	4	yes	maximal
5	#19134752	C₂	1	1	yes	maximal, characteristic
6	#19134750	C₂	1	2	yes	maximal
7	#106×Dic₄₃	C₂	1	2	yes	maximal
8	C₄×#106×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₈₆	—	2	2	no	—
10	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
11	C₈×Dic₈₆	C₄	1	8	yes	complemented
12	C₂×#13063257	C₂²	1	2	yes	—
13	C₂×#13063257	C₂²	1	2	yes	—
14	C₂×#13063256	C₂²	1	4	yes	—
15	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₃⋊C₈	C₂²	1	2	yes	—
17	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
18	#14582797	—	2	2	no	—
19	#14582796	—	2	4	no	—
20	#14582769	C₂²	1	2	yes	—
21	#14582768	C₂²	1	4	yes	—
22	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
23	#14582661	C₂²	1	4	yes	—
24	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
28	C₄₃×#340	C₂²	1	2	yes	—
29	C₂²×Dic₈₆	D₄	1	1	yes	characteristic
30	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
31	C₄×Dic₈₆	—	4	8	no	—
32	C₄×Dic₈₆	C₂×C₄	1	4	yes	—
33	C₄×Dic₈₆	Q₈	1	4	yes	—
34	C₂×C₄⋊Dic₄₃	—	2	2	no	—
35	C₂×C₄⋊Dic₄₃	—	2	2	no	—
36	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
37	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
38	C₂×Dic₄₃⋊C₄	—	2	4	no	—
39	C₂×Dic₄₃⋊C₄	—	2	4	no	—
40	C₂×Dic₄₃⋊C₄	D₄	1	2	yes	—
41	C₂×C₄×Dic₄₃	—	2	4	no	—
42	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
43	#13063257	—	2	4	no	—
44	#13063257	—	4	2	no	—
45	#13063256	—	4	4	no	—
46	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
47	Dic₄₃⋊C₈	C₂×C₄	1	8	yes	—
48	C₈×Dic₄₃	C₂×C₄	1	8	yes	—
49	C₄₃⋊#95	—	2	4	no	—
50	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
54	C₂×C₄²×C₄₃	—	2	2	no	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	#106×C₄₃	—	2	2	no	—
57	#106×C₄₃	Q₈	1	4	yes	complemented
58	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
59	#95×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	—	2	2	no	—
61	C₂²×C₄×C₄₃	—	2	2	no	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₄₃	—	2	4	no	—
66	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
67	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
68	C₂×Dic₈₆	C₄⋊C₄	1	4	yes	—
69	C₂×Dic₈₆	SD₁₆	1	8	yes	—
70	C₂×C₈×C₄₃	—	2	1	no	—
71	C₂×C₈×C₄₃	—	2	1	no	—
72	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₄₃	#41	1	2	yes	—
76	C₄⋊C₄×C₄₃	—	2	2	no	—
77	C₄⋊C₄×C₄₃	—	2	2	no	—
78	C₄⋊C₄×C₄₃	—	2	2	no	—
79	C₄⋊C₄×C₄₃	C₂×Q₈	1	4	yes	—
80	C₄²×C₄₃	—	4	4	no	—
81	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
83	C₄⋊Dic₄₃	—	2	4	no	—
84	C₄⋊Dic₄₃	—	2	4	no	—
85	C₄⋊Dic₄₃	—	4	4	no	—
86	C₄⋊Dic₄₃	—	8	2	no	—
87	C₄⋊Dic₄₃	C₄⋊C₄	1	4	yes	—
88	Dic₄₃⋊C₄	—	4	8	no	—
89	Dic₄₃⋊C₄	—	8	4	no	—
90	Dic₄₃⋊C₄	C₄⋊C₄	1	8	yes	—
91	C₄×Dic₄₃	—	4	8	no	—
92	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
93	C₄×Dic₄₃	C₂×Q₈	1	4	yes	—
94	C₂×C₄₃⋊C₈	—	2	2	no	—
95	C₂×C₄₃⋊C₈	—	2	2	no	—
96	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
97	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	4	no	—
100	C₂×C₄×C₄₃	—	4	2	no	—
101	C₂×C₄×C₄₃	—	4	2	no	—
102	C₂×C₄×C₄₃	—	4	2	no	—
103	C₂×C₄×C₄₃	—	4	2	no	—
104	C₂×C₄×C₄₃	—	4	2	no	—
105	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₄₃	C₂×SD₁₆	1	4	yes	—
108	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×Dic₄₃	—	2	4	no	—
115	C₂×Dic₄₃	—	4	4	no	—
116	C₂×Dic₄₃	—	4	4	no	—
117	C₂×Dic₄₃	—	4	4	no	—
118	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×Dic₄₃	C₂×SD₁₆	1	8	yes	—
121	Dic₈₆	#106	1	16	yes	direct product factor
122	C₈×C₄₃	—	2	2	no	—
123	C₈×C₄₃	C₄×Q₈	1	4	yes	complemented
124	C₄₃⋊C₈	—	2	4	no	—
125	Q₈×#106	—	43	1	no	maximal, Sylow
126	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₄₃	D₄×Q₈	1	1	yes	characteristic
128	C₂²×C₄₃	#550	1	1	yes	characteristic
129	C₂²×C₄₃	C₂²×SD₁₆	1	2	yes	—
130	C₂²×C₄₃	#577	1	2	yes	—
131	C₄×C₄₃	—	2	2	no	—
132	C₄×C₄₃	—	4	4	no	—
133	C₄×C₄₃	—	8	2	no	—
134	C₄×C₄₃	C₂×#106	1	4	yes	—
135	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
136	C₄×C₄₃	#520	1	1	yes	characteristic
137	C₄×C₄₃	#557	1	1	yes	characteristic
138	C₄×C₄₃	Q₈²	1	1	yes	characteristic
139	Dic₄₃	—	2	4	no	—
140	Dic₄₃	—	8	4	no	—
141	Dic₄₃	C₂×#106	1	8	yes	—
142	C₂×C₈×Q₈	—	43	1	no	—
143	Q₈×C₄⋊C₄	—	43	2	no	—
144	#1948	—	43	2	no	—
145	#1948	—	43	4	no	—
146	#1844	—	43	2	no	—
147	#1844	—	43	1	no	—
148	C₄×#106	—	43	2	no	—
149	C₄×#106	D₄₃	1	1	yes	characteristic
150	C₂×C₄₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄₃	C₂²×#106	1	1	yes	characteristic
152	C₂×C₄₃	#2912	1	1	yes	characteristic
153	C₂×C₄₃	Q₈×SD₁₆	1	2	yes	—
154	C₂×C₄₃	#3392	1	2	yes	—
155	C₂×C₄×Q₈	—	43	1	no	—
156	C₂×C₄×Q₈	—	86	2	no	—
157	C₈×Q₈	—	43	8	no	—
158	C₂×#106	—	43	2	no	—
159	C₂×#106	—	43	2	no	—
160	C₂×#106	—	43	4	no	—
161	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
162	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
163	C₂×C₄⋊C₈	—	43	2	no	—
164	C₂×C₄⋊C₈	—	43	1	no	—
165	C₂×C₄×C₈	—	43	2	no	—
166	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
167	#389	—	86	4	no	—
168	#389	—	86	2	no	—
169	#384	—	43	4	no	—
170	#384	—	43	2	no	—
171	C₄×C₄⋊C₄	—	43	4	no	—
172	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
173	#340	—	43	4	no	—
174	#340	C₂×D₄₃	1	2	yes	—
175	C₄₃	Q₈×#106	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	43	1	no	—
177	C₂²×C₈	—	43	2	no	—
178	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
180	C₄×Q₈	—	43	4	no	—
181	C₄×Q₈	—	43	4	no	—
182	C₄×Q₈	—	172	8	no	—
183	C₂×C₄⋊C₄	—	43	2	no	—
184	C₂×C₄⋊C₄	—	43	1	no	—
185	C₂×C₄⋊C₄	—	43	4	no	—
186	C₂×C₄⋊C₄	—	86	4	no	—
187	C₂×C₄⋊C₄	—	86	2	no	—
188	C₂×C₄⋊C₄	—	86	4	no	—
189	C₂×C₄⋊C₄	—	86	2	no	—
190	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
191	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	43	2	no	—
194	C₂×C₄²	—	86	4	no	—
195	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
196	#106	—	2	2	no	—
197	#106	—	86	4	no	—
198	#106	—	172	4	no	—
199	#106	—	172	2	no	—
200	#106	Dic₈₆	1	4	yes	direct product factor
201	C₄⋊C₈	—	43	8	no	—
202	C₄⋊C₈	—	43	4	no	—
203	C₄×C₈	—	43	8	no	—
204	C₄×C₈	C₄×D₄₃	1	4	yes	—
205	#95	—	2	2	no	—
206	#95	—	86	4	no	—
207	C₂×Q₈	—	43	4	no	—
208	C₂×Q₈	—	43	8	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	43	2	no	—
212	C₂²×C₄	—	43	2	no	—
213	C₂²×C₄	—	86	4	no	—
214	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
216	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	43	4	no	—
220	C₂×C₈	—	86	2	no	—
221	C₂×C₈	—	86	2	no	—
222	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
223	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
224	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
225	C₂×C₈	#10621898	1	2	yes	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	43	8	no	—
230	C₄⋊C₄	—	43	4	no	—
231	C₄⋊C₄	—	86	4	no	—
232	C₄⋊C₄	—	86	4	no	—
233	C₄⋊C₄	—	344	2	no	—
234	C₄⋊C₄	—	344	4	no	—
235	C₄⋊C₄	—	172	8	no	—
236	C₄⋊C₄	—	172	4	no	—
237	C₄⋊C₄	C₂×Dic₈₆	1	4	yes	—
238	C₄²	—	4	4	no	—
239	C₄²	—	43	4	no	—
240	C₄²	—	43	4	no	—
241	C₄²	—	172	8	no	—
242	C₄²	C₂×C₄×D₄₃	1	2	yes	—
243	C₄²	Q₈×D₄₃	1	2	yes	—
244	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
245	Q₈	—	43	16	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	43	4	no	—
254	C₂×C₄	—	43	8	no	—
255	C₂×C₄	—	43	4	no	—
256	C₂×C₄	—	86	4	no	—
257	C₂×C₄	—	172	4	no	—
258	C₂×C₄	—	172	4	no	—
259	C₂×C₄	—	172	4	no	—
260	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
261	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
262	C₂×C₄	SD₁₆×D₄₃	1	4	yes	—
263	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
264	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
265	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
266	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
267	C₂×C₄	#13063371	1	1	yes	characteristic
268	C₂×C₄	#13063372	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	86	4	no	—
271	C₈	C₄×Dic₈₆	1	4	yes	complemented
272	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
273	C₂²	D₄×Dic₈₆	1	1	yes	characteristic
274	C₂²	#14583570	1	1	yes	characteristic
275	C₂²	C₂×SD₁₆×D₄₃	1	2	yes	—
276	C₂²	#14583706	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	4	4	no	—
279	C₄	—	8	2	no	—
280	C₄	—	43	8	no	—
281	C₄	—	86	4	no	—
282	C₄	—	344	4	no	—
283	C₄	#106×D₄₃	1	4	yes	—
284	C₄	C₂×C₄×Dic₈₆	1	1	yes	characteristic
285	C₄	#14583379	1	1	yes	characteristic
286	C₄	#14583627	1	1	yes	characteristic
287	C₄	Q₈×Dic₈₆	1	1	yes	characteristic
288	C₂	C₄⋊C₄×Dic₈₆	1	1	yes	characteristic
289	C₂	C₂×#106×D₄₃	1	1	yes	characteristic
290	C₂	#19143187	1	1	yes	characteristic
291	C₂	SD₁₆×Dic₈₆	1	2	yes	—
292	C₂	#19147574	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#106×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#106	D₄₃
hypocenter	C₄₃	Q₈×#106
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
11. C₈×Dic₈₆	C₄	semidirect
57. #106×C₄₃	Q₈	semidirect
121. Dic₈₆	#106	direct (N × H)
123. C₈×C₄₃	C₄×Q₈	semidirect
175. C₄₃	Q₈×#106	semidirect
200. #106	Dic₈₆	direct (N × H)
271. C₈	C₄×Dic₈₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₈×Dic₈₆	2
C₄⋊C₄×Dic₈₆	2
#19135358	2
#19135357	2
#19134752	2
#19134750	2
#106×Dic₄₃	2
C₄×#106×C₄₃	2
Q₈×#106	43

Subgroups — #106×Dic86 #25115423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #106×Dic₈₆ #25115423