Subgroups — #24987205

Order 11008.

598 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582631	C₂	1	2	yes	maximal
4	#19132435	C₂	1	2	yes	maximal
5	C₄₃×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₈₆	C₄	1	2	yes	—
7	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
10	C₄×Dic₄₃⋊C₄	—	2	1	no	—
11	C₄×Dic₄₃⋊C₄	—	2	2	no	—
12	C₄×C₄⋊Dic₄₃	C₄	1	2	yes	—
13	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
14	C₄²×Dic₄₃	—	2	1	no	—
15	#14582654	—	2	1	no	—
16	#14582631	C₄	1	8	yes	complemented
17	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
18	C₄³×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#336	—	2	2	no	—
20	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄₃⋊C₈	—	2	2	no	—
22	C₄×Dic₈₆	—	2	4	no	—
23	C₄×Dic₈₆	—	2	2	no	—
24	C₄×Dic₈₆	C₈	1	8	yes	complemented
25	C₄×Dic₈₆	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
28	C₂×Dic₄₃⋊C₄	—	2	2	no	—
29	C₂×Dic₄₃⋊C₄	—	2	1	no	—
30	C₂×C₄×Dic₄₃	—	2	2	no	—
31	C₂×C₄×Dic₄₃	—	2	1	no	—
32	C₂×C₄×Dic₄₃	—	2	2	no	—
33	C₂×C₄×Dic₄₃	—	2	1	no	—
34	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
35	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
36	C₄×C₄₃⋊C₈	—	4	2	no	—
37	C₄₃⋊#95	—	2	2	no	—
38	C₄₃⋊#95	—	2	1	no	—
39	C₂²×C₈×C₄₃	—	2	2	no	—
40	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	D₄	1	2	yes	—
44	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
45	C₄×C₈×C₄₃	—	2	2	no	—
46	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
47	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₄₃	—	2	1	no	—
53	C₂²×Dic₄₃	—	2	1	no	—
54	C₂²×Dic₄₃	—	2	2	no	—
55	C₂×Dic₈₆	—	2	1	no	—
56	C₂×Dic₈₆	—	2	2	no	—
57	C₂×Dic₈₆	C₄²	1	2	yes	—
58	C₂×Dic₈₆	C₂×C₈	1	4	yes	—
59	C₂×C₈×C₄₃	—	2	4	no	—
60	C₂×C₈×C₄₃	—	4	4	no	—
61	C₄²×C₄₃	—	2	1	no	—
62	C₄²×C₄₃	—	2	1	no	—
63	C₄²×C₄₃	—	2	2	no	—
64	C₄²×C₄₃	—	4	2	no	—
65	C₄²×C₄₃	C₂×C₈	1	4	yes	—
66	C₄²×C₄₃	M₁₆	1	4	yes	—
67	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
68	C₄²×C₄₃	#41	1	2	yes	—
69	C₄⋊Dic₄₃	—	2	1	no	—
70	C₄⋊Dic₄₃	—	2	2	no	—
71	C₄⋊Dic₄₃	C₄²	1	2	yes	—
72	C₄⋊Dic₄₃	C₂×C₈	1	4	yes	—
73	Dic₄₃⋊C₄	—	2	8	no	—
74	Dic₄₃⋊C₄	—	2	4	no	—
75	C₄×Dic₄₃	—	2	8	no	—
76	C₄×Dic₄₃	—	2	4	no	—
77	C₄×Dic₄₃	—	4	4	no	—
78	C₄×Dic₄₃	—	4	1	no	—
79	C₄×Dic₄₃	—	4	1	no	—
80	C₂×C₄₃⋊C₈	—	2	4	no	—
81	C₂×C₄₃⋊C₈	—	4	2	no	—
82	C₂×C₄₃⋊C₈	—	4	2	no	—
83	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	—	2	2	no	—
86	C₂×C₄×C₄₃	—	2	2	no	—
87	C₂×C₄×C₄₃	—	2	2	no	—
88	C₂×C₄×C₄₃	—	2	2	no	—
89	C₂×C₄×C₄₃	C₂²⋊C₈	1	4	yes	—
90	C₂×C₄×C₄₃	C₂²⋊C₈	1	4	yes	—
91	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₄₃	#137	1	2	yes	—
99	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
100	C₂×Dic₄₃	—	2	2	no	—
101	C₂×Dic₄₃	—	2	4	no	—
102	C₂×Dic₄₃	—	2	2	no	—
103	C₂×Dic₄₃	—	2	4	no	—
104	C₂×Dic₄₃	—	4	2	no	—
105	C₂×Dic₄₃	—	4	2	no	—
106	C₂×Dic₄₃	—	4	2	no	—
107	Dic₈₆	—	2	2	no	—
108	Dic₈₆	—	2	4	no	—
109	Dic₈₆	C₄×C₈	1	4	yes	complemented
110	C₈×C₄₃	—	4	4	no	—
111	C₄₃⋊C₈	—	4	4	no	—
112	#8171	—	43	1	no	maximal, Sylow
113	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₄₃	C₂×C₂²⋊C₈	1	2	yes	—
115	C₂²×C₄₃	#417	1	1	yes	characteristic
116	C₂²×C₄₃	#419	1	1	yes	characteristic
117	C₂²×C₄₃	#421	1	2	yes	—
118	C₄×C₄₃	—	2	4	no	—
119	C₄×C₄₃	—	2	4	no	—
120	C₄×C₄₃	—	2	2	no	—
121	C₄×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
122	C₄×C₄₃	C₄×M₁₆	1	1	yes	characteristic
123	C₄×C₄₃	#431	1	2	yes	—
124	C₄×C₄₃	#440	1	2	yes	—
125	C₄×C₄₃	#441	1	2	yes	—
126	Dic₄₃	—	2	4	no	—
127	Dic₄₃	—	4	4	no	—
128	Dic₄₃	—	4	2	no	—
129	C₄²×Q₈	—	43	1	no	—
130	C₄×C₄⋊C₈	—	43	1	no	—
131	#1744	D₄₃	1	1	yes	characteristic
132	C₂×Q₈⋊C₈	—	43	2	no	—
133	#1279	—	43	2	no	—
134	C₂×C₄₃	#1477	1	2	yes	—
135	C₂×C₄₃	#1479	1	2	yes	—
136	C₂×C₄₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
137	C₂×C₄₃	#1759	1	1	yes	characteristic
138	C₂×C₄₃	#1763	1	1	yes	characteristic
139	C₂×C₄×Q₈	—	43	1	no	—
140	C₂×C₄×Q₈	—	43	2	no	—
141	C₂×C₄⋊C₈	—	43	2	no	—
142	C₂×C₄×C₈	—	86	1	no	—
143	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
144	C₄×C₄⋊C₄	—	43	1	no	—
145	C₄×C₄⋊C₄	—	43	2	no	—
146	C₄×C₄⋊C₄	—	86	1	no	—
147	C₄×C₄⋊C₄	—	86	2	no	—
148	C₄³	—	86	1	no	—
149	C₄³	C₂×D₄₃	1	1	yes	characteristic
150	#336	—	2	2	no	—
151	#336	—	86	1	no	—
152	Q₈⋊C₈	—	43	8	no	—
153	C₄₃	#8171	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	—	43	1	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	86	2	no	—
157	C₄×Q₈	—	43	8	no	—
158	C₄×Q₈	—	43	4	no	—
159	C₄×Q₈	—	86	4	no	—
160	C₄×Q₈	—	86	2	no	—
161	C₂×C₄⋊C₄	—	43	1	no	—
162	C₂×C₄⋊C₄	—	43	2	no	—
163	C₂×C₄⋊C₄	—	86	2	no	—
164	C₂×C₄⋊C₄	—	86	1	no	—
165	C₂×C₄²	—	86	2	no	—
166	C₂×C₄²	—	86	1	no	—
167	C₂×C₄²	—	86	2	no	—
168	C₂×C₄²	—	86	1	no	—
169	C₂×C₄²	C₄×D₄₃	1	2	yes	—
170	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
171	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
173	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
174	C₄⋊C₈	—	43	4	no	—
175	C₄⋊C₈	—	86	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	172	2	no	—
178	C₄×C₈	C₄×D₄₃	1	4	yes	—
179	#95	—	86	2	no	—
180	#95	—	86	1	no	—
181	C₂×Q₈	—	43	2	no	—
182	C₂×Q₈	—	43	4	no	—
183	C₂×Q₈	—	86	1	no	—
184	C₂×Q₈	—	86	2	no	—
185	C₂²×C₄	—	86	1	no	—
186	C₂²×C₄	—	86	2	no	—
187	C₂²×C₄	—	86	1	no	—
188	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
189	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
190	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	86	4	no	—
196	C₂×C₈	—	172	2	no	—
197	C₂×C₈	—	172	2	no	—
198	C₄⋊C₄	—	43	2	no	—
199	C₄⋊C₄	—	43	4	no	—
200	C₄⋊C₄	—	86	8	no	—
201	C₄⋊C₄	—	86	1	no	—
202	C₄⋊C₄	—	86	4	no	—
203	C₄⋊C₄	—	86	2	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	4	2	no	—
208	C₄²	—	86	8	no	—
209	C₄²	—	86	4	no	—
210	C₄²	—	172	4	no	—
211	C₄²	—	172	1	no	—
212	C₄²	—	172	1	no	—
213	C₄²	C₈×D₄₃	1	4	yes	—
214	C₄²	#10621872	1	4	yes	—
215	C₄²	C₂×C₄×D₄₃	1	2	yes	—
216	C₄²	#10621898	1	2	yes	—
217	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
218	Q₈	—	43	4	no	—
219	Q₈	—	86	2	no	—
220	Q₈	—	86	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	86	4	no	—
227	C₂×C₄	—	86	2	no	—
228	C₂×C₄	—	86	4	no	—
229	C₂×C₄	—	86	2	no	—
230	C₂×C₄	—	172	2	no	—
231	C₂×C₄	—	172	2	no	—
232	C₂×C₄	—	172	2	no	—
233	C₂×C₄	(C₂×D₄₃)⋊C₈	1	4	yes	—
234	C₂×C₄	(C₂×D₄₃)⋊C₈	1	4	yes	—
235	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
236	C₂×C₄	#13063282	1	1	yes	characteristic
237	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
238	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
239	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
240	C₂×C₄	C₂×#10621872	1	2	yes	—
241	C₂×C₄	#13063353	1	1	yes	characteristic
242	C₂×C₄	#13063358	1	2	yes	—
243	C₂×C₄	#13063358	1	1	yes	characteristic
244	C₈	—	4	4	no	—
245	C₈	—	172	4	no	—
246	C₂²	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
247	C₂²	C₂×(C₂×D₄₃)⋊C₈	1	2	yes	—
248	C₂²	#14582901	1	1	yes	characteristic
249	C₂²	#14582909	1	1	yes	characteristic
250	C₂²	#14582917	1	2	yes	—
251	C₄	—	2	4	no	—
252	C₄	—	2	4	no	—
253	C₄	—	2	2	no	—
254	C₄	—	86	4	no	—
255	C₄	—	172	4	no	—
256	C₄	—	172	2	no	—
257	C₄	C₄×C₈×D₄₃	1	1	yes	characteristic
258	C₄	C₄×#10621872	1	1	yes	characteristic
259	C₄	#14582950	1	2	yes	—
260	C₄	#14583001	1	2	yes	—
261	C₄	#14583007	1	2	yes	—
262	C₂	#19132983	1	2	yes	—
263	C₂	#19132991	1	2	yes	—
264	C₂	C₄×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
265	C₂	#19134406	1	1	yes	characteristic
266	C₂	#19134426	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₄×C₄₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1744	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1744	D₄₃
hypocenter	C₄₃	#8171
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
16. #14582631	C₄	semidirect
24. C₄×Dic₈₆	C₈	semidirect
109. Dic₈₆	C₄×C₈	semidirect
153. C₄₃	#8171	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Dic₈₆	2
C₄×C₄⋊(C₄₃⋊C₈)	2
C₂×#14582631	2
#19132435	2
C₄₃×#1744	2
#8171	43