Subgroups — #24987217

Order 11008.

610 conjugacy classes of subgroups.

The subgroup lattice has 270 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×#13063236	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582629	C₂	1	2	yes	maximal
4	#19132434	C₂	1	2	yes	maximal
5	C₄₃×#1767	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×#13063236	—	2	2	no	—
9	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₄₃⋊C₄	—	2	2	no	—
11	C₄×Dic₄₃⋊C₄	—	2	1	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	#14582629	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
17	C₂×#97×C₄₃	—	2	1	no	—
18	C₄³×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#336	—	2	1	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	—	2	4	no	—
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	#13063236	—	2	8	no	—
35	C₄×C₄₃⋊C₈	—	2	2	no	—
36	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
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	2	yes	—
42	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
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	#97×C₄₃	—	4	2	no	—
48	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₄₃	—	2	1	no	—
54	C₂²×Dic₄₃	—	2	1	no	—
55	C₂²×Dic₄₃	—	2	2	no	—
56	C₂×Dic₈₆	—	2	1	no	—
57	C₂×Dic₈₆	—	2	2	no	—
58	C₂×Dic₈₆	C₄²	1	2	yes	—
59	C₂×Dic₈₆	M₁₆	1	4	yes	—
60	C₂×C₈×C₄₃	—	2	4	no	—
61	C₂×C₈×C₄₃	—	4	2	no	—
62	C₂×C₈×C₄₃	—	4	2	no	—
63	C₄²×C₄₃	—	2	4	no	—
64	C₄²×C₄₃	—	2	2	no	—
65	C₄²×C₄₃	—	2	2	no	—
66	C₄²×C₄₃	—	2	1	no	—
67	C₄²×C₄₃	—	2	2	no	—
68	C₄²×C₄₃	—	2	1	no	—
69	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₃	#41	1	2	yes	—
71	C₄⋊Dic₄₃	—	2	1	no	—
72	C₄⋊Dic₄₃	—	2	2	no	—
73	C₄⋊Dic₄₃	C₄²	1	2	yes	—
74	C₄⋊Dic₄₃	M₁₆	1	4	yes	—
75	Dic₄₃⋊C₄	—	2	8	no	—
76	Dic₄₃⋊C₄	—	2	4	no	—
77	C₄×Dic₄₃	—	2	8	no	—
78	C₄×Dic₄₃	—	2	4	no	—
79	C₄×Dic₄₃	—	4	4	no	—
80	C₄×Dic₄₃	—	4	1	no	—
81	C₄×Dic₄₃	—	4	1	no	—
82	C₂×C₄₃⋊C₈	—	2	8	no	—
83	C₂×C₄₃⋊C₈	—	2	4	no	—
84	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
90	C₂×C₄×C₄₃	—	2	2	no	—
91	C₂×C₄×C₄₃	#104	1	4	yes	—
92	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
94	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₄₃	#137	1	2	yes	—
100	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
101	C₂×Dic₄₃	—	2	2	no	—
102	C₂×Dic₄₃	—	2	4	no	—
103	C₂×Dic₄₃	—	2	2	no	—
104	C₂×Dic₄₃	—	2	4	no	—
105	C₂×Dic₄₃	—	4	2	no	—
106	C₂×Dic₄₃	—	4	2	no	—
107	C₂×Dic₄₃	—	4	2	no	—
108	Dic₈₆	—	2	2	no	—
109	Dic₈₆	—	2	4	no	—
110	Dic₈₆	#97	1	4	yes	complemented
111	C₈×C₄₃	—	4	4	no	—
112	C₄₃⋊C₈	—	2	8	no	—
113	#8173	—	43	1	no	maximal, Sylow
114	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₄₃	#407	1	2	yes	—
116	C₂²×C₄₃	#417	1	1	yes	characteristic
117	C₂²×C₄₃	#419	1	1	yes	characteristic
118	C₂²×C₄₃	C₂×#104	1	2	yes	—
119	C₄×C₄₃	—	2	4	no	—
120	C₄×C₄₃	—	2	2	no	—
121	C₄×C₄₃	—	4	2	no	—
122	C₄×C₄₃	C₂×#97	1	1	yes	characteristic
123	C₄×C₄₃	C₄×M₁₆	1	1	yes	characteristic
124	C₄×C₄₃	#432	1	2	yes	—
125	C₄×C₄₃	#440	1	2	yes	—
126	C₄×C₄₃	#441	1	2	yes	—
127	Dic₄₃	—	2	4	no	—
128	Dic₄₃	—	4	4	no	—
129	Dic₄₃	—	4	2	no	—
130	C₄²×Q₈	—	43	1	no	—
131	#1767	D₄₃	1	1	yes	characteristic
132	C₄×#97	—	43	1	no	—
133	C₂×Q₈⋊C₈	—	43	2	no	—
134	#1279	—	43	2	no	—
135	C₂×C₄₃	#1489	1	2	yes	—
136	C₂×C₄₃	#1491	1	2	yes	—
137	C₂×C₄₃	#1749	1	1	yes	characteristic
138	C₂×C₄₃	C₄×#104	1	1	yes	characteristic
139	C₂×C₄₃	#1763	1	1	yes	characteristic
140	C₂×C₄×Q₈	—	43	1	no	—
141	C₂×C₄×Q₈	—	43	2	no	—
142	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
143	C₂×#97	—	2	1	no	—
144	C₂×#97	—	86	2	no	—
145	C₂×C₄×C₈	—	43	2	no	—
146	C₄×C₄⋊C₄	—	43	1	no	—
147	C₄×C₄⋊C₄	—	43	2	no	—
148	C₄×C₄⋊C₄	—	86	2	no	—
149	C₄×C₄⋊C₄	—	86	1	no	—
150	C₄³	—	86	1	no	—
151	C₄³	C₂×D₄₃	1	1	yes	characteristic
152	#336	—	2	1	no	—
153	Q₈⋊C₈	—	43	8	no	—
154	C₄₃	#8173	1	1	yes	characteristic, complemented, Sylow
155	C₂²×Q₈	—	43	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	86	2	no	—
158	C₄×Q₈	—	43	4	no	—
159	C₄×Q₈	—	86	4	no	—
160	C₄×Q₈	—	86	4	no	—
161	C₄×Q₈	—	86	2	no	—
162	C₂×C₄⋊C₄	—	43	1	no	—
163	C₂×C₄⋊C₄	—	43	2	no	—
164	C₂×C₄⋊C₄	—	86	2	no	—
165	C₂×C₄⋊C₄	—	86	1	no	—
166	C₂×C₄²	—	86	2	no	—
167	C₂×C₄²	—	86	1	no	—
168	C₂×C₄²	—	86	2	no	—
169	C₂×C₄²	—	86	1	no	—
170	C₂×C₄²	C₄×D₄₃	1	2	yes	—
171	C₂×C₄²	D₁₇₂	1	2	yes	—
172	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	C₄×D₄₃	1	4	yes	—
177	#97	—	4	2	no	—
178	#97	—	86	8	no	—
179	C₄×C₈	—	43	4	no	—
180	C₄×C₈	—	86	2	no	—
181	#95	—	86	2	no	—
182	#95	—	86	1	no	—
183	C₂×Q₈	—	43	2	no	—
184	C₂×Q₈	—	43	4	no	—
185	C₂×Q₈	—	86	1	no	—
186	C₂×Q₈	—	86	2	no	—
187	C₂²×C₄	—	86	1	no	—
188	C₂²×C₄	—	86	2	no	—
189	C₂²×C₄	—	86	1	no	—
190	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
191	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
192	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	86	8	no	—
199	C₂×C₈	—	86	4	no	—
200	C₄⋊C₄	—	43	2	no	—
201	C₄⋊C₄	—	43	4	no	—
202	C₄⋊C₄	—	86	8	no	—
203	C₄⋊C₄	—	86	1	no	—
204	C₄⋊C₄	—	86	4	no	—
205	C₄⋊C₄	—	86	2	no	—
206	C₄²	—	2	4	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	86	8	no	—
213	C₄²	—	86	4	no	—
214	C₄²	—	172	4	no	—
215	C₄²	—	172	1	no	—
216	C₄²	—	172	1	no	—
217	C₄²	C₂×C₄×D₄₃	1	2	yes	—
218	C₄²	#10621898	1	2	yes	—
219	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
220	Q₈	—	43	4	no	—
221	Q₈	—	86	2	no	—
222	Q₈	—	86	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	86	4	no	—
230	C₂×C₄	—	86	2	no	—
231	C₂×C₄	—	86	4	no	—
232	C₂×C₄	—	86	2	no	—
233	C₂×C₄	—	172	2	no	—
234	C₂×C₄	—	172	2	no	—
235	C₂×C₄	—	172	2	no	—
236	C₂×C₄	#13063251	1	4	yes	—
237	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
238	C₂×C₄	#13063282	1	1	yes	characteristic
239	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
240	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
241	C₂×C₄	M₁₆×D₄₃	1	2	yes	—
242	C₂×C₄	M₁₆×D₄₃	1	2	yes	—
243	C₂×C₄	#13063351	1	1	yes	characteristic
244	C₂×C₄	#13063355	1	2	yes	—
245	C₂×C₄	#13063356	1	1	yes	characteristic
246	C₈	—	4	4	no	—
247	C₈	—	86	8	no	—
248	C₂²	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
249	C₂²	#14582862	1	2	yes	—
250	C₂²	#14582898	1	1	yes	characteristic
251	C₂²	#14582907	1	1	yes	characteristic
252	C₂²	C₂×#13063251	1	2	yes	—
253	C₄	—	2	4	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	2	no	—
256	C₄	—	86	4	no	—
257	C₄	—	172	4	no	—
258	C₄	—	172	2	no	—
259	C₄	#97×D₄₃	1	1	yes	characteristic
260	C₄	#14582851	1	1	yes	characteristic
261	C₄	#14582954	1	2	yes	—
262	C₄	#14582999	1	2	yes	—
263	C₄	#14583004	1	2	yes	—
264	C₂	#19133032	1	2	yes	—
265	C₂	#19133041	1	2	yes	—
266	C₂	#19134358	1	1	yes	characteristic
267	C₂	C₄×#13063251	1	1	yes	characteristic
268	C₂	#19134428	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1767	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1767	D₄₃
hypocenter	C₄₃	#8173
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
15. #14582629	C₄	semidirect
110. Dic₈₆	#97	semidirect
154. C₄₃	#8173	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×#13063236	2
C₂×#14582629	2
#19132434	2
C₄₃×#1767	2
#8173	43