Subgroups — #24983166

Order 11008.

352 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134747	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
5	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₄₃⋊C₈)	C₄	1	2	yes	—
7	C₂×C₄⋊(C₄₃⋊C₈)	C₄	1	2	yes	—
8	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
9	#14582654	—	2	1	no	—
10	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
11	C₂×C₄×C₈×C₄₃	—	2	1	no	—
12	C₄³×C₄₃	C₂²	1	1	yes	characteristic
13	C₄₃×#336	C₄	1	2	yes	—
14	C₂²×C₄₃⋊C₈	—	2	2	no	—
15	C₂²×C₄₃⋊C₈	—	2	2	no	—
16	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
17	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
18	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
19	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
20	C₄⋊(C₄₃⋊C₈)	—	4	1	no	—
21	C₄⋊(C₄₃⋊C₈)	—	4	2	no	—
22	C₄⋊(C₄₃⋊C₈)	C₈	1	4	yes	complemented
23	C₄×C₄₃⋊C₈	—	4	1	no	—
24	C₄×C₄₃⋊C₈	—	4	1	no	—
25	C₂²×C₈×C₄₃	—	2	2	no	—
26	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₃	Q₈	1	1	yes	characteristic
33	C₄⋊C₈×C₄₃	—	2	4	no	—
34	C₄×C₈×C₄₃	—	4	1	no	—
35	C₄×C₈×C₄₃	—	4	1	no	—
36	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₃	—	2	4	no	—
44	C₂×C₈×C₄₃	—	4	2	no	—
45	C₂×C₈×C₄₃	—	4	2	no	—
46	C₄²×C₄₃	—	2	1	no	—
47	C₄²×C₄₃	—	2	1	no	—
48	C₄²×C₄₃	—	2	2	no	—
49	C₄²×C₄₃	—	2	2	no	—
50	C₄²×C₄₃	—	4	1	no	—
51	C₄²×C₄₃	C₂×C₈	1	2	yes	—
52	C₄²×C₄₃	M₁₆	1	2	yes	—
53	C₄²×C₄₃	D₈	1	1	yes	characteristic
54	C₄²×C₄₃	D₈	1	1	yes	characteristic
55	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
56	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
57	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
58	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
59	C₄²×C₄₃	Q₁₆	1	1	yes	characteristic
60	C₄²×C₄₃	Q₁₆	1	1	yes	characteristic
61	C₂×C₄₃⋊C₈	—	2	4	no	—
62	C₂×C₄₃⋊C₈	—	4	2	no	—
63	C₂×C₄₃⋊C₈	—	4	2	no	—
64	C₂×C₄₃⋊C₈	—	4	2	no	—
65	C₂×C₄₃⋊C₈	—	4	2	no	—
66	C₂×C₄₃⋊C₈	—	4	2	no	—
67	C₂³×C₄₃	#95	1	1	yes	characteristic
68	C₂×C₄×C₄₃	—	2	1	no	—
69	C₂×C₄×C₄₃	—	2	1	no	—
70	C₂×C₄×C₄₃	—	2	1	no	—
71	C₂×C₄×C₄₃	—	2	1	no	—
72	C₂×C₄×C₄₃	—	2	1	no	—
73	C₂×C₄×C₄₃	—	2	1	no	—
74	C₂×C₄×C₄₃	—	2	1	no	—
75	C₂×C₄×C₄₃	—	2	1	no	—
76	C₂×C₄×C₄₃	—	2	1	no	—
77	C₂×C₄×C₄₃	C₄×C₈	1	1	yes	characteristic
78	C₂×C₄×C₄₃	#97	1	1	yes	characteristic
79	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
80	C₂×C₄×C₄₃	#100	1	1	yes	characteristic
81	C₂×C₄×C₄₃	#101	1	1	yes	characteristic
82	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
84	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
87	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
88	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
89	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
90	C₂×C₄×C₄₃	#104	1	2	yes	—
91	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
93	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
94	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
95	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
96	C₂×C₄×C₄₃	#108	1	2	yes	—
97	C₈×C₄₃	—	4	4	no	—
98	C₄₃⋊C₈	—	4	4	no	—
99	C₄₃⋊C₈	—	8	2	no	—
100	#7031	—	43	1	no	maximal, Sylow
101	C₂²×C₄₃	#336	1	1	yes	characteristic
102	C₂²×C₄₃	#339	1	1	yes	characteristic
103	C₂²×C₄₃	#340	1	1	yes	characteristic
104	C₂²×C₄₃	#340	1	1	yes	characteristic
105	C₂²×C₄₃	#341	1	1	yes	characteristic
106	C₂²×C₄₃	#343	1	1	yes	characteristic
107	C₂²×C₄₃	#344	1	1	yes	characteristic
108	C₄×C₄₃	—	2	1	no	—
109	C₄×C₄₃	—	2	1	no	—
110	C₄×C₄₃	—	2	1	no	—
111	C₄×C₄₃	—	2	1	no	—
112	C₄×C₄₃	—	2	1	no	—
113	C₄×C₄₃	—	2	1	no	—
114	C₄×C₄₃	—	4	1	no	—
115	C₄×C₄₃	D₄⋊C₈	1	2	yes	—
116	C₄×C₄₃	Q₈⋊C₈	1	2	yes	—
117	C₄×C₄₃	#331	1	1	yes	characteristic
118	C₄×C₄₃	#332	1	1	yes	characteristic
119	C₄×C₄₃	#332	1	1	yes	characteristic
120	C₄×C₄₃	#333	1	1	yes	characteristic
121	C₄×C₄₃	#334	1	2	yes	—
122	C₄×C₄₃	#335	1	2	yes	—
123	#1842	—	43	1	no	—
124	C₄×C₄⋊C₈	—	43	1	no	—
125	C₄×C₄⋊C₈	D₄₃	1	1	yes	characteristic
126	C₂×C₄₃	#1277	1	1	yes	characteristic
127	C₂×C₄₃	#1279	1	1	yes	characteristic
128	C₂×C₄₃	#1283	1	1	yes	characteristic
129	C₂×C₄₃	#1296	1	1	yes	characteristic
130	C₂×C₄₃	#1299	1	1	yes	characteristic
131	C₂×C₄₃	#1301	1	1	yes	characteristic
132	C₂×C₄₃	#1303	1	1	yes	characteristic
133	C₂×C₄⋊C₈	—	2	1	no	—
134	C₂×C₄⋊C₈	—	43	2	no	—
135	C₂×C₄⋊C₈	—	43	2	no	—
136	C₂×C₄⋊C₈	—	86	1	no	—
137	C₂×C₄⋊C₈	—	86	1	no	—
138	C₂×C₄×C₈	—	2	1	no	—
139	C₂×C₄×C₈	—	86	1	no	—
140	C₄³	C₂×D₄₃	1	1	yes	characteristic
141	#336	—	86	1	no	—
142	#336	Dic₄₃	1	2	yes	—
143	C₄₃	#7031	1	1	yes	characteristic, complemented, Sylow
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	86	2	no	—
146	C₂²×C₈	—	86	2	no	—
147	C₂×C₄²	Dic₈₆	1	1	yes	characteristic
148	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
149	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
150	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
152	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
153	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
154	C₄⋊C₈	—	2	4	no	—
155	C₄⋊C₈	—	43	4	no	—
156	C₄⋊C₈	—	86	2	no	—
157	C₄⋊C₈	—	172	2	no	—
158	C₄⋊C₈	—	172	1	no	—
159	C₄⋊C₈	—	172	1	no	—
160	C₄⋊C₈	—	172	1	no	—
161	C₄⋊C₈	—	172	1	no	—
162	C₄×C₈	—	4	1	no	—
163	C₄×C₈	—	4	1	no	—
164	C₄×C₈	—	172	1	no	—
165	C₄×C₈	—	172	1	no	—
166	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
167	C₂²×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
169	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
170	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
171	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
172	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	86	4	no	—
177	C₂×C₈	—	172	2	no	—
178	C₂×C₈	—	172	2	no	—
179	C₂×C₈	—	172	2	no	—
180	C₂×C₈	—	172	2	no	—
181	C₂×C₈	—	172	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	4	1	no	—
187	C₄²	C₈×D₄₃	1	2	yes	—
188	C₄²	#10621872	1	2	yes	—
189	C₄²	#10621873	1	1	yes	characteristic
190	C₄²	#10621873	1	1	yes	characteristic
191	C₄²	D₃₄₄	1	1	yes	characteristic
192	C₄²	Dic₁₇₂	1	1	yes	characteristic
193	C₄²	#10621882	1	1	yes	characteristic
194	C₄²	#10621883	1	1	yes	characteristic
195	C₄²	#10621884	1	1	yes	characteristic
196	C₄²	#10621885	1	1	yes	characteristic
197	C₂³	C₄₃⋊#95	1	1	yes	characteristic
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	C₈×Dic₄₃	1	1	yes	characteristic
208	C₂×C₄	#13063235	1	1	yes	characteristic
209	C₂×C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
210	C₂×C₄	#13063241	1	1	yes	characteristic
211	C₂×C₄	#13063243	1	1	yes	characteristic
212	C₂×C₄	#13063245	1	1	yes	characteristic
213	C₂×C₄	#13063246	1	1	yes	characteristic
214	C₂×C₄	#13063247	1	1	yes	characteristic
215	C₂×C₄	#13063247	1	1	yes	characteristic
216	C₂×C₄	#13063248	1	1	yes	characteristic
217	C₂×C₄	#13063249	1	1	yes	characteristic
218	C₂×C₄	#13063250	1	1	yes	characteristic
219	C₂×C₄	#13063250	1	1	yes	characteristic
220	C₂×C₄	#13063253	1	2	yes	—
221	C₂×C₄	Dic₄₃⋊C₈	1	2	yes	—
222	C₂×C₄	#13063256	1	1	yes	characteristic
223	C₂×C₄	#13063257	1	1	yes	characteristic
224	C₂×C₄	#13063258	1	1	yes	characteristic
225	C₂×C₄	#13063259	1	1	yes	characteristic
226	C₂×C₄	#13063260	1	2	yes	—
227	C₈	—	4	4	no	—
228	C₈	—	344	2	no	—
229	C₈	—	172	4	no	—
230	C₂²	#14582653	1	1	yes	characteristic
231	C₂²	#14582659	1	1	yes	characteristic
232	C₂²	#14582661	1	1	yes	characteristic
233	C₂²	#14582662	1	1	yes	characteristic
234	C₂²	#14582663	1	1	yes	characteristic
235	C₂²	#14582667	1	1	yes	characteristic
236	C₂²	#14582669	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	1	no	—
244	C₄	#14582628	1	2	yes	—
245	C₄	#14582631	1	2	yes	—
246	C₄	#14582642	1	1	yes	characteristic
247	C₄	#14582644	1	1	yes	characteristic
248	C₄	#14582646	1	1	yes	characteristic
249	C₄	#14582647	1	1	yes	characteristic
250	C₄	#14582649	1	2	yes	—
251	C₄	#14582651	1	2	yes	—
252	C₂	#19132429	1	1	yes	characteristic
253	C₂	#19132435	1	1	yes	characteristic
254	C₂	#19132444	1	1	yes	characteristic
255	C₂	#19132479	1	1	yes	characteristic
256	C₂	#19132485	1	1	yes	characteristic
257	C₂	#19132491	1	1	yes	characteristic
258	C₂	#19132498	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₃)⋊C₄
commutator	C₂×C₄×C₄₃	C₄×C₈
Frattini	C₄³	C₂×D₄₃
Fitting	C₄×C₄⋊C₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	#95
cosocle	C₄³×C₄₃	C₂²
hypercenter	C₄×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#7031
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
22. C₄⋊(C₄₃⋊C₈)	C₈	semidirect
143. C₄₃	#7031	semidirect

Maximal subgroups

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

subgroup	index
#19134747	2
C₄×C₄⋊(C₄₃⋊C₈)	2
C₄×C₄⋊C₈×C₄₃	2
#7031	43