Subgroups — #25101406

Order 11008.

854 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	C₂×#14582952	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10621877	C₂	1	1	yes	maximal, characteristic
3	#19135314	C₂	1	4	yes	maximal, complemented
4	#19134780	C₂	1	4	yes	maximal
5	#19134452	C₂	1	4	yes	maximal, complemented
6	C₄₃×#2283	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10621877	C₂²	1	2	yes	—
8	#14582952	—	2	4	no	—
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	8	yes	—
11	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
12	C₄×#10621877	—	2	4	no	—
13	C₂×#13063236	C₂²	1	2	yes	—
14	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
16	#14582654	C₂²	1	8	yes	—
17	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#383	C₄	1	2	yes	—
19	C₄₃×#382	C₄	1	2	yes	—
20	C₄₃×#381	C₄	1	2	yes	—
21	C₄₃×#380	C₄	1	2	yes	—
22	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
23	C₄×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
24	C₄₃×#376	C₂²	1	2	yes	—
25	C₂×#10621877	—	2	4	no	—
26	C₂×#10621877	—	2	8	no	—
27	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
28	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
29	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
30	C₄⋊(C₄₃⋊C₈)	—	2	16	no	—
31	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
32	#13063236	—	2	4	no	—
33	C₄×C₄₃⋊C₈	—	2	4	no	—
34	C₄×C₄₃⋊C₈	—	2	4	no	—
35	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
39	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₃	D₄	1	4	yes	—
43	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
44	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
45	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
47	#95×C₄₃	C₂×C₄	1	4	yes	—
48	#95×C₄₃	C₂×C₄	1	4	yes	—
49	#95×C₄₃	C₂×C₄	1	4	yes	—
50	#95×C₄₃	C₂×C₄	1	4	yes	—
51	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	—	2	4	no	—
53	C₂²×C₄×C₄₃	—	2	4	no	—
54	C₂²×C₄×C₄₃	—	4	2	no	—
55	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₄₃	—	2	8	no	—
67	C₂²⋊C₄×C₄₃	—	4	4	no	—
68	C₄²×C₄₃	—	2	8	no	—
69	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
70	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
71	C₄²×C₄₃	C₂×D₄	1	2	yes	—
72	C₄²×C₄₃	C₂×D₄	1	2	yes	—
73	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
74	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
75	#10621877	—	2	16	no	—
76	C₂×C₄₃⋊C₈	—	2	4	no	—
77	C₂×C₄₃⋊C₈	—	2	4	no	—
78	C₂×C₄₃⋊C₈	—	2	4	no	—
79	C₂×C₄₃⋊C₈	—	2	8	no	—
80	C₂×C₄₃⋊C₈	#41	1	8	yes	—
81	C₂³×C₄₃	—	2	2	no	—
82	C₂³×C₄₃	—	2	2	no	—
83	C₂³×C₄₃	—	2	2	no	—
84	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
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	4	no	—
91	C₂×C₄×C₄₃	—	2	4	no	—
92	C₂×C₄×C₄₃	—	2	8	no	—
93	C₂×C₄×C₄₃	—	2	8	no	—
94	C₂×C₄×C₄₃	—	2	8	no	—
95	C₂×C₄×C₄₃	—	2	8	no	—
96	C₂×C₄×C₄₃	—	4	4	no	—
97	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₄₃	#131	1	4	yes	—
103	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
106	C₄₃⋊C₈	—	2	8	no	—
107	C₄₃⋊C₈	—	4	4	no	—
108	#21354	—	43	1	no	maximal, Sylow
109	C₂²×C₄₃	—	2	2	no	—
110	C₂²×C₄₃	—	2	2	no	—
111	C₂²×C₄₃	—	2	2	no	—
112	C₂²×C₄₃	—	4	4	no	—
113	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₄₃	C₄×#41	1	2	yes	—
115	C₂²×C₄₃	C₂×#131	1	2	yes	—
116	C₂²×C₄₃	#568	1	2	yes	—
117	C₄×C₄₃	—	2	2	no	—
118	C₄×C₄₃	—	2	2	no	—
119	C₄×C₄₃	—	2	2	no	—
120	C₄×C₄₃	—	2	2	no	—
121	C₄×C₄₃	—	4	8	no	—
122	C₄×C₄₃	#424	1	4	yes	—
123	C₄×C₄₃	#525	1	2	yes	—
124	C₄×C₄₃	#533	1	2	yes	—
125	C₂×#431	—	43	1	no	—
126	C₂×C₄×M₁₆	—	43	1	no	—
127	#2283	D₄₃	1	1	yes	characteristic
128	#1939	—	43	4	no	—
129	#1851	—	43	4	no	—
130	#1768	—	43	4	no	—
131	C₂×C₄₃	—	4	2	no	—
132	C₂×C₄₃	#2296	1	1	yes	characteristic
133	C₂×C₄₃	C₂×#424	1	1	yes	characteristic
134	C₂×C₄₃	#2905	1	1	yes	characteristic
135	C₂×C₄₃	#2969	1	4	yes	—
136	C₂²×M₁₆	—	43	2	no	—
137	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
138	#431	—	86	4	no	—
139	C₂×C₄⋊C₈	—	43	8	no	—
140	C₂×C₄⋊C₈	—	43	2	no	—
141	C₂×C₂²⋊C₈	—	43	2	no	—
142	C₄×M₁₆	—	86	4	no	—
143	C₂×#97	—	43	2	no	—
144	C₂×C₄×C₈	—	43	2	no	—
145	C₂×C₄×C₈	—	43	2	no	—
146	#383	Dic₄₃	1	2	yes	—
147	#382	Dic₄₃	1	2	yes	—
148	#381	Dic₄₃	1	2	yes	—
149	#380	Dic₄₃	1	2	yes	—
150	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
151	C₄×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
152	#376	C₂×D₄₃	1	2	yes	—
153	#336	—	43	8	no	—
154	C₄₃	#21354	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
156	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
157	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
158	C₂×M₁₆	—	86	8	no	—
159	C₂×M₁₆	—	86	4	no	—
160	C₂²×C₈	—	43	4	no	—
161	C₂²×C₈	—	43	4	no	—
162	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
163	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	4	yes	—
164	C₂×C₄²	Dic₈₆	1	4	yes	—
165	C₂×C₄²	D₁₇₂	1	4	yes	—
166	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
167	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
168	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
169	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
171	C₄⋊C₈	—	86	16	no	—
172	C₄⋊C₈	—	86	4	no	—
173	C₂²⋊C₈	—	86	4	no	—
174	#97	—	86	4	no	—
175	C₄×C₈	—	86	4	no	—
176	C₄×C₈	—	86	4	no	—
177	#95	C₂×Dic₄₃	1	4	yes	—
178	#95	C₂×Dic₄₃	1	4	yes	—
179	#95	C₂×Dic₄₃	1	4	yes	—
180	#95	C₂×Dic₄₃	1	4	yes	—
181	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
186	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
187	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
188	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
189	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
190	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
195	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
196	M₁₆	—	86	16	no	—
197	C₂×C₈	—	43	8	no	—
198	C₂×C₈	—	86	4	no	—
199	C₂×C₈	—	86	4	no	—
200	C₂×C₈	—	86	4	no	—
201	C₂×C₈	—	86	8	no	—
202	C₄⋊C₄	—	2	8	no	—
203	C₂²⋊C₄	—	4	4	no	—
204	C₄²	—	2	8	no	—
205	C₄²	C₄⋊Dic₄₃	1	4	yes	—
206	C₄²	C₄⋊Dic₄₃	1	4	yes	—
207	C₄²	C₂×Dic₈₆	1	2	yes	—
208	C₄²	C₂×Dic₈₆	1	2	yes	—
209	C₄²	C₂×D₁₇₂	1	2	yes	—
210	C₄²	C₂×D₁₇₂	1	2	yes	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
215	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
230	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
232	C₂×C₄	#13063332	1	4	yes	—
233	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
234	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
235	C₂×C₄	#41×D₄₃	1	4	yes	—
236	C₈	—	86	8	no	—
237	C₈	—	172	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	4	no	—
242	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
243	C₂²	#41×Dic₄₃	1	2	yes	—
244	C₂²	C₂×#13063332	1	2	yes	—
245	C₂²	#14583666	1	2	yes	—
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	8	no	—
251	C₄	#14582926	1	4	yes	—
252	C₄	#14583411	1	2	yes	—
253	C₄	#14583452	1	2	yes	—
254	C₂	—	4	2	no	—
255	C₂	#19137079	1	1	yes	characteristic
256	C₂	C₂×#14582926	1	1	yes	characteristic
257	C₂	#19143159	1	1	yes	characteristic
258	C₂	#19143600	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂³×C₄
Fitting	C₄₃×#2283	C₂
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2283	D₄₃
hypocenter	C₄₃	#21354
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2283

Complements

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

N	H ≅ G/N	type
3. #19135314	C₂	semidirect
5. #19134452	C₂	semidirect
154. C₄₃	#21354	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582952	2
C₂×C₄×#10621877	2
#19135314	2
#19134780	2
#19134452	2
C₄₃×#2283	2
#21354	43