Subgroups — #25094011

Order 11008.

2158 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583382	C₂	1	2	yes	maximal
2	#19134315	C₂	1	6	yes	maximal, complemented
3	#19134312	C₂	1	3	yes	maximal, complemented
4	C₂×#14582662	C₂	1	3	yes	maximal, complemented
5	C₂²×#131×C₄₃	C₂	1	1	yes	maximal, characteristic
6	#14583382	—	2	8	no	—
7	#14583382	C₄	1	16	yes	complemented
8	C₂×Q₈×Dic₄₃	—	2	2	no	—
9	C₂×D₄×Dic₄₃	—	2	6	no	—
10	C₂×#13063284	C₂²	1	6	yes	—
11	C₂²×C₄⋊Dic₄₃	C₂²	1	6	yes	—
12	#14582662	C₂²	1	3	yes	—
13	#14582662	C₂²	1	6	yes	—
14	#14582662	C₂²	1	1	yes	characteristic
15	#14582662	C₂²	1	6	yes	—
16	C₂²×#41×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#131×C₄₃	—	2	8	no	—
18	C₂×#131×C₄₃	C₄	1	8	yes	complemented
19	C₂²×M₁₆×C₄₃	C₂²	1	3	yes	—
20	C₂³×C₈×C₄₃	C₂²	1	3	yes	—
21	C₂³×Dic₄₃	—	2	6	no	—
22	Q₈×Dic₄₃	—	2	16	no	—
23	D₄×Dic₄₃	—	2	48	no	—
24	#13063284	—	2	12	no	—
25	#13063284	C₂×C₄	1	24	yes	complemented
26	C₂×C₄⋊Dic₄₃	—	2	6	no	—
27	C₂×C₄⋊Dic₄₃	—	2	12	no	—
28	C₂×C₄⋊Dic₄₃	C₂×C₄	1	24	yes	complemented
29	C₂×C₄⋊Dic₄₃	C₂³	1	6	yes	—
30	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
31	C₂×C₂²⋊Dic₄₃	—	2	6	no	—
32	C₂×C₄×Dic₄₃	—	2	6	no	—
33	C₂×#41×C₄₃	C₂×C₄	1	8	yes	—
34	C₂×#41×C₄₃	C₂×C₄	1	4	yes	—
35	C₂×#41×C₄₃	D₄	1	8	yes	—
36	C₂×#41×C₄₃	Q₈	1	4	yes	—
37	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
38	C₂²×D₄×C₄₃	D₄	1	3	yes	—
39	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
40	#131×C₄₃	—	2	32	no	—
41	C₂×M₁₆×C₄₃	—	2	12	no	—
42	C₂×M₁₆×C₄₃	C₂×C₄	1	12	yes	complemented
43	C₂²×C₈×C₄₃	—	2	12	no	—
44	C₂²×C₈×C₄₃	C₂×C₄	1	12	yes	complemented
45	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
47	C₂⁴×C₄₃	C₂×D₄	1	3	yes	—
48	#41×C₄₃	C₄²	1	16	yes	—
49	#41×C₄₃	C₂²⋊C₄	1	16	yes	—
50	#41×C₄₃	C₄⋊C₄	1	32	yes	—
51	C₂×Q₈×C₄₃	C₂²⋊C₄	1	8	yes	—
52	C₂×Q₈×C₄₃	C₄⋊C₄	1	4	yes	—
53	C₂×D₄×C₄₃	C₂²⋊C₄	1	24	yes	—
54	C₂×D₄×C₄₃	C₄⋊C₄	1	12	yes	—
55	C₂²×C₄×C₄₃	C₂²×C₄	1	12	yes	—
56	C₂²×C₄×C₄₃	C₂²×C₄	1	6	yes	—
57	C₂²×C₄×C₄₃	C₂×D₄	1	12	yes	—
58	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
59	C₂²×C₄×C₄₃	C₂×Q₈	1	6	yes	—
60	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₄₃	—	2	24	no	—
62	C₂²×Dic₄₃	—	2	2	no	—
63	C₂²×Dic₄₃	—	2	6	no	—
64	C₂²×Dic₄₃	—	8	6	no	—
65	C₂²×Dic₄₃	—	8	6	no	—
66	M₁₆×C₄₃	—	2	24	no	—
67	C₂×C₈×C₄₃	—	2	24	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	—	2	6	no	—
70	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₄₃	C₂²×C₄	1	6	yes	—
72	C₂²⋊Dic₄₃	—	2	24	no	—
73	C₄⋊Dic₄₃	—	2	24	no	—
74	C₄⋊Dic₄₃	—	2	6	no	—
75	C₄⋊Dic₄₃	—	2	2	no	—
76	C₄⋊Dic₄₃	C₂²×C₄	1	12	yes	—
77	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
78	C₄×Dic₄₃	—	2	24	no	—
79	C₂³×C₄₃	—	2	3	no	—
80	C₂³×C₄₃	—	2	6	no	—
81	C₂³×C₄₃	—	2	3	no	—
82	C₂³×C₄₃	C₂×C₂²⋊C₄	1	12	yes	—
83	C₂³×C₄₃	C₂×C₄⋊C₄	1	6	yes	—
84	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
85	Q₈×C₄₃	#95	1	16	yes	—
86	D₄×C₄₃	#95	1	48	yes	—
87	C₂×C₄×C₄₃	C₂×C₄²	1	12	yes	—
88	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	12	yes	—
89	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	12	yes	—
90	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	24	yes	—
91	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	6	yes	—
92	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
93	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
95	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂×Dic₄₃	—	2	4	no	—
97	C₂×Dic₄₃	—	2	12	no	—
98	C₂×Dic₄₃	—	4	6	no	—
99	C₂×Dic₄₃	—	4	2	no	—
100	C₂×Dic₄₃	—	4	6	no	—
101	C₂×Dic₄₃	—	4	2	no	—
102	C₂×Dic₄₃	—	8	24	no	—
103	C₈×C₄₃	—	2	2	no	—
104	C₈×C₄₃	—	2	6	no	—
105	#20348	—	43	1	no	maximal, Sylow
106	C₂²×C₄₃	—	2	12	no	—
107	C₂²×C₄₃	—	2	12	no	—
108	C₂²×C₄₃	—	2	6	no	—
109	C₂²×C₄₃	—	2	6	no	—
110	C₂²×C₄₃	C₂×#95	1	12	yes	—
111	C₂²×C₄₃	C₂²×C₂²⋊C₄	1	2	yes	—
112	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₄₃	#576	1	1	yes	characteristic
114	C₂²×C₄₃	#577	1	2	yes	—
115	C₂²×C₄₃	#578	1	1	yes	characteristic
116	C₄×C₄₃	C₂×#95	1	12	yes	—
117	C₄×C₄₃	C₂²×C₄²	1	1	yes	characteristic
118	C₄×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₄₃	C₂²×C₄⋊C₄	1	2	yes	—
120	Dic₄₃	—	4	12	no	—
121	Dic₄₃	—	4	4	no	—
122	C₂²×#131	D₄₃	1	1	yes	characteristic
123	C₂×#520	—	43	2	no	—
124	#1736	—	43	6	no	—
125	#1735	—	43	3	no	—
126	C₂×#340	—	43	3	no	—
127	C₂×C₄₃	—	2	12	no	—
128	C₂×C₄₃	C₂²×#95	1	1	yes	characteristic
129	C₂×C₄₃	#2897	1	2	yes	—
130	C₂×C₄₃	#2898	1	2	yes	—
131	C₂×C₄₃	#2912	1	1	yes	characteristic
132	C₂×C₄₃	#2913	1	1	yes	characteristic
133	C₂²×#41	C₂×D₄₃	1	1	yes	characteristic
134	C₂×#131	—	2	8	no	—
135	C₂×#131	Dic₄₃	1	8	yes	complemented
136	C₂²×M₁₆	C₂×D₄₃	1	3	yes	—
137	C₂³×C₈	C₂×D₄₃	1	3	yes	—
138	#520	—	43	16	no	—
139	#520	—	86	8	no	—
140	C₂×C₄×Q₈	—	86	2	no	—
141	C₂×C₄×D₄	—	86	6	no	—
142	C₂×#117	—	43	6	no	—
143	C₂²×C₄⋊C₄	—	43	6	no	—
144	#340	—	43	3	no	—
145	#340	—	43	6	no	—
146	#340	—	43	1	no	—
147	#340	—	43	6	no	—
148	C₄₃	#20348	1	1	yes	characteristic, complemented, Sylow
149	C₂×#41	Dic₈₆	1	4	yes	—
150	C₂×#41	C₄×D₄₃	1	8	yes	—
151	C₂×#41	C₂×Dic₄₃	1	4	yes	—
152	C₂×#41	C₂²⋊D₄₃	1	8	yes	—
153	C₂²×Q₈	D₁₇₂	1	1	yes	characteristic
154	C₂²×D₄	D₁₇₂	1	3	yes	—
155	C₂³×C₄	—	86	6	no	—
156	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
157	#131	—	2	32	no	—
158	C₂×M₁₆	—	2	12	no	—
159	C₂×M₁₆	C₂×Dic₄₃	1	12	yes	complemented
160	C₂²×C₈	—	2	12	no	—
161	C₂²×C₈	C₂×Dic₄₃	1	12	yes	complemented
162	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
164	C₄×Q₈	—	86	16	no	—
165	C₄×D₄	—	86	48	no	—
166	#117	—	43	24	no	—
167	#117	—	86	12	no	—
168	C₂×C₄⋊C₄	—	43	24	no	—
169	C₂×C₄⋊C₄	—	43	6	no	—
170	C₂×C₄⋊C₄	—	43	2	no	—
171	C₂×C₄⋊C₄	—	86	6	no	—
172	C₂×C₄⋊C₄	—	86	12	no	—
173	C₂×C₂²⋊C₄	—	86	6	no	—
174	C₂×C₄²	—	86	6	no	—
175	C₂⁴	C₂×D₁₇₂	1	3	yes	—
176	#41	C₄×Dic₄₃	1	16	yes	—
177	#41	Dic₄₃⋊C₄	1	32	yes	—
178	#41	C₂²⋊Dic₄₃	1	16	yes	—
179	C₂×Q₈	C₄⋊Dic₄₃	1	4	yes	—
180	C₂×Q₈	(C₂×D₄₃)⋊C₄	1	8	yes	—
181	C₂×D₄	C₄⋊Dic₄₃	1	12	yes	—
182	C₂×D₄	(C₂×D₄₃)⋊C₄	1	24	yes	—
183	C₂²×C₄	—	86	24	no	—
184	C₂²×C₄	—	86	2	no	—
185	C₂²×C₄	—	86	6	no	—
186	C₂²×C₄	—	344	6	no	—
187	C₂²×C₄	—	344	6	no	—
188	C₂²×C₄	C₂×Dic₈₆	1	6	yes	—
189	C₂²×C₄	C₂×C₄×D₄₃	1	12	yes	—
190	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
191	C₂²×C₄	C₂²×Dic₄₃	1	6	yes	—
192	C₂²×C₄	C₂×C₂²⋊D₄₃	1	12	yes	—
193	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
194	M₁₆	—	2	24	no	—
195	C₂×C₈	—	2	24	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	6	no	—
198	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
199	C₂×C₈	C₂²×Dic₄₃	1	6	yes	—
200	C₄⋊C₄	—	43	12	no	—
201	C₄⋊C₄	—	43	4	no	—
202	C₄⋊C₄	—	86	24	no	—
203	C₄⋊C₄	—	86	6	no	—
204	C₄⋊C₄	—	86	2	no	—
205	C₂²⋊C₄	—	86	24	no	—
206	C₄²	—	86	24	no	—
207	C₂³	—	2	3	no	—
208	C₂³	—	2	6	no	—
209	C₂³	—	2	3	no	—
210	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	12	yes	—
211	C₂³	C₂×C₄⋊Dic₄₃	1	6	yes	—
212	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
213	Q₈	C₄₃⋊#95	1	16	yes	—
214	D₄	C₄₃⋊#95	1	48	yes	—
215	C₂×C₄	—	86	4	no	—
216	C₂×C₄	—	86	12	no	—
217	C₂×C₄	—	344	24	no	—
218	C₂×C₄	—	172	6	no	—
219	C₂×C₄	—	172	6	no	—
220	C₂×C₄	—	172	2	no	—
221	C₂×C₄	—	172	2	no	—
222	C₂×C₄	C₂×C₄×Dic₄₃	1	12	yes	—
223	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	12	yes	—
224	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	12	yes	—
225	C₂×C₄	C₂×Dic₄₃⋊C₄	1	24	yes	—
226	C₂×C₄	C₂×C₄⋊Dic₄₃	1	6	yes	—
227	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
228	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
229	C₂×C₄	C₂²×C₂²⋊D₄₃	1	2	yes	—
230	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	2	6	no	—
233	C₂²	—	2	6	no	—
234	C₂²	—	2	6	no	—
235	C₂²	—	2	12	no	—
236	C₂²	—	2	12	no	—
237	C₂²	C₂×C₄₃⋊#95	1	12	yes	—
238	C₂²	C₂²×(C₂×D₄₃)⋊C₄	1	2	yes	—
239	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
240	C₂²	#14583704	1	1	yes	characteristic
241	C₂²	#14583711	1	2	yes	—
242	C₂²	#14583716	1	1	yes	characteristic
243	C₄	—	172	12	no	—
244	C₄	—	172	4	no	—
245	C₄	C₂×C₄₃⋊#95	1	12	yes	—
246	C₄	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
247	C₄	C₂²×C₂²⋊Dic₄₃	1	1	yes	characteristic
248	C₄	C₂²×Dic₄₃⋊C₄	1	2	yes	—
249	C₂	—	2	12	no	—
250	C₂	C₂²×C₄₃⋊#95	1	1	yes	characteristic
251	C₂	#19143122	1	2	yes	—
252	C₂	#19143129	1	2	yes	—
253	C₂	#19143190	1	1	yes	characteristic
254	C₂	#19143195	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₄×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂²×#131×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×#41×C₄₃	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂²×#131	D₄₃
hypocenter	C₄₃	#20348
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
2. #19134315	C₂	semidirect
3. #19134312	C₂	semidirect
4. C₂×#14582662	C₂	semidirect
7. #14583382	C₄	semidirect
18. C₂×#131×C₄₃	C₄	semidirect
25. #13063284	C₂×C₄	semidirect
28. C₂×C₄⋊Dic₄₃	C₂×C₄	semidirect
42. C₂×M₁₆×C₄₃	C₂×C₄	semidirect
44. C₂²×C₈×C₄₃	C₂×C₄	semidirect
135. C₂×#131	Dic₄₃	semidirect
148. C₄₃	#20348	semidirect
159. C₂×M₁₆	C₂×Dic₄₃	semidirect
161. C₂²×C₈	C₂×Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583382	2
#19134315	2
#19134312	2
C₂×#14582662	2
C₂²×#131×C₄₃	2
#20348	43