Subgroups — #25093971

Order 11008.

2036 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583666	C₂	1	1	yes	maximal, characteristic
2	#19147798	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582655	C₂	1	12	yes	maximal, complemented
4	C₄₃×#3428	C₂	1	1	yes	maximal, characteristic
5	#14583666	—	2	4	no	—
6	#14583666	—	2	4	no	—
7	C₂×#13063332	C₄	1	8	yes	complemented
8	C₂²×#10621877	C₂²	1	6	yes	—
9	#14583382	—	2	4	no	—
10	#14583377	—	2	1	no	—
11	#14583374	—	2	3	no	—
12	#41×Dic₄₃	C₄	1	8	yes	complemented
13	C₂×#13063284	C₂²	1	6	yes	—
14	#14582655	C₂²	1	16	yes	—
15	C₂²×#41×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#520	—	2	4	no	—
17	C₄₃×#519	—	2	1	no	—
18	C₄₃×#518	—	2	3	no	—
19	C₄×#41×C₄₃	C₄	1	8	yes	complemented
20	C₂×#117×C₄₃	C₂²	1	6	yes	—
21	#13063332	—	2	32	no	—
22	C₂×#10621877	—	2	12	no	—
23	C₂×#10621877	—	2	12	no	—
24	C₂×#10621877	C₂×C₄	1	12	yes	complemented
25	C₂×#10621877	C₂³	1	4	yes	—
26	C₂²×C₄₃⋊C₈	C₂×C₄	1	12	yes	complemented
27	Q₈×Dic₄₃	—	2	8	no	—
28	D₄×Dic₄₃	—	2	24	no	—
29	#13063284	—	2	6	no	—
30	#13063284	—	2	6	no	—
31	#13063284	C₂×C₄	1	12	yes	complemented
32	#13063284	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₄₃	—	2	6	no	—
34	C₂×C₂²⋊Dic₄₃	—	2	6	no	—
35	C₂×C₄×Dic₄₃	C₂×C₄	1	12	yes	complemented
36	C₂×#41×C₄₃	C₂×C₄	1	4	yes	—
37	C₂×#41×C₄₃	C₂×C₄	1	4	yes	—
38	C₂×#41×C₄₃	C₂×C₄	1	4	yes	—
39	C₂×#41×C₄₃	D₄	1	4	yes	—
40	C₂×#41×C₄₃	D₄	1	4	yes	—
41	C₂×#41×C₄₃	Q₈	1	4	yes	—
42	C₂²×Q₈×C₄₃	D₄	1	1	yes	characteristic
43	C₂²×D₄×C₄₃	D₄	1	3	yes	—
44	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
45	C₄×Q₈×C₄₃	—	2	8	no	—
46	C₄×D₄×C₄₃	—	2	24	no	—
47	#117×C₄₃	—	2	6	no	—
48	#117×C₄₃	—	2	6	no	—
49	#117×C₄₃	C₂×C₄	1	12	yes	complemented
50	#117×C₄₃	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₄₃	—	2	6	no	—
52	C₂×C₂²⋊C₄×C₄₃	—	2	6	no	—
53	C₂×C₄²×C₄₃	C₂×C₄	1	12	yes	complemented
54	C₂⁴×C₄₃	C₂×D₄	1	3	yes	—
55	#41×C₄₃	C₄²	1	16	yes	—
56	#41×C₄₃	C₂²⋊C₄	1	16	yes	—
57	#41×C₄₃	C₄⋊C₄	1	16	yes	—
58	#41×C₄₃	C₄⋊C₄	1	16	yes	—
59	C₂×Q₈×C₄₃	C₂²⋊C₄	1	4	yes	—
60	C₂×Q₈×C₄₃	C₂²⋊C₄	1	4	yes	—
61	C₂×Q₈×C₄₃	C₄⋊C₄	1	4	yes	—
62	C₂×D₄×C₄₃	C₂²⋊C₄	1	12	yes	—
63	C₂×D₄×C₄₃	C₂²⋊C₄	1	12	yes	—
64	C₂×D₄×C₄₃	C₄⋊C₄	1	12	yes	—
65	C₂²×C₄×C₄₃	—	2	12	no	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	6	yes	—
67	C₂²×C₄×C₄₃	C₂²×C₄	1	6	yes	—
68	C₂²×C₄×C₄₃	C₂²×C₄	1	6	yes	—
69	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
70	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
71	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
72	C₂²×C₄×C₄₃	C₂×Q₈	1	6	yes	—
73	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₃	—	2	12	no	—
75	C₄⋊C₄×C₄₃	—	2	12	no	—
76	C₄⋊C₄×C₄₃	—	2	4	no	—
77	C₂²⋊C₄×C₄₃	—	2	12	no	—
78	C₂²⋊C₄×C₄₃	—	2	4	no	—
79	C₄²×C₄₃	—	2	12	no	—
80	C₄²×C₄₃	C₂²×C₄	1	8	yes	—
81	C₂²⋊Dic₄₃	—	2	12	no	—
82	C₂²⋊Dic₄₃	—	2	4	no	—
83	C₄⋊Dic₄₃	—	2	12	no	—
84	C₄⋊Dic₄₃	—	2	4	no	—
85	C₄×Dic₄₃	—	2	12	no	—
86	C₄×Dic₄₃	C₂²×C₄	1	8	yes	—
87	#10621877	—	2	24	no	—
88	#10621877	—	2	4	no	—
89	#10621877	—	2	4	no	—
90	C₂×C₄₃⋊C₈	—	2	24	no	—
91	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
92	C₂³×C₄₃	—	8	3	no	—
93	C₂³×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
94	C₂³×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
95	C₂³×C₄₃	C₂×C₄⋊C₄	1	6	yes	—
96	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₄₃	#95	1	16	yes	—
98	D₄×C₄₃	#95	1	48	yes	—
99	C₂×C₄×C₄₃	—	2	8	no	—
100	C₂×C₄×C₄₃	—	8	12	no	—
101	C₂×C₄×C₄₃	C₂×C₄²	1	12	yes	—
102	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	12	yes	—
103	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
104	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
105	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	12	yes	—
106	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	6	yes	—
107	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	12	yes	—
108	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
114	C₂×Dic₄₃	—	2	8	no	—
115	C₂×Dic₄₃	—	8	12	no	—
116	C₄₃⋊C₈	—	2	8	no	—
117	#20338	—	43	1	no	maximal, Sylow
118	C₂²×C₄₃	—	4	1	no	—
119	C₂²×C₄₃	—	4	6	no	—
120	C₂²×C₄₃	—	4	6	no	—
121	C₂²×C₄₃	—	4	6	no	—
122	C₂²×C₄₃	C₂×#95	1	12	yes	—
123	C₂²×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₄₃	—	4	8	no	—
127	C₄×C₄₃	C₂×#95	1	12	yes	—
128	C₄×C₄₃	C₂²×C₄²	1	1	yes	characteristic
129	C₄×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	Dic₄₃	—	4	8	no	—
133	C₂×#568	—	43	1	no	—
134	#3428	—	43	1	no	—
135	#3428	D₄₃	1	1	yes	characteristic
136	C₂×H(3,ℤ/4)	—	43	12	no	—
137	C₂×C₄₃	—	2	1	no	—
138	C₂×C₄₃	—	2	1	no	—
139	C₂×C₄₃	—	2	1	no	—
140	C₂×C₄₃	—	2	12	no	—
141	C₂×C₄₃	C₂²×#95	1	1	yes	characteristic
142	C₂²×#41	C₂×D₄₃	1	1	yes	characteristic
143	#568	—	86	4	no	—
144	#568	—	86	4	no	—
145	C₂×#131	—	43	8	no	—
146	C₂²×M₁₆	—	43	6	no	—
147	#520	—	2	4	no	—
148	#520	—	86	4	no	—
149	#519	—	2	1	no	—
150	#519	—	86	1	no	—
151	#518	—	2	3	no	—
152	#518	—	86	3	no	—
153	C₄×#41	—	43	8	no	—
154	C₄×#41	Dic₄₃	1	8	yes	complemented
155	C₂×#117	—	43	6	no	—
156	C₂×#117	C₂×D₄₃	1	6	yes	—
157	H(3,ℤ/4)	—	43	16	no	—
158	C₄₃	#20338	1	1	yes	characteristic, complemented, Sylow
159	C₂×#41	Dic₈₆	1	4	yes	—
160	C₂×#41	C₄×D₄₃	1	4	yes	—
161	C₂×#41	C₄×D₄₃	1	4	yes	—
162	C₂×#41	D₁₇₂	1	4	yes	—
163	C₂×#41	C₂×Dic₄₃	1	4	yes	—
164	C₂×#41	C₂²⋊D₄₃	1	4	yes	—
165	C₂²×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
166	C₂²×D₄	C₂²⋊D₄₃	1	3	yes	—
167	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
168	#131	—	86	32	no	—
169	C₂×M₁₆	—	43	12	no	—
170	C₂×M₁₆	—	43	4	no	—
171	C₂×M₁₆	—	86	12	no	—
172	C₂×M₁₆	—	86	12	no	—
173	C₂²×C₈	—	43	12	no	—
174	C₄×Q₈	—	2	8	no	—
175	C₄×Q₈	—	86	8	no	—
176	C₄×D₄	—	2	24	no	—
177	C₄×D₄	—	86	24	no	—
178	#117	—	2	6	no	—
179	#117	—	2	6	no	—
180	#117	—	43	12	no	—
181	#117	—	43	4	no	—
182	#117	—	86	6	no	—
183	#117	—	86	6	no	—
184	#117	C₂×Dic₄₃	1	12	yes	complemented
185	#117	C₂²×D₄₃	1	4	yes	—
186	C₂×C₄⋊C₄	—	2	6	no	—
187	C₂×C₄⋊C₄	—	86	6	no	—
188	C₂×C₂²⋊C₄	—	2	6	no	—
189	C₂×C₂²⋊C₄	—	86	6	no	—
190	C₂×C₄²	—	43	12	no	—
191	C₂×C₄²	C₂×Dic₄₃	1	12	yes	complemented
192	C₂⁴	C₂×C₂²⋊D₄₃	1	3	yes	—
193	#41	C₄×Dic₄₃	1	16	yes	—
194	#41	Dic₄₃⋊C₄	1	16	yes	—
195	#41	C₄⋊Dic₄₃	1	16	yes	—
196	#41	(C₂×D₄₃)⋊C₄	1	16	yes	—
197	C₂×Q₈	Dic₄₃⋊C₄	1	4	yes	—
198	C₂×Q₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
199	C₂×Q₈	C₂²⋊Dic₄₃	1	4	yes	—
200	C₂×D₄	Dic₄₃⋊C₄	1	12	yes	—
201	C₂×D₄	(C₂×D₄₃)⋊C₄	1	12	yes	—
202	C₂×D₄	C₂²⋊Dic₄₃	1	12	yes	—
203	C₂²×C₄	—	2	12	no	—
204	C₂²×C₄	—	86	12	no	—
205	C₂²×C₄	C₂×Dic₈₆	1	6	yes	—
206	C₂²×C₄	C₂×C₄×D₄₃	1	6	yes	—
207	C₂²×C₄	C₂×C₄×D₄₃	1	6	yes	—
208	C₂²×C₄	C₂×D₁₇₂	1	6	yes	—
209	C₂²×C₄	C₂²×Dic₄₃	1	6	yes	—
210	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
211	C₂²×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
212	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
213	M₁₆	—	86	24	no	—
214	M₁₆	—	86	4	no	—
215	M₁₆	—	86	4	no	—
216	C₂×C₈	—	43	8	no	—
217	C₂×C₈	—	86	24	no	—
218	C₄⋊C₄	—	2	12	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	86	12	no	—
221	C₄⋊C₄	—	86	4	no	—
222	C₂²⋊C₄	—	2	12	no	—
223	C₂²⋊C₄	—	2	4	no	—
224	C₂²⋊C₄	—	86	12	no	—
225	C₂²⋊C₄	—	86	4	no	—
226	C₄²	—	2	12	no	—
227	C₄²	—	43	8	no	—
228	C₄²	—	86	12	no	—
229	C₄²	C₂²×Dic₄₃	1	8	yes	—
230	C₂³	—	8	3	no	—
231	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	6	yes	—
232	C₂³	C₂×C₂²⋊Dic₄₃	1	6	yes	—
233	C₂³	C₂×Dic₄₃⋊C₄	1	6	yes	—
234	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
235	Q₈	C₄₃⋊#95	1	16	yes	—
236	D₄	C₄₃⋊#95	1	48	yes	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	8	12	no	—
239	C₂×C₄	—	86	8	no	—
240	C₂×C₄	—	344	12	no	—
241	C₂×C₄	C₂×C₄×Dic₄₃	1	12	yes	—
242	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	12	yes	—
243	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	6	yes	—
244	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	6	yes	—
245	C₂×C₄	C₂×Dic₄₃⋊C₄	1	12	yes	—
246	C₂×C₄	C₂×Dic₄₃⋊C₄	1	6	yes	—
247	C₂×C₄	C₂×C₄⋊Dic₄₃	1	12	yes	—
248	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
249	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
250	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
253	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
254	C₈	—	86	8	no	—
255	C₂²	—	4	6	no	—
256	C₂²	—	4	6	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	6	no	—
259	C₂²	C₂×C₄₃⋊#95	1	12	yes	—
260	C₂²	C₂²×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
261	C₂²	C₂²×C₂²⋊Dic₄₃	1	1	yes	characteristic
262	C₂²	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
263	C₄	—	4	8	no	—
264	C₄	—	172	8	no	—
265	C₄	C₂×C₄₃⋊#95	1	12	yes	—
266	C₄	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
267	C₄	C₂²×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
268	C₄	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
269	C₄	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	—	2	12	no	—
274	C₂	C₂²×C₄₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₄₃
commutator	C₄×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#3428	C₂
socle	C₂×C₄₃	C₂²×#95
prime radical	C₂²×#41×C₄₃	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#3428	D₄₃
hypocenter	C₄₃	#20338
LCS	G ▷ C₄×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ #3428

Complements

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

N	H ≅ G/N	type
3. C₂×#14582655	C₂	semidirect
7. C₂×#13063332	C₄	semidirect
12. #41×Dic₄₃	C₄	semidirect
19. C₄×#41×C₄₃	C₄	semidirect
24. C₂×#10621877	C₂×C₄	semidirect
26. C₂²×C₄₃⋊C₈	C₂×C₄	semidirect
31. #13063284	C₂×C₄	semidirect
35. C₂×C₄×Dic₄₃	C₂×C₄	semidirect
49. #117×C₄₃	C₂×C₄	semidirect
53. C₂×C₄²×C₄₃	C₂×C₄	semidirect
154. C₄×#41	Dic₄₃	semidirect
158. C₄₃	#20338	semidirect
184. #117	C₂×Dic₄₃	semidirect
191. C₂×C₄²	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×#14583666	2
#19147798	2
C₂×#14582655	2
C₄₃×#3428	2
#20338	43