Subgroups — #25256708

Order 11008.

3006 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	C₂×#14583382	C₂	1	2	yes	maximal
2	#19137516	C₂	1	4	yes	maximal, complemented
3	#19137505	C₂	1	12	yes	maximal, complemented
4	#19137218	C₂	1	6	yes	maximal, complemented
5	#19137002	C₂	1	3	yes	maximal, complemented
6	C₂×#14582765	C₂	1	3	yes	maximal, complemented
7	C₂×C₄×#41×C₄₃	C₂	1	1	yes	maximal, characteristic
8	#14583382	—	2	16	no	—
9	C₂×Q₈×Dic₄₃	C₂²	1	4	yes	—
10	C₂×D₄×Dic₄₃	C₂²	1	12	yes	—
11	C₂×#13063284	C₂²	1	6	yes	—
12	C₂²×C₄⋊Dic₄₃	C₂²	1	6	yes	—
13	#14582798	C₂²	1	24	yes	—
14	#14582774	C₂²	1	24	yes	—
15	#14582769	C₂²	1	6	yes	—
16	#14582765	C₂²	1	6	yes	—
17	#14582765	C₂²	1	3	yes	—
18	#14582765	C₂²	1	12	yes	—
19	#14582765	C₂²	1	1	yes	characteristic
20	#14582759	C₂²	1	12	yes	—
21	C₄×C₄⋊Dic₄₃	C₂²	1	12	yes	—
22	C₂×C₄₃⋊#95	C₂²	1	12	yes	—
23	C₂²×#41×C₄₃	C₂²	1	1	yes	characteristic
24	C₄×#41×C₄₃	C₄	1	16	yes	complemented
25	C₂×C₄×Q₈×C₄₃	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₄₃	C₂²	1	6	yes	—
27	C₂×#117×C₄₃	C₂²	1	3	yes	—
28	C₂²×C₄²×C₄₃	C₂²	1	3	yes	—
29	C₂³×Dic₄₃	C₂³	1	12	yes	—
30	Q₈×Dic₄₃	—	2	16	no	—
31	D₄×Dic₄₃	—	2	48	no	—
32	#13063284	—	2	24	no	—
33	C₂×C₄⋊Dic₄₃	—	2	24	no	—
34	C₂×C₄⋊Dic₄₃	C₂³	1	12	yes	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	6	yes	—
36	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
37	C₂×C₄⋊Dic₄₃	C₂³	1	24	yes	—
38	C₂×C₂²⋊Dic₄₃	C₂³	1	12	yes	—
39	C₂×C₄×Dic₄₃	C₂³	1	12	yes	—
40	C₄₃⋊#95	C₂³	1	24	yes	—
41	C₄₃⋊#95	C₂³	1	12	yes	—
42	C₄₃⋊#95	C₂³	1	4	yes	—
43	C₂×#41×C₄₃	C₂×C₄	1	8	yes	—
44	C₂×#41×C₄₃	D₄	1	8	yes	—
45	C₂×#41×C₄₃	Q₈	1	8	yes	—
46	C₂²×Q₈×C₄₃	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₄₃	C₂³	1	3	yes	—
48	C₂³×C₄×C₄₃	C₂³	1	6	yes	—
49	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
50	C₄×Q₈×C₄₃	C₂×C₄	1	16	yes	complemented
51	C₄×D₄×C₄₃	C₂×C₄	1	48	yes	complemented
52	#117×C₄₃	C₂×C₄	1	24	yes	complemented
53	C₂×C₄⋊C₄×C₄₃	C₂³	1	6	yes	—
54	C₂×C₂²⋊C₄×C₄₃	C₂³	1	6	yes	—
55	C₂×C₄²×C₄₃	C₂×C₄	1	24	yes	complemented
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
58	C₂×C₄²×C₄₃	C₂³	1	6	yes	—
59	C₂⁴×C₄₃	C₂⁴	1	3	yes	—
60	#41×C₄₃	C₄⋊C₄	1	64	yes	complemented
61	C₂×Q₈×C₄₃	C₂²×C₄	1	4	yes	—
62	C₂×Q₈×C₄₃	C₂×D₄	1	4	yes	—
63	C₂×Q₈×C₄₃	C₂×Q₈	1	4	yes	—
64	C₂×D₄×C₄₃	C₂²×C₄	1	12	yes	—
65	C₂×D₄×C₄₃	C₂×D₄	1	12	yes	—
66	C₂×D₄×C₄₃	C₂×Q₈	1	12	yes	—
67	C₂²×C₄×C₄₃	—	4	12	no	—
68	C₂²×C₄×C₄₃	C₂²×C₄	1	24	yes	—
69	C₂²×C₄×C₄₃	C₂²×C₄	1	12	yes	—
70	C₂²×C₄×C₄₃	C₂×D₄	1	12	yes	—
71	C₂²×C₄×C₄₃	C₂×Q₈	1	12	yes	—
72	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
73	C₂²×C₄×C₄₃	C₂⁴	1	6	yes	—
74	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
76	C₂²×Dic₄₃	—	2	24	no	—
77	C₂²×Dic₄₃	—	4	24	no	—
78	C₂²×Dic₄₃	C₂⁴	1	12	yes	—
79	C₂²×Dic₄₃	C₂⁴	1	4	yes	—
80	C₄⋊C₄×C₄₃	C₂²×C₄	1	24	yes	—
81	C₂²⋊C₄×C₄₃	C₂²×C₄	1	24	yes	—
82	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
83	C₄²×C₄₃	C₂²×C₄	1	12	yes	—
84	C₄²×C₄₃	C₂²×C₄	1	24	yes	—
85	C₂²⋊Dic₄₃	—	2	24	no	—
86	C₄⋊Dic₄₃	—	2	12	no	—
87	C₄⋊Dic₄₃	—	2	24	no	—
88	C₄⋊Dic₄₃	—	2	4	no	—
89	C₄⋊Dic₄₃	—	4	24	no	—
90	C₄×Dic₄₃	—	2	24	no	—
91	C₂³×C₄₃	—	2	12	no	—
92	C₂³×C₄₃	C₂³×C₄	1	6	yes	—
93	C₂³×C₄₃	C₂²×D₄	1	6	yes	—
94	C₂³×C₄₃	C₂²×Q₈	1	6	yes	—
95	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
96	Q₈×C₄₃	C₂×C₄⋊C₄	1	16	yes	—
97	D₄×C₄₃	C₂×C₄⋊C₄	1	48	yes	—
98	C₂×C₄×C₄₃	—	2	12	no	—
99	C₂×C₄×C₄₃	—	2	4	no	—
100	C₂×C₄×C₄₃	—	4	24	no	—
101	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	48	yes	—
102	C₂×C₄×C₄₃	C₂³×C₄	1	4	yes	—
103	C₂×C₄×C₄₃	C₂³×C₄	1	12	yes	—
104	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
105	C₂×C₄×C₄₃	C₂³×C₄	1	6	yes	—
106	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
107	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
109	C₂×C₄×C₄₃	C₂²×Q₈	1	6	yes	—
110	C₂×Dic₄₃	—	2	4	no	—
111	C₂×Dic₄₃	—	2	12	no	—
112	C₂×Dic₄₃	—	2	8	no	—
113	C₂×Dic₄₃	—	2	24	no	—
114	C₂×Dic₄₃	—	8	24	no	—
115	#34184	—	43	1	no	maximal, Sylow
116	C₂²×C₄₃	—	2	12	no	—
117	C₂²×C₄₃	—	2	12	no	—
118	C₂²×C₄₃	—	2	12	no	—
119	C₂²×C₄₃	C₂²×C₄⋊C₄	1	12	yes	—
120	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
121	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
122	C₂²×C₄₃	C₂³×Q₈	1	1	yes	characteristic
123	C₂²×C₄₃	#585	1	4	yes	—
124	C₄×C₄₃	—	2	4	no	—
125	C₄×C₄₃	—	2	12	no	—
126	C₄×C₄₃	C₂²×C₄⋊C₄	1	4	yes	—
127	C₄×C₄₃	C₂²×C₄⋊C₄	1	12	yes	—
128	Dic₄₃	—	4	4	no	—
129	Dic₄₃	—	4	12	no	—
130	C₂×#520	—	43	2	no	—
131	C₂×C₄×#41	D₄₃	1	1	yes	characteristic
132	#2351	—	43	4	no	—
133	#2349	—	43	12	no	—
134	#2317	—	43	6	no	—
135	#2285	—	43	3	no	—
136	C₂×#382	—	43	3	no	—
137	C₂×C₄₃	—	2	12	no	—
138	C₂×C₄₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄₃	#3428	1	2	yes	—
140	C₂×C₄₃	#3500	1	2	yes	—
141	C₂×C₄₃	#3504	1	2	yes	—
142	C₂²×#41	C₂×D₄₃	1	1	yes	characteristic
143	#520	—	86	16	no	—
144	C₄×#41	Dic₄₃	1	16	yes	complemented
145	C₂×C₄×Q₈	—	43	4	no	—
146	C₂×C₄×Q₈	C₂×D₄₃	1	2	yes	—
147	C₂×C₄×D₄	—	43	12	no	—
148	C₂×C₄×D₄	C₂×D₄₃	1	6	yes	—
149	C₂×#117	—	43	6	no	—
150	C₂×#117	C₂×D₄₃	1	3	yes	—
151	C₂²×C₄⋊C₄	—	43	6	no	—
152	C₂²×C₄²	C₂×D₄₃	1	3	yes	—
153	#389	—	43	24	no	—
154	#385	—	43	24	no	—
155	#384	—	43	6	no	—
156	#382	—	43	6	no	—
157	#382	—	43	12	no	—
158	#382	—	43	3	no	—
159	#382	—	43	1	no	—
160	#380	—	43	12	no	—
161	C₄×C₄⋊C₄	—	43	12	no	—
162	C₂×#95	—	43	12	no	—
163	C₄₃	#34184	1	1	yes	characteristic, complemented, Sylow
164	C₂×#41	Dic₈₆	1	8	yes	—
165	C₂×#41	D₁₇₂	1	8	yes	—
166	C₂×#41	C₂×Dic₄₃	1	8	yes	—
167	C₂²×Q₈	C₂²×D₄₃	1	1	yes	characteristic
168	C₂²×D₄	C₂²×D₄₃	1	3	yes	—
169	C₂³×C₄	—	43	12	no	—
170	C₂³×C₄	C₂²×D₄₃	1	6	yes	—
171	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
172	C₄×Q₈	—	86	16	no	—
173	C₄×Q₈	C₂×Dic₄₃	1	16	yes	complemented
174	C₄×D₄	—	86	48	no	—
175	C₄×D₄	C₂×Dic₄₃	1	48	yes	complemented
176	#117	—	86	24	no	—
177	#117	C₂×Dic₄₃	1	24	yes	complemented
178	C₂×C₄⋊C₄	—	43	6	no	—
179	C₂×C₄⋊C₄	—	43	12	no	—
180	C₂×C₄⋊C₄	—	43	2	no	—
181	C₂×C₄⋊C₄	—	43	24	no	—
182	C₂×C₄⋊C₄	—	86	24	no	—
183	C₂×C₄⋊C₄	C₂²×D₄₃	1	6	yes	—
184	C₂×C₂²⋊C₄	—	43	12	no	—
185	C₂×C₂²⋊C₄	C₂²×D₄₃	1	6	yes	—
186	C₂×C₄²	—	43	12	no	—
187	C₂×C₄²	C₂×Dic₄₃	1	24	yes	complemented
188	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
189	C₂×C₄²	C₂²×D₄₃	1	6	yes	—
190	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
191	#95	—	43	12	no	—
192	#95	—	43	24	no	—
193	#95	—	43	4	no	—
194	C₂⁴	C₂³×D₄₃	1	3	yes	—
195	#41	C₄⋊Dic₄₃	1	64	yes	complemented
196	C₂×Q₈	C₂×Dic₈₆	1	4	yes	—
197	C₂×Q₈	C₂×D₁₇₂	1	4	yes	—
198	C₂×Q₈	C₂²×Dic₄₃	1	4	yes	—
199	C₂×D₄	C₂×Dic₈₆	1	12	yes	—
200	C₂×D₄	C₂×D₁₇₂	1	12	yes	—
201	C₂×D₄	C₂²×Dic₄₃	1	12	yes	—
202	C₂²×C₄	—	4	12	no	—
203	C₂²×C₄	—	43	12	no	—
204	C₂²×C₄	—	43	4	no	—
205	C₂²×C₄	—	86	24	no	—
206	C₂²×C₄	—	172	24	no	—
207	C₂²×C₄	C₂×Dic₈₆	1	12	yes	—
208	C₂²×C₄	C₂×D₁₇₂	1	12	yes	—
209	C₂²×C₄	C₂²×Dic₄₃	1	24	yes	—
210	C₂²×C₄	C₂²×Dic₄₃	1	12	yes	—
211	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
212	C₂²×C₄	C₂³×D₄₃	1	6	yes	—
213	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
214	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
215	C₄⋊C₄	—	86	12	no	—
216	C₄⋊C₄	—	86	24	no	—
217	C₄⋊C₄	—	86	4	no	—
218	C₄⋊C₄	—	172	24	no	—
219	C₄⋊C₄	C₂²×Dic₄₃	1	24	yes	—
220	C₂²⋊C₄	—	86	24	no	—
221	C₂²⋊C₄	C₂²×Dic₄₃	1	24	yes	—
222	C₄²	—	86	24	no	—
223	C₄²	C₂²×Dic₄₃	1	4	yes	—
224	C₄²	C₂²×Dic₄₃	1	12	yes	—
225	C₄²	C₂²×Dic₄₃	1	24	yes	—
226	C₂³	—	2	12	no	—
227	C₂³	C₂³×Dic₄₃	1	6	yes	—
228	C₂³	C₂²×D₁₇₂	1	6	yes	—
229	C₂³	C₂²×Dic₈₆	1	6	yes	—
230	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
231	Q₈	C₂×C₄⋊Dic₄₃	1	16	yes	—
232	D₄	C₂×C₄⋊Dic₄₃	1	48	yes	—
233	C₂×C₄	—	2	12	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	24	no	—
236	C₂×C₄	—	86	4	no	—
237	C₂×C₄	—	86	12	no	—
238	C₂×C₄	—	86	8	no	—
239	C₂×C₄	—	86	24	no	—
240	C₂×C₄	—	344	24	no	—
241	C₂×C₄	C₂×C₄⋊Dic₄₃	1	48	yes	—
242	C₂×C₄	C₂³×Dic₄₃	1	4	yes	—
243	C₂×C₄	C₂³×Dic₄₃	1	12	yes	—
244	C₂×C₄	C₂³×Dic₄₃	1	2	yes	—
245	C₂×C₄	C₂³×Dic₄₃	1	6	yes	—
246	C₂×C₄	C₂²×D₁₇₂	1	6	yes	—
247	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
248	C₂×C₄	C₂²×Dic₈₆	1	2	yes	—
249	C₂×C₄	C₂²×Dic₈₆	1	6	yes	—
250	C₂²	—	2	12	no	—
251	C₂²	—	2	12	no	—
252	C₂²	—	2	12	no	—
253	C₂²	C₂²×C₄⋊Dic₄₃	1	12	yes	—
254	C₂²	C₂⁴×Dic₄₃	1	1	yes	characteristic
255	C₂²	C₂³×D₁₇₂	1	1	yes	characteristic
256	C₂²	C₂³×Dic₈₆	1	1	yes	characteristic
257	C₂²	#14583735	1	4	yes	—
258	C₄	—	2	4	no	—
259	C₄	—	2	12	no	—
260	C₄	—	172	12	no	—
261	C₄	—	172	4	no	—
262	C₄	C₂²×C₄⋊Dic₄₃	1	4	yes	—
263	C₄	C₂²×C₄⋊Dic₄₃	1	12	yes	—
264	C₂	—	2	12	no	—
265	C₂	C₂³×C₄⋊Dic₄₃	1	1	yes	characteristic
266	C₂	#19147798	1	2	yes	—
267	C₂	#19148388	1	2	yes	—
268	C₂	#19148421	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₃
commutator	C₂²×C₄₃	C₂⁴×C₄
Fitting	C₂×C₄×#41×C₄₃	C₂
socle, cosocle	C₂³×C₄₃	C₂⁵
prime radical	C₂²×#41×C₄₃	C₂²
hypercenter	C₂×C₄×#41	D₄₃
hypocenter	C₄₃	#34184
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
2. #19137516	C₂	semidirect
3. #19137505	C₂	semidirect
4. #19137218	C₂	semidirect
5. #19137002	C₂	semidirect
6. C₂×#14582765	C₂	semidirect
24. C₄×#41×C₄₃	C₄	semidirect
50. C₄×Q₈×C₄₃	C₂×C₄	semidirect
51. C₄×D₄×C₄₃	C₂×C₄	semidirect
52. #117×C₄₃	C₂×C₄	semidirect
55. C₂×C₄²×C₄₃	C₂×C₄	semidirect
60. #41×C₄₃	C₄⋊C₄	semidirect
144. C₄×#41	Dic₄₃	semidirect
163. C₄₃	#34184	semidirect
173. C₄×Q₈	C₂×Dic₄₃	semidirect
175. C₄×D₄	C₂×Dic₄₃	semidirect
177. #117	C₂×Dic₄₃	semidirect
187. C₂×C₄²	C₂×Dic₄₃	semidirect
195. #41	C₄⋊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
#19137516	2
#19137505	2
#19137218	2
#19137002	2
C₂×#14582765	2
C₂×C₄×#41×C₄₃	2
#34184	43