Subgroups — #25002425

Order 11008.

814 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582923	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582663	C₂	1	4	yes	maximal
4	C₂×C₄₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10621877	C₂²	1	2	yes	—
6	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
7	#14582923	—	2	2	no	—
8	#14582923	C₄	1	4	yes	—
9	C₂×C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
10	C₂×C₄×C₄₃⋊C₈	—	2	2	no	—
11	C₂×C₄×C₄₃⋊C₈	—	2	2	no	—
12	C₂×C₄×C₄₃⋊C₈	C₄	1	2	yes	—
13	C₂×C₄×C₄₃⋊C₈	C₄	1	2	yes	—
14	#14582663	—	2	8	no	—
15	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#423	—	2	2	no	—
18	C₄₃×#423	C₄	1	4	yes	—
19	C₂×C₄⋊C₈×C₄₃	—	2	2	no	—
20	C₂×#10621877	—	2	2	no	—
21	C₂×#10621877	—	2	4	no	—
22	C₂×#10621877	—	2	4	no	—
23	C₂×#10621877	C₂×C₄	1	4	yes	—
24	C₂²×C₄₃⋊C₈	—	2	2	no	—
25	C₂²×C₄₃⋊C₈	—	2	2	no	—
26	C₂²×C₄₃⋊C₈	—	2	2	no	—
27	C₂²×C₄₃⋊C₈	—	2	2	no	—
28	C₂²×C₄₃⋊C₈	—	2	2	no	—
29	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₄₃⋊C₈	D₄	1	4	yes	—
31	C₂²×C₄₃⋊C₈	Q₈	1	4	yes	—
32	C₄⋊(C₄₃⋊C₈)	—	2	8	no	—
33	C₄×C₄₃⋊C₈	—	2	4	no	—
34	C₄×C₄₃⋊C₈	—	2	4	no	—
35	C₂³×C₄×C₄₃	Q₈	1	2	yes	—
36	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₄₃	—	2	2	no	—
38	C₂×M₁₆×C₄₃	—	2	4	no	—
39	C₂×M₁₆×C₄₃	—	2	4	no	—
40	C₂×M₁₆×C₄₃	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₄₃	—	2	2	no	—
42	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₃	D₄	1	2	yes	—
47	C₂×C₄²×C₄₃	D₄	1	2	yes	—
48	C₂×C₄²×C₄₃	D₄	1	2	yes	—
49	C₄⋊C₈×C₄₃	—	2	8	no	—
50	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	—	2	8	no	—
52	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₄₃	#41	1	2	yes	—
65	C₂²×C₄×C₄₃	#41	1	2	yes	—
66	M₁₆×C₄₃	—	2	8	no	—
67	M₁₆×C₄₃	—	4	4	no	—
68	C₂×C₈×C₄₃	—	2	4	no	—
69	C₂×C₈×C₄₃	—	4	2	no	—
70	C₂×C₈×C₄₃	—	4	2	no	—
71	C₄²×C₄₃	C₄²	1	2	yes	—
72	C₄²×C₄₃	C₄²	1	2	yes	—
73	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₄₃	C₂²⋊C₄	1	4	yes	—
76	C₄²×C₄₃	C₂²⋊C₄	1	4	yes	—
77	#10621877	—	2	8	no	—
78	#10621877	—	4	4	no	—
79	C₂×C₄₃⋊C₈	—	2	4	no	—
80	C₂×C₄₃⋊C₈	—	2	4	no	—
81	C₂×C₄₃⋊C₈	—	2	4	no	—
82	C₂×C₄₃⋊C₈	—	2	4	no	—
83	C₂×C₄₃⋊C₈	—	2	4	no	—
84	C₂×C₄₃⋊C₈	—	2	4	no	—
85	C₂×C₄₃⋊C₈	—	2	4	no	—
86	C₂×C₄₃⋊C₈	—	4	2	no	—
87	C₂×C₄₃⋊C₈	—	4	2	no	—
88	C₂×C₄₃⋊C₈	C₄⋊C₄	1	8	yes	—
89	C₂³×C₄₃	—	2	4	no	—
90	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₄₃	#125	1	2	yes	—
94	C₂³×C₄₃	C₄⋊Q₈	1	2	yes	—
95	C₂×C₄×C₄₃	—	2	4	no	—
96	C₂×C₄×C₄₃	—	2	8	no	—
97	C₂×C₄×C₄₃	—	2	8	no	—
98	C₂×C₄×C₄₃	#95	1	4	yes	—
99	C₂×C₄×C₄₃	#95	1	4	yes	—
100	C₂×C₄×C₄₃	#108	1	8	yes	—
101	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₄₃	#117	1	2	yes	—
107	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₄₃	#121	1	2	yes	—
112	C₂×C₄×C₄₃	#121	1	2	yes	—
113	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
119	C₈×C₄₃	—	4	4	no	—
120	C₄₃⋊C₈	—	2	8	no	—
121	C₄₃⋊C₈	—	4	4	no	—
122	#10844	—	43	1	no	maximal, Sylow
123	C₂²×C₄₃	—	2	4	no	—
124	C₂²×C₄₃	—	2	4	no	—
125	C₂²×C₄₃	—	2	2	no	—
126	C₂²×C₄₃	—	2	2	no	—
127	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
128	C₂²×C₄₃	#382	1	1	yes	characteristic
129	C₂²×C₄₃	#384	1	1	yes	characteristic
130	C₂²×C₄₃	#389	1	2	yes	—
131	C₂²×C₄₃	#389	1	2	yes	—
132	C₂²×C₄₃	C₂×#108	1	4	yes	—
133	C₄×C₄₃	—	2	4	no	—
134	C₄×C₄₃	#341	1	8	yes	complemented
135	C₄×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
136	C₄×C₄₃	#380	1	2	yes	—
137	C₄×C₄₃	#390	1	1	yes	characteristic
138	C₄×C₄₃	#390	1	1	yes	characteristic
139	C₄×C₄₃	#391	1	1	yes	characteristic
140	C₄×C₄₃	#391	1	1	yes	characteristic
141	C₂×#423	—	43	1	no	—
142	C₂×#423	D₄₃	1	1	yes	characteristic
143	C₂²×C₄×C₈	—	43	1	no	—
144	C₂×#341	—	43	4	no	—
145	C₂×C₄₃	—	2	2	no	—
146	C₂×C₄₃	—	2	2	no	—
147	C₂×C₄₃	C₄⋊#95	1	1	yes	characteristic
148	C₂×C₄₃	C₂×#341	1	2	yes	—
149	C₂×C₄₃	#1837	1	2	yes	—
150	C₂×C₄₃	#1848	1	2	yes	—
151	C₂²×M₁₆	—	43	2	no	—
152	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
153	C₂³×C₈	—	43	2	no	—
154	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
155	#423	—	2	2	no	—
156	#423	—	43	4	no	—
157	#423	—	86	2	no	—
158	#423	Dic₄₃	1	4	yes	—
159	C₂×C₄⋊C₈	—	2	2	no	—
160	C₂×C₄⋊C₈	—	86	2	no	—
161	C₂×C₄×C₈	—	43	2	no	—
162	C₂×C₄×C₈	—	43	2	no	—
163	C₂×C₄×C₈	—	86	2	no	—
164	C₂×C₄×C₈	—	86	2	no	—
165	#341	—	86	8	no	—
166	C₄₃	#10844	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	Dic₈₆	1	2	yes	—
168	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	—	43	4	no	—
173	C₂×M₁₆	—	86	2	no	—
174	C₂×M₁₆	—	86	4	no	—
175	C₂×M₁₆	—	86	4	no	—
176	C₂×M₁₆	C₂×Dic₄₃	1	4	yes	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	43	4	no	—
179	C₂²×C₈	—	43	4	no	—
180	C₂²×C₈	—	43	4	no	—
181	C₂²×C₈	—	86	2	no	—
182	C₂²×C₈	—	86	2	no	—
183	C₂²×C₈	—	86	2	no	—
184	C₂²×C₈	—	86	2	no	—
185	C₂²×C₈	—	86	2	no	—
186	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
187	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₄₃	1	2	yes	—
189	C₂×C₄²	D₁₇₂	1	2	yes	—
190	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
192	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
193	C₄⋊C₈	—	2	8	no	—
194	C₄⋊C₈	—	86	8	no	—
195	C₄×C₈	—	86	4	no	—
196	C₄×C₈	—	86	4	no	—
197	C₂⁴	C₂×Dic₈₆	1	1	yes	characteristic
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
200	C₂²×C₄	Dic₄₃⋊C₄	1	4	yes	—
201	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
202	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
203	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₄₃	1	2	yes	—
207	C₂²×C₄	#10621900	1	2	yes	—
208	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
209	C₂²×C₄	#10621902	1	2	yes	—
210	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
213	M₁₆	—	2	8	no	—
214	M₁₆	—	4	4	no	—
215	M₁₆	—	86	8	no	—
216	M₁₆	—	172	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	43	8	no	—
221	C₂×C₈	—	86	4	no	—
222	C₂×C₈	—	86	4	no	—
223	C₂×C₈	—	86	4	no	—
224	C₂×C₈	—	86	4	no	—
225	C₂×C₈	—	86	4	no	—
226	C₂×C₈	—	86	4	no	—
227	C₂×C₈	—	86	4	no	—
228	C₂×C₈	—	172	2	no	—
229	C₂×C₈	—	172	2	no	—
230	C₄²	C₄×Dic₄₃	1	2	yes	—
231	C₄²	C₄×Dic₄₃	1	2	yes	—
232	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
233	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
234	C₄²	(C₂×D₄₃)⋊C₄	1	4	yes	—
235	C₄²	C₂²⋊Dic₄₃	1	4	yes	—
236	C₂³	—	2	4	no	—
237	C₂³	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
238	C₂³	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
239	C₂³	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
240	C₂³	#13063316	1	2	yes	—
241	C₂³	#13063324	1	2	yes	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	C₄₃⋊#95	1	4	yes	—
246	C₂×C₄	C₄₃⋊#95	1	4	yes	—
247	C₂×C₄	#13063260	1	8	yes	—
248	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
249	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
250	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
252	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
253	C₂×C₄	#13063283	1	2	yes	—
254	C₂×C₄	#13063287	1	2	yes	—
255	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
256	C₂×C₄	#13063290	1	2	yes	—
257	C₂×C₄	Q₈×Dic₄₃	1	2	yes	—
258	C₂×C₄	#13063296	1	2	yes	—
259	C₂×C₄	#13063300	1	2	yes	—
260	C₂×C₄	#13063301	1	2	yes	—
261	C₂×C₄	#13063305	1	2	yes	—
262	C₂×C₄	#13063312	1	1	yes	characteristic
263	C₂×C₄	#13063314	1	1	yes	characteristic
264	C₂×C₄	#13063321	1	1	yes	characteristic
265	C₂×C₄	#13063323	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₈	—	86	8	no	—
268	C₈	—	172	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
274	C₂²	#14582764	1	1	yes	characteristic
275	C₂²	#14582768	1	1	yes	characteristic
276	C₂²	#14582795	1	2	yes	—
277	C₂²	#14582798	1	2	yes	—
278	C₂²	C₂×#13063260	1	4	yes	—
279	C₄	—	2	4	no	—
280	C₄	#14582663	1	8	yes	complemented
281	C₄	C₄⋊C₄×Dic₄₃	1	2	yes	—
282	C₄	#14582756	1	2	yes	—
283	C₄	#14582800	1	1	yes	characteristic
284	C₄	#14582802	1	1	yes	characteristic
285	C₄	#14582804	1	1	yes	characteristic
286	C₄	#14582806	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	2	no	—
289	C₂	#19132824	1	1	yes	characteristic
290	C₂	C₂×#14582663	1	2	yes	—
291	C₂	#19134732	1	2	yes	—
292	C₂	#19134766	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×#423	C₂
socle	C₂³×C₄₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₃	C₂×Q₈
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×#423	D₄₃
hypocenter	C₄₃	#10844
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
134. C₄×C₄₃	#341	semidirect
166. C₄₃	#10844	semidirect
280. C₄	#14582663	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582923	2
C₂²×C₄×C₄₃⋊C₈	2
C₂×#14582663	2
C₂×C₄₃×#423	2
#10844	43