Subgroups — #25002418

Order 11008.

950 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582765	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582661	C₂	1	4	yes	maximal
4	C₂×C₄₃×#382	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₄₃	C₂²	1	2	yes	—
7	C₂×C₄×C₄₃⋊C₈	—	2	2	no	—
8	C₂×C₄×C₄₃⋊C₈	—	2	2	no	—
9	C₂×C₄×C₄₃⋊C₈	C₄	1	4	yes	complemented
10	#14582765	—	2	2	no	—
11	#14582765	C₄	1	4	yes	complemented
12	C₂×C₄₃⋊#95	—	2	2	no	—
13	#14582661	—	2	8	no	—
14	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#382	—	2	2	no	—
17	C₄₃×#382	C₄	1	4	yes	complemented
18	C₂×#95×C₄₃	—	2	2	no	—
19	C₂³×Dic₄₃	—	2	2	no	—
20	C₂²×C₄₃⋊C₈	—	2	2	no	—
21	C₂²×C₄₃⋊C₈	—	2	2	no	—
22	C₂²×C₄₃⋊C₈	—	2	2	no	—
23	C₂²×C₄₃⋊C₈	—	2	2	no	—
24	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₄₃⋊C₈	D₄	1	4	yes	—
26	C₂²×C₄₃⋊C₈	Q₈	1	4	yes	—
27	C₂×C₄⋊Dic₄₃	—	2	4	no	—
28	C₂×C₄⋊Dic₄₃	—	2	4	no	—
29	C₂×C₄⋊Dic₄₃	—	2	2	no	—
30	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
31	C₄×C₄₃⋊C₈	—	2	8	no	—
32	C₄₃⋊#95	—	2	8	no	—
33	C₂³×C₄×C₄₃	—	2	2	no	—
34	C₂³×C₄×C₄₃	D₄	1	2	yes	—
35	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
37	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
38	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₃	D₄	1	2	yes	—
44	C₂×C₄²×C₄₃	D₄	1	2	yes	—
45	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
46	#95×C₄₃	—	2	8	no	—
47	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	—	2	4	no	—
49	C₂²×C₄×C₄₃	—	2	8	no	—
50	C₂²×C₄×C₄₃	—	4	2	no	—
51	C₂²×C₄×C₄₃	—	4	2	no	—
52	C₂²×C₄×C₄₃	—	4	8	no	—
53	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
54	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	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	2	yes	—
62	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
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	C₂²×Dic₄₃	—	2	4	no	—
67	C₂²×Dic₄₃	—	4	2	no	—
68	C₂²×Dic₄₃	—	4	2	no	—
69	C₂²×Dic₄₃	—	4	8	no	—
70	C₄⋊C₄×C₄₃	—	2	8	no	—
71	C₄⋊C₄×C₄₃	—	4	4	no	—
72	C₄²×C₄₃	C₄²	1	4	yes	—
73	C₄²×C₄₃	C₂²⋊C₄	1	4	yes	—
74	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
75	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
76	C₄⋊Dic₄₃	—	2	8	no	—
77	C₄⋊Dic₄₃	—	4	4	no	—
78	C₂×C₄₃⋊C₈	—	2	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	C₄⋊C₄	1	8	yes	—
85	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂³×C₄₃	#124	1	1	yes	characteristic
89	C₂³×C₄₃	#124	1	1	yes	characteristic
90	C₂³×C₄₃	#127	1	1	yes	characteristic
91	C₂³×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
92	C₂³×C₄₃	#135	1	8	yes	—
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	—	4	8	no	—
98	C₂×C₄×C₄₃	—	8	4	no	—
99	C₂×C₄×C₄₃	—	8	4	no	—
100	C₂×C₄×C₄₃	#95	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×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	#125	1	2	yes	—
116	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
117	C₂×Dic₄₃	—	4	4	no	—
118	C₂×Dic₄₃	—	4	8	no	—
119	C₂×Dic₄₃	—	8	4	no	—
120	C₂×Dic₄₃	—	8	4	no	—
121	C₄₃⋊C₈	—	2	8	no	—
122	#10842	—	43	1	no	maximal, Sylow
123	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
124	C₂²×C₄₃	#382	1	1	yes	characteristic
125	C₂²×C₄₃	#384	1	1	yes	characteristic
126	C₂²×C₄₃	#390	1	1	yes	characteristic
127	C₂²×C₄₃	#390	1	1	yes	characteristic
128	C₂²×C₄₃	#391	1	1	yes	characteristic
129	C₂²×C₄₃	#391	1	1	yes	characteristic
130	C₂²×C₄₃	#416	1	4	yes	—
131	C₂²×C₄₃	#416	1	4	yes	—
132	C₂²×C₄₃	#427	1	4	yes	—
133	C₂²×C₄₃	#488	1	4	yes	—
134	C₂²×C₄₃	#488	1	4	yes	—
135	C₂²×C₄₃	#495	1	4	yes	—
136	C₂²×C₄₃	#499	1	4	yes	—
137	C₄×C₄₃	—	2	8	no	—
138	C₄×C₄₃	—	8	4	no	—
139	C₄×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
140	C₄×C₄₃	#380	1	2	yes	—
141	C₄×C₄₃	#389	1	2	yes	—
142	C₄×C₄₃	#389	1	2	yes	—
143	Dic₄₃	—	8	4	no	—
144	C₂²×C₄×C₈	—	43	1	no	—
145	C₂×#382	—	43	1	no	—
146	C₂×#382	D₄₃	1	1	yes	characteristic
147	C₂×#340	—	43	4	no	—
148	C₂×C₄₃	C₄⋊#95	1	1	yes	characteristic
149	C₂×C₄₃	#1735	1	2	yes	—
150	C₂×C₄₃	#1835	1	2	yes	—
151	C₂×C₄₃	#1846	1	2	yes	—
152	C₂×C₄₃	#1958	1	2	yes	—
153	C₂×C₄₃	#1958	1	2	yes	—
154	C₂×C₄₃	#1991	1	2	yes	—
155	C₂×C₄₃	#1991	1	2	yes	—
156	C₂³×C₈	—	43	2	no	—
157	C₂²×C₄⋊C₄	—	43	2	no	—
158	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
159	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
160	C₂×C₄×C₈	—	43	4	no	—
161	C₂×C₄×C₈	—	86	2	no	—
162	C₂×C₄×C₈	—	86	2	no	—
163	#382	—	2	2	no	—
164	#382	—	43	4	no	—
165	#382	—	86	2	no	—
166	#382	Dic₄₃	1	4	yes	complemented
167	C₂×#95	—	2	2	no	—
168	C₂×#95	—	86	2	no	—
169	#340	—	86	8	no	—
170	C₄₃	#10842	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₄	—	2	2	no	—
172	C₂³×C₄	—	86	2	no	—
173	C₂³×C₄	C₂²⋊D₄₃	1	2	yes	—
174	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
175	C₂²×C₈	—	43	4	no	—
176	C₂²×C₈	—	43	4	no	—
177	C₂²×C₈	—	43	4	no	—
178	C₂²×C₈	—	86	2	no	—
179	C₂²×C₈	—	86	2	no	—
180	C₂²×C₈	—	86	2	no	—
181	C₂²×C₈	—	86	2	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	43	4	no	—
186	C₂×C₄⋊C₄	—	86	4	no	—
187	C₂×C₄⋊C₄	—	86	4	no	—
188	C₂×C₄⋊C₄	—	86	2	no	—
189	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
190	C₂×C₄²	Dic₈₆	1	2	yes	—
191	C₂×C₄²	C₄×D₄₃	1	2	yes	—
192	C₂×C₄²	C₄×D₄₃	1	2	yes	—
193	C₂×C₄²	D₁₇₂	1	2	yes	—
194	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
195	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
196	C₄×C₈	—	86	8	no	—
197	#95	—	2	8	no	—
198	#95	—	86	8	no	—
199	C₂⁴	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	4	8	no	—
205	C₂²×C₄	—	86	4	no	—
206	C₂²×C₄	—	172	2	no	—
207	C₂²×C₄	—	172	2	no	—
208	C₂²×C₄	—	172	8	no	—
209	C₂²×C₄	Dic₄₃⋊C₄	1	4	yes	—
210	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
211	C₂²×C₄	C₂²⋊Dic₄₃	1	4	yes	—
212	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₄₃	1	2	yes	—
217	C₂²×C₄	#10621900	1	2	yes	—
218	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
219	C₂²×C₄	#10621902	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
222	C₂×C₈	—	43	8	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	—	86	4	no	—
229	C₄⋊C₄	—	2	8	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	86	8	no	—
232	C₄⋊C₄	—	172	4	no	—
233	C₄²	C₄×Dic₄₃	1	4	yes	—
234	C₄²	Dic₄₃⋊C₄	1	4	yes	—
235	C₄²	C₄⋊Dic₄₃	1	4	yes	—
236	C₄²	(C₂×D₄₃)⋊C₄	1	4	yes	—
237	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
238	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
239	C₂³	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
240	C₂³	#13063312	1	1	yes	characteristic
241	C₂³	#13063314	1	1	yes	characteristic
242	C₂³	#13063321	1	1	yes	characteristic
243	C₂³	#13063323	1	1	yes	characteristic
244	C₂³	#13063345	1	8	yes	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	8	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	—	8	4	no	—
252	C₂×C₄	—	344	4	no	—
253	C₂×C₄	—	344	4	no	—
254	C₂×C₄	—	172	4	no	—
255	C₂×C₄	—	172	8	no	—
256	C₂×C₄	C₄₃⋊#95	1	8	yes	—
257	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
258	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
260	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
262	C₂×C₄	#13063283	1	2	yes	—
263	C₂×C₄	#13063287	1	2	yes	—
264	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
265	C₂×C₄	#13063290	1	2	yes	—
266	C₂×C₄	Q₈×Dic₄₃	1	2	yes	—
267	C₂×C₄	#13063296	1	2	yes	—
268	C₂×C₄	#13063300	1	2	yes	—
269	C₂×C₄	#13063301	1	2	yes	—
270	C₂×C₄	#13063305	1	2	yes	—
271	C₂×C₄	#13063316	1	2	yes	—
272	C₂×C₄	#13063324	1	2	yes	—
273	C₈	—	86	8	no	—
274	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
275	C₂²	#14582764	1	1	yes	characteristic
276	C₂²	#14582768	1	1	yes	characteristic
277	C₂²	#14582800	1	1	yes	characteristic
278	C₂²	#14582802	1	1	yes	characteristic
279	C₂²	#14582804	1	1	yes	characteristic
280	C₂²	#14582806	1	1	yes	characteristic
281	C₂²	#14582893	1	4	yes	—
282	C₂²	#14582895	1	4	yes	—
283	C₂²	#14582935	1	4	yes	—
284	C₂²	#14583253	1	4	yes	—
285	C₂²	#14583256	1	4	yes	—
286	C₂²	#14583281	1	4	yes	—
287	C₂²	#14583298	1	4	yes	—
288	C₄	—	2	8	no	—
289	C₄	—	8	4	no	—
290	C₄	—	344	4	no	—
291	C₄	C₄⋊C₄×Dic₄₃	1	2	yes	—
292	C₄	#14582756	1	2	yes	—
293	C₄	#14582795	1	2	yes	—
294	C₄	#14582798	1	2	yes	—
295	C₂	#19132824	1	1	yes	characteristic
296	C₂	#19134311	1	2	yes	—
297	C₂	#19134725	1	2	yes	—
298	C₂	#19134757	1	2	yes	—
299	C₂	#19135404	1	2	yes	—
300	C₂	#19135407	1	2	yes	—
301	C₂	#19135559	1	2	yes	—
302	C₂	#19135561	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×#382	C₂
socle, prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×#382	D₄₃
hypocenter	C₄₃	#10842
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#382

Complements

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

N	H ≅ G/N	type
9. C₂×C₄×C₄₃⋊C₈	C₄	semidirect
11. #14582765	C₄	semidirect
17. C₄₃×#382	C₄	semidirect
166. #382	Dic₄₃	semidirect
170. C₄₃	#10842	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₄₃⋊C₈	2
C₂×#14582765	2
C₂×#14582661	2
C₂×C₄₃×#382	2
#10842	43