Subgroups — #25065889

Order 11008.

616 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143720	C₂	1	2	yes	maximal
2	#19133384	C₂	1	2	yes	maximal, complemented
3	#19133380	C₂	1	4	yes	maximal, complemented
4	#19133358	C₂	1	2	yes	maximal, complemented
5	#19133351	C₂	1	1	yes	maximal, characteristic, complemented
6	#19133343	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14582650	C₂	1	2	yes	maximal, complemented
8	C₈×#41×C₄₃	C₂	1	1	yes	maximal, characteristic
9	#14583030	—	2	2	no	—
10	#14582971	—	2	4	no	—
11	D₄×C₄₃⋊C₈	—	2	2	no	—
12	#14582956	C₂²	1	4	yes	—
13	#14582952	C₂²	1	2	yes	—
14	#14582923	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
16	#14582652	C₂²	1	2	yes	—
17	#14582652	C₂²	1	4	yes	—
18	#14582650	C₂²	1	1	yes	characteristic
19	#14582650	C₂²	1	2	yes	—
20	#14582650	C₂²	1	4	yes	—
21	#14582650	C₂²	1	1	yes	characteristic
22	#14582650	C₂²	1	2	yes	—
23	C₄×#41×C₄₃	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₄₃	C₄	1	1	yes	characteristic
25	C₈×Q₈×C₄₃	C₄	1	1	yes	characteristic
26	C₈×D₄×C₄₃	C₄	1	2	yes	—
27	C₈×D₄×C₄₃	C₄	1	1	yes	characteristic
28	C₈×D₄×C₄₃	C₄	1	2	yes	—
29	C₈×D₄×C₄₃	C₄	1	1	yes	characteristic
30	C₄₃×#431	C₂²	1	2	yes	—
31	C₄₃×#431	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×#10621877	—	2	2	no	—
35	C₂²×C₄₃⋊C₈	—	2	4	no	—
36	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
37	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
38	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
39	C₄⋊(C₄₃⋊C₈)	C₂³	1	4	yes	—
40	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
41	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
42	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
43	#13063236	—	2	4	no	—
44	C₄×C₄₃⋊C₈	—	2	2	no	—
45	C₂×#41×C₄₃	Q₈	1	1	yes	characteristic
46	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₄×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₄×Q₈×C₄₃	D₄	1	1	yes	characteristic
52	C₄×D₄×C₄₃	C₂×C₄	1	2	yes	—
53	C₄×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
54	C₄×D₄×C₄₃	D₄	1	1	yes	characteristic
55	C₄×D₄×C₄₃	D₄	1	2	yes	—
56	#117×C₄₃	Q₈	1	1	yes	characteristic
57	#117×C₄₃	Q₈	1	2	yes	—
58	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
59	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
61	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
62	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
63	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
66	C₂²⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₄₃	C₂³	1	2	yes	—
73	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	#41×C₄₃	—	2	4	no	—
75	C₂×Q₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂×D₄×C₄₃	C₄⋊C₄	1	2	yes	—
77	C₂×D₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₄₃	—	2	4	no	—
85	C₂×C₈×C₄₃	—	2	2	no	—
86	C₂×C₈×C₄₃	—	4	2	no	—
87	C₂×C₈×C₄₃	—	4	1	no	—
88	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
94	C₄⋊C₄×C₄₃	C₄⋊C₄	1	2	yes	—
95	C₄⋊C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄⋊C₄×C₄₃	C₄⋊C₄	1	2	yes	—
97	C₄⋊C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
98	C₂²⋊C₄×C₄₃	C₄⋊C₄	1	2	yes	—
99	C₂²⋊C₄×C₄₃	C₄⋊C₄	1	2	yes	—
100	C₂²⋊C₄×C₄₃	C₄⋊C₄	1	2	yes	—
101	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₄₃	C₂×D₄	1	2	yes	—
105	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
106	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
107	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
108	#10621877	—	8	2	no	—
109	C₂×C₄₃⋊C₈	—	2	2	no	—
110	C₂×C₄₃⋊C₈	—	2	2	no	—
111	C₂×C₄₃⋊C₈	—	2	4	no	—
112	C₂×C₄₃⋊C₈	—	8	4	no	—
113	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
115	Q₈×C₄₃	—	2	2	no	—
116	D₄×C₄₃	—	2	4	no	—
117	D₄×C₄₃	#108	1	4	yes	—
118	C₂×C₄×C₄₃	—	2	4	no	—
119	C₂×C₄×C₄₃	—	2	2	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	2	4	no	—
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	—	2	2	no	—
126	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
135	C₈×C₄₃	—	2	2	no	—
136	C₈×C₄₃	—	2	1	no	—
137	C₈×C₄₃	—	2	1	no	—
138	C₈×C₄₃	—	2	2	no	—
139	C₈×C₄₃	—	2	1	no	—
140	C₈×C₄₃	#131	1	2	yes	—
141	C₄₃⋊C₈	—	4	2	no	—
142	C₄₃⋊C₈	—	4	2	no	—
143	C₄₃⋊C₈	—	4	4	no	—
144	#17834	—	43	1	no	maximal, Sylow
145	C₂²×C₄₃	—	2	2	no	—
146	C₂²×C₄₃	—	2	2	no	—
147	C₂²×C₄₃	—	2	1	no	—
148	C₂²×C₄₃	—	2	1	no	—
149	C₂²×C₄₃	—	2	1	no	—
150	C₂²×C₄₃	C₂×#108	1	4	yes	—
151	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₄₃	—	2	2	no	—
153	C₄×C₄₃	—	2	1	no	—
154	C₄×C₄₃	—	2	2	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	2	1	no	—
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	C₂×#108	1	2	yes	—
159	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
160	C₄×C₄₃	#568	1	1	yes	characteristic
161	C₄×C₄₃	#577	1	2	yes	—
162	#2990	—	43	2	no	—
163	C₈×#41	D₄₃	1	1	yes	characteristic
164	#1574	—	43	2	no	—
165	#1573	—	43	4	no	—
166	#1568	—	43	2	no	—
167	#1566	—	43	1	no	—
168	#1564	—	43	1	no	—
169	C₂×#334	—	43	2	no	—
170	C₂×C₄₃	—	2	4	no	—
171	C₂×C₄₃	—	4	1	no	—
172	C₂×C₄₃	#2906	1	1	yes	characteristic
173	C₂×C₄₃	#2912	1	1	yes	characteristic
174	C₂×C₄₃	C₂²×#108	1	1	yes	characteristic
175	C₄×#41	C₂×D₄₃	1	1	yes	characteristic
176	#446	—	86	2	no	—
177	C₈×Q₈	Dic₄₃	1	1	yes	characteristic
178	C₈×Q₈	Dic₄₃	1	1	yes	characteristic
179	#435	—	86	4	no	—
180	C₈×D₄	—	86	2	no	—
181	C₈×D₄	Dic₄₃	1	2	yes	—
182	C₈×D₄	Dic₄₃	1	1	yes	characteristic
183	C₈×D₄	Dic₄₃	1	2	yes	—
184	C₈×D₄	Dic₄₃	1	1	yes	characteristic
185	#432	—	43	4	no	—
186	#431	—	43	2	no	—
187	#431	C₂×D₄₃	1	2	yes	—
188	#431	C₂×D₄₃	1	1	yes	characteristic
189	#423	—	43	2	no	—
190	C₂×C₄⋊C₈	—	43	4	no	—
191	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
192	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
193	#335	—	43	2	no	—
194	#335	—	43	4	no	—
195	#334	—	43	1	no	—
196	#334	—	43	2	no	—
197	#334	—	43	4	no	—
198	#334	—	43	1	no	—
199	#334	—	43	2	no	—
200	C₄₃	#17834	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	Dic₈₆	1	1	yes	characteristic
202	C₂×M₁₆	—	86	2	no	—
203	C₂²×C₈	—	86	4	no	—
204	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
205	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
206	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
207	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
208	C₄×Q₈	D₁₇₂	1	1	yes	characteristic
209	C₄×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
210	C₄×D₄	D₁₇₂	1	1	yes	characteristic
211	C₄×D₄	D₁₇₂	1	2	yes	—
212	C₄×D₄	C₂×Dic₄₃	1	2	yes	—
213	C₄×D₄	C₂×Dic₄₃	1	1	yes	characteristic
214	#117	Dic₈₆	1	1	yes	characteristic
215	#117	Dic₈₆	1	2	yes	—
216	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
217	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
218	C₄⋊C₈	—	43	2	no	—
219	C₄⋊C₈	—	43	4	no	—
220	C₄⋊C₈	—	43	2	no	—
221	C₄⋊C₈	—	86	4	no	—
222	C₄⋊C₈	—	86	2	no	—
223	C₄⋊C₈	C₂×Dic₄₃	1	2	yes	—
224	C₄⋊C₈	C₂×Dic₄₃	1	2	yes	—
225	C₄⋊C₈	C₂×Dic₄₃	1	1	yes	characteristic
226	C₄⋊C₈	C₂×Dic₄₃	1	1	yes	characteristic
227	C₂²⋊C₈	—	86	2	no	—
228	C₂²⋊C₈	—	86	4	no	—
229	C₂²⋊C₈	C₂×Dic₄₃	1	2	yes	—
230	C₂²⋊C₈	C₂×Dic₄₃	1	2	yes	—
231	C₂²⋊C₈	C₂×Dic₄₃	1	2	yes	—
232	#97	—	86	4	no	—
233	C₄×C₈	—	86	2	no	—
234	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
235	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
236	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
237	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₄₃	1	2	yes	—
240	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
241	#41	—	2	4	no	—
242	C₂×Q₈	C₄⋊Dic₄₃	1	1	yes	characteristic
243	C₂×D₄	C₄⋊Dic₄₃	1	2	yes	—
244	C₂×D₄	C₄⋊Dic₄₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
246	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
248	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
251	M₁₆	—	344	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	86	2	no	—
257	C₂×C₈	—	86	2	no	—
258	C₂×C₈	—	86	4	no	—
259	C₂×C₈	—	344	4	no	—
260	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
261	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
262	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
263	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
264	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
265	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
266	C₄⋊C₄	C₄⋊Dic₄₃	1	2	yes	—
267	C₄⋊C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
268	C₄⋊C₄	C₄⋊Dic₄₃	1	2	yes	—
269	C₄⋊C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
270	C₂²⋊C₄	C₄⋊Dic₄₃	1	2	yes	—
271	C₂²⋊C₄	C₄⋊Dic₄₃	1	2	yes	—
272	C₂²⋊C₄	C₄⋊Dic₄₃	1	2	yes	—
273	C₄²	C₂×Dic₈₆	1	2	yes	—
274	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
275	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
276	C₄²	C₂×D₁₇₂	1	2	yes	—
277	C₄²	C₂²×Dic₄₃	1	2	yes	—
278	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
279	C₄²	C₂³×D₄₃	1	1	yes	characteristic
280	C₂³	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
281	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
282	Q₈	—	2	2	no	—
283	D₄	—	2	4	no	—
284	D₄	#13063261	1	4	yes	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
295	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
296	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
297	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
298	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
299	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
301	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	2	no	—
306	C₈	—	2	1	no	—
307	C₈	—	172	2	no	—
308	C₈	—	172	4	no	—
309	C₈	—	172	2	no	—
310	C₈	#13063332	1	2	yes	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	C₂×#13063261	1	4	yes	—
317	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	C₂×#13063261	1	2	yes	—
325	C₄	C₂×#13063332	1	1	yes	characteristic
326	C₄	#14583666	1	1	yes	characteristic
327	C₄	#14583711	1	2	yes	—
328	C₂	—	2	4	no	—
329	C₂	—	4	1	no	—
330	C₂	#19143163	1	1	yes	characteristic
331	C₂	#19143190	1	1	yes	characteristic
332	C₂	C₂²×#13063261	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₈×#41×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₄₃	Q₈
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	C₈×#41	D₄₃
hypocenter	C₄₃	#17834
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #19133384	C₂	semidirect
3. #19133380	C₂	semidirect
4. #19133358	C₂	semidirect
5. #19133351	C₂	semidirect
6. #19133343	C₂	semidirect
7. C₂×#14582650	C₂	semidirect
200. C₄₃	#17834	semidirect

Maximal subgroups

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

subgroup	index
#19143720	2
#19133384	2
#19133380	2
#19133358	2
#19133351	2
#19133343	2
C₂×#14582650	2
C₈×#41×C₄₃	2
#17834	43