Subgroups — #24987527

Order 11008.

572 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134344	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13063236	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582639	C₂	1	2	yes	maximal
4	#19132476	C₂	1	2	yes	maximal, complemented
5	C₄×#124×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063236	—	2	1	no	—
7	C₂×#13063236	C₂²	1	2	yes	—
8	C₂×#13063236	C₂²	1	2	yes	—
9	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
10	#14582654	—	2	1	no	—
11	#14582654	—	2	1	no	—
12	#14582639	C₄	1	8	yes	complemented
13	C₂×#124×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₄₃	C₄	1	1	yes	characteristic
15	C₂×C₄×D₄×C₄₃	C₄	1	1	yes	characteristic
16	C₄₃×#391	C₄	1	1	yes	characteristic
17	C₄₃×#390	C₄	1	1	yes	characteristic
18	C₄₃×#388	—	2	1	no	—
19	C₄₃×#388	—	2	1	no	—
20	C₄₃×#382	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊C₄×C₄₃	—	2	1	no	—
22	C₄×C₂²⋊C₄×C₄₃	—	2	1	no	—
23	C₄³×C₄₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄₃⋊C₈	—	2	2	no	—
25	C₂²×C₄₃⋊C₈	—	2	2	no	—
26	#13063236	—	2	4	no	—
27	#13063236	—	2	2	no	—
28	#13063236	—	2	2	no	—
29	#13063236	C₂×C₄	1	4	yes	complemented
30	#13063236	C₂×C₄	1	4	yes	complemented
31	C₄×C₄₃⋊C₈	—	2	4	no	—
32	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₃	—	2	1	no	—
35	#124×C₄₃	—	2	2	no	—
36	#124×C₄₃	C₂×C₄	1	4	yes	—
37	C₄×Q₈×C₄₃	—	4	2	no	—
38	C₄×D₄×C₄₃	—	4	2	no	—
39	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
46	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
47	C₂×C₄²×C₄₃	—	2	1	no	—
48	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	#95×C₄₃	—	2	1	no	—
56	#95×C₄₃	—	2	1	no	—
57	#95×C₄₃	—	2	1	no	—
58	#95×C₄₃	—	2	1	no	—
59	C₂⁴×C₄₃	—	2	1	no	—
60	C₂×Q₈×C₄₃	—	2	1	no	—
61	C₂×Q₈×C₄₃	—	4	1	no	—
62	C₂×Q₈×C₄₃	—	4	1	no	—
63	C₂×Q₈×C₄₃	C₄²	1	2	yes	—
64	C₂×D₄×C₄₃	—	2	1	no	—
65	C₂×D₄×C₄₃	—	4	1	no	—
66	C₂×D₄×C₄₃	—	4	1	no	—
67	C₂×D₄×C₄₃	C₄²	1	2	yes	—
68	C₂²×C₄×C₄₃	—	2	1	no	—
69	C₂²×C₄×C₄₃	—	2	1	no	—
70	C₂²×C₄×C₄₃	—	2	1	no	—
71	C₂²×C₄×C₄₃	—	4	2	no	—
72	C₂²×C₄×C₄₃	—	4	2	no	—
73	C₂²×C₄×C₄₃	—	4	2	no	—
74	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₄₃	—	4	1	no	—
82	C₄⋊C₄×C₄₃	—	4	1	no	—
83	C₄⋊C₄×C₄₃	—	4	2	no	—
84	C₂²⋊C₄×C₄₃	—	2	4	no	—
85	C₂²⋊C₄×C₄₃	—	2	4	no	—
86	C₂²⋊C₄×C₄₃	—	4	2	no	—
87	C₂²⋊C₄×C₄₃	—	4	2	no	—
88	C₂²⋊C₄×C₄₃	—	4	2	no	—
89	C₄²×C₄₃	—	2	2	no	—
90	C₄²×C₄₃	—	2	2	no	—
91	C₄²×C₄₃	—	2	2	no	—
92	C₄²×C₄₃	—	4	1	no	—
93	C₄²×C₄₃	—	4	1	no	—
94	C₄²×C₄₃	—	4	1	no	—
95	C₄²×C₄₃	—	4	2	no	—
96	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₄₃	#41	1	2	yes	—
98	C₂×C₄₃⋊C₈	—	2	4	no	—
99	C₂×C₄₃⋊C₈	—	2	4	no	—
100	C₂×C₄₃⋊C₈	—	2	4	no	—
101	C₂×C₄₃⋊C₈	—	2	4	no	—
102	C₂×C₄₃⋊C₈	—	4	2	no	—
103	C₂×C₄₃⋊C₈	—	4	2	no	—
104	C₂³×C₄₃	—	2	2	no	—
105	C₂³×C₄₃	—	4	1	no	—
106	C₂³×C₄₃	—	4	1	no	—
107	C₂³×C₄₃	—	4	1	no	—
108	C₂³×C₄₃	—	4	1	no	—
109	C₂³×C₄₃	—	4	1	no	—
110	C₂³×C₄₃	—	4	1	no	—
111	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	Q₈×C₄₃	—	4	2	no	—
113	Q₈×C₄₃	—	4	2	no	—
114	D₄×C₄₃	—	4	2	no	—
115	D₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	2	1	no	—
126	C₂×C₄×C₄₃	—	2	1	no	—
127	C₂×C₄×C₄₃	—	2	1	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	2	1	no	—
132	C₂×C₄×C₄₃	—	4	1	no	—
133	C₂×C₄×C₄₃	—	4	1	no	—
134	C₂×C₄×C₄₃	—	4	1	no	—
135	C₂×C₄×C₄₃	—	4	1	no	—
136	C₂×C₄×C₄₃	—	4	1	no	—
137	C₂×C₄×C₄₃	—	4	1	no	—
138	C₂×C₄×C₄₃	—	4	1	no	—
139	C₂×C₄×C₄₃	—	4	1	no	—
140	C₂×C₄×C₄₃	—	8	4	no	—
141	C₂×C₄×C₄₃	#100	1	2	yes	—
142	C₂×C₄×C₄₃	#104	1	2	yes	—
143	C₂×C₄×C₄₃	#104	1	2	yes	—
144	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
145	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
147	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
148	C₄₃⋊C₈	—	2	8	no	—
149	C₄₃⋊C₈	—	4	4	no	—
150	#8226	—	43	1	no	maximal, Sylow
151	C₂²×C₄₃	—	4	2	no	—
152	C₂²×C₄₃	—	4	2	no	—
153	C₂²×C₄₃	—	4	2	no	—
154	C₂²×C₄₃	—	8	1	no	—
155	C₂²×C₄₃	—	8	1	no	—
156	C₂²×C₄₃	—	8	1	no	—
157	C₂²×C₄₃	—	8	1	no	—
158	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄₃	C₂×#100	1	1	yes	characteristic
160	C₂²×C₄₃	#413	1	1	yes	characteristic
161	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
162	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
163	C₂²×C₄₃	#421	1	1	yes	characteristic
164	C₂²×C₄₃	#421	1	1	yes	characteristic
165	C₄×C₄₃	—	2	1	no	—
166	C₄×C₄₃	—	2	1	no	—
167	C₄×C₄₃	—	2	2	no	—
168	C₄×C₄₃	—	2	2	no	—
169	C₄×C₄₃	—	2	2	no	—
170	C₄×C₄₃	—	2	2	no	—
171	C₄×C₄₃	—	2	2	no	—
172	C₄×C₄₃	—	4	2	no	—
173	C₄×C₄₃	—	4	1	no	—
174	C₄×C₄₃	—	8	1	no	—
175	C₄×C₄₃	—	8	1	no	—
176	C₄×#124	D₄₃	1	1	yes	characteristic
177	#1745	—	43	1	no	—
178	C₄×#97	—	43	1	no	—
179	C₂×#329	—	43	2	no	—
180	#1295	—	43	2	no	—
181	C₂×C₄₃	—	8	2	no	—
182	C₂×C₄₃	#1527	1	1	yes	characteristic
183	C₂×C₄₃	#1529	1	1	yes	characteristic
184	C₂×C₄₃	#1530	1	1	yes	characteristic
185	C₂×C₄₃	#1532	1	1	yes	characteristic
186	C₂×C₄₃	C₄×#100	1	1	yes	characteristic
187	C₂×C₄₃	C₄×#104	1	1	yes	characteristic
188	C₂×C₄₃	#1759	1	1	yes	characteristic
189	C₂×#124	C₂×D₄₃	1	1	yes	characteristic
190	C₂×C₄×Q₈	Dic₄₃	1	1	yes	characteristic
191	C₂×C₄×D₄	Dic₄₃	1	1	yes	characteristic
192	C₂×#97	—	43	2	no	—
193	C₂×#97	—	43	2	no	—
194	C₂×#97	—	86	1	no	—
195	C₂×C₄×C₈	—	86	1	no	—
196	#391	Dic₄₃	1	1	yes	characteristic
197	#390	Dic₄₃	1	1	yes	characteristic
198	#388	—	2	1	no	—
199	#388	—	2	1	no	—
200	#382	C₂×D₄₃	1	1	yes	characteristic
201	C₄×C₂²⋊C₄	—	2	1	no	—
202	C₄×C₂²⋊C₄	—	2	1	no	—
203	C₄³	C₂×D₄₃	1	1	yes	characteristic
204	#336	—	86	1	no	—
205	#336	—	86	1	no	—
206	#329	—	43	8	no	—
207	C₄₃	#8226	1	1	yes	characteristic, complemented, Sylow
208	C₂²×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
209	C₂²×D₄	C₂×Dic₄₃	1	1	yes	characteristic
210	C₂³×C₄	—	2	1	no	—
211	C₂²×C₈	—	86	2	no	—
212	C₂²×C₈	—	86	2	no	—
213	#124	—	2	2	no	—
214	#124	C₄×D₄₃	1	4	yes	—
215	C₄×Q₈	—	4	2	no	—
216	C₄×D₄	—	4	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	2	1	no	—
222	C₂×C₂²⋊C₄	—	2	1	no	—
223	C₂×C₂²⋊C₄	—	2	1	no	—
224	C₂×C₂²⋊C₄	—	2	1	no	—
225	C₂×C₄²	—	2	1	no	—
226	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
227	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
228	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
229	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
230	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
231	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
233	#97	—	43	4	no	—
234	#97	—	43	4	no	—
235	#97	—	86	4	no	—
236	#97	—	86	2	no	—
237	#97	—	86	2	no	—
238	C₄×C₈	—	86	4	no	—
239	#95	—	2	1	no	—
240	#95	—	2	1	no	—
241	#95	—	2	1	no	—
242	#95	—	2	1	no	—
243	C₂⁴	—	2	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	C₄×Dic₄₃	1	2	yes	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	C₄×Dic₄₃	1	2	yes	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
259	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
260	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
261	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
262	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
265	C₂×C₈	—	86	4	no	—
266	C₂×C₈	—	86	4	no	—
267	C₂×C₈	—	86	4	no	—
268	C₂×C₈	—	86	4	no	—
269	C₂×C₈	—	172	2	no	—
270	C₂×C₈	—	172	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	2	4	no	—
275	C₂²⋊C₄	—	2	4	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	4	1	no	—
283	C₄²	—	4	1	no	—
284	C₄²	—	4	1	no	—
285	C₄²	—	4	2	no	—
286	C₄²	C₂×C₄×D₄₃	1	2	yes	—
287	C₄²	#10621898	1	2	yes	—
288	C₂³	—	2	2	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	4	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	4	1	no	—
294	C₂³	—	4	1	no	—
295	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
296	Q₈	—	4	2	no	—
297	Q₈	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	8	4	no	—
325	C₂×C₄	#13063242	1	2	yes	—
326	C₂×C₄	#13063252	1	2	yes	—
327	C₂×C₄	#13063252	1	2	yes	—
328	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
329	C₂×C₄	#13063284	1	1	yes	characteristic
330	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
331	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
332	C₈	—	86	8	no	—
333	C₈	—	172	4	no	—
334	C₂²	—	4	2	no	—
335	C₂²	—	4	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	—	8	1	no	—
340	C₂²	—	8	1	no	—
341	C₂²	C₄×C₂²⋊Dic₄₃	1	1	yes	characteristic
342	C₂²	C₂×#13063242	1	1	yes	characteristic
343	C₂²	#14582883	1	1	yes	characteristic
344	C₂²	C₂×#13063252	1	1	yes	characteristic
345	C₂²	C₂×#13063252	1	1	yes	characteristic
346	C₂²	#14582916	1	1	yes	characteristic
347	C₂²	#14582916	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	2	no	—
357	C₄	—	8	1	no	—
358	C₄	—	8	1	no	—
359	C₂	—	8	2	no	—
360	C₂	#19133193	1	1	yes	characteristic
361	C₂	#19133202	1	1	yes	characteristic
362	C₂	#19133206	1	1	yes	characteristic
363	C₂	#19133215	1	1	yes	characteristic
364	C₂	C₄×#13063242	1	1	yes	characteristic
365	C₂	C₄×#13063252	1	1	yes	characteristic
366	C₂	#19134405	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×#124×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₃	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×#124	D₄₃
hypocenter	C₄₃	#8226
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#124

Complements

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

N	H ≅ G/N	type
1. #19134344	C₂	semidirect
2. C₄×#13063236	C₂	semidirect
4. #19132476	C₂	semidirect
12. #14582639	C₄	semidirect
29. #13063236	C₂×C₄	semidirect
30. #13063236	C₂×C₄	semidirect
207. C₄₃	#8226	semidirect

Maximal subgroups

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

subgroup	index
#19134344	2
C₄×#13063236	2
C₂×#14582639	2
#19132476	2
C₄×#124×C₄₃	2
#8226	43