Subgroups — #24984571

Order 11008.

684 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13063322	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	#19132415	C₂	1	4	yes	maximal, complemented
4	C₄×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13063322	C₄	1	1	yes	characteristic
6	C₂×C₄×D₁₇₂	—	2	1	no	—
7	C₂×C₄×D₁₇₂	—	2	1	no	—
8	C₂×C₄×D₁₇₂	C₄	1	2	yes	—
9	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
10	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
11	#14582801	—	2	1	no	—
12	#14582801	—	2	1	no	—
13	#14582801	C₂²	1	2	yes	—
14	#14582769	C₄	1	1	yes	characteristic
15	#14582654	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
17	C₂×C₄×C₈×C₄₃	—	2	1	no	—
18	C₄³×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#336	C₂²	1	2	yes	—
20	C₂²×D₁₇₂	—	2	1	no	—
21	C₂²×D₁₇₂	—	2	1	no	—
22	C₂²×D₁₇₂	C₂×C₄	1	2	yes	—
23	C₂²×C₄×D₄₃	—	2	2	no	—
24	C₂²×C₄₃⋊C₈	—	2	2	no	—
25	#13063322	—	2	2	no	—
26	#13063322	C₈	1	4	yes	complemented
27	C₄×D₁₇₂	—	4	4	no	—
28	C₄×D₁₇₂	—	4	2	no	—
29	C₄×D₁₇₂	—	4	2	no	—
30	C₄×D₁₇₂	—	4	2	no	—
31	C₄×D₁₇₂	—	4	2	no	—
32	C₂×C₄⋊Dic₄₃	—	2	1	no	—
33	C₂×C₄⋊Dic₄₃	—	2	1	no	—
34	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
35	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
36	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
37	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
38	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
39	C₄×C₄₃⋊C₈	—	4	2	no	—
40	C₂²×C₈×C₄₃	—	2	2	no	—
41	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₄₃	—	2	4	no	—
49	C₄×C₈×C₄₃	—	4	2	no	—
50	C₂³×D₄₃	—	2	2	no	—
51	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₄₃	—	2	2	no	—
59	C₂×D₁₇₂	—	2	4	no	—
60	C₂×D₁₇₂	—	2	4	no	—
61	C₂×D₁₇₂	—	2	2	no	—
62	C₂×D₁₇₂	—	4	2	no	—
63	C₂×D₁₇₂	—	4	2	no	—
64	C₂×D₁₇₂	—	4	2	no	—
65	C₂×D₁₇₂	—	4	2	no	—
66	C₂×D₁₇₂	—	4	2	no	—
67	C₂×D₁₇₂	—	4	2	no	—
68	C₂×D₁₇₂	C₂×C₈	1	4	yes	—
69	C₂×C₄×D₄₃	—	4	4	no	—
70	C₂×C₄×D₄₃	—	4	4	no	—
71	C₂×C₄×D₄₃	—	4	4	no	—
72	C₂×C₈×C₄₃	—	2	4	no	—
73	C₂×C₈×C₄₃	—	4	2	no	—
74	C₂×C₈×C₄₃	—	4	2	no	—
75	C₄²×C₄₃	—	2	2	no	—
76	C₄²×C₄₃	—	2	2	no	—
77	C₄²×C₄₃	—	2	2	no	—
78	C₄²×C₄₃	—	4	1	no	—
79	C₄²×C₄₃	C₂×C₈	1	2	yes	—
80	C₄²×C₄₃	M₁₆	1	2	yes	—
81	C₄²×C₄₃	D₈	1	2	yes	—
82	C₄²×C₄₃	D₈	1	2	yes	—
83	C₄²×C₄₃	SD₁₆	1	2	yes	—
84	C₄²×C₄₃	SD₁₆	1	2	yes	—
85	(C₂×D₄₃)⋊C₄	—	4	4	no	—
86	(C₂×D₄₃)⋊C₄	—	4	2	no	—
87	(C₂×D₄₃)⋊C₄	—	4	2	no	—
88	(C₂×D₄₃)⋊C₄	—	4	2	no	—
89	(C₂×D₄₃)⋊C₄	—	4	2	no	—
90	C₄⋊Dic₄₃	—	4	2	no	—
91	C₄⋊Dic₄₃	—	4	2	no	—
92	C₄⋊Dic₄₃	—	4	2	no	—
93	C₂×C₄₃⋊C₈	—	2	4	no	—
94	C₂×C₄₃⋊C₈	—	4	2	no	—
95	C₂×C₄₃⋊C₈	—	4	2	no	—
96	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×D₄₃	—	2	4	no	—
98	C₂²×D₄₃	—	4	2	no	—
99	C₂²×D₄₃	—	4	2	no	—
100	C₂²×D₄₃	—	4	2	no	—
101	C₂²×D₄₃	—	4	2	no	—
102	C₂²×D₄₃	—	4	2	no	—
103	C₂²×D₄₃	—	4	2	no	—
104	C₂×C₄×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	2	1	no	—
106	C₂×C₄×C₄₃	—	2	1	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
114	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
115	C₂×C₄×C₄₃	#100	1	2	yes	—
116	C₂×C₄×C₄₃	#102	1	2	yes	—
117	C₂×C₄×C₄₃	#102	1	2	yes	—
118	C₂×C₄×C₄₃	#102	1	2	yes	—
119	C₂×C₄×C₄₃	#102	1	2	yes	—
120	C₂×C₄×C₄₃	#104	1	2	yes	—
121	C₂×C₄×C₄₃	#104	1	2	yes	—
122	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×Dic₄₃	—	4	2	no	—
129	C₂×Dic₄₃	—	4	2	no	—
130	C₂×Dic₄₃	—	4	2	no	—
131	D₁₇₂	—	4	4	no	—
132	D₁₇₂	—	4	4	no	—
133	D₁₇₂	—	4	8	no	—
134	D₁₇₂	—	4	8	no	—
135	C₄×D₄₃	—	8	8	no	—
136	C₈×C₄₃	—	4	4	no	—
137	C₄₃⋊C₈	—	4	4	no	—
138	#7569	—	43	1	no	maximal, Sylow
139	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
140	C₂²×C₄₃	#410	1	1	yes	characteristic
141	C₂²×C₄₃	C₂×#100	1	1	yes	characteristic
142	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
143	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
144	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
145	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
146	C₂×D₄₃	—	4	4	no	—
147	C₂×D₄₃	—	4	4	no	—
148	C₂×D₄₃	—	4	4	no	—
149	C₂×D₄₃	—	8	2	no	—
150	C₂×D₄₃	—	8	2	no	—
151	C₂×D₄₃	—	8	2	no	—
152	C₂×D₄₃	—	8	2	no	—
153	C₄×C₄₃	—	2	1	no	—
154	C₄×C₄₃	—	2	1	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	2	1	no	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	—	4	1	no	—
160	C₄×C₄₃	D₄⋊C₈	1	4	yes	—
161	C₄×C₄₃	D₄⋊C₈	1	4	yes	—
162	C₄×C₄₃	#331	1	4	yes	—
163	Dic₄₃	—	8	2	no	—
164	C₄×#127	—	43	1	no	—
165	C₄×C₄⋊C₈	—	43	1	no	—
166	C₄×C₄⋊C₈	D₄₃	1	1	yes	characteristic, complemented
167	#1270	—	43	4	no	—
168	C₂×C₄₃	#1460	1	1	yes	characteristic
169	C₂×C₄₃	C₂×D₄⋊C₈	1	1	yes	characteristic
170	C₂×C₄₃	C₂×D₄⋊C₈	1	1	yes	characteristic
171	C₂×C₄₃	#1502	1	1	yes	characteristic
172	C₂×C₄₃	#1502	1	1	yes	characteristic
173	C₂×C₄₃	#1525	1	1	yes	characteristic
174	C₂×C₄₃	C₂×#331	1	1	yes	characteristic
175	D₄₃	—	8	4	no	—
176	C₂×#127	—	43	1	no	—
177	C₂×C₄×D₄	—	43	2	no	—
178	C₂×C₄×D₄	—	86	1	no	—
179	C₂×C₄×D₄	—	86	1	no	—
180	C₂×C₄⋊C₈	—	2	1	no	—
181	C₂×C₄⋊C₈	—	86	1	no	—
182	C₂×C₄×C₈	—	2	1	no	—
183	C₂×C₄×C₈	—	86	1	no	—
184	#390	—	43	2	no	—
185	#390	—	86	1	no	—
186	#390	—	86	1	no	—
187	#384	—	43	1	no	—
188	C₄³	C₂×D₄₃	1	1	yes	characteristic
189	#336	—	43	2	no	—
190	#336	C₂×D₄₃	1	2	yes	—
191	C₄₃	#7569	1	1	yes	characteristic, complemented, Sylow
192	C₂²×D₄	—	43	2	no	—
193	C₂²×D₄	—	86	1	no	—
194	C₂²×D₄	—	86	1	no	—
195	C₂³×C₄	—	86	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	86	2	no	—
198	#127	—	43	4	no	—
199	#127	—	86	2	no	—
200	C₄×D₄	—	172	4	no	—
201	C₄×D₄	—	172	2	no	—
202	C₄×D₄	—	172	2	no	—
203	C₄×D₄	—	172	2	no	—
204	C₄×D₄	—	172	2	no	—
205	C₂×C₄⋊C₄	—	43	2	no	—
206	C₂×C₄⋊C₄	—	86	1	no	—
207	C₂×C₄⋊C₄	—	86	1	no	—
208	C₂×C₂²⋊C₄	—	86	2	no	—
209	C₂×C₂²⋊C₄	—	86	2	no	—
210	C₂×C₂²⋊C₄	—	86	2	no	—
211	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
212	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
213	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
214	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
215	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
217	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
218	C₄⋊C₈	—	2	4	no	—
219	C₄⋊C₈	—	86	4	no	—
220	C₄×C₈	—	4	2	no	—
221	C₄×C₈	—	172	2	no	—
222	C₂⁴	—	86	2	no	—
223	C₂×D₄	—	43	4	no	—
224	C₂×D₄	—	86	4	no	—
225	C₂×D₄	—	86	4	no	—
226	C₂×D₄	—	86	2	no	—
227	C₂×D₄	—	172	2	no	—
228	C₂×D₄	—	172	2	no	—
229	C₂×D₄	—	172	2	no	—
230	C₂×D₄	—	172	2	no	—
231	C₂×D₄	—	172	2	no	—
232	C₂×D₄	—	172	2	no	—
233	C₂²×C₄	—	86	2	no	—
234	C₂²×C₄	—	172	4	no	—
235	C₂²×C₄	—	172	4	no	—
236	C₂²×C₄	—	172	4	no	—
237	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
238	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
239	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
240	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	86	4	no	—
248	C₂×C₈	—	172	2	no	—
249	C₂×C₈	—	172	2	no	—
250	C₄⋊C₄	—	172	2	no	—
251	C₄⋊C₄	—	172	2	no	—
252	C₄⋊C₄	—	172	2	no	—
253	C₂²⋊C₄	—	172	4	no	—
254	C₂²⋊C₄	—	172	2	no	—
255	C₂²⋊C₄	—	172	2	no	—
256	C₂²⋊C₄	—	172	2	no	—
257	C₂²⋊C₄	—	172	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	4	1	no	—
262	C₄²	C₈×D₄₃	1	2	yes	—
263	C₄²	#10621872	1	2	yes	—
264	C₄²	#10621873	1	2	yes	—
265	C₄²	D₃₄₄	1	2	yes	—
266	C₄²	#10621882	1	2	yes	—
267	C₄²	#10621884	1	2	yes	—
268	C₂³	—	86	4	no	—
269	C₂³	—	172	2	no	—
270	C₂³	—	172	2	no	—
271	C₂³	—	172	2	no	—
272	C₂³	—	172	2	no	—
273	C₂³	—	172	2	no	—
274	C₂³	—	172	2	no	—
275	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
276	D₄	—	172	4	no	—
277	D₄	—	172	4	no	—
278	D₄	—	172	8	no	—
279	D₄	—	172	8	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	344	8	no	—
290	C₂×C₄	—	172	2	no	—
291	C₂×C₄	—	172	2	no	—
292	C₂×C₄	—	172	2	no	—
293	C₂×C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
294	C₂×C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
295	C₂×C₄	#13063241	1	2	yes	—
296	C₂×C₄	#13063245	1	2	yes	—
297	C₂×C₄	#13063245	1	2	yes	—
298	C₂×C₄	#13063247	1	2	yes	—
299	C₂×C₄	#13063247	1	2	yes	—
300	C₂×C₄	#13063251	1	2	yes	—
301	C₂×C₄	#13063253	1	2	yes	—
302	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
303	C₂×C₄	C₂×#10621872	1	1	yes	characteristic
304	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
305	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
306	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
307	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
308	C₈	—	4	4	no	—
309	C₈	—	172	4	no	—
310	C₂²	—	344	2	no	—
311	C₂²	—	344	2	no	—
312	C₂²	—	344	2	no	—
313	C₂²	—	344	2	no	—
314	C₂²	—	172	4	no	—
315	C₂²	—	172	4	no	—
316	C₂²	—	172	4	no	—
317	C₂²	C₂×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
318	C₂²	#14582872	1	1	yes	characteristic
319	C₂²	C₂×#13063241	1	1	yes	characteristic
320	C₂²	C₂×#13063245	1	1	yes	characteristic
321	C₂²	C₂×#13063247	1	1	yes	characteristic
322	C₂²	C₂×#13063251	1	1	yes	characteristic
323	C₂²	C₂×#13063253	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	344	2	no	—
332	C₄	#14582626	1	4	yes	—
333	C₄	#14582628	1	4	yes	—
334	C₄	#14582642	1	4	yes	—
335	C₂	—	344	4	no	—
336	C₂	#19132915	1	1	yes	characteristic
337	C₂	C₂×#14582626	1	1	yes	characteristic
338	C₂	C₂×#14582628	1	1	yes	characteristic
339	C₂	#19133089	1	1	yes	characteristic
340	C₂	#19133091	1	1	yes	characteristic
341	C₂	#19133186	1	1	yes	characteristic
342	C₂	C₂×#14582642	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₃)⋊C₄
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×C₄⋊C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂×#13063322	C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#7569
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊(C₄₃⋊C₈)	C₂	semidirect
3. #19132415	C₂	semidirect
4. C₄×C₄⋊C₈×C₄₃	C₂	semidirect
26. #13063322	C₈	semidirect
166. C₄×C₄⋊C₈	D₄₃	semidirect
191. C₄₃	#7569	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13063322	2
C₄×C₄⋊(C₄₃⋊C₈)	2
#19132415	2
C₄×C₄⋊C₈×C₄₃	2
#7569	43