Subgroups — #25101001

Order 11008.

722 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×Dic₄₃	C₂	1	1	yes	maximal, characteristic
2	#19143242	C₂	1	2	yes	maximal, complemented
3	#19135484	C₂	1	1	yes	maximal, characteristic
4	#19135343	C₂	1	1	yes	maximal, characteristic, complemented
5	#19135341	C₂	1	1	yes	maximal, characteristic, complemented
6	#19135309	C₂	1	2	yes	maximal, complemented
7	#19135303	C₂	1	2	yes	maximal, complemented
8	#108×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#19134407	C₂	1	2	yes	maximal, complemented
10	#19134281	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#1974	C₂	1	1	yes	maximal, characteristic
12	#41×Dic₄₃	C₂²	1	2	yes	—
13	Q₁₆×Dic₄₃	C₄	1	2	yes	complemented
14	SD₁₆×Dic₄₃	C₄	1	4	yes	complemented
15	D₈×Dic₄₃	C₄	1	2	yes	complemented
16	#14582957	—	2	4	no	—
17	#14582953	—	2	4	no	—
18	C₂×#13063261	C₂²	1	1	yes	characteristic
19	#14582937	C₂²	1	1	yes	characteristic
20	#14582927	C₂²	1	2	yes	—
21	#14582925	C₂²	1	2	yes	—
22	C₂×#13063252	C₂²	1	2	yes	—
23	#14582895	C₂²	1	2	yes	—
24	#14582866	C₂²	1	2	yes	—
25	M₁₆×Dic₄₃	C₂²	1	2	yes	—
26	C₂×#13063236	C₂²	1	1	yes	characteristic
27	C₂×#13063235	C₂²	1	1	yes	characteristic
28	C₂×#13063235	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
31	#14582745	C₂²	1	1	yes	characteristic
32	#14582663	C₂²	1	2	yes	—
33	#14582660	C₂²	1	2	yes	—
34	#14582659	C₂²	1	2	yes	—
35	C₂×#135×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×#104×C₄₃	C₂²	1	2	yes	—
38	C₄₃×#408	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
40	C₂×#10621877	C₂³	1	2	yes	—
41	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
43	Q₈×Dic₄₃	C₂×C₄	1	4	yes	—
44	D₄×Dic₄₃	—	2	4	no	—
45	D₄×Dic₄₃	C₂×C₄	1	4	yes	—
46	#13063284	—	2	2	no	—
47	#13063284	C₂³	1	2	yes	—
48	C₂×C₄×Dic₄₃	—	2	2	no	—
49	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
50	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
51	#13063261	—	2	2	no	—
52	#13063259	C₂×C₄	1	2	yes	—
53	#13063257	C₂×C₄	1	2	yes	—
54	Dic₄₃⋊C₈	—	2	4	no	—
55	Dic₄₃⋊C₈	—	2	4	no	—
56	#13063252	—	2	4	no	—
57	#13063249	C₂×C₄	1	4	yes	—
58	#13063246	C₂×C₄	1	4	yes	—
59	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
60	#13063236	—	2	2	no	—
61	#13063235	—	2	2	no	—
62	#13063235	—	2	2	no	—
63	#13063235	D₄	1	2	yes	—
64	#13063235	D₄	1	2	yes	—
65	C₈×Dic₄₃	—	2	2	no	—
66	C₈×Dic₄₃	—	2	2	no	—
67	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
68	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
69	C₄₃⋊#95	—	2	2	no	—
70	C₂×#41×C₄₃	C₂³	1	2	yes	—
71	#135×C₄₃	—	2	2	no	—
72	#135×C₄₃	—	2	2	no	—
73	C₂×Q₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
74	C₂×SD₁₆×C₄₃	C₂×C₄	1	2	yes	—
75	C₂×D₈×C₄₃	C₂×C₄	1	1	yes	characteristic
76	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
80	#108×C₄₃	—	2	2	no	—
81	#104×C₄₃	—	2	4	no	—
82	C₂²⋊C₈×C₄₃	—	2	4	no	—
83	C₄×C₈×C₄₃	—	2	2	no	—
84	#41×C₄₃	—	2	2	no	—
85	#41×C₄₃	—	2	2	no	—
86	#41×C₄₃	—	2	2	no	—
87	#41×C₄₃	—	2	2	no	—
88	C₂×Q₈×C₄₃	C₂²×C₄	1	2	yes	—
89	C₂×D₄×C₄₃	—	2	2	no	—
90	C₂×D₄×C₄₃	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₄₃	—	2	2	no	—
92	C₂²×C₄×C₄₃	—	2	1	no	—
93	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
94	C₂²×Dic₄₃	—	2	4	no	—
95	C₂²×Dic₄₃	—	2	1	no	—
96	C₂²×Dic₄₃	#41	1	2	yes	—
97	Q₁₆×C₄₃	—	2	2	no	—
98	SD₁₆×C₄₃	—	2	4	no	—
99	D₈×C₄₃	—	2	2	no	—
100	M₁₆×C₄₃	—	2	2	no	—
101	M₁₆×C₄₃	—	2	2	no	—
102	C₂×C₈×C₄₃	—	2	1	no	—
103	C₂×C₈×C₄₃	—	2	1	no	—
104	C₂×C₈×C₄₃	—	2	2	no	—
105	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₄₃	#41	1	2	yes	—
109	C₂×C₈×C₄₃	#41	1	2	yes	—
110	C₄²×C₄₃	—	2	1	no	—
111	C₄²×C₄₃	#41	1	2	yes	—
112	C₂²⋊Dic₄₃	—	2	4	no	—
113	C₂²⋊Dic₄₃	—	2	2	no	—
114	C₄⋊Dic₄₃	—	2	4	no	—
115	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
116	C₄×Dic₄₃	—	2	2	no	—
117	C₄×Dic₄₃	—	2	1	no	—
118	C₄×Dic₄₃	—	2	4	no	—
119	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
120	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₄₃	#41	1	2	yes	—
123	#10621877	—	2	2	no	—
124	#10621877	—	2	2	no	—
125	C₂×C₄₃⋊C₈	—	2	1	no	—
126	C₂×C₄₃⋊C₈	—	2	1	no	—
127	C₂×C₄₃⋊C₈	—	2	1	no	—
128	C₂×C₄₃⋊C₈	—	2	1	no	—
129	C₂×C₄₃⋊C₈	—	2	2	no	—
130	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
131	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
132	C₂³×C₄₃	—	2	2	no	—
133	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
134	Q₈×C₄₃	—	2	2	no	—
135	Q₈×C₄₃	—	2	2	no	—
136	D₄×C₄₃	—	2	2	no	—
137	D₄×C₄₃	—	2	2	no	—
138	D₄×C₄₃	—	4	2	no	—
139	D₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	—	2	2	no	—
141	C₂×C₄×C₄₃	—	2	2	no	—
142	C₂×C₄×C₄₃	—	4	2	no	—
143	C₂×C₄×C₄₃	—	4	2	no	—
144	C₂×C₄×C₄₃	—	4	2	no	—
145	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
151	C₂×Dic₄₃	—	2	4	no	—
152	C₂×Dic₄₃	—	2	2	no	—
153	C₂×Dic₄₃	—	4	2	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	8	4	no	—
156	C₂×Dic₄₃	C₄×D₄	1	2	yes	—
157	C₂×Dic₄₃	C₄×D₄	1	2	yes	—
158	C₈×C₄₃	—	2	2	no	—
159	C₈×C₄₃	—	4	2	no	—
160	C₈×C₄₃	#117	1	2	yes	—
161	C₈×C₄₃	#117	1	2	yes	—
162	C₄₃⋊C₈	—	2	2	no	—
163	C₄₃⋊C₈	—	2	2	no	—
164	C₄₃⋊C₈	—	4	2	no	—
165	#21294	—	43	1	no	maximal, Sylow
166	C₂²×C₄₃	—	2	2	no	—
167	C₂²×C₄₃	—	4	4	no	—
168	C₂²×C₄₃	—	4	2	no	—
169	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×C₄₃	#529	1	1	yes	characteristic
171	C₂²×C₄₃	#529	1	1	yes	characteristic
172	C₄×C₄₃	—	4	2	no	—
173	C₄×C₄₃	—	4	2	no	—
174	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
175	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
176	C₄×C₄₃	#532	1	1	yes	characteristic
177	C₄×C₄₃	#532	1	1	yes	characteristic
178	Dic₄₃	—	2	2	no	—
179	Dic₄₃	—	2	2	no	—
180	Dic₄₃	—	4	2	no	—
181	Dic₄₃	—	4	4	no	—
182	C₄×#135	—	43	1	no	—
183	#2923	—	43	2	no	—
184	#1974	—	43	1	no	—
185	#1974	D₄₃	1	1	yes	characteristic
186	#1945	—	43	1	no	—
187	#1945	—	43	1	no	—
188	#1938	—	43	2	no	—
189	#1938	—	43	2	no	—
190	C₄×#108	—	43	1	no	—
191	#1759	—	43	2	no	—
192	#1726	—	43	1	no	—
193	C₂×C₄₃	—	2	1	no	—
194	C₂×C₄₃	—	2	1	no	—
195	C₂×C₄₃	—	8	2	no	—
196	C₂×C₄₃	#2293	1	1	yes	characteristic
197	C₂×C₄₃	#2957	1	2	yes	—
198	C₂×#135	C₂×D₄₃	1	1	yes	characteristic
199	C₄×#41	—	43	2	no	—
200	C₄×Q₁₆	—	43	2	no	—
201	C₄×SD₁₆	—	43	4	no	—
202	C₄×D₈	—	43	2	no	—
203	#433	—	86	4	no	—
204	#432	—	86	4	no	—
205	C₂×#108	—	43	1	no	—
206	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
207	#427	—	43	1	no	—
208	#424	—	43	2	no	—
209	#424	—	43	2	no	—
210	C₂×#104	—	43	2	no	—
211	C₂×#104	C₂×D₄₃	1	2	yes	—
212	#416	—	43	2	no	—
213	#408	—	43	2	no	—
214	#408	C₂×D₄₃	1	2	yes	—
215	C₄×M₁₆	—	43	2	no	—
216	C₂×#97	—	43	1	no	—
217	C₂×#97	—	43	1	no	—
218	C₂×#97	—	43	1	no	—
219	C₂×C₄×C₈	—	43	1	no	—
220	C₂×C₄×C₈	—	43	1	no	—
221	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
222	#376	—	43	1	no	—
223	#341	—	43	2	no	—
224	#339	—	43	2	no	—
225	#339	—	43	2	no	—
226	C₄₃	#21294	1	1	yes	characteristic, complemented, Sylow
227	C₂×#41	C₂²×D₄₃	1	2	yes	—
228	#135	—	2	2	no	—
229	#135	—	2	2	no	—
230	C₂×Q₁₆	C₄×D₄₃	1	1	yes	characteristic
231	C₂×SD₁₆	C₄×D₄₃	1	2	yes	—
232	C₂×D₈	C₄×D₄₃	1	1	yes	characteristic
233	C₂×M₁₆	—	43	2	no	—
234	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
235	C₂²×C₈	—	43	1	no	—
236	C₂²×C₈	—	43	1	no	—
237	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
238	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
239	C₄×Q₈	—	43	4	no	—
240	C₄×D₄	—	43	4	no	—
241	C₄×D₄	—	86	4	no	—
242	#117	—	43	2	no	—
243	#117	—	86	2	no	—
244	C₂×C₄²	—	43	1	no	—
245	C₂×C₄²	—	43	1	no	—
246	C₂×C₄²	—	86	2	no	—
247	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
248	#108	—	2	2	no	—
249	#108	—	86	2	no	—
250	#107	—	43	2	no	—
251	#106	—	43	2	no	—
252	C₄⋊C₈	—	86	4	no	—
253	C₄⋊C₈	—	86	4	no	—
254	#104	—	2	4	no	—
255	#104	—	86	4	no	—
256	#103	—	43	4	no	—
257	#102	—	43	4	no	—
258	C₂²⋊C₈	—	2	4	no	—
259	C₂²⋊C₈	—	86	4	no	—
260	#97	—	43	2	no	—
261	#97	—	43	2	no	—
262	#97	—	86	2	no	—
263	#97	—	86	2	no	—
264	#97	—	86	2	no	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	43	2	no	—
267	C₄×C₈	—	43	2	no	—
268	C₄×C₈	—	86	2	no	—
269	C₄×C₈	—	86	2	no	—
270	#95	—	86	2	no	—
271	#41	—	2	2	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	#41	—	2	2	no	—
275	C₂×Q₈	C₂×C₄×D₄₃	1	2	yes	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	C₂×C₄×D₄₃	1	2	yes	—
278	C₂²×C₄	—	2	1	no	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	43	2	no	—
281	C₂²×C₄	—	86	4	no	—
282	C₂²×C₄	—	86	1	no	—
283	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
284	Q₁₆	—	2	2	no	—
285	SD₁₆	—	2	4	no	—
286	D₈	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	86	2	no	—
290	M₁₆	—	86	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	43	2	no	—
295	C₂×C₈	—	43	2	no	—
296	C₂×C₈	—	86	1	no	—
297	C₂×C₈	—	86	1	no	—
298	C₂×C₈	—	86	1	no	—
299	C₂×C₈	—	86	1	no	—
300	C₂×C₈	—	86	2	no	—
301	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
303	C₂×C₈	D₄×D₄₃	1	2	yes	—
304	C₂×C₈	#10621900	1	2	yes	—
305	C₂×C₈	#10621900	1	2	yes	—
306	C₄⋊C₄	—	43	4	no	—
307	C₄⋊C₄	—	86	4	no	—
308	C₂²⋊C₄	—	86	4	no	—
309	C₂²⋊C₄	—	86	2	no	—
310	C₄²	—	2	1	no	—
311	C₄²	—	43	2	no	—
312	C₄²	—	43	1	no	—
313	C₄²	—	43	1	no	—
314	C₄²	—	43	2	no	—
315	C₄²	—	86	4	no	—
316	C₄²	—	86	2	no	—
317	C₄²	—	86	1	no	—
318	C₄²	#10621900	1	2	yes	—
319	C₂³	—	2	2	no	—
320	C₂³	#41×D₄₃	1	1	yes	characteristic
321	Q₈	—	2	2	no	—
322	Q₈	—	2	2	no	—
323	D₄	—	2	2	no	—
324	D₄	—	2	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	43	2	no	—
333	C₂×C₄	—	43	2	no	—
334	C₂×C₄	—	86	4	no	—
335	C₂×C₄	—	86	2	no	—
336	C₂×C₄	—	344	4	no	—
337	C₂×C₄	—	172	2	no	—
338	C₂×C₄	—	172	2	no	—
339	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
341	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
342	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
343	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
344	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	—	4	2	no	—
347	C₈	—	86	2	no	—
348	C₈	—	86	2	no	—
349	C₈	—	172	2	no	—
350	C₈	#13063281	1	2	yes	—
351	C₈	#13063281	1	2	yes	—
352	C₂²	—	2	2	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	4	no	—
355	C₂²	C₄×D₄×D₄₃	1	1	yes	characteristic
356	C₂²	#14583428	1	1	yes	characteristic
357	C₂²	#14583430	1	1	yes	characteristic
358	C₄	—	4	2	no	—
359	C₄	—	4	2	no	—
360	C₄	—	86	2	no	—
361	C₄	—	86	2	no	—
362	C₄	—	172	2	no	—
363	C₄	—	172	4	no	—
364	C₄	C₂×#13063281	1	1	yes	characteristic
365	C₄	C₄×#10621900	1	1	yes	characteristic
366	C₄	#14583444	1	1	yes	characteristic
367	C₄	#14583447	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	8	2	no	—
371	C₂	#19137057	1	1	yes	characteristic
372	C₂	#19143507	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1974	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂×#135×C₄₃	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1974	D₄₃
hypocenter	C₄₃	#21294
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1974

Complements

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

N	H ≅ G/N	type
2. #19143242	C₂	semidirect
4. #19135343	C₂	semidirect
5. #19135341	C₂	semidirect
6. #19135309	C₂	semidirect
7. #19135303	C₂	semidirect
8. #108×Dic₄₃	C₂	semidirect
9. #19134407	C₂	semidirect
10. #19134281	C₂	semidirect
13. Q₁₆×Dic₄₃	C₄	semidirect
14. SD₁₆×Dic₄₃	C₄	semidirect
15. D₈×Dic₄₃	C₄	semidirect
226. C₄₃	#21294	semidirect

Maximal subgroups

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

subgroup	index
#135×Dic₄₃	2
#19143242	2
#19135484	2
#19135343	2
#19135341	2
#19135309	2
#19135303	2
#108×Dic₄₃	2
#19134407	2
#19134281	2
C₄₃×#1974	2
#21294	43