Subgroups — #25014774

Order 11008.

438 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583014	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582719	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582684	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582683	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136208	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#2113	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063345	C₄	1	1	yes	characteristic
9	C₂×#13063331	C₂²	1	1	yes	characteristic
10	C₂×#13063265	C₂²	1	1	yes	characteristic
11	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	#14583014	—	2	2	no	—
13	#14583014	—	2	2	no	—
14	C₂×#13063260	C₄	1	1	yes	characteristic
15	C₂×#13063252	C₄	1	1	yes	characteristic
16	C₂×#13063251	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
18	#14582719	—	2	2	no	—
19	#14582684	—	2	2	no	—
20	#14582683	—	2	2	no	—
21	(C₂×D₄₃)⋊C₁₆	—	2	2	no	—
22	C₁₆×Dic₄₃	—	2	2	no	—
23	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×C₂²⋊C₁₆	—	2	2	no	—
27	C₂×#10621898	C₂×C₄	1	1	yes	characteristic
28	#13063345	—	2	2	no	—
29	#13063345	—	2	2	no	—
30	C₂×#10621885	C₈	1	1	yes	characteristic
31	C₂×#10621883	C₈	1	1	yes	characteristic
32	C₂×#10621884	C₈	1	1	yes	characteristic
33	C₂×#10621882	C₈	1	1	yes	characteristic
34	#13063331	—	2	1	no	—
35	#13063331	—	2	1	no	—
36	#13063331	—	2	1	no	—
37	#13063331	—	2	1	no	—
38	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
39	C₂×#10621872	—	2	1	no	—
40	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×D₄₃	—	2	1	no	—
42	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
43	#13063265	—	2	1	no	—
44	#13063265	—	2	1	no	—
45	C₂×C₄₃⋊C₁₆	—	2	1	no	—
46	C₂×C₄₃⋊C₁₆	—	2	1	no	—
47	C₂×C₄₃⋊C₁₆	—	2	1	no	—
48	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	—
49	#13063260	—	2	2	no	—
50	#13063252	—	2	2	no	—
51	#13063251	—	2	2	no	—
52	C₈×Dic₄₃	D₄	1	2	yes	—
53	C₈×Dic₄₃	D₄	1	2	yes	—
54	C₂×#41×C₄₃	C₂×C₄	1	1	yes	characteristic
55	#131×C₄₃	—	2	1	no	—
56	#131×C₄₃	—	2	1	no	—
57	#131×C₄₃	—	2	1	no	—
58	#131×C₄₃	—	2	1	no	—
59	C₂×M₁₆×C₄₃	—	2	1	no	—
60	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₄₃	—	2	1	no	—
62	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
63	M₃₂×C₄₃	—	2	1	no	—
64	M₃₂×C₄₃	—	2	1	no	—
65	C₂×C₁₆×C₄₃	—	2	1	no	—
66	C₂×C₁₆×C₄₃	—	2	1	no	—
67	C₂×C₁₆×C₄₃	—	2	1	no	—
68	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
69	#41×C₄₃	—	2	1	no	—
70	#41×C₄₃	—	2	1	no	—
71	#41×C₄₃	—	2	1	no	—
72	#41×C₄₃	—	2	1	no	—
73	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
74	C₂×D₄×C₄₃	—	2	1	no	—
75	C₂×D₄×C₄₃	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	—	2	1	no	—
77	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₄₃	—	2	1	no	—
79	C₂²×Dic₄₃	—	2	1	no	—
80	#10621898	—	2	1	no	—
81	#10621898	—	2	1	no	—
82	#10621898	—	2	1	no	—
83	#10621898	—	2	1	no	—
84	C₂×D₁₇₂	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₄₃	—	2	1	no	—
86	C₂×Dic₈₆	C₂×C₈	1	1	yes	characteristic
87	M₁₆×C₄₃	—	2	1	no	—
88	M₁₆×C₄₃	—	2	1	no	—
89	M₁₆×C₄₃	—	4	1	no	—
90	M₁₆×C₄₃	—	4	1	no	—
91	C₂×C₈×C₄₃	—	2	1	no	—
92	C₂×C₈×C₄₃	—	4	1	no	—
93	C₂×C₈×C₄₃	—	4	1	no	—
94	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
100	#10621885	—	2	2	no	—
101	#10621884	—	2	2	no	—
102	#10621883	—	2	2	no	—
103	#10621882	—	2	2	no	—
104	C₄×Dic₄₃	—	2	1	no	—
105	C₄×Dic₄₃	C₂²⋊C₄	1	2	yes	—
106	#10621877	—	2	1	no	—
107	#10621877	—	2	1	no	—
108	C₂×C₄₃⋊C₈	—	2	1	no	—
109	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
110	C₂×C₄₃⋊C₈	C₂×C₈	1	1	yes	characteristic
111	C₂×C₄₃⋊C₈	C₂×C₈	1	1	yes	characteristic
112	C₂×C₄₃⋊C₈	M₁₆	1	2	yes	—
113	#10621872	—	4	1	no	—
114	#10621872	—	4	1	no	—
115	C₈×D₄₃	—	4	1	no	—
116	C₈×D₄₃	—	4	1	no	—
117	C₁₆×C₄₃	—	2	2	no	—
118	C₁₆×C₄₃	—	4	1	no	—
119	C₄₃⋊C₁₆	—	2	2	no	—
120	C₄₃⋊C₁₆	—	4	1	no	—
121	C₂³×C₄₃	—	2	1	no	—
122	C₂³×C₄₃	#131	1	1	yes	characteristic
123	C₂²×D₄₃	—	2	1	no	—
124	Q₈×C₄₃	—	2	1	no	—
125	Q₈×C₄₃	—	2	1	no	—
126	D₄×C₄₃	—	2	1	no	—
127	D₄×C₄₃	—	2	1	no	—
128	D₄×C₄₃	—	4	1	no	—
129	D₄×C₄₃	—	4	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	—	4	1	no	—
133	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
139	C₂²⋊D₄₃	—	4	1	no	—
140	C₂²⋊D₄₃	—	4	1	no	—
141	C₂×Dic₄₃	—	2	4	no	—
142	C₂×Dic₄₃	—	2	2	no	—
143	C₂×Dic₄₃	—	2	1	no	—
144	D₁₇₂	—	2	1	no	—
145	D₁₇₂	—	2	1	no	—
146	C₄×D₄₃	—	4	1	no	—
147	C₄×D₄₃	—	4	1	no	—
148	Dic₈₆	—	2	1	no	—
149	Dic₈₆	—	2	1	no	—
150	C₈×C₄₃	—	4	1	no	—
151	C₈×C₄₃	#121	1	1	yes	characteristic
152	C₈×C₄₃	#121	1	1	yes	characteristic
153	C₈×C₄₃	#124	1	1	yes	characteristic
154	C₈×C₄₃	#127	1	1	yes	characteristic
155	C₄₃⋊C₈	—	4	1	no	—
156	C₄₃⋊C₈	C₂²⋊C₈	1	2	yes	—
157	C₄₃⋊C₈	C₂²⋊C₈	1	2	yes	—
158	#13195	—	43	1	no	maximal, Sylow
159	C₂²×C₄₃	—	2	2	no	—
160	C₂²×C₄₃	—	4	2	no	—
161	C₂²×C₄₃	—	4	1	no	—
162	C₂²×C₄₃	#408	1	1	yes	characteristic
163	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
164	C₂²×C₄₃	#436	1	1	yes	characteristic
165	C₂×D₄₃	—	4	2	no	—
166	C₂×D₄₃	—	4	1	no	—
167	C₄×C₄₃	—	4	1	no	—
168	C₄×C₄₃	#390	1	1	yes	characteristic
169	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
170	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
171	C₄×C₄₃	#436	1	1	yes	characteristic
172	Dic₄₃	—	2	4	no	—
173	Dic₄₃	—	4	1	no	—
174	C₂×#443	—	43	1	no	—
175	C₂×#364	—	43	1	no	—
176	C₂×#350	—	43	1	no	—
177	C₂×#350	—	43	1	no	—
178	#2113	—	43	1	no	—
179	#2113	D₄₃	1	1	yes	characteristic, complemented
180	C₂×C₄×C₁₆	—	43	1	no	—
181	C₂×C₄₃	—	2	1	no	—
182	C₂×C₄₃	—	2	1	no	—
183	C₂×C₄₃	—	8	1	no	—
184	C₂×C₄₃	#1954	1	1	yes	characteristic
185	C₂×C₄₃	#2170	1	2	yes	—
186	D₄₃	—	8	1	no	—
187	C₂×#135	—	43	1	no	—
188	C₂×#131	—	43	1	no	—
189	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
190	C₂×M₃₂	—	43	1	no	—
191	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
192	C₂²×C₁₆	—	43	1	no	—
193	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
194	#443	—	86	2	no	—
195	#443	—	86	2	no	—
196	C₂×#108	—	43	1	no	—
197	C₂×#104	—	43	1	no	—
198	C₂×#104	—	43	1	no	—
199	C₂×C₄×C₈	—	43	1	no	—
200	#364	—	86	2	no	—
201	#350	—	86	2	no	—
202	#350	—	86	2	no	—
203	C₂²⋊C₁₆	—	2	2	no	—
204	C₂²⋊C₁₆	—	86	2	no	—
205	C₄×C₁₆	—	86	2	no	—
206	C₄₃	#13195	1	1	yes	characteristic, complemented, Sylow
207	C₂×#41	—	43	1	no	—
208	C₂×#41	C₄×D₄₃	1	1	yes	characteristic
209	#135	—	86	2	no	—
210	#135	—	86	2	no	—
211	C₂×Q₁₆	—	43	1	no	—
212	C₂×SD₁₆	—	43	1	no	—
213	C₂×SD₁₆	—	43	1	no	—
214	C₂×D₈	—	43	1	no	—
215	#131	—	2	1	no	—
216	#131	—	2	1	no	—
217	#131	—	2	1	no	—
218	#131	—	2	1	no	—
219	#131	—	86	1	no	—
220	#131	—	86	1	no	—
221	#131	—	86	1	no	—
222	#131	—	86	1	no	—
223	C₂×M₁₆	—	2	1	no	—
224	C₂×M₁₆	—	43	1	no	—
225	C₂×M₁₆	—	86	1	no	—
226	C₂×M₁₆	C₄×D₄₃	1	1	yes	characteristic
227	C₂²×C₈	—	2	1	no	—
228	C₂²×C₈	—	43	1	no	—
229	C₂²×C₈	—	86	1	no	—
230	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
231	C₂×C₄²	—	43	1	no	—
232	M₃₂	—	2	1	no	—
233	M₃₂	—	2	1	no	—
234	M₃₂	—	86	1	no	—
235	M₃₂	—	86	1	no	—
236	C₂×C₁₆	—	2	1	no	—
237	C₂×C₁₆	—	2	1	no	—
238	C₂×C₁₆	—	2	1	no	—
239	C₂×C₁₆	—	43	2	no	—
240	C₂×C₁₆	—	86	1	no	—
241	C₂×C₁₆	—	86	1	no	—
242	C₂×C₁₆	—	86	1	no	—
243	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
244	#108	—	86	2	no	—
245	#104	—	86	2	no	—
246	#104	—	86	2	no	—
247	C₄×C₈	—	43	2	no	—
248	C₄×C₈	—	43	2	no	—
249	#41	—	2	1	no	—
250	#41	—	2	1	no	—
251	#41	—	2	1	no	—
252	#41	—	2	1	no	—
253	#41	—	86	1	no	—
254	#41	—	86	1	no	—
255	#41	—	86	1	no	—
256	#41	—	86	1	no	—
257	C₂×Q₈	—	43	1	no	—
258	C₂×Q₈	C₈×D₄₃	1	1	yes	characteristic
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	43	1	no	—
261	C₂×D₄	—	86	1	no	—
262	C₂×D₄	C₈×D₄₃	1	1	yes	characteristic
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	86	1	no	—
265	C₂²×C₄	—	86	1	no	—
266	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
267	Q₁₆	—	86	2	no	—
268	SD₁₆	—	86	2	no	—
269	SD₁₆	—	86	2	no	—
270	D₈	—	86	2	no	—
271	M₁₆	—	2	1	no	—
272	M₁₆	—	2	1	no	—
273	M₁₆	—	4	1	no	—
274	M₁₆	—	4	1	no	—
275	M₁₆	—	86	1	no	—
276	M₁₆	—	86	1	no	—
277	M₁₆	—	172	1	no	—
278	M₁₆	—	172	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	43	2	no	—
283	C₂×C₈	—	43	1	no	—
284	C₂×C₈	—	43	1	no	—
285	C₂×C₈	—	43	2	no	—
286	C₂×C₈	—	86	1	no	—
287	C₂×C₈	—	172	1	no	—
288	C₂×C₈	—	172	1	no	—
289	C₂×C₈	#10621898	1	1	yes	characteristic
290	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
291	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
292	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
293	C₂×C₈	#10621900	1	1	yes	characteristic
294	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
295	C₄²	—	43	2	no	—
296	C₄²	—	86	1	no	—
297	C₁₆	—	2	2	no	—
298	C₁₆	—	4	1	no	—
299	C₁₆	—	86	2	no	—
300	C₁₆	—	172	1	no	—
301	C₂³	—	2	1	no	—
302	C₂³	—	86	1	no	—
303	C₂³	#13063331	1	1	yes	characteristic
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	86	1	no	—
307	Q₈	—	86	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	4	1	no	—
311	D₄	—	4	1	no	—
312	D₄	—	86	1	no	—
313	D₄	—	86	1	no	—
314	D₄	—	172	1	no	—
315	D₄	—	172	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	86	4	no	—
320	C₂×C₄	—	86	2	no	—
321	C₂×C₄	—	86	1	no	—
322	C₂×C₄	—	172	1	no	—
323	C₂×C₄	—	172	1	no	—
324	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
325	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
326	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
327	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
328	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
329	C₂×C₄	#13063333	1	1	yes	characteristic
330	C₈	—	4	1	no	—
331	C₈	—	43	2	no	—
332	C₈	—	43	2	no	—
333	C₈	—	172	1	no	—
334	C₈	#13063297	1	1	yes	characteristic
335	C₈	#13063298	1	1	yes	characteristic
336	C₈	#13063311	1	1	yes	characteristic
337	C₈	#13063321	1	1	yes	characteristic
338	C₂²	—	2	2	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	172	1	no	—
342	C₂²	—	172	2	no	—
343	C₂²	#14582864	1	1	yes	characteristic
344	C₂²	C₈×C₂²⋊D₄₃	1	1	yes	characteristic
345	C₂²	#14582974	1	1	yes	characteristic
346	C₄	—	4	1	no	—
347	C₄	—	86	4	no	—
348	C₄	—	172	1	no	—
349	C₄	#14582799	1	1	yes	characteristic
350	C₄	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
351	C₄	(C₂²⋊D₄₃)⋊C₈	1	1	yes	characteristic
352	C₄	#14582975	1	1	yes	characteristic
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	8	1	no	—
356	C₂	—	344	1	no	—
357	C₂	#19135387	1	1	yes	characteristic
358	C₂	#19136430	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#2113	C₂
socle	C₂²×C₄₃	C₈×D₄
prime radical	C₂×#13063345	C₄
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#2113	D₄₃
hypocenter	C₄₃	#13195
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ #2113

Complements

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

N	H ≅ G/N	type
2. C₂×#14582719	C₂	semidirect
3. C₂×#14582684	C₂	semidirect
4. C₂×#14582683	C₂	semidirect
5. #19136208	C₂	semidirect
6. C₂×C₁₆×Dic₄₃	C₂	semidirect
7. C₄₃×#2113	C₂	semidirect
179. #2113	D₄₃	semidirect
206. C₄₃	#13195	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583014	2
C₂×#14582719	2
C₂×#14582684	2
C₂×#14582683	2
#19136208	2
C₂×C₁₆×Dic₄₃	2
C₄₃×#2113	2
#13195	43