Subgroups — #25013473

Order 11008.

470 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583315	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583017	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582682	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582682	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136208	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132677	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	C₄	1	1	yes	characteristic
9	C₂×#13063331	C₂²	1	1	yes	characteristic
10	C₂×#13063331	C₂²	1	1	yes	characteristic
11	#14583315	—	2	2	no	—
12	#14583315	—	2	2	no	—
13	C₂×#13063265	C₂²	1	1	yes	characteristic
14	C₂×#13063265	C₂²	1	1	yes	characteristic
15	C₂×#13063264	—	2	1	no	—
16	C₂×C₁₆×D₄₃	—	2	1	no	—
17	#14583017	—	2	2	no	—
18	#14583017	—	2	2	no	—
19	C₂×#13063261	C₄	1	1	yes	characteristic
20	C₂×#13063253	C₄	1	1	yes	characteristic
21	C₂×#13063253	C₄	1	1	yes	characteristic
22	#14582682	—	2	2	no	—
23	#14582682	—	2	2	no	—
24	(C₂×D₄₃)⋊C₁₆	—	2	1	no	—
25	(C₂×D₄₃)⋊C₁₆	—	2	1	no	—
26	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×#10621898	C₂×C₄	1	1	yes	characteristic
30	C₂×#10621898	C₂×C₄	1	1	yes	characteristic
31	#13063347	—	2	2	no	—
32	#13063347	—	2	2	no	—
33	C₂×Dic₁₇₂	C₈	1	1	yes	characteristic
34	C₂×#10621873	C₈	1	1	yes	characteristic
35	C₂×#10621873	C₈	1	1	yes	characteristic
36	C₂×D₃₄₄	C₈	1	1	yes	characteristic
37	#13063331	—	2	1	no	—
38	#13063331	—	2	1	no	—
39	#13063331	—	2	1	no	—
40	#13063331	—	2	1	no	—
41	#13063331	—	2	1	no	—
42	#13063331	—	2	1	no	—
43	#13063331	D₄	1	2	yes	—
44	#13063331	D₄	1	2	yes	—
45	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
46	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
47	C₂×#10621872	—	2	1	no	—
48	C₂×#10621872	—	2	1	no	—
49	C₂×C₈×D₄₃	—	2	1	no	—
50	C₂×C₈×D₄₃	—	2	1	no	—
51	#13063265	—	2	1	no	—
52	#13063265	—	2	1	no	—
53	#13063265	—	2	1	no	—
54	#13063265	D₄	1	2	yes	—
55	#13063264	—	2	4	no	—
56	C₂×C₄₃⋊C₁₆	—	2	1	no	—
57	C₂×C₄₃⋊C₁₆	—	2	1	no	—
58	C₁₆×D₄₃	—	2	4	no	—
59	#13063261	—	2	2	no	—
60	#13063253	—	2	2	no	—
61	#13063253	—	2	2	no	—
62	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
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	—	2	1	no	—
69	C₂×C₁₆×C₄₃	—	2	1	no	—
70	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
71	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
72	C₄×C₈×C₄₃	—	2	1	no	—
73	C₄×C₈×C₄₃	—	2	1	no	—
74	C₂²×C₄×C₄₃	—	2	1	no	—
75	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₄₃	—	2	1	no	—
77	C₂×C₂²⋊D₄₃	—	2	1	no	—
78	#10621898	—	2	1	no	—
79	#10621898	—	2	1	no	—
80	#10621898	—	2	1	no	—
81	#10621898	—	2	1	no	—
82	#10621898	—	2	1	no	—
83	#10621898	—	2	1	no	—
84	#10621898	C₂²⋊C₄	1	2	yes	—
85	#10621898	M₁₆	1	2	yes	—
86	C₂×D₁₇₂	C₂×C₈	1	1	yes	characteristic
87	C₂×D₁₇₂	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₄₃	—	2	1	no	—
89	C₂×C₄×D₄₃	—	2	1	no	—
90	C₂×Dic₈₆	C₂×C₈	1	1	yes	characteristic
91	C₂×Dic₈₆	C₂×C₈	1	1	yes	characteristic
92	C₂×C₈×C₄₃	—	2	1	no	—
93	C₂×C₈×C₄₃	—	2	1	no	—
94	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
95	C₂×C₈×C₄₃	C₂×C₈	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	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
101	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
102	C₄²×C₄₃	—	2	1	no	—
103	C₄²×C₄₃	—	2	1	no	—
104	#10621877	—	2	1	no	—
105	#10621877	—	2	1	no	—
106	#10621877	—	2	1	no	—
107	#10621877	C₂²⋊C₄	1	2	yes	—
108	C₂×C₄₃⋊C₈	—	2	1	no	—
109	C₂×C₄₃⋊C₈	—	2	1	no	—
110	Dic₁₇₂	—	2	2	no	—
111	D₃₄₄	—	2	2	no	—
112	#10621873	—	2	2	no	—
113	#10621873	—	2	2	no	—
114	#10621872	—	2	2	no	—
115	#10621872	—	2	2	no	—
116	#10621872	—	4	1	no	—
117	#10621872	—	4	1	no	—
118	C₈×D₄₃	—	2	2	no	—
119	C₈×D₄₃	—	2	2	no	—
120	C₈×D₄₃	—	4	1	no	—
121	C₈×D₄₃	—	4	1	no	—
122	C₁₆×C₄₃	—	2	1	no	—
123	C₁₆×C₄₃	—	2	1	no	—
124	C₁₆×C₄₃	—	4	1	no	—
125	C₄₃⋊C₁₆	—	2	2	no	—
126	C₄₃⋊C₁₆	—	4	1	no	—
127	C₂³×C₄₃	#131	1	1	yes	characteristic
128	C₂²×D₄₃	—	2	1	no	—
129	C₂²×D₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	2	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	—	2	2	no	—
140	C₂²⋊D₄₃	—	2	2	no	—
141	C₂²⋊D₄₃	—	4	1	no	—
142	C₂²⋊D₄₃	—	4	1	no	—
143	C₂×Dic₄₃	—	2	1	no	—
144	C₂×Dic₄₃	—	2	1	no	—
145	D₁₇₂	—	2	1	no	—
146	D₁₇₂	—	2	1	no	—
147	D₁₇₂	—	2	1	no	—
148	D₁₇₂	C₂²⋊C₈	1	2	yes	—
149	C₄×D₄₃	—	2	2	no	—
150	C₄×D₄₃	—	2	2	no	—
151	C₄×D₄₃	—	4	1	no	—
152	C₄×D₄₃	—	4	1	no	—
153	Dic₈₆	—	2	1	no	—
154	Dic₈₆	—	2	1	no	—
155	Dic₈₆	—	2	1	no	—
156	Dic₈₆	C₂²⋊C₈	1	2	yes	—
157	C₈×C₄₃	—	2	1	no	—
158	C₈×C₄₃	—	2	1	no	—
159	C₈×C₄₃	#120	1	1	yes	characteristic
160	C₈×C₄₃	#121	1	1	yes	characteristic
161	C₈×C₄₃	#121	1	1	yes	characteristic
162	C₈×C₄₃	#123	1	1	yes	characteristic
163	C₄₃⋊C₈	—	2	2	no	—
164	C₄₃⋊C₈	—	4	1	no	—
165	#12980	—	43	1	no	maximal, Sylow
166	C₂²×C₄₃	—	2	1	no	—
167	C₂²×C₄₃	—	2	1	no	—
168	C₂²×C₄₃	#408	1	1	yes	characteristic
169	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
170	C₂²×C₄₃	#435	1	1	yes	characteristic
171	C₂×D₄₃	—	2	2	no	—
172	C₂×D₄₃	—	4	1	no	—
173	C₂×D₄₃	—	4	1	no	—
174	C₂×D₄₃	—	4	1	no	—
175	C₂×D₄₃	—	4	1	no	—
176	C₂×D₄₃	—	4	1	no	—
177	C₄×C₄₃	—	4	2	no	—
178	C₄×C₄₃	#386	1	1	yes	characteristic
179	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
180	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
181	C₄×C₄₃	#435	1	1	yes	characteristic
182	Dic₄₃	—	2	2	no	—
183	Dic₄₃	—	4	1	no	—
184	C₂×#504	—	43	1	no	—
185	C₂×#443	—	43	1	no	—
186	C₂×#350	—	43	1	no	—
187	C₂×#350	—	43	1	no	—
188	#2113	—	43	1	no	—
189	#1376	—	43	1	no	—
190	#1374	D₄₃	1	1	yes	characteristic, complemented
191	C₂×C₄₃	—	2	1	no	—
192	C₂×C₄₃	—	2	1	no	—
193	C₂×C₄₃	#1869	1	1	yes	characteristic
194	C₂×C₄₃	#2170	1	1	yes	characteristic
195	C₂×C₄₃	#2171	1	1	yes	characteristic
196	D₄₃	—	4	2	no	—
197	D₄₃	—	8	1	no	—
198	C₂×#135	—	43	1	no	—
199	C₂×#131	—	43	1	no	—
200	C₂×#131	—	43	1	no	—
201	#504	—	86	2	no	—
202	#504	—	86	2	no	—
203	C₂×M₃₂	—	43	1	no	—
204	C₂×M₃₂	—	43	1	no	—
205	C₂×M₃₂	—	86	1	no	—
206	C₂²×C₁₆	—	86	1	no	—
207	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
208	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
209	#443	—	86	2	no	—
210	#443	—	86	2	no	—
211	C₂×#108	—	43	1	no	—
212	C₂×#104	—	43	1	no	—
213	C₂×#104	—	43	1	no	—
214	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
215	#350	—	86	2	no	—
216	#350	—	86	2	no	—
217	C₂²⋊C₁₆	—	86	1	no	—
218	C₂²⋊C₁₆	—	86	1	no	—
219	C₄₃	#12980	1	1	yes	characteristic, complemented, Sylow
220	C₂×#41	—	43	1	no	—
221	C₂×#41	—	43	1	no	—
222	#135	—	86	2	no	—
223	#135	—	86	2	no	—
224	C₂×Q₁₆	—	43	1	no	—
225	C₂×SD₁₆	—	43	1	no	—
226	C₂×SD₁₆	—	43	1	no	—
227	C₂×D₈	—	43	1	no	—
228	#131	—	43	2	no	—
229	#131	—	43	2	no	—
230	#131	—	86	1	no	—
231	#131	—	86	1	no	—
232	#131	—	86	1	no	—
233	#131	—	86	1	no	—
234	#131	—	86	1	no	—
235	#131	—	86	1	no	—
236	C₂×M₁₆	—	43	1	no	—
237	C₂×M₁₆	—	43	1	no	—
238	C₂×M₁₆	—	86	1	no	—
239	C₂×M₁₆	—	86	1	no	—
240	C₂²×C₈	—	86	1	no	—
241	C₂²×C₈	—	86	1	no	—
242	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
243	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
244	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
245	M₃₂	—	43	2	no	—
246	M₃₂	—	86	4	no	—
247	M₃₂	—	86	1	no	—
248	M₃₂	—	86	1	no	—
249	M₃₂	—	86	1	no	—
250	C₂×C₁₆	—	2	1	no	—
251	C₂×C₁₆	—	2	1	no	—
252	C₂×C₁₆	—	2	1	no	—
253	C₂×C₁₆	—	2	1	no	—
254	C₂×C₁₆	—	2	1	no	—
255	C₂×C₁₆	—	86	4	no	—
256	C₂×C₁₆	—	86	1	no	—
257	C₂×C₁₆	—	86	1	no	—
258	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
259	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
260	#108	—	86	2	no	—
261	#104	—	86	2	no	—
262	#104	—	86	2	no	—
263	C₄×C₈	—	2	1	no	—
264	C₄×C₈	—	2	1	no	—
265	#41	—	43	2	no	—
266	#41	—	43	2	no	—
267	#41	—	86	1	no	—
268	#41	—	86	1	no	—
269	#41	—	86	1	no	—
270	#41	—	86	1	no	—
271	#41	—	86	1	no	—
272	#41	—	86	1	no	—
273	C₂×Q₈	—	43	1	no	—
274	C₂×Q₈	—	43	1	no	—
275	C₂×D₄	—	43	1	no	—
276	C₂×D₄	—	43	1	no	—
277	C₂×D₄	—	86	1	no	—
278	C₂×D₄	—	86	1	no	—
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	86	1	no	—
281	C₂²×C₄	—	86	1	no	—
282	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
283	Q₁₆	—	86	2	no	—
284	SD₁₆	—	86	2	no	—
285	SD₁₆	—	86	2	no	—
286	D₈	—	86	2	no	—
287	M₁₆	—	43	2	no	—
288	M₁₆	—	86	1	no	—
289	M₁₆	—	86	2	no	—
290	M₁₆	—	86	2	no	—
291	M₁₆	—	86	1	no	—
292	M₁₆	—	86	1	no	—
293	M₁₆	—	172	1	no	—
294	M₁₆	—	172	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	86	1	no	—
298	C₂×C₈	—	86	2	no	—
299	C₂×C₈	—	86	2	no	—
300	C₂×C₈	—	86	1	no	—
301	C₂×C₈	—	172	1	no	—
302	C₂×C₈	—	172	1	no	—
303	C₂×C₈	C₈×D₄₃	1	1	yes	characteristic
304	C₂×C₈	C₈×D₄₃	1	1	yes	characteristic
305	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
306	C₂×C₈	#10621898	1	1	yes	characteristic
307	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
308	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
309	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
310	C₂×C₈	#10621902	1	1	yes	characteristic
311	C₄²	—	2	1	no	—
312	C₄²	—	2	1	no	—
313	C₁₆	—	2	1	no	—
314	C₁₆	—	2	1	no	—
315	C₁₆	—	4	1	no	—
316	C₁₆	—	86	2	no	—
317	C₁₆	—	172	1	no	—
318	C₂³	—	86	1	no	—
319	C₂³	—	86	1	no	—
320	C₂³	#13063331	1	1	yes	characteristic
321	Q₈	—	43	2	no	—
322	Q₈	—	86	1	no	—
323	Q₈	—	86	1	no	—
324	Q₈	—	86	1	no	—
325	D₄	—	43	2	no	—
326	D₄	—	86	2	no	—
327	D₄	—	86	2	no	—
328	D₄	—	86	1	no	—
329	D₄	—	86	1	no	—
330	D₄	—	86	1	no	—
331	D₄	—	172	1	no	—
332	D₄	—	172	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	86	1	no	—
337	C₂×C₄	—	86	2	no	—
338	C₂×C₄	—	86	2	no	—
339	C₂×C₄	—	86	1	no	—
340	C₂×C₄	—	172	1	no	—
341	C₂×C₄	—	172	1	no	—
342	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
343	C₂×C₄	#13063287	1	1	yes	characteristic
344	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
345	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
346	C₂×C₄	M₁₆×D₄₃	1	1	yes	characteristic
347	C₂×C₄	#13063333	1	1	yes	characteristic
348	C₈	—	2	1	no	—
349	C₈	—	2	1	no	—
350	C₈	—	86	2	no	—
351	C₈	—	172	1	no	—
352	C₈	#13063295	1	1	yes	characteristic
353	C₈	#13063298	1	1	yes	characteristic
354	C₈	#13063300	1	1	yes	characteristic
355	C₈	#13063310	1	1	yes	characteristic
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	86	2	no	—
359	C₂²	—	172	1	no	—
360	C₂²	—	172	1	no	—
361	C₂²	—	172	1	no	—
362	C₂²	—	172	1	no	—
363	C₂²	—	172	1	no	—
364	C₂²	#14582864	1	1	yes	characteristic
365	C₂²	C₈×D₁₇₂	1	1	yes	characteristic
366	C₂²	#14582969	1	1	yes	characteristic
367	C₄	—	4	2	no	—
368	C₄	—	86	2	no	—
369	C₄	—	172	1	no	—
370	C₄	#14582777	1	1	yes	characteristic
371	C₄	C₂²⋊C₈×D₄₃	1	1	yes	characteristic
372	C₄	#14582964	1	1	yes	characteristic
373	C₄	#14582972	1	1	yes	characteristic
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	—	344	1	no	—
377	C₂	—	172	2	no	—
378	C₂	#19134886	1	1	yes	characteristic
379	C₂	#19136433	1	1	yes	characteristic
380	C₂	#19136438	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1374	C₂
socle	C₂²×C₄₃	C₈×D₄
prime radical	C₂×#13063347	C₄
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1374	D₄₃
hypocenter	C₄₃	#12980
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ #1374

Complements

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

N	H ≅ G/N	type
1. C₂×#14583315	C₂	semidirect
3. C₂×#14582682	C₂	semidirect
4. C₂×#14582682	C₂	semidirect
5. #19136208	C₂	semidirect
6. #19132677	C₂	semidirect
7. C₄₃×#1374	C₂	semidirect
190. #1374	D₄₃	semidirect
219. C₄₃	#12980	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583315	2
C₂×#14583017	2
C₂×#14582682	2
#19136208	2
#19132677	2
C₄₃×#1374	2
#12980	43