Subgroups — #25253242

Order 11008.

674 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583341	C₂	1	2	yes	maximal, complemented
2	#19146354	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136851	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136709	C₂	1	2	yes	maximal, complemented
6	#19136645	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136635	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136538	C₂	1	2	yes	maximal, complemented
9	#19136513	C₂	1	2	yes	maximal, complemented
10	#19136428	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063348	C₂²	1	2	yes	—
13	C₂×Q₁₆×D₄₃	C₂²	1	2	yes	—
14	#14583571	—	2	1	no	—
15	Q₈×D₁₇₂	—	2	1	no	—
16	#14583341	—	2	8	no	—
17	C₂×#13063271	C₂²	1	2	yes	—
18	C₂×#13063270	C₂²	1	2	yes	—
19	C₂×#13063270	C₂²	1	2	yes	—
20	C₂×#13063268	C₂²	1	2	yes	—
21	C₂×#13063264	C₂²	1	2	yes	—
22	#14583279	C₂²	1	1	yes	characteristic
23	#14583276	C₂²	1	1	yes	characteristic
24	#14583141	C₂²	1	2	yes	—
25	#14583136	C₂²	1	2	yes	—
26	#14583109	—	2	1	no	—
27	#14583099	—	2	1	no	—
28	#14583056	—	2	2	no	—
29	#14583047	—	2	2	no	—
30	C₈×D₁₇₂	C₂²	1	1	yes	characteristic
31	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
32	#14582702	C₂²	1	2	yes	—
33	#14582699	C₂²	1	2	yes	—
34	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	—
35	C₂×Q₃₂×C₄₃	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×#366	C₂²	1	1	yes	characteristic
39	C₄₃×#358	C₂²	1	1	yes	characteristic
40	C₄₃×#358	C₂²	1	1	yes	characteristic
41	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
42	C₂×#10621902	—	2	2	no	—
43	C₂×Q₈×D₄₃	—	2	2	no	—
44	#13063348	—	2	4	no	—
45	#13063348	—	4	2	no	—
46	C₂×Dic₁₇₂	—	2	1	no	—
47	C₂×Dic₁₇₂	C₂³	1	2	yes	—
48	C₂×#10621885	—	2	2	no	—
49	Q₁₆×D₄₃	—	2	4	no	—
50	Q₁₆×D₄₃	—	4	2	no	—
51	C₂×#10621884	—	2	2	no	—
52	C₂×D₃₄₄	—	2	1	no	—
53	C₂×D₃₄₄	C₂³	1	2	yes	—
54	C₂×C₈×D₄₃	C₂³	1	2	yes	—
55	#13063325	—	2	1	no	—
56	#13063325	—	4	1	no	—
57	#13063322	—	2	1	no	—
58	#13063322	—	4	1	no	—
59	#13063305	—	2	2	no	—
60	#13063305	—	4	2	no	—
61	#13063300	—	2	2	no	—
62	#13063300	—	4	2	no	—
63	C₄×D₁₇₂	—	4	1	no	—
64	C₄×D₁₇₂	—	4	1	no	—
65	C₄×D₁₇₂	D₄	1	1	yes	characteristic
66	#13063271	—	2	4	no	—
67	#13063270	—	2	4	no	—
68	#13063270	—	4	2	no	—
69	#13063268	—	2	4	no	—
70	#13063264	—	2	4	no	—
71	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
72	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
73	#13063248	—	2	2	no	—
74	#13063245	—	2	2	no	—
75	(C₂×D₄₃)⋊C₈	D₄	1	2	yes	—
76	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
77	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
78	C₄×Q₈×C₄₃	—	2	1	no	—
79	C₄×Q₈×C₄₃	—	2	1	no	—
80	Q₃₂×C₄₃	D₄	1	4	yes	complemented
81	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
83	#107×C₄₃	C₂³	1	1	yes	characteristic
84	#107×C₄₃	C₂³	1	1	yes	characteristic
85	#103×C₄₃	—	2	1	no	—
86	#103×C₄₃	—	2	1	no	—
87	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
88	C₂×Q₈×C₄₃	—	2	1	no	—
89	C₂×Q₈×C₄₃	—	2	1	no	—
90	#10621902	—	4	4	no	—
91	#10621902	—	8	2	no	—
92	Q₈×D₄₃	—	4	4	no	—
93	Q₈×D₄₃	—	8	2	no	—
94	C₂×D₁₇₂	—	2	2	no	—
95	C₂×D₁₇₂	—	4	1	no	—
96	C₂×D₁₇₂	—	4	1	no	—
97	C₂×D₁₇₂	—	4	2	no	—
98	C₂×D₁₇₂	—	4	2	no	—
99	C₂×D₁₇₂	D₈	1	1	yes	characteristic
100	C₂×C₄×D₄₃	—	4	2	no	—
101	C₂×C₄×D₄₃	—	4	2	no	—
102	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
103	C₂×Dic₈₆	—	2	2	no	—
104	C₂×Dic₈₆	—	4	2	no	—
105	Q₁₆×C₄₃	—	2	1	no	—
106	Q₁₆×C₄₃	—	2	1	no	—
107	Q₁₆×C₄₃	C₂×D₄	1	2	yes	—
108	Q₁₆×C₄₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₄₃	—	2	1	no	—
112	C₄⋊C₄×C₄₃	—	2	1	no	—
113	C₄⋊C₄×C₄₃	—	4	1	no	—
114	C₄⋊C₄×C₄₃	—	4	1	no	—
115	C₄²×C₄₃	—	4	1	no	—
116	C₄²×C₄₃	—	4	1	no	—
117	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
118	#10621885	—	4	4	no	—
119	#10621884	—	4	4	no	—
120	(C₂×D₄₃)⋊C₄	—	4	2	no	—
121	(C₂×D₄₃)⋊C₄	—	4	2	no	—
122	(C₂×D₄₃)⋊C₄	D₈	1	2	yes	—
123	C₄⋊Dic₄₃	—	4	1	no	—
124	C₄⋊Dic₄₃	—	4	1	no	—
125	C₄⋊Dic₄₃	—	4	2	no	—
126	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
127	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
128	Dic₁₇₂	—	2	2	no	—
129	Dic₁₇₂	—	2	2	no	—
130	Dic₁₇₂	—	8	1	no	—
131	D₃₄₄	—	2	2	no	—
132	D₃₄₄	—	2	2	no	—
133	D₃₄₄	—	8	1	no	—
134	C₈×D₄₃	—	2	2	no	—
135	C₈×D₄₃	—	2	2	no	—
136	C₁₆×C₄₃	—	2	1	no	—
137	C₁₆×C₄₃	C₂×D₄	1	2	yes	—
138	C₄₃⋊C₁₆	—	2	2	no	—
139	C₂²×D₄₃	—	4	2	no	—
140	C₂²×D₄₃	C₂×D₈	1	2	yes	—
141	Q₈×C₄₃	—	2	2	no	—
142	Q₈×C₄₃	—	2	2	no	—
143	Q₈×C₄₃	—	4	1	no	—
144	Q₈×C₄₃	—	4	1	no	—
145	C₂×C₄×C₄₃	—	4	1	no	—
146	C₂×C₄×C₄₃	—	4	1	no	—
147	C₂×C₄×C₄₃	—	4	1	no	—
148	C₂×C₄×C₄₃	—	4	1	no	—
149	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×Dic₄₃	—	4	2	no	—
153	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
154	D₁₇₂	—	2	2	no	—
155	D₁₇₂	—	4	2	no	—
156	D₁₇₂	—	4	2	no	—
157	D₁₇₂	—	8	4	no	—
158	D₁₇₂	—	8	2	no	—
159	D₁₇₂	—	16	1	no	—
160	D₁₇₂	—	16	2	no	—
161	C₄×D₄₃	—	2	2	no	—
162	C₄×D₄₃	—	2	2	no	—
163	C₄×D₄₃	—	8	4	no	—
164	C₄×D₄₃	—	8	4	no	—
165	Dic₈₆	—	4	2	no	—
166	Dic₈₆	—	4	2	no	—
167	Dic₈₆	—	8	4	no	—
168	C₈×C₄₃	—	2	1	no	—
169	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
170	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
171	C₄₃⋊C₈	—	2	2	no	—
172	#33844	—	43	1	no	maximal, Sylow
173	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
174	C₂×D₄₃	—	2	2	no	—
175	C₂×D₄₃	—	2	2	no	—
176	C₂×D₄₃	—	2	2	no	—
177	C₂×D₄₃	—	8	2	no	—
178	C₂×D₄₃	—	8	2	no	—
179	C₂×D₄₃	—	8	2	no	—
180	C₄×C₄₃	—	2	1	no	—
181	C₄×C₄₃	—	4	2	no	—
182	C₄×C₄₃	—	4	2	no	—
183	C₄×C₄₃	—	8	1	no	—
184	C₄×C₄₃	—	8	1	no	—
185	C₄×C₄₃	#510	1	2	yes	complemented
186	C₄×C₄₃	D₄²	1	1	yes	characteristic
187	C₄×C₄₃	#576	1	1	yes	characteristic
188	Dic₄₃	—	2	2	no	—
189	Dic₄₃	—	8	2	no	—
190	C₂×#510	—	43	2	no	—
191	#3287	—	43	1	no	—
192	D₄×Q₁₆	—	43	1	no	—
193	#2248	—	43	1	no	—
194	#2220	—	43	2	no	—
195	#2209	—	43	1	no	—
196	#2207	—	43	1	no	—
197	#2189	—	43	2	no	—
198	#2185	—	43	2	no	—
199	C₄×Q₃₂	D₄₃	1	1	yes	characteristic, complemented
200	#2169	—	43	1	no	—
201	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
202	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
203	C₂×C₄₃	#3416	1	1	yes	characteristic
204	D₄₃	—	4	2	no	—
205	D₄₃	—	16	2	no	—
206	C₂×#135	—	43	2	no	—
207	C₂²×Q₁₆	—	43	2	no	—
208	#550	—	86	1	no	—
209	D₄×Q₈	—	86	1	no	—
210	#510	—	86	8	no	—
211	C₂×Q₃₂	—	43	2	no	—
212	C₂×Q₃₂	C₂×D₄₃	1	1	yes	characteristic
213	C₂×SD₃₂	—	43	2	no	—
214	C₂×SD₃₂	—	43	2	no	—
215	C₂×SD₃₂	—	43	2	no	—
216	C₂×M₃₂	—	43	2	no	—
217	#494	—	43	1	no	—
218	#493	—	43	1	no	—
219	#467	—	43	2	no	—
220	#466	—	43	2	no	—
221	#462	—	86	1	no	—
222	#460	—	86	1	no	—
223	#451	—	86	2	no	—
224	#449	—	86	2	no	—
225	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
226	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
227	C₈×D₄	—	43	1	no	—
228	#366	C₂×D₄₃	1	1	yes	characteristic
229	C₄⋊C₁₆	—	43	1	no	—
230	#358	—	43	2	no	—
231	#358	C₂×D₄₃	1	1	yes	characteristic
232	#358	C₂×D₄₃	1	1	yes	characteristic
233	#357	—	43	2	no	—
234	C₂²⋊C₁₆	—	43	2	no	—
235	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
236	C₄₃	#33844	1	1	yes	characteristic, complemented, Sylow
237	C₂×#41	—	86	2	no	—
238	C₂²×Q₈	—	86	2	no	—
239	#135	—	86	4	no	—
240	#135	—	172	2	no	—
241	C₂×Q₁₆	—	43	2	no	—
242	C₂×Q₁₆	—	86	2	no	—
243	C₂×Q₁₆	—	86	4	no	—
244	C₂×Q₁₆	—	86	1	no	—
245	C₂×Q₁₆	—	172	2	no	—
246	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
247	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
248	C₂×SD₁₆	—	86	2	no	—
249	C₂×D₈	—	43	2	no	—
250	C₂×D₈	—	86	1	no	—
251	C₂²×C₈	—	43	2	no	—
252	C₄⋊Q₈	—	86	1	no	—
253	C₄⋊Q₈	—	172	1	no	—
254	#127	—	86	1	no	—
255	#127	—	172	1	no	—
256	C₂²⋊Q₈	—	86	2	no	—
257	C₂²⋊Q₈	—	172	2	no	—
258	#121	—	86	2	no	—
259	#121	—	172	2	no	—
260	C₄×Q₈	—	2	1	no	—
261	C₄×Q₈	—	2	1	no	—
262	C₄×D₄	—	43	1	no	—
263	C₄×D₄	—	172	1	no	—
264	C₄×D₄	—	172	1	no	—
265	Q₃₂	—	86	4	no	—
266	Q₃₂	D₁₇₂	1	4	yes	complemented
267	SD₃₂	—	86	4	no	—
268	SD₃₂	—	86	4	no	—
269	SD₃₂	—	172	2	no	—
270	M₃₂	—	86	4	no	—
271	C₂×C₁₆	—	43	2	no	—
272	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
273	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
274	#107	C₂²×D₄₃	1	1	yes	characteristic
275	#107	C₂²×D₄₃	1	1	yes	characteristic
276	C₄⋊C₈	—	43	1	no	—
277	#103	—	2	1	no	—
278	#103	—	2	1	no	—
279	#103	—	86	2	no	—
280	#102	—	86	2	no	—
281	C₂²⋊C₈	—	43	2	no	—
282	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
283	#41	—	344	2	no	—
284	#41	—	172	4	no	—
285	C₂×Q₈	—	2	1	no	—
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	—	86	2	no	—
288	C₂×Q₈	—	344	2	no	—
289	C₂×Q₈	—	172	2	no	—
290	C₂×Q₈	—	172	4	no	—
291	C₂×D₄	—	43	1	no	—
292	C₂×D₄	—	86	2	no	—
293	C₂×D₄	—	172	2	no	—
294	C₂×D₄	—	172	2	no	—
295	C₂×D₄	—	172	1	no	—
296	C₂×D₄	—	172	1	no	—
297	C₂²×C₄	—	43	2	no	—
298	C₂²×C₄	—	172	2	no	—
299	C₂²×C₄	—	172	2	no	—
300	Q₁₆	—	2	1	no	—
301	Q₁₆	—	2	1	no	—
302	Q₁₆	—	86	2	no	—
303	Q₁₆	—	86	2	no	—
304	Q₁₆	—	344	1	no	—
305	Q₁₆	—	172	4	no	—
306	Q₁₆	C₂×D₁₇₂	1	2	yes	—
307	Q₁₆	C₂×D₁₇₂	1	2	yes	—
308	SD₁₆	—	172	4	no	—
309	D₈	—	86	2	no	—
310	D₈	—	86	2	no	—
311	D₈	—	344	1	no	—
312	C₂×C₈	—	43	2	no	—
313	C₂×C₈	—	86	2	no	—
314	C₂×C₈	—	86	2	no	—
315	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
316	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	4	1	no	—
321	C₄⋊C₄	—	43	1	no	—
322	C₄⋊C₄	—	172	2	no	—
323	C₄⋊C₄	—	172	1	no	—
324	C₄⋊C₄	—	172	1	no	—
325	C₂²⋊C₄	—	43	2	no	—
326	C₂²⋊C₄	—	172	2	no	—
327	C₂²⋊C₄	—	172	2	no	—
328	C₄²	—	4	1	no	—
329	C₄²	—	4	1	no	—
330	C₄²	D₄×D₄₃	1	1	yes	characteristic
331	C₁₆	—	2	1	no	—
332	C₁₆	—	86	2	no	—
333	C₁₆	C₂×D₁₇₂	1	2	yes	—
334	C₂³	—	43	2	no	—
335	C₂³	—	172	2	no	—
336	Q₈	—	2	2	no	—
337	Q₈	—	2	2	no	—
338	Q₈	—	4	1	no	—
339	Q₈	—	4	1	no	—
340	Q₈	—	344	4	no	—
341	Q₈	—	172	2	no	—
342	Q₈	—	172	2	no	—
343	D₄	—	86	2	no	—
344	D₄	—	344	4	no	—
345	D₄	—	344	2	no	—
346	D₄	—	172	2	no	—
347	D₄	—	172	2	no	—
348	D₄	—	688	1	no	—
349	D₄	—	688	2	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	43	2	no	—
355	C₂×C₄	—	86	2	no	—
356	C₂×C₄	—	86	2	no	—
357	C₂×C₄	—	344	4	no	—
358	C₂×C₄	—	344	4	no	—
359	C₂×C₄	—	172	2	no	—
360	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
361	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
362	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
363	C₈	—	2	1	no	—
364	C₈	—	86	2	no	—
365	C₈	C₂²×D₁₇₂	1	1	yes	characteristic
366	C₈	#13063371	1	1	yes	characteristic
367	C₂²	—	86	2	no	—
368	C₂²	—	86	2	no	—
369	C₂²	—	86	2	no	—
370	C₂²	—	344	2	no	—
371	C₂²	—	344	2	no	—
372	C₂²	—	344	2	no	—
373	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	4	2	no	—
377	C₄	—	8	1	no	—
378	C₄	—	8	1	no	—
379	C₄	—	86	2	no	—
380	C₄	—	344	2	no	—
381	C₄	#14583341	1	2	yes	complemented
382	C₄	D₄×D₁₇₂	1	1	yes	characteristic
383	C₄	#14583701	1	1	yes	characteristic
384	C₂	—	172	2	no	—
385	C₂	—	688	2	no	—
386	C₂	D₈×D₁₇₂	1	1	yes	characteristic
387	C₂	C₂×#14583341	1	1	yes	characteristic
388	C₂	#19147738	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄×Q₃₂×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19146354	C₂
hypercenter	C₄×Q₃₂	D₄₃
hypocenter	C₄₃	#33844
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#14583341	C₂	semidirect
3. Q₁₆×D₁₇₂	C₂	semidirect
4. #19136851	C₂	semidirect
5. #19136709	C₂	semidirect
6. #19136645	C₂	semidirect
7. #19136635	C₂	semidirect
8. #19136538	C₂	semidirect
9. #19136513	C₂	semidirect
10. #19136428	C₂	semidirect
11. C₄×Q₃₂×C₄₃	C₂	semidirect
80. Q₃₂×C₄₃	D₄	semidirect
185. C₄×C₄₃	#510	semidirect
199. C₄×Q₃₂	D₄₃	semidirect
236. C₄₃	#33844	semidirect
266. Q₃₂	D₁₇₂	semidirect
381. C₄	#14583341	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583341	2
#19146354	2
Q₁₆×D₁₇₂	2
#19136851	2
#19136709	2
#19136645	2
#19136635	2
#19136538	2
#19136513	2
#19136428	2
C₄×Q₃₂×C₄₃	2
#33844	43