Subgroups — #25253237

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×#14583340	C₂	1	2	yes	maximal, complemented
2	#19146355	C₂	1	1	yes	maximal, characteristic
3	#19136852	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136710	C₂	1	2	yes	maximal, complemented
5	#19136642	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136635	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136533	C₂	1	2	yes	maximal, complemented
8	#19136512	C₂	1	2	yes	maximal, complemented
9	C₄×#13063271	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136427	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×Q₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063347	C₂²	1	2	yes	—
13	#14583570	—	2	1	no	—
14	#14583340	—	2	8	no	—
15	C₂×#13063271	C₂²	1	2	yes	—
16	C₂×#13063271	C₂²	1	1	yes	characteristic
17	C₂×#13063268	C₂²	1	2	yes	—
18	C₂×#13063268	C₂²	1	2	yes	—
19	C₂×#13063268	C₂²	1	2	yes	—
20	C₂×#13063265	C₂²	1	2	yes	—
21	#14583276	C₂²	1	1	yes	characteristic
22	#14583135	C₂²	1	2	yes	—
23	#14583100	—	2	1	no	—
24	#14583046	—	2	2	no	—
25	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
26	#14582724	C₂²	1	1	yes	characteristic
27	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
28	#14582703	C₂²	1	1	yes	characteristic
29	#14582703	C₂²	1	2	yes	—
30	#14582702	C₂²	1	1	yes	characteristic
31	#14582699	C₂²	1	2	yes	—
32	C₂²⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
33	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₂²×Q₁₆×C₄₃	C₂²	1	2	yes	—
35	D₄×Q₈×C₄₃	—	2	1	no	—
36	C₄₃×#494	C₂²	1	1	yes	characteristic
37	C₄₃×#467	C₂²	1	2	yes	—
38	C₄₃×#462	—	2	1	no	—
39	C₄₃×#451	—	2	2	no	—
40	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
42	C₂×#10621898	—	2	2	no	—
43	#13063347	—	2	4	no	—
44	#13063347	—	4	2	no	—
45	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
46	C₂×#10621873	—	2	2	no	—
47	C₂×D₃₄₄	—	2	1	no	—
48	C₂×D₃₄₄	C₂³	1	2	yes	—
49	#13063322	—	2	1	no	—
50	#13063321	—	4	1	no	—
51	#13063299	—	2	2	no	—
52	#13063298	—	4	2	no	—
53	C₄×Dic₈₆	—	2	1	no	—
54	#13063287	—	4	1	no	—
55	#13063271	—	2	4	no	—
56	#13063271	D₄	1	4	yes	complemented
57	#13063268	—	2	4	no	—
58	#13063268	—	2	4	no	—
59	#13063268	—	4	2	no	—
60	#13063265	—	2	4	no	—
61	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
62	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
64	#13063259	C₂³	1	1	yes	characteristic
65	#13063250	—	2	1	no	—
66	#13063247	—	2	2	no	—
67	C₂²×Q₈×C₄₃	—	2	2	no	—
68	C₂×Q₁₆×C₄₃	—	2	4	no	—
69	C₂×Q₁₆×C₄₃	—	2	2	no	—
70	C₂×Q₁₆×C₄₃	—	2	1	no	—
71	C₂×Q₁₆×C₄₃	—	4	2	no	—
72	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
73	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₄₃	—	2	1	no	—
76	C₄⋊Q₈×C₄₃	—	4	1	no	—
77	C₂²⋊Q₈×C₄₃	—	2	2	no	—
78	C₂²⋊Q₈×C₄₃	—	4	2	no	—
79	C₄×Q₈×C₄₃	—	2	1	no	—
80	C₄×D₄×C₄₃	—	4	1	no	—
81	C₄×D₄×C₄₃	D₄	1	1	yes	characteristic
82	#107×C₄₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₄₃	D₄	1	1	yes	characteristic
84	#103×C₄₃	—	2	1	no	—
85	#103×C₄₃	—	2	2	no	—
86	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
87	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
88	C₂×Q₈×C₄₃	—	2	1	no	—
89	C₂×Q₈×C₄₃	—	2	2	no	—
90	C₂×Q₈×C₄₃	—	4	4	no	—
91	C₂×Q₈×C₄₃	—	4	2	no	—
92	C₂×Q₈×C₄₃	—	8	2	no	—
93	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₄₃	—	4	2	no	—
95	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
96	C₂×C₂²⋊D₄₃	—	4	2	no	—
97	C₂×C₂²⋊D₄₃	—	4	2	no	—
98	#10621898	—	4	4	no	—
99	#10621898	—	8	2	no	—
100	C₂×D₁₇₂	—	2	2	no	—
101	C₂×D₁₇₂	—	4	1	no	—
102	C₂×D₁₇₂	—	4	1	no	—
103	C₂×C₄×D₄₃	—	4	2	no	—
104	C₂×Dic₈₆	—	2	1	no	—
105	Q₁₆×C₄₃	—	2	2	no	—
106	Q₁₆×C₄₃	—	2	1	no	—
107	Q₁₆×C₄₃	—	2	2	no	—
108	Q₁₆×C₄₃	—	4	4	no	—
109	Q₁₆×C₄₃	—	8	1	no	—
110	Q₁₆×C₄₃	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₄₃	—	2	2	no	—
112	C₂×C₈×C₄₃	—	2	2	no	—
113	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₄₃	—	2	1	no	—
117	C₄⋊C₄×C₄₃	—	4	1	no	—
118	C₄⋊C₄×C₄₃	—	4	1	no	—
119	C₄⋊C₄×C₄₃	—	4	2	no	—
120	C₄⋊C₄×C₄₃	—	4	1	no	—
121	C₄⋊C₄×C₄₃	D₈	1	1	yes	characteristic
122	C₂²⋊C₄×C₄₃	—	4	2	no	—
123	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
124	C₄²×C₄₃	—	4	1	no	—
125	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
126	(C₂×D₄₃)⋊C₄	—	4	2	no	—
127	C₄⋊Dic₄₃	—	2	1	no	—
128	Dic₄₃⋊C₄	—	4	1	no	—
129	C₄×Dic₄₃	—	4	1	no	—
130	Dic₁₇₂	—	2	1	no	—
131	Dic₁₇₂	C₂×D₄	1	2	yes	—
132	D₃₄₄	—	2	2	no	—
133	D₃₄₄	—	2	2	no	—
134	D₃₄₄	—	8	1	no	—
135	#10621873	—	4	4	no	—
136	C₄₃⋊C₁₆	—	2	2	no	—
137	C₄₃⋊C₁₆	—	2	1	no	—
138	C₄₃⋊C₁₆	C₂×D₄	1	2	yes	—
139	C₂³×C₄₃	C₂×D₈	1	2	yes	—
140	C₂²×D₄₃	—	4	2	no	—
141	Q₈×C₄₃	—	2	2	no	—
142	Q₈×C₄₃	—	4	1	no	—
143	Q₈×C₄₃	—	4	2	no	—
144	Q₈×C₄₃	—	4	2	no	—
145	Q₈×C₄₃	—	8	4	no	—
146	D₄×C₄₃	—	2	2	no	—
147	C₂×C₄×C₄₃	—	2	2	no	—
148	C₂×C₄×C₄₃	—	2	2	no	—
149	C₂×C₄×C₄₃	—	4	1	no	—
150	C₂×C₄×C₄₃	—	4	1	no	—
151	C₂×C₄×C₄₃	—	4	2	no	—
152	C₂×C₄×C₄₃	—	8	4	no	—
153	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
155	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
157	C₂²⋊D₄₃	—	8	4	no	—
158	C₂²⋊D₄₃	—	16	2	no	—
159	C₂×Dic₄₃	—	4	1	no	—
160	C₂×Dic₄₃	—	4	1	no	—
161	D₁₇₂	—	4	2	no	—
162	D₁₇₂	—	4	2	no	—
163	D₁₇₂	—	8	2	no	—
164	D₁₇₂	—	16	1	no	—
165	C₄×D₄₃	—	8	4	no	—
166	Dic₈₆	—	2	2	no	—
167	Dic₈₆	—	4	1	no	—
168	C₈×C₄₃	—	2	2	no	—
169	C₈×C₄₃	—	2	1	no	—
170	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
171	C₈×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
172	#33844	—	43	1	no	maximal, Sylow
173	C₂²×C₄₃	—	2	2	no	—
174	C₂²×C₄₃	—	2	2	no	—
175	C₂²×C₄₃	—	2	2	no	—
176	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
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	—	2	2	no	—
182	C₄×C₄₃	—	4	2	no	—
183	C₄×C₄₃	—	8	1	no	—
184	C₄×C₄₃	—	8	2	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	—	4	2	no	—
189	Dic₄₃	—	8	1	no	—
190	C₂×#510	—	43	2	no	—
191	#3287	—	43	1	no	—
192	D₄×Q₁₆	D₄₃	1	1	yes	characteristic, complemented
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	—	43	1	no	—
200	#2169	—	43	1	no	—
201	C₂×C₄₃	—	4	2	no	—
202	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
203	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
204	C₂×C₄₃	#3416	1	1	yes	characteristic
205	D₄₃	—	16	2	no	—
206	C₂×#135	—	43	2	no	—
207	C₂²×Q₁₆	C₂×D₄₃	1	2	yes	—
208	#550	—	86	1	no	—
209	D₄×Q₈	—	2	1	no	—
210	#510	—	86	8	no	—
211	C₂×Q₃₂	—	43	1	no	—
212	C₂×Q₃₂	—	43	2	no	—
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	C₂×D₄₃	1	1	yes	characteristic
218	#493	—	43	1	no	—
219	#467	C₂×D₄₃	1	2	yes	—
220	#466	—	43	2	no	—
221	#462	—	2	1	no	—
222	#460	—	86	1	no	—
223	#451	—	2	2	no	—
224	#449	—	86	2	no	—
225	C₄×Q₁₆	—	43	1	no	—
226	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
227	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
228	#366	—	43	1	no	—
229	C₄⋊C₁₆	—	43	1	no	—
230	#358	—	43	1	no	—
231	#358	—	43	2	no	—
232	#358	—	43	1	no	—
233	#357	—	43	2	no	—
234	C₂²⋊C₁₆	—	43	2	no	—
235	C₄×C₁₆	—	43	1	no	—
236	C₄₃	#33844	1	1	yes	characteristic, complemented, Sylow
237	C₂×#41	—	86	2	no	—
238	C₂²×Q₈	—	2	2	no	—
239	#135	—	86	4	no	—
240	#135	—	172	2	no	—
241	C₂×Q₁₆	—	2	4	no	—
242	C₂×Q₁₆	—	2	2	no	—
243	C₂×Q₁₆	—	2	1	no	—
244	C₂×Q₁₆	—	4	2	no	—
245	C₂×Q₁₆	—	43	1	no	—
246	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
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	C₂²×D₄₃	1	2	yes	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	4	1	no	—
254	#127	—	86	1	no	—
255	#127	—	172	1	no	—
256	C₂²⋊Q₈	—	2	2	no	—
257	C₂²⋊Q₈	—	4	2	no	—
258	#121	—	86	2	no	—
259	#121	—	172	2	no	—
260	C₄×Q₈	—	2	1	no	—
261	C₄×Q₈	—	86	1	no	—
262	C₄×D₄	—	4	1	no	—
263	C₄×D₄	—	172	1	no	—
264	C₄×D₄	C₂²⋊D₄₃	1	1	yes	characteristic
265	Q₃₂	—	43	4	no	—
266	Q₃₂	—	86	4	no	—
267	SD₃₂	—	86	4	no	—
268	SD₃₂	—	86	4	no	—
269	SD₃₂	—	172	2	no	—
270	M₃₂	—	86	4	no	—
271	C₂×C₁₆	—	43	1	no	—
272	C₂×C₁₆	—	43	1	no	—
273	C₂×C₁₆	—	43	2	no	—
274	#107	—	43	1	no	—
275	#107	C₂²×D₄₃	1	1	yes	characteristic
276	C₄⋊C₈	C₂²⋊D₄₃	1	1	yes	characteristic
277	#103	—	2	1	no	—
278	#103	—	2	2	no	—
279	#103	—	86	1	no	—
280	#102	—	86	2	no	—
281	C₂²⋊C₈	C₂²⋊D₄₃	1	2	yes	—
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	2	no	—
287	C₂×Q₈	—	4	4	no	—
288	C₂×Q₈	—	4	2	no	—
289	C₂×Q₈	—	8	2	no	—
290	C₂×Q₈	—	86	1	no	—
291	C₂×D₄	—	86	2	no	—
292	C₂×D₄	—	172	2	no	—
293	C₂×D₄	—	172	2	no	—
294	C₂×D₄	—	172	1	no	—
295	C₂×D₄	—	172	1	no	—
296	C₂×D₄	#10621882	1	1	yes	characteristic
297	C₂²×C₄	—	4	2	no	—
298	C₂²×C₄	—	172	2	no	—
299	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
300	Q₁₆	—	2	2	no	—
301	Q₁₆	—	2	1	no	—
302	Q₁₆	—	2	2	no	—
303	Q₁₆	—	4	4	no	—
304	Q₁₆	—	8	1	no	—
305	Q₁₆	—	43	2	no	—
306	Q₁₆	—	86	1	no	—
307	Q₁₆	D₄×D₄₃	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	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
315	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
316	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	—	4	1	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄⋊C₄	—	4	2	no	—
321	C₄⋊C₄	—	4	1	no	—
322	C₄⋊C₄	—	86	1	no	—
323	C₄⋊C₄	—	172	1	no	—
324	C₄⋊C₄	#10621882	1	1	yes	characteristic
325	C₂²⋊C₄	—	4	2	no	—
326	C₂²⋊C₄	—	172	2	no	—
327	C₂²⋊C₄	#10621882	1	2	yes	—
328	C₄²	—	4	1	no	—
329	C₄²	—	172	1	no	—
330	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
331	C₁₆	—	43	2	no	—
332	C₁₆	—	86	2	no	—
333	C₁₆	—	86	1	no	—
334	C₂³	—	172	2	no	—
335	C₂³	C₂×#10621882	1	2	yes	—
336	Q₈	—	2	2	no	—
337	Q₈	—	4	1	no	—
338	Q₈	—	4	2	no	—
339	Q₈	—	4	2	no	—
340	Q₈	—	8	4	no	—
341	Q₈	—	86	2	no	—
342	Q₈	—	172	1	no	—
343	D₄	—	2	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	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	8	4	no	—
356	C₂×C₄	—	344	4	no	—
357	C₂×C₄	—	172	1	no	—
358	C₂×C₄	—	172	1	no	—
359	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
360	C₂×C₄	C₂×#10621882	1	2	yes	—
361	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
362	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
363	C₈	—	2	1	no	—
364	C₈	—	2	2	no	—
365	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
366	C₈	#13063370	1	1	yes	characteristic
367	C₂²	—	2	2	no	—
368	C₂²	—	2	2	no	—
369	C₂²	—	2	2	no	—
370	C₂²	—	344	2	no	—
371	C₂²	—	344	2	no	—
372	C₂²	—	344	2	no	—
373	C₂²	C₂²×#10621882	1	1	yes	characteristic
374	C₄	—	2	2	no	—
375	C₄	—	2	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	8	2	no	—
378	C₄	—	8	1	no	—
379	C₄	—	344	1	no	—
380	C₄	—	172	2	no	—
381	C₄	#14583340	1	2	yes	complemented
382	C₄	D₄×C₂²⋊D₄₃	1	1	yes	characteristic
383	C₄	#14583702	1	1	yes	characteristic
384	C₂	—	4	2	no	—
385	C₂	—	688	2	no	—
386	C₂	D₄×#10621882	1	1	yes	characteristic
387	C₂	C₂×#14583340	1	1	yes	characteristic
388	C₂	#19147739	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	D₄×Q₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19146355	C₂
hypercenter	D₄×Q₁₆	D₄₃
hypocenter	C₄₃	#33844
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#14583340	C₂	semidirect
3. #19136852	C₂	semidirect
4. #19136710	C₂	semidirect
5. #19136642	C₂	semidirect
6. #19136635	C₂	semidirect
7. #19136533	C₂	semidirect
8. #19136512	C₂	semidirect
9. C₄×#13063271	C₂	semidirect
10. #19136427	C₂	semidirect
11. D₄×Q₁₆×C₄₃	C₂	semidirect
56. #13063271	D₄	semidirect
185. C₄×C₄₃	#510	semidirect
192. D₄×Q₁₆	D₄₃	semidirect
236. C₄₃	#33844	semidirect
381. C₄	#14583340	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583340	2
#19146355	2
#19136852	2
#19136710	2
#19136642	2
#19136635	2
#19136533	2
#19136512	2
C₄×#13063271	2
#19136427	2
D₄×Q₁₆×C₄₃	2
#33844	43