Subgroups — #25249905

Order 11008.

682 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583329	C₂	1	1	yes	maximal, characteristic
2	C₄×#13063347	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136469	C₂	1	2	yes	maximal, complemented
4	#19136464	C₂	1	2	yes	maximal, complemented
5	#19136459	C₂	1	2	yes	maximal, complemented
6	#19136457	C₂	1	2	yes	maximal, complemented
7	#19136377	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136302	C₂	1	1	yes	maximal, characteristic, complemented
9	#19136300	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14582675	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#135×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13063347	C₂²	1	1	yes	characteristic
13	C₄×#10621898	—	2	1	no	—
14	#14583329	C₄	1	8	yes	complemented
15	C₂×#13063271	C₂²	1	1	yes	characteristic
16	C₂×#13063269	C₂²	1	1	yes	characteristic
17	C₂×#13063268	C₂²	1	1	yes	characteristic
18	C₂×#13063266	C₂²	1	1	yes	characteristic
19	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₇₂	C₂²	1	2	yes	—
22	C₄×#10621873	—	2	2	no	—
23	C₄×D₃₄₄	C₂²	1	2	yes	—
24	#14582937	C₂²	1	1	yes	characteristic
25	#14582896	—	2	1	no	—
26	#14582726	C₂²	1	2	yes	—
27	#14582724	C₂²	1	2	yes	—
28	#14582703	C₂²	1	2	yes	—
29	#14582702	C₂²	1	2	yes	—
30	#14582700	C₂²	1	2	yes	—
31	#14582699	C₂²	1	2	yes	—
32	#14582675	C₂²	1	2	yes	—
33	#14582675	C₂²	1	2	yes	—
34	C₂×#135×C₄₃	C₂²	1	1	yes	characteristic
35	C₄×#41×C₄₃	—	2	1	no	—
36	C₄×Q₁₆×C₄₃	C₂²	1	2	yes	—
37	C₄×SD₁₆×C₄₃	—	2	2	no	—
38	C₄×D₈×C₄₃	C₂²	1	2	yes	—
39	C₄₃×#427	C₂²	1	1	yes	characteristic
40	C₄₃×#416	—	2	1	no	—
41	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
42	C₂×#10621898	—	2	1	no	—
43	#13063347	—	2	2	no	—
44	#13063347	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
46	C₂×#10621873	—	2	1	no	—
47	C₂×D₃₄₄	C₂³	1	1	yes	characteristic
48	C₄×Dic₈₆	—	2	2	no	—
49	C₄×D₁₇₂	—	2	2	no	—
50	C₄×C₂²⋊D₄₃	—	4	2	no	—
51	#13063284	—	2	1	no	—
52	#13063282	—	4	1	no	—
53	C₄²×D₄₃	—	4	1	no	—
54	#13063271	C₂×C₄	1	4	yes	complemented
55	#13063269	C₂×C₄	1	4	yes	complemented
56	#13063268	C₂×C₄	1	4	yes	complemented
57	#13063266	C₂×C₄	1	4	yes	complemented
58	C₂×C₄₃⋊C₁₆	—	2	2	no	—
59	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	complemented
60	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
61	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
63	#13063259	C₂³	1	2	yes	—
64	#13063257	—	2	1	no	—
65	#13063250	—	2	2	no	—
66	#13063247	—	2	2	no	—
67	C₂×#41×C₄₃	—	2	1	no	—
68	#135×C₄₃	—	2	2	no	—
69	#135×C₄₃	C₂×C₄	1	4	yes	complemented
70	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×SD₁₆×C₄₃	—	2	1	no	—
72	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
74	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₄×Q₈×C₄₃	—	2	2	no	—
76	C₄×D₄×C₄₃	—	2	2	no	—
77	C₄×D₄×C₄₃	—	4	2	no	—
78	#117×C₄₃	—	2	1	no	—
79	#117×C₄₃	—	4	1	no	—
80	C₂×C₄²×C₄₃	—	4	1	no	—
81	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
82	#107×C₄₃	C₂³	1	2	yes	—
83	#106×C₄₃	—	2	1	no	—
84	#103×C₄₃	—	2	2	no	—
85	#102×C₄₃	—	2	2	no	—
86	C₄×C₈×C₄₃	D₄	1	2	yes	—
87	C₄×C₈×C₄₃	C₂³	1	2	yes	—
88	#41×C₄₃	—	2	4	no	—
89	#41×C₄₃	—	4	2	no	—
90	C₂×Q₈×C₄₃	—	2	1	no	—
91	C₂×D₄×C₄₃	—	2	1	no	—
92	C₂×D₄×C₄₃	—	4	1	no	—
93	C₂²×C₄×C₄₃	—	4	1	no	—
94	C₂²×C₄×C₄₃	—	4	2	no	—
95	C₂²×C₄×C₄₃	D₈	1	2	yes	—
96	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₂²⋊D₄₃	—	4	1	no	—
98	#10621898	—	2	4	no	—
99	#10621898	—	4	2	no	—
100	C₂×D₁₇₂	—	2	1	no	—
101	C₂×C₄×D₄₃	—	4	1	no	—
102	C₂×C₄×D₄₃	—	4	2	no	—
103	C₂×Dic₈₆	—	2	1	no	—
104	Q₁₆×C₄₃	—	2	1	no	—
105	Q₁₆×C₄₃	C₂²×C₄	1	2	yes	—
106	SD₁₆×C₄₃	—	2	4	no	—
107	D₈×C₄₃	—	2	1	no	—
108	D₈×C₄₃	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₄₃	—	2	2	no	—
110	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₄₃	#41	1	2	yes	—
114	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₄₃	—	2	2	no	—
116	C₄⋊C₄×C₄₃	—	4	2	no	—
117	C₂²⋊C₄×C₄₃	—	4	2	no	—
118	C₂²⋊C₄×C₄₃	—	4	1	no	—
119	C₄²×C₄₃	—	4	1	no	—
120	C₄²×C₄₃	—	4	2	no	—
121	C₄²×C₄₃	D₈	1	1	yes	characteristic
122	C₄²×C₄₃	D₈	1	1	yes	characteristic
123	C₄²×C₄₃	C₂×D₄	1	2	yes	—
124	C₂²⋊Dic₄₃	—	4	1	no	—
125	(C₂×D₄₃)⋊C₄	—	4	2	no	—
126	C₄⋊Dic₄₃	—	2	2	no	—
127	Dic₄₃⋊C₄	—	4	2	no	—
128	C₄×Dic₄₃	—	4	1	no	—
129	C₄×Dic₄₃	—	4	2	no	—
130	Dic₁₇₂	—	2	1	no	—
131	Dic₁₇₂	C₂²×C₄	1	2	yes	—
132	D₃₄₄	—	2	1	no	—
133	D₃₄₄	C₂²×C₄	1	2	yes	—
134	#10621873	—	2	4	no	—
135	C₄₃⋊C₁₆	—	2	2	no	—
136	C₄₃⋊C₁₆	C₂²×C₄	1	2	yes	—
137	C₄₃⋊C₁₆	C₂²×C₄	1	2	yes	—
138	C₂³×C₄₃	—	4	1	no	—
139	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
140	C₂²×D₄₃	—	4	1	no	—
141	Q₈×C₄₃	—	2	2	no	—
142	Q₈×C₄₃	—	4	1	no	—
143	D₄×C₄₃	—	2	2	no	—
144	D₄×C₄₃	—	4	4	no	—
145	D₄×C₄₃	—	4	1	no	—
146	C₂×C₄×C₄₃	—	4	2	no	—
147	C₂×C₄×C₄₃	—	4	1	no	—
148	C₂×C₄×C₄₃	—	4	4	no	—
149	C₂×C₄×C₄₃	—	4	2	no	—
150	C₂×C₄×C₄₃	—	8	4	no	—
151	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
152	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
153	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
154	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₄₃	#135	1	2	yes	—
156	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
157	C₂²⋊D₄₃	—	4	4	no	—
158	C₂×Dic₄₃	—	4	1	no	—
159	C₂×Dic₄₃	—	4	2	no	—
160	D₁₇₂	—	2	2	no	—
161	D₁₇₂	—	4	1	no	—
162	C₄×D₄₃	—	4	4	no	—
163	C₄×D₄₃	—	8	4	no	—
164	Dic₈₆	—	2	2	no	—
165	Dic₈₆	—	4	1	no	—
166	C₈×C₄₃	—	2	2	no	—
167	C₈×C₄₃	C₄×D₄	1	2	yes	—
168	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
169	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
170	#33451	—	43	1	no	maximal, Sylow
171	C₂²×C₄₃	—	2	2	no	—
172	C₂²×C₄₃	—	4	2	no	—
173	C₂²×C₄₃	—	8	2	no	—
174	C₂²×C₄₃	C₄×D₈	1	2	yes	—
175	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
176	C₂×D₄₃	—	4	2	no	—
177	C₂×D₄₃	—	8	2	no	—
178	C₄×C₄₃	—	2	2	no	—
179	C₄×C₄₃	—	2	2	no	—
180	C₄×C₄₃	—	4	2	no	—
181	C₄×C₄₃	—	8	2	no	—
182	C₄×C₄₃	C₄×D₈	1	2	yes	—
183	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
184	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
185	Dic₄₃	—	4	2	no	—
186	Dic₄₃	—	8	2	no	—
187	C₂×#508	—	43	1	no	—
188	C₄×#135	—	43	1	no	—
189	C₄×#135	D₄₃	1	1	yes	characteristic, complemented
190	#2177	—	43	2	no	—
191	#2176	—	43	2	no	—
192	#2175	—	43	2	no	—
193	#2175	—	43	2	no	—
194	#2158	—	43	1	no	—
195	#2138	—	43	1	no	—
196	#2138	—	43	1	no	—
197	C₂×#346	—	43	1	no	—
198	C₂×C₄₃	—	2	2	no	—
199	C₂×C₄₃	—	8	2	no	—
200	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
201	C₂×C₄₃	#3414	1	2	yes	—
202	D₄₃	—	8	2	no	—
203	C₂×#135	—	43	1	no	—
204	C₂×#135	C₂×D₄₃	1	1	yes	characteristic
205	C₄×#41	—	2	1	no	—
206	C₄×#41	—	86	1	no	—
207	#508	—	43	8	no	—
208	C₂×Q₃₂	—	43	1	no	—
209	C₂×SD₃₂	—	43	1	no	—
210	C₂×SD₃₂	—	43	1	no	—
211	C₂×D₁₆	—	43	1	no	—
212	C₂²×C₁₆	—	43	1	no	—
213	C₂²×C₁₆	—	43	1	no	—
214	C₄×Q₁₆	—	43	2	no	—
215	C₄×Q₁₆	C₂×D₄₃	1	2	yes	—
216	C₄×SD₁₆	—	2	2	no	—
217	C₄×SD₁₆	—	86	2	no	—
218	C₄×D₈	—	43	2	no	—
219	C₄×D₈	C₂×D₄₃	1	2	yes	—
220	#427	—	43	1	no	—
221	#427	C₂×D₄₃	1	1	yes	characteristic
222	#416	—	2	1	no	—
223	#416	—	86	1	no	—
224	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
225	#367	—	43	2	no	—
226	#366	—	43	2	no	—
227	#358	—	43	2	no	—
228	#358	—	43	2	no	—
229	#357	—	43	2	no	—
230	#357	—	43	2	no	—
231	#346	—	43	2	no	—
232	#346	—	43	2	no	—
233	C₄₃	#33451	1	1	yes	characteristic, complemented, Sylow
234	C₂×#41	—	2	1	no	—
235	C₂×#41	—	86	1	no	—
236	#135	—	2	2	no	—
237	#135	—	43	4	no	—
238	#135	—	86	2	no	—
239	#135	C₄×D₄₃	1	4	yes	complemented
240	C₂×Q₁₆	—	43	1	no	—
241	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
242	C₂×SD₁₆	—	2	1	no	—
243	C₂×SD₁₆	—	86	1	no	—
244	C₂×D₈	—	43	1	no	—
245	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
246	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
247	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
248	C₄×Q₈	—	2	2	no	—
249	C₄×Q₈	—	86	2	no	—
250	C₄×D₄	—	2	2	no	—
251	C₄×D₄	—	4	2	no	—
252	C₄×D₄	—	86	2	no	—
253	C₄×D₄	—	172	2	no	—
254	#117	—	2	1	no	—
255	#117	—	4	1	no	—
256	#117	—	86	1	no	—
257	#117	—	172	1	no	—
258	C₂×C₄²	—	4	1	no	—
259	C₂×C₄²	—	172	1	no	—
260	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
261	Q₃₂	—	43	4	no	—
262	SD₃₂	—	43	4	no	—
263	SD₃₂	—	43	4	no	—
264	D₁₆	—	43	4	no	—
265	C₂×C₁₆	—	43	4	no	—
266	C₂×C₁₆	—	43	1	no	—
267	C₂×C₁₆	—	43	1	no	—
268	C₂×C₁₆	—	43	2	no	—
269	C₂×C₁₆	—	86	2	no	—
270	#107	—	43	2	no	—
271	#107	C₂²×D₄₃	1	2	yes	—
272	#106	—	2	1	no	—
273	#106	—	86	1	no	—
274	#103	—	2	2	no	—
275	#103	—	86	2	no	—
276	#102	—	2	2	no	—
277	#102	—	86	2	no	—
278	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
279	C₄×C₈	C₂²×D₄₃	1	2	yes	—
280	#41	—	2	4	no	—
281	#41	—	4	2	no	—
282	#41	—	86	4	no	—
283	#41	—	172	2	no	—
284	C₂×Q₈	—	2	1	no	—
285	C₂×Q₈	—	86	1	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	—	86	1	no	—
289	C₂×D₄	—	172	1	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	4	2	no	—
292	C₂²×C₄	—	172	2	no	—
293	C₂²×C₄	—	172	1	no	—
294	C₂²×C₄	#10621882	1	2	yes	—
295	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
296	Q₁₆	—	2	1	no	—
297	Q₁₆	—	43	2	no	—
298	Q₁₆	—	86	1	no	—
299	Q₁₆	C₂×C₄×D₄₃	1	2	yes	—
300	SD₁₆	—	2	4	no	—
301	SD₁₆	—	86	4	no	—
302	D₈	—	2	1	no	—
303	D₈	—	43	2	no	—
304	D₈	—	86	1	no	—
305	D₈	C₂×C₄×D₄₃	1	2	yes	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
308	C₂×C₈	#10621898	1	2	yes	—
309	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
310	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
311	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	4	2	no	—
314	C₄⋊C₄	—	86	2	no	—
315	C₄⋊C₄	—	172	2	no	—
316	C₂²⋊C₄	—	4	1	no	—
317	C₂²⋊C₄	—	4	2	no	—
318	C₂²⋊C₄	—	172	1	no	—
319	C₂²⋊C₄	—	172	2	no	—
320	C₄²	—	4	2	no	—
321	C₄²	—	4	1	no	—
322	C₄²	—	172	2	no	—
323	C₄²	—	172	1	no	—
324	C₄²	#10621882	1	1	yes	characteristic
325	C₄²	#10621882	1	1	yes	characteristic
326	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
327	C₁₆	—	43	2	no	—
328	C₁₆	—	43	2	no	—
329	C₁₆	—	86	2	no	—
330	C₂³	—	4	1	no	—
331	C₂³	—	172	1	no	—
332	C₂³	C₂×#10621882	1	1	yes	characteristic
333	Q₈	—	2	2	no	—
334	Q₈	—	4	1	no	—
335	Q₈	—	86	2	no	—
336	Q₈	—	172	1	no	—
337	D₄	—	2	2	no	—
338	D₄	—	4	4	no	—
339	D₄	—	4	1	no	—
340	D₄	—	86	2	no	—
341	D₄	—	172	4	no	—
342	D₄	—	172	1	no	—
343	C₂×C₄	—	4	2	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	4	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	8	4	no	—
348	C₂×C₄	—	344	4	no	—
349	C₂×C₄	—	172	2	no	—
350	C₂×C₄	—	172	1	no	—
351	C₂×C₄	—	172	4	no	—
352	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
353	C₂×C₄	C₂×#10621882	1	2	yes	—
354	C₂×C₄	C₂×#10621882	1	2	yes	—
355	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
356	C₂×C₄	#13063345	1	2	yes	—
357	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
358	C₈	—	2	2	no	—
359	C₈	C₄×C₂²⋊D₄₃	1	2	yes	—
360	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
361	C₈	C₂×#10621898	1	1	yes	characteristic
362	C₂²	—	2	2	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	344	2	no	—
366	C₂²	—	172	2	no	—
367	C₂²	C₄×#10621882	1	2	yes	—
368	C₂²	C₂²×#10621882	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	2	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	8	2	no	—
373	C₄	—	344	2	no	—
374	C₄	—	172	2	no	—
375	C₄	C₄×#10621882	1	2	yes	—
376	C₄	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
377	C₄	C₂×#13063345	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	8	2	no	—
380	C₂	—	344	2	no	—
381	C₂	C₂×C₄×#10621882	1	1	yes	characteristic
382	C₂	#19147729	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#10621882
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄×#135×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#14583329	C₂
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₄×#135	D₄₃
hypocenter	C₄₃	#33451
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. C₄×#13063347	C₂	semidirect
3. #19136469	C₂	semidirect
4. #19136464	C₂	semidirect
5. #19136459	C₂	semidirect
6. #19136457	C₂	semidirect
7. #19136377	C₂	semidirect
8. #19136302	C₂	semidirect
9. #19136300	C₂	semidirect
10. C₂×#14582675	C₂	semidirect
11. C₄×#135×C₄₃	C₂	semidirect
14. #14583329	C₄	semidirect
44. #13063347	C₂×C₄	semidirect
54. #13063271	C₂×C₄	semidirect
55. #13063269	C₂×C₄	semidirect
56. #13063268	C₂×C₄	semidirect
57. #13063266	C₂×C₄	semidirect
59. C₂×C₄₃⋊C₁₆	C₂×C₄	semidirect
69. #135×C₄₃	C₂×C₄	semidirect
189. C₄×#135	D₄₃	semidirect
233. C₄₃	#33451	semidirect
239. #135	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583329	2
C₄×#13063347	2
#19136469	2
#19136464	2
#19136459	2
#19136457	2
#19136377	2
#19136302	2
#19136300	2
C₂×#14582675	2
C₄×#135×C₄₃	2
#33451	43