Subgroups — #25196552

Order 11008.

612 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145109	C₂	1	1	yes	maximal, characteristic, complemented
2	#19135767	C₂	1	4	yes	maximal, complemented
3	#19135494	C₂	1	2	yes	maximal
4	#19134720	C₂	1	2	yes	maximal, complemented
5	#19134649	C₂	1	2	yes	maximal, complemented
6	#19134641	C₂	1	2	yes	maximal, complemented
7	#19134474	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#443	C₂	1	1	yes	maximal, characteristic
9	#14583456	C₂²	1	2	yes	—
10	#14583304	—	2	1	no	—
11	#14583301	—	2	1	no	—
12	#14583301	—	2	1	no	—
13	#14583297	—	2	1	no	—
14	C₂×#13063261	C₂²	1	1	yes	characteristic
15	#14582937	C₂²	1	2	yes	—
16	#14582937	C₂²	1	1	yes	characteristic
17	#14582937	C₂²	1	2	yes	—
18	#14582916	C₂²	1	4	yes	—
19	#14582873	C₂²	1	4	yes	—
20	#14582854	C₂²	1	2	yes	—
21	#14582671	C₂²	1	4	yes	—
22	#14582659	C₂²	1	2	yes	—
23	#14582656	C₂²	1	4	yes	—
24	C₂×#135×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#131×C₄₃	C₂²	1	2	yes	—
26	C₄₃×#443	—	2	1	no	—
27	C₄₃×#443	—	2	1	no	—
28	C₄₃×#443	—	2	1	no	—
29	C₄₃×#443	—	2	1	no	—
30	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×#104×C₄₃	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×#10621877	C₂³	1	2	yes	—
34	#13063325	—	2	2	no	—
35	#13063317	—	2	2	no	—
36	#13063304	—	4	2	no	—
37	C₄×Dic₈₆	—	4	2	no	—
38	#13063284	C₂³	1	2	yes	—
39	#13063284	C₂³	1	4	yes	—
40	#13063261	—	2	1	no	—
41	#13063261	—	2	1	no	—
42	#13063259	—	2	2	no	—
43	#13063259	—	2	2	no	—
44	#13063259	—	2	1	no	—
45	#13063257	—	2	2	no	—
46	#13063257	—	2	2	no	—
47	#13063257	—	2	1	no	—
48	#13063252	—	4	4	no	—
49	#13063240	—	4	4	no	—
50	#13063235	—	2	2	no	—
51	C₈×Dic₄₃	—	2	2	no	—
52	C₂×#41×C₄₃	C₂³	1	2	yes	—
53	#135×C₄₃	—	2	1	no	—
54	#135×C₄₃	—	2	1	no	—
55	#135×C₄₃	—	2	1	no	—
56	#135×C₄₃	—	2	1	no	—
57	C₂×Q₁₆×C₄₃	Q₈	1	1	yes	characteristic, complemented
58	C₂×SD₁₆×C₄₃	Q₈	1	2	yes	complemented
59	C₂×D₈×C₄₃	Q₈	1	1	yes	characteristic, complemented
60	#131×C₄₃	—	2	2	no	—
61	#131×C₄₃	—	2	2	no	—
62	#131×C₄₃	—	2	2	no	—
63	#131×C₄₃	—	2	2	no	—
64	C₂×M₁₆×C₄₃	—	2	2	no	—
65	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
66	C₂²×C₈×C₄₃	—	2	2	no	—
67	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
70	#108×C₄₃	—	2	1	no	—
71	#108×C₄₃	—	2	1	no	—
72	#104×C₄₃	—	2	2	no	—
73	#104×C₄₃	—	2	2	no	—
74	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
75	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
78	#41×C₄₃	—	2	2	no	—
79	#41×C₄₃	—	2	2	no	—
80	#41×C₄₃	—	2	2	no	—
81	#41×C₄₃	—	2	2	no	—
82	C₂×Q₈×C₄₃	C₂×Q₈	1	2	yes	—
83	C₂×D₄×C₄₃	—	2	2	no	—
84	C₂×D₄×C₄₃	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₄₃	—	2	2	no	—
86	C₂²×C₄×C₄₃	—	2	1	no	—
87	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₈₆	—	4	2	no	—
89	Q₁₆×C₄₃	—	2	1	no	—
90	Q₁₆×C₄₃	—	2	1	no	—
91	SD₁₆×C₄₃	—	2	2	no	—
92	SD₁₆×C₄₃	—	2	2	no	—
93	D₈×C₄₃	—	2	1	no	—
94	D₈×C₄₃	—	2	1	no	—
95	M₁₆×C₄₃	—	2	2	no	—
96	M₁₆×C₄₃	—	2	2	no	—
97	M₁₆×C₄₃	—	4	2	no	—
98	M₁₆×C₄₃	—	4	2	no	—
99	C₂×C₈×C₄₃	—	2	2	no	—
100	C₂×C₈×C₄₃	—	4	2	no	—
101	C₂×C₈×C₄₃	—	4	2	no	—
102	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
107	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
108	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
109	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
110	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
111	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
112	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
113	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
114	C₄²×C₄₃	—	2	1	no	—
115	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₂²⋊Dic₄₃	—	2	2	no	—
118	C₂²⋊Dic₄₃	—	2	4	no	—
119	C₄⋊Dic₄₃	—	2	4	no	—
120	C₄⋊Dic₄₃	—	2	2	no	—
121	C₄⋊Dic₄₃	—	4	2	no	—
122	Dic₄₃⋊C₄	—	4	4	no	—
123	C₄×Dic₄₃	—	2	2	no	—
124	C₄×Dic₄₃	—	2	4	no	—
125	#10621877	—	2	2	no	—
126	#10621877	—	2	2	no	—
127	C₂×C₄₃⋊C₈	—	2	2	no	—
128	C₂³×C₄₃	—	2	2	no	—
129	C₂³×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
130	Q₈×C₄₃	—	2	2	no	—
131	Q₈×C₄₃	—	2	2	no	—
132	D₄×C₄₃	—	2	2	no	—
133	D₄×C₄₃	—	2	2	no	—
134	D₄×C₄₃	—	4	2	no	—
135	D₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	—	2	1	no	—
137	C₂×C₄×C₄₃	—	2	1	no	—
138	C₂×C₄×C₄₃	—	2	2	no	—
139	C₂×C₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	—	4	2	no	—
141	C₂×C₄×C₄₃	—	4	2	no	—
142	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×Dic₄₃	—	4	4	no	—
149	C₂×Dic₄₃	—	4	2	no	—
150	Dic₈₆	—	8	4	no	—
151	C₈×C₄₃	—	4	2	no	—
152	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
153	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
154	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
155	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
156	C₈×C₄₃	#124	1	1	yes	characteristic
157	C₈×C₄₃	#124	1	1	yes	characteristic
158	C₈×C₄₃	#124	1	1	yes	characteristic
159	C₈×C₄₃	#124	1	1	yes	characteristic
160	C₄₃⋊C₈	—	4	2	no	—
161	#29154	—	43	1	no	maximal, Sylow
162	C₂²×C₄₃	—	4	1	no	—
163	C₂²×C₄₃	—	4	2	no	—
164	C₂²×C₄₃	—	8	2	no	—
165	C₂²×C₄₃	#535	1	1	yes	characteristic
166	C₂²×C₄₃	#554	1	1	yes	characteristic
167	C₂²×C₄₃	#555	1	1	yes	characteristic
168	C₄×C₄₃	—	2	2	no	—
169	C₄×C₄₃	—	4	1	no	—
170	C₄×C₄₃	—	4	2	no	—
171	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
173	C₄×C₄₃	#554	1	1	yes	characteristic
174	C₄×C₄₃	#555	1	1	yes	characteristic
175	Dic₄₃	—	8	4	no	—
176	Dic₄₃	—	8	2	no	—
177	#3160	—	43	1	no	—
178	C₂×#443	D₄₃	1	1	yes	characteristic
179	#2031	—	43	4	no	—
180	#1976	—	43	2	no	—
181	#1833	—	43	2	no	—
182	#1814	—	43	2	no	—
183	#1812	—	43	2	no	—
184	#1772	—	43	1	no	—
185	C₂×C₄₃	—	2	1	no	—
186	C₂×C₄₃	—	2	1	no	—
187	C₂×C₄₃	—	2	1	no	—
188	C₂×C₄₃	—	8	2	no	—
189	C₂×C₄₃	#2563	1	1	yes	characteristic
190	C₂×#135	C₂×D₄₃	1	1	yes	characteristic
191	C₂×#131	C₂×D₄₃	1	2	yes	—
192	#533	—	43	2	no	—
193	#500	—	86	1	no	—
194	#499	—	86	1	no	—
195	#499	—	86	1	no	—
196	#498	—	86	1	no	—
197	#443	—	2	1	no	—
198	#443	—	2	1	no	—
199	#443	—	2	1	no	—
200	#443	—	2	1	no	—
201	C₂×#108	—	43	1	no	—
202	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
203	#427	—	43	1	no	—
204	#427	—	43	2	no	—
205	#427	—	43	2	no	—
206	#421	—	43	4	no	—
207	C₂×#104	C₂×D₄₃	1	2	yes	—
208	#410	—	43	4	no	—
209	#405	—	43	2	no	—
210	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
211	#344	—	43	4	no	—
212	#339	—	43	2	no	—
213	H(3,ℤ/4)	—	43	4	no	—
214	C₄₃	#29154	1	1	yes	characteristic, complemented, Sylow
215	C₂×#41	C₂²×D₄₃	1	2	yes	—
216	#135	—	2	1	no	—
217	#135	—	2	1	no	—
218	#135	—	2	1	no	—
219	#135	—	2	1	no	—
220	C₂×Q₁₆	Dic₈₆	1	1	yes	characteristic, complemented
221	C₂×SD₁₆	Dic₈₆	1	2	yes	complemented
222	C₂×D₈	Dic₈₆	1	1	yes	characteristic, complemented
223	#131	—	2	2	no	—
224	#131	—	2	2	no	—
225	#131	—	2	2	no	—
226	#131	—	2	2	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	—	43	2	no	—
229	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
230	C₂²×C₈	—	2	2	no	—
231	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
233	C₄⋊Q₈	—	86	2	no	—
234	#125	—	86	2	no	—
235	C₂²⋊Q₈	—	172	2	no	—
236	C₄×Q₈	—	172	2	no	—
237	#117	—	43	4	no	—
238	#117	—	43	2	no	—
239	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
240	#108	—	2	1	no	—
241	#108	—	2	1	no	—
242	#108	—	86	1	no	—
243	#108	—	86	1	no	—
244	#107	—	86	1	no	—
245	#107	—	86	2	no	—
246	#107	—	86	2	no	—
247	#106	—	86	1	no	—
248	#106	—	86	2	no	—
249	#106	—	86	2	no	—
250	#104	—	2	2	no	—
251	#104	—	2	2	no	—
252	#104	—	172	4	no	—
253	#99	—	172	4	no	—
254	#97	—	86	2	no	—
255	C₄×C₈	—	86	2	no	—
256	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
257	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
258	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
259	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
260	#41	—	2	2	no	—
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	2	2	no	—
264	C₂×Q₈	—	172	2	no	—
265	C₂×Q₈	C₂×Dic₈₆	1	2	yes	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	C₂×Dic₈₆	1	2	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	1	no	—
273	SD₁₆	—	2	2	no	—
274	SD₁₆	—	2	2	no	—
275	D₈	—	2	1	no	—
276	D₈	—	2	1	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	4	2	no	—
280	M₁₆	—	4	2	no	—
281	M₁₆	—	86	2	no	—
282	M₁₆	—	86	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	2	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	86	2	no	—
287	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
288	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
289	C₂×C₈	#10621898	1	1	yes	characteristic
290	C₂×C₈	#10621898	1	1	yes	characteristic
291	C₂×C₈	#10621900	1	1	yes	characteristic
292	C₂×C₈	#10621900	1	1	yes	characteristic
293	C₂×C₈	#10621900	1	1	yes	characteristic
294	C₂×C₈	#10621900	1	1	yes	characteristic
295	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
296	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
297	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
298	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
299	C₄⋊C₄	—	86	4	no	—
300	C₄⋊C₄	—	86	2	no	—
301	C₄⋊C₄	—	172	4	no	—
302	C₄⋊C₄	—	172	2	no	—
303	C₂²⋊C₄	—	86	2	no	—
304	C₂²⋊C₄	—	86	4	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	86	2	no	—
307	C₄²	—	86	4	no	—
308	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
309	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	#13063371	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	344	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	172	2	no	—
326	C₂×C₄	—	172	4	no	—
327	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
329	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
330	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
331	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
332	C₂×C₄	#13063371	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	172	2	no	—
335	C₈	#13063304	1	1	yes	characteristic
336	C₈	#13063304	1	1	yes	characteristic
337	C₈	#13063304	1	1	yes	characteristic
338	C₈	#13063304	1	1	yes	characteristic
339	C₈	#13063312	1	1	yes	characteristic
340	C₈	#13063312	1	1	yes	characteristic
341	C₈	#13063312	1	1	yes	characteristic
342	C₈	#13063312	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	#14583463	1	1	yes	characteristic
347	C₂²	#14583606	1	1	yes	characteristic
348	C₂²	#14583614	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	344	4	no	—
353	C₄	—	344	2	no	—
354	C₄	C₂×#13063304	1	1	yes	characteristic
355	C₄	C₂×#13063312	1	1	yes	characteristic
356	C₄	#14583606	1	1	yes	characteristic
357	C₄	#14583614	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	#19139768	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14583614
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#443	C₂
socle	C₂×C₄₃	#2563
prime radical	C₂×#135×C₄₃	C₂²
hypercenter	C₂×#443	D₄₃
hypocenter	C₄₃	#29154
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. #19145109	C₂	semidirect
2. #19135767	C₂	semidirect
4. #19134720	C₂	semidirect
5. #19134649	C₂	semidirect
6. #19134641	C₂	semidirect
7. #19134474	C₂	semidirect
57. C₂×Q₁₆×C₄₃	Q₈	semidirect
58. C₂×SD₁₆×C₄₃	Q₈	semidirect
59. C₂×D₈×C₄₃	Q₈	semidirect
214. C₄₃	#29154	semidirect
220. C₂×Q₁₆	Dic₈₆	semidirect
221. C₂×SD₁₆	Dic₈₆	semidirect
222. C₂×D₈	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
#19145109	2
#19135767	2
#19135494	2
#19134720	2
#19134649	2
#19134641	2
#19134474	2
C₂×C₄₃×#443	2
#29154	43