Subgroups — #24992868

Order 11008.

604 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144627	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#19135423	C₂	1	1	yes	maximal, characteristic
4	#19133091	C₂	1	1	yes	maximal, characteristic
5	#19133025	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132497	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583449	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₈₆	—	2	1	no	—
10	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
11	#14583218	—	2	1	no	—
12	#14583185	C₄	1	2	yes	complemented
13	#14583100	C₄	1	2	yes	complemented
14	#14583040	—	2	1	no	—
15	C₄×#10621873	—	2	2	no	—
16	#14582923	C₂²	1	1	yes	characteristic
17	C₂×#13063253	—	2	1	no	—
18	C₂×#13063253	—	2	1	no	—
19	C₄×Dic₄₃⋊C₄	—	2	2	no	—
20	C₄×Dic₄₃⋊C₄	—	2	1	no	—
21	C₄×C₄⋊Dic₄₃	—	2	1	no	—
22	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
23	C₄²×Dic₄₃	—	2	1	no	—
24	#14582634	—	2	2	no	—
25	#14582631	—	2	2	no	—
26	C₄₃×#431	C₂²	1	1	yes	characteristic
27	C₄₃×#431	C₂²	1	1	yes	characteristic
28	C₄³×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×#10621898	—	2	1	no	—
30	C₂²×Dic₈₆	D₄	1	1	yes	characteristic
31	C₂×#10621873	—	2	2	no	—
32	C₂×#10621877	—	2	1	no	—
33	#13063325	C₂×C₄	1	1	yes	characteristic
34	#13063322	C₂×C₄	1	1	yes	characteristic
35	#13063313	—	2	1	no	—
36	#13063299	—	2	1	no	—
37	#13063299	—	2	1	no	—
38	C₄×Dic₈₆	—	2	8	no	—
39	C₄×Dic₈₆	—	2	1	no	—
40	C₄×Dic₈₆	—	2	1	no	—
41	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
42	C₄×Dic₈₆	D₄	1	2	yes	—
43	C₄×D₁₇₂	—	2	1	no	—
44	C₄×D₁₇₂	—	2	1	no	—
45	C₂×C₄⋊Dic₄₃	—	2	1	no	—
46	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
47	C₂×Dic₄₃⋊C₄	—	2	2	no	—
48	C₂×Dic₄₃⋊C₄	—	2	1	no	—
49	C₂×C₄×Dic₄₃	—	2	2	no	—
50	C₂×C₄×Dic₄₃	—	2	1	no	—
51	C₂×C₄×Dic₄₃	—	2	2	no	—
52	C₂×C₄×Dic₄₃	—	2	1	no	—
53	#13063257	—	2	2	no	—
54	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
55	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
56	#13063253	—	4	2	no	—
57	#13063253	—	4	2	no	—
58	#13063250	—	2	2	no	—
59	#13063247	—	2	2	no	—
60	C₄₃⋊#95	—	2	2	no	—
61	C₄₃⋊#95	D₄	1	2	yes	—
62	C₂×C₄²×C₄₃	—	2	1	no	—
63	C₂×C₄²×C₄₃	—	2	1	no	—
64	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₄₃	—	2	1	no	—
68	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
69	C₂²⋊C₈×C₄₃	—	2	1	no	—
70	C₂²⋊C₈×C₄₃	—	2	1	no	—
71	C₄×C₈×C₄₃	—	2	1	no	—
72	C₄×C₈×C₄₃	—	2	1	no	—
73	C₂²×C₄×C₄₃	—	2	1	no	—
74	C₂²×C₄×C₄₃	—	2	1	no	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₄₃	—	4	1	no	—
79	C₂²×Dic₄₃	—	2	1	no	—
80	C₂²×Dic₄₃	—	2	1	no	—
81	C₂²×Dic₄₃	—	2	2	no	—
82	#10621898	—	4	1	no	—
83	#10621898	—	4	1	no	—
84	#10621898	—	4	1	no	—
85	#10621898	—	4	1	no	—
86	C₂×D₁₇₂	—	2	1	no	—
87	C₂×D₁₇₂	—	2	1	no	—
88	C₂×D₁₇₂	—	2	1	no	—
89	C₂×C₄×D₄₃	—	4	1	no	—
90	C₂×Dic₈₆	—	2	1	no	—
91	C₂×Dic₈₆	—	2	4	no	—
92	C₂×Dic₈₆	—	2	1	no	—
93	C₂×Dic₈₆	C₂²⋊C₄	1	2	yes	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	—	4	1	no	—
96	C₄²×C₄₃	—	2	4	no	—
97	C₄²×C₄₃	—	2	1	no	—
98	C₄²×C₄₃	—	2	1	no	—
99	C₄²×C₄₃	—	2	1	no	—
100	C₄²×C₄₃	—	4	2	no	—
101	C₄²×C₄₃	SD₁₆	1	2	yes	complemented
102	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₄₃	#41	1	1	yes	characteristic
105	C₄²×C₄₃	#41	1	1	yes	characteristic
106	(C₂×D₄₃)⋊C₄	—	4	1	no	—
107	(C₂×D₄₃)⋊C₄	—	4	1	no	—
108	C₄⋊Dic₄₃	—	2	4	no	—
109	C₄⋊Dic₄₃	—	2	1	no	—
110	C₄⋊Dic₄₃	—	2	1	no	—
111	C₄⋊Dic₄₃	—	2	1	no	—
112	C₄⋊Dic₄₃	C₂²⋊C₄	1	2	yes	—
113	Dic₄₃⋊C₄	—	2	8	no	—
114	Dic₄₃⋊C₄	—	2	2	no	—
115	Dic₄₃⋊C₄	—	2	2	no	—
116	C₄×Dic₄₃	—	2	8	no	—
117	C₄×Dic₄₃	—	2	2	no	—
118	C₄×Dic₄₃	—	2	2	no	—
119	C₄×Dic₄₃	—	4	4	no	—
120	C₄×Dic₄₃	—	4	1	no	—
121	C₄×Dic₄₃	—	4	1	no	—
122	#10621877	—	4	1	no	—
123	#10621877	—	4	1	no	—
124	C₂×C₄₃⋊C₈	—	4	1	no	—
125	#10621873	—	4	4	no	—
126	C₂³×C₄₃	#120	1	1	yes	characteristic
127	C₂²×D₄₃	—	4	1	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	2	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	—	2	2	no	—
133	C₂×C₄×C₄₃	—	4	1	no	—
134	C₂×C₄×C₄₃	—	4	1	no	—
135	C₂×C₄×C₄₃	—	4	1	no	—
136	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
144	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
146	C₂²⋊D₄₃	—	8	1	no	—
147	C₂²⋊D₄₃	—	8	1	no	—
148	C₂×Dic₄₃	—	2	4	no	—
149	C₂×Dic₄₃	—	2	2	no	—
150	C₂×Dic₄₃	—	2	4	no	—
151	C₂×Dic₄₃	—	2	2	no	—
152	C₂×Dic₄₃	—	4	2	no	—
153	C₂×Dic₄₃	—	4	2	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	4	1	no	—
156	D₁₇₂	—	4	2	no	—
157	D₁₇₂	—	4	1	no	—
158	D₁₇₂	—	4	1	no	—
159	D₁₇₂	—	8	1	no	—
160	C₄×D₄₃	—	8	1	no	—
161	C₄×D₄₃	—	8	1	no	—
162	Dic₈₆	—	2	4	no	—
163	Dic₈₆	—	2	2	no	—
164	Dic₈₆	—	4	1	no	—
165	Dic₈₆	—	4	1	no	—
166	Dic₈₆	#104	1	4	yes	complemented
167	C₈×C₄₃	—	4	2	no	—
168	C₈×C₄₃	—	8	1	no	—
169	C₄₃⋊C₈	—	8	1	no	—
170	#9234	—	43	1	no	maximal, Sylow
171	C₂²×C₄₃	—	2	1	no	—
172	C₂²×C₄₃	—	2	1	no	—
173	C₂²×C₄₃	#386	1	1	yes	characteristic
174	C₂²×C₄₃	#448	1	1	yes	characteristic
175	C₂²×C₄₃	#449	1	1	yes	characteristic
176	C₂×D₄₃	—	8	1	no	—
177	C₂×D₄₃	—	8	1	no	—
178	C₂×D₄₃	—	8	1	no	—
179	C₄×C₄₃	—	2	4	no	—
180	C₄×C₄₃	—	2	1	no	—
181	C₄×C₄₃	—	2	1	no	—
182	C₄×C₄₃	—	2	1	no	—
183	C₄×C₄₃	—	4	2	no	—
184	C₄×C₄₃	C₂×#104	1	2	yes	—
185	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
186	C₄×C₄₃	#440	1	1	yes	characteristic
187	C₄×C₄₃	#460	1	1	yes	characteristic
188	C₄×C₄₃	#464	1	1	yes	characteristic
189	Dic₄₃	—	2	4	no	—
190	Dic₄₃	—	4	4	no	—
191	Dic₄₃	—	4	2	no	—
192	Dic₄₃	—	8	1	no	—
193	#3103	—	43	1	no	—
194	C₄²×Q₈	—	43	1	no	—
195	#1962	—	43	1	no	—
196	#1502	—	43	1	no	—
197	#1487	—	43	1	no	—
198	#1303	—	43	1	no	—
199	#1276	D₄₃	1	1	yes	characteristic, complemented
200	C₂×C₄₃	—	2	1	no	—
201	C₂×C₄₃	—	2	1	no	—
202	C₂×C₄₃	#1880	1	1	yes	characteristic
203	C₂×C₄₃	#1896	1	1	yes	characteristic
204	C₂×C₄₃	#1906	1	1	yes	characteristic
205	D₄₃	—	16	1	no	—
206	#532	—	43	1	no	—
207	C₂×C₄×Q₈	—	43	1	no	—
208	C₂×C₄×Q₈	—	86	1	no	—
209	#481	—	86	1	no	—
210	#475	—	43	2	no	—
211	#460	—	43	2	no	—
212	#448	—	86	1	no	—
213	C₄×SD₁₆	—	86	2	no	—
214	#431	C₂×D₄₃	1	1	yes	characteristic
215	#431	C₂×D₄₃	1	1	yes	characteristic
216	#423	—	43	1	no	—
217	C₂×#104	—	86	1	no	—
218	C₂×#104	—	86	1	no	—
219	C₄×C₄⋊C₄	—	43	1	no	—
220	C₄×C₄⋊C₄	—	86	2	no	—
221	C₄×C₄⋊C₄	—	86	1	no	—
222	C₄×C₄⋊C₄	—	86	1	no	—
223	C₄³	—	86	1	no	—
224	C₄³	C₂×D₄₃	1	1	yes	characteristic
225	#327	—	86	2	no	—
226	Q₈⋊C₈	—	86	2	no	—
227	C₄₃	#9234	1	1	yes	characteristic, complemented, Sylow
228	C₂×#41	—	86	1	no	—
229	C₂²×Q₈	—	43	1	no	—
230	C₂×SD₁₆	—	86	2	no	—
231	C₂×M₁₆	—	86	1	no	—
232	C₄⋊Q₈	—	43	1	no	—
233	#127	—	43	1	no	—
234	#124	—	86	1	no	—
235	#121	—	86	1	no	—
236	#121	—	86	1	no	—
237	C₄×Q₈	—	43	2	no	—
238	C₄×Q₈	—	43	2	no	—
239	C₄×Q₈	—	86	8	no	—
240	C₄×Q₈	—	86	1	no	—
241	C₄×Q₈	—	86	1	no	—
242	C₄×D₄	—	86	1	no	—
243	C₄×D₄	—	86	1	no	—
244	C₂×C₄⋊C₄	—	43	1	no	—
245	C₂×C₄⋊C₄	—	86	2	no	—
246	C₂×C₄⋊C₄	—	86	1	no	—
247	C₂×C₄⋊C₄	—	86	1	no	—
248	C₂×C₄²	—	2	1	no	—
249	C₂×C₄²	—	2	1	no	—
250	C₂×C₄²	—	86	2	no	—
251	C₂×C₄²	—	86	1	no	—
252	C₂×C₄²	—	86	2	no	—
253	C₂×C₄²	—	86	1	no	—
254	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
255	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
256	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
257	#106	—	86	2	no	—
258	C₄⋊C₈	—	2	1	no	—
259	C₄⋊C₈	—	86	1	no	—
260	C₄⋊C₈	—	86	1	no	—
261	C₄⋊C₈	C₄×D₄₃	1	2	yes	—
262	#104	—	172	2	no	—
263	#104	—	172	2	no	—
264	#103	—	86	2	no	—
265	#102	—	86	2	no	—
266	C₂²⋊C₈	—	2	1	no	—
267	C₂²⋊C₈	—	2	1	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	2	1	no	—
270	#95	—	43	2	no	—
271	#95	—	86	2	no	—
272	#41	—	172	1	no	—
273	#41	—	172	1	no	—
274	#41	—	172	1	no	—
275	#41	—	172	1	no	—
276	C₂×Q₈	—	43	2	no	—
277	C₂×Q₈	—	86	4	no	—
278	C₂×Q₈	—	86	1	no	—
279	C₂×Q₈	—	86	1	no	—
280	C₂×D₄	—	86	1	no	—
281	C₂×D₄	—	86	1	no	—
282	C₂×D₄	—	86	1	no	—
283	C₂×D₄	—	172	1	no	—
284	C₂²×C₄	—	2	1	no	—
285	C₂²×C₄	—	2	1	no	—
286	C₂²×C₄	—	86	1	no	—
287	C₂²×C₄	—	86	2	no	—
288	C₂²×C₄	—	86	1	no	—
289	C₂²×C₄	—	172	1	no	—
290	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
291	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
293	SD₁₆	—	172	4	no	—
294	M₁₆	—	172	1	no	—
295	M₁₆	—	172	1	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	4	1	no	—
298	C₂×C₈	—	172	1	no	—
299	C₄⋊C₄	—	43	2	no	—
300	C₄⋊C₄	—	86	8	no	—
301	C₄⋊C₄	—	86	4	no	—
302	C₄⋊C₄	—	86	2	no	—
303	C₄⋊C₄	—	86	2	no	—
304	C₄⋊C₄	—	86	1	no	—
305	C₄⋊C₄	—	86	1	no	—
306	C₄⋊C₄	—	86	1	no	—
307	C₂²⋊C₄	—	172	1	no	—
308	C₂²⋊C₄	—	172	1	no	—
309	C₄²	—	2	4	no	—
310	C₄²	—	2	1	no	—
311	C₄²	—	2	1	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	4	2	no	—
314	C₄²	—	86	8	no	—
315	C₄²	—	86	2	no	—
316	C₄²	—	86	2	no	—
317	C₄²	—	172	4	no	—
318	C₄²	—	172	1	no	—
319	C₄²	—	172	1	no	—
320	C₄²	#10621873	1	2	yes	complemented
321	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
322	C₄²	#10621898	1	1	yes	characteristic
323	C₄²	D₄×D₄₃	1	1	yes	characteristic
324	C₄²	#10621902	1	1	yes	characteristic
325	C₂³	—	172	1	no	—
326	C₂³	#13063295	1	1	yes	characteristic
327	Q₈	—	43	4	no	—
328	Q₈	—	86	4	no	—
329	Q₈	—	86	2	no	—
330	Q₈	—	172	1	no	—
331	Q₈	—	172	1	no	—
332	D₄	—	344	1	no	—
333	D₄	—	344	1	no	—
334	D₄	—	344	1	no	—
335	D₄	—	172	2	no	—
336	D₄	—	172	1	no	—
337	D₄	—	172	1	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	86	4	no	—
347	C₂×C₄	—	86	2	no	—
348	C₂×C₄	—	86	4	no	—
349	C₂×C₄	—	86	2	no	—
350	C₂×C₄	—	344	1	no	—
351	C₂×C₄	—	344	1	no	—
352	C₂×C₄	—	172	1	no	—
353	C₂×C₄	—	172	2	no	—
354	C₂×C₄	—	172	2	no	—
355	C₂×C₄	—	172	2	no	—
356	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
357	C₂×C₄	#13063287	1	1	yes	characteristic
358	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
359	C₂×C₄	#13063298	1	1	yes	characteristic
360	C₂×C₄	#13063300	1	1	yes	characteristic
361	C₂×C₄	#13063310	1	1	yes	characteristic
362	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
363	C₂×C₄	#13063347	1	1	yes	characteristic
364	C₂×C₄	#13063353	1	1	yes	characteristic
365	C₂×C₄	#13063358	1	1	yes	characteristic
366	C₈	—	4	2	no	—
367	C₈	—	8	1	no	—
368	C₈	—	344	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	344	1	no	—
372	C₂²	—	344	1	no	—
373	C₂²	—	344	1	no	—
374	C₂²	#14582777	1	1	yes	characteristic
375	C₂²	#14583040	1	1	yes	characteristic
376	C₂²	#14583046	1	1	yes	characteristic
377	C₄	—	2	4	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	4	2	no	—
382	C₄	—	86	4	no	—
383	C₄	—	344	1	no	—
384	C₄	—	172	4	no	—
385	C₄	—	172	2	no	—
386	C₄	#104×D₄₃	1	2	yes	—
387	C₄	C₄×#10621873	1	1	yes	characteristic
388	C₄	#14583001	1	1	yes	characteristic
389	C₄	#14583100	1	1	yes	characteristic
390	C₄	#14583122	1	1	yes	characteristic
391	C₂	—	2	1	no	—
392	C₂	—	2	1	no	—
393	C₂	—	688	1	no	—
394	C₂	#19134949	1	1	yes	characteristic
395	C₂	#19135037	1	1	yes	characteristic
396	C₂	#19135095	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063300
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1276	C₂
socle	C₂²×C₄₃	#386
prime radical	#14583449	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1276	D₄₃
hypocenter	C₄₃	#9234
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1276

Complements

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

N	H ≅ G/N	type
2. C₄²×Dic₈₆	C₂	semidirect
5. #19133025	C₂	semidirect
6. #19132497	C₂	semidirect
7. C₄₃×#1276	C₂	semidirect
12. #14583185	C₄	semidirect
13. #14583100	C₄	semidirect
101. C₄²×C₄₃	SD₁₆	semidirect
166. Dic₈₆	#104	semidirect
199. #1276	D₄₃	semidirect
227. C₄₃	#9234	semidirect
320. C₄²	#10621873	semidirect

Maximal subgroups

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

subgroup	index
#19144627	2
C₄²×Dic₈₆	2
#19135423	2
#19133091	2
#19133025	2
#19132497	2
C₄₃×#1276	2
#9234	43