Subgroups — #26003461

Order 11008.

7226 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19148029	C₂	1	12	yes	maximal, complemented
2	#19148018	C₂	1	12	yes	maximal, complemented
3	#19148007	C₂	1	8	yes	maximal, complemented
4	#19147987	C₂	1	12	yes	maximal, complemented
5	#19147928	C₂	1	3	yes	maximal, complemented
6	#19147892	C₂	1	12	yes	maximal, complemented
7	C₂×#14583367	C₂	1	3	yes	maximal, complemented
8	C₄₃×#3458	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10621898	C₂²	1	3	yes	—
10	#14583523	C₂²	1	48	yes	complemented
11	#14583512	C₂²	1	24	yes	complemented
12	#14583505	C₂²	1	24	yes	complemented
13	#14583498	C₂²	1	48	yes	complemented
14	#14583489	C₂²	1	48	yes	complemented
15	#14583480	C₂²	1	12	yes	—
16	#14583475	C₂²	1	12	yes	complemented
17	#14583468	C₂²	1	12	yes	complemented
18	#14583461	C₂²	1	12	yes	—
19	#14583454	C₂²	1	24	yes	complemented
20	#14583450	C₂²	1	24	yes	complemented
21	#14583449	C₂²	1	12	yes	—
22	#14583436	C₂²	1	48	yes	complemented
23	#14583435	C₂²	1	48	yes	complemented
24	#14583434	C₂²	1	48	yes	complemented
25	#14583412	C₂²	1	48	yes	complemented
26	#14583411	C₂²	1	24	yes	complemented
27	C₄×#10621902	C₂²	1	16	yes	complemented
28	#14583367	C₂²	1	12	yes	complemented
29	#14583367	C₂²	1	12	yes	complemented
30	#14583367	C₂²	1	24	yes	complemented
31	C₂×#13063290	C₂²	1	6	yes	complemented
32	C₂×#13063287	C₂²	1	6	yes	complemented
33	C₂×(C₂²⋊D₄₃)⋊C₄	C₂²	1	12	yes	—
34	C₂×#13063282	C₂²	1	6	yes	complemented
35	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
36	C₂×C₄²×D₄₃	C₂²	1	6	yes	complemented
37	C₄₃×#544	C₂²	1	12	yes	complemented
38	C₄₃×#543	C₂²	1	4	yes	—
39	C₄₃×#533	C₂²	1	12	yes	complemented
40	C₂×#117×C₄₃	C₂²	1	3	yes	—
41	C₂×#10621902	C₂³	1	16	yes	—
42	C₂×#10621898	—	2	24	no	—
43	C₂×#10621898	C₂³	1	12	yes	—
44	C₂×#10621898	C₂³	1	12	yes	—
45	C₂²×Dic₈₆	C₂³	1	3	yes	—
46	C₂²×D₁₇₂	C₂³	1	3	yes	—
47	C₂²×C₂²⋊D₄₃	C₂³	1	6	yes	—
48	C₂³×Dic₄₃	C₂³	1	2	yes	—
49	C₂²×C₄×D₄₃	C₂³	1	6	yes	—
50	C₂²×C₄×D₄₃	C₂³	1	12	yes	—
51	#13063325	C₂³	1	12	yes	—
52	#13063322	C₂³	1	12	yes	—
53	#13063320	C₂³	1	96	yes	—
54	#13063316	C₂³	1	48	yes	—
55	#13063314	C₂³	1	48	yes	—
56	#13063313	C₂³	1	24	yes	—
57	#13063310	C₂³	1	96	yes	—
58	#13063309	C₂³	1	48	yes	—
59	#13063305	C₂³	1	48	yes	—
60	#13063301	C₂³	1	48	yes	—
61	#13063300	C₂³	1	48	yes	—
62	#13063300	C₂³	1	48	yes	—
63	#13063299	C₂³	1	48	yes	—
64	#13063295	C₂³	1	48	yes	—
65	Q₈×Dic₄₃	C₂³	1	16	yes	—
66	#13063290	C₂³	1	24	yes	—
67	#13063290	C₂³	1	24	yes	—
68	C₄×D₁₇₂	C₂³	1	48	yes	—
69	#13063287	C₂³	1	24	yes	—
70	#13063287	C₂³	1	24	yes	—
71	#13063287	C₂³	1	48	yes	—
72	(C₂²⋊D₄₃)⋊C₄	C₂³	1	48	yes	—
73	(C₂²⋊D₄₃)⋊C₄	C₂³	1	48	yes	—
74	#13063283	C₂³	1	48	yes	—
75	#13063282	C₂³	1	24	yes	—
76	#13063282	C₂³	1	24	yes	—
77	C₂×Dic₄₃⋊C₄	C₂³	1	12	yes	—
78	C₂×(C₂×D₄₃)⋊C₄	C₂³	1	12	yes	—
79	C₂×C₄×Dic₄₃	C₂³	1	3	yes	—
80	C₂×C₄×Dic₄₃	C₂³	1	12	yes	—
81	C₂×C₄×Dic₄₃	C₂³	1	3	yes	—
82	C₂×C₄×Dic₄₃	C₂³	1	6	yes	—
83	C₄²×D₄₃	C₂³	1	12	yes	—
84	C₄²×D₄₃	C₂³	1	24	yes	—
85	C₄²×D₄₃	C₂³	1	12	yes	—
86	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
87	C₄⋊Q₈×C₄₃	C₂³	1	24	yes	—
88	#125×C₄₃	C₂³	1	24	yes	—
89	C₂²⋊Q₈×C₄₃	C₂³	1	48	yes	—
90	C₄×Q₈×C₄₃	C₂³	1	16	yes	—
91	#117×C₄₃	C₂³	1	12	yes	—
92	#117×C₄₃	C₂³	1	12	yes	—
93	C₂×C₄⋊C₄×C₄₃	C₂³	1	6	yes	—
94	C₂×C₂²⋊C₄×C₄₃	C₂³	1	6	yes	—
95	C₂×C₄²×C₄₃	C₂³	1	6	yes	—
96	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
97	C₂³×D₄₃	C₂⁴	1	6	yes	—
98	C₂×Q₈×C₄₃	C₂⁴	1	16	yes	—
99	C₂²×C₄×C₄₃	#41	1	8	yes	complemented
100	C₂²×C₄×C₄₃	C₂⁴	1	12	yes	—
101	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
102	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
103	C₂×C₂²⋊D₄₃	—	2	24	no	—
104	C₂×C₂²⋊D₄₃	C₂⁴	1	24	yes	—
105	C₂²×Dic₄₃	—	4	4	no	—
106	C₂²×Dic₄₃	C₂⁴	1	6	yes	—
107	C₂²×Dic₄₃	C₂⁴	1	6	yes	—
108	C₂²×Dic₄₃	C₂⁴	1	6	yes	—
109	#10621902	—	4	32	no	—
110	#10621898	—	2	48	no	—
111	#10621898	—	2	48	no	—
112	C₂×D₁₇₂	—	2	12	no	—
113	C₂×D₁₇₂	C₂⁴	1	12	yes	—
114	C₂×D₁₇₂	C₂⁴	1	48	yes	—
115	C₂×C₄×D₄₃	—	2	24	no	—
116	C₂×C₄×D₄₃	—	4	24	no	—
117	C₂×C₄×D₄₃	C₂⁴	1	12	yes	—
118	C₂×C₄×D₄₃	C₂⁴	1	12	yes	—
119	C₂×C₄×D₄₃	C₂⁴	1	24	yes	—
120	C₂×C₄×D₄₃	C₂⁴	1	24	yes	—
121	C₂×Dic₈₆	—	2	12	no	—
122	C₂×Dic₈₆	C₂⁴	1	12	yes	—
123	C₄⋊C₄×C₄₃	C₂⁴	1	24	yes	—
124	C₄⋊C₄×C₄₃	C₂⁴	1	48	yes	—
125	C₂²⋊C₄×C₄₃	C₂⁴	1	24	yes	—
126	C₄²×C₄₃	C₂⁴	1	12	yes	—
127	C₄²×C₄₃	C₂⁴	1	12	yes	—
128	(C₂×D₄₃)⋊C₄	C₂⁴	1	48	yes	—
129	(C₂×D₄₃)⋊C₄	C₂⁴	1	96	yes	—
130	C₄⋊Dic₄₃	C₂⁴	1	48	yes	—
131	Dic₄₃⋊C₄	C₂⁴	1	48	yes	—
132	C₄×Dic₄₃	C₂⁴	1	1	yes	characteristic
133	C₄×Dic₄₃	C₂⁴	1	3	yes	—
134	C₄×Dic₄₃	C₂⁴	1	6	yes	—
135	C₄×Dic₄₃	C₂⁴	1	24	yes	—
136	C₄×Dic₄₃	C₂⁴	1	24	yes	—
137	C₄×Dic₄₃	C₂⁴	1	6	yes	—
138	C₂³×C₄₃	—	4	2	no	—
139	C₂³×C₄₃	C₂×#41	1	4	yes	—
140	C₂³×C₄₃	C₂⁵	1	3	yes	—
141	C₂²×D₄₃	—	2	12	no	—
142	C₂²×D₄₃	—	4	12	no	—
143	C₂²×D₄₃	C₂⁵	1	12	yes	—
144	Q₈×C₄₃	—	4	16	no	—
145	C₂×C₄×C₄₃	—	4	12	no	—
146	C₂×C₄×C₄₃	C₂×#41	1	12	yes	—
147	C₂×C₄×C₄₃	C₂×#41	1	12	yes	—
148	C₂×C₄×C₄₃	C₂⁵	1	24	yes	—
149	C₂×C₄×C₄₃	C₂⁵	1	3	yes	—
150	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
151	C₂²⋊D₄₃	—	2	48	no	—
152	C₂×Dic₄₃	—	2	12	no	—
153	C₂×Dic₄₃	—	2	12	no	—
154	C₂×Dic₄₃	—	4	6	no	—
155	C₂×Dic₄₃	C₂⁵	1	6	yes	—
156	C₂×Dic₄₃	C₂⁵	1	12	yes	—
157	C₂×Dic₄₃	C₂⁵	1	2	yes	—
158	D₁₇₂	—	2	24	no	—
159	D₁₇₂	—	4	48	no	—
160	C₄×D₄₃	—	2	24	no	—
161	C₄×D₄₃	—	2	24	no	—
162	C₄×D₄₃	—	2	48	no	—
163	C₄×D₄₃	—	4	24	no	—
164	Dic₈₆	—	2	24	no	—
165	#62686	—	43	1	no	maximal, Sylow
166	C₂²×C₄₃	—	2	6	no	—
167	C₂²×C₄₃	—	4	4	no	—
168	C₂²×C₄₃	C₂²×#41	1	6	yes	—
169	C₂²×C₄₃	C₂⁶	1	1	yes	characteristic
170	C₂×D₄₃	—	2	12	no	—
171	C₂×D₄₃	—	2	24	no	—
172	C₂×D₄₃	—	4	24	no	—
173	C₄×C₄₃	—	2	24	no	—
174	C₄×C₄₃	C₂²×#41	1	6	yes	—
175	C₄×C₄₃	C₂²×#41	1	2	yes	—
176	Dic₄₃	—	2	12	no	—
177	Dic₄₃	—	2	6	no	—
178	Dic₄₃	#585	1	4	yes	—
179	#3461	—	43	12	no	—
180	#3460	—	43	12	no	—
181	#3459	—	43	8	no	—
182	#3458	D₄₃	1	1	yes	characteristic, complemented
183	#3455	—	43	12	no	—
184	#3449	—	43	3	no	—
185	#3444	—	43	12	no	—
186	C₂×C₄×#41	—	43	3	no	—
187	C₂×C₄₃	—	2	6	no	—
188	C₂×C₄₃	C₂³×#41	1	1	yes	characteristic
189	C₂×C₄₃	C₂×#585	1	2	yes	—
190	D₄₃	—	4	12	no	—
191	C₂²×#41	—	43	3	no	—
192	#544	C₂×D₄₃	1	12	yes	complemented
193	#543	C₂×D₄₃	1	4	yes	—
194	#542	—	43	48	no	—
195	#541	—	43	24	no	—
196	#540	—	43	24	no	—
197	#539	—	43	48	no	—
198	#538	—	43	48	no	—
199	#537	—	43	12	no	—
200	#536	—	43	12	no	—
201	#535	—	43	12	no	—
202	#534	—	43	12	no	—
203	#533	—	43	24	no	—
204	#533	C₂×D₄₃	1	12	yes	complemented
205	#532	—	43	24	no	—
206	#532	—	43	12	no	—
207	#529	—	43	48	no	—
208	#529	—	43	48	no	—
209	#529	—	43	48	no	—
210	#525	—	43	48	no	—
211	#525	—	43	24	no	—
212	C₄×#41	—	43	16	no	—
213	C₄×#41	—	43	12	no	—
214	C₄×#41	—	43	12	no	—
215	C₄×#41	—	43	24	no	—
216	C₂×C₄×Q₈	—	43	6	no	—
217	C₂×C₄×D₄	—	43	6	no	—
218	C₂×C₄×D₄	—	43	12	no	—
219	C₂×#117	—	43	6	no	—
220	C₂×#117	C₂×D₄₃	1	3	yes	—
221	C₂²×C₄²	—	43	1	no	—
222	C₂²×C₄²	—	43	6	no	—
223	C₄₃	#62686	1	1	yes	characteristic, complemented, Sylow
224	C₂×#41	—	43	16	no	—
225	C₂×#41	—	43	12	no	—
226	C₂×#41	—	43	12	no	—
227	C₂×#41	—	86	24	no	—
228	C₂²×Q₈	—	43	3	no	—
229	C₂²×D₄	—	43	6	no	—
230	C₂²×D₄	—	43	3	no	—
231	C₂³×C₄	—	43	2	no	—
232	C₂³×C₄	—	43	6	no	—
233	C₂³×C₄	—	43	12	no	—
234	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
235	C₄⋊Q₈	—	43	12	no	—
236	C₄⋊Q₈	C₂²×D₄₃	1	24	yes	—
237	#127	—	43	12	no	—
238	#126	—	43	96	no	—
239	#125	—	43	48	no	—
240	#125	C₂²×D₄₃	1	24	yes	—
241	#124	—	43	48	no	—
242	#124	—	43	24	no	—
243	#123	—	43	96	no	—
244	#123	—	43	48	no	—
245	C₂²⋊Q₈	—	43	48	no	—
246	C₂²⋊Q₈	—	43	48	no	—
247	C₂²⋊Q₈	C₂²×D₄₃	1	48	yes	—
248	#121	—	43	48	no	—
249	#121	—	43	48	no	—
250	#121	—	43	48	no	—
251	#120	—	43	48	no	—
252	C₄×Q₈	—	43	16	no	—
253	C₄×Q₈	—	43	24	no	—
254	C₄×Q₈	—	43	24	no	—
255	C₄×Q₈	C₂²×D₄₃	1	16	yes	—
256	C₄×D₄	—	43	24	no	—
257	C₄×D₄	—	43	24	no	—
258	C₄×D₄	—	43	48	no	—
259	C₄×D₄	—	43	48	no	—
260	C₄×D₄	—	43	48	no	—
261	C₄×D₄	—	43	48	no	—
262	#117	—	43	48	no	—
263	#117	—	43	24	no	—
264	#117	—	43	24	no	—
265	#117	C₂²×D₄₃	1	12	yes	—
266	#117	C₂²×D₄₃	1	12	yes	—
267	C₂×C₄⋊C₄	—	43	12	no	—
268	C₂×C₄⋊C₄	C₂²×D₄₃	1	6	yes	—
269	C₂×C₂²⋊C₄	—	43	12	no	—
270	C₂×C₂²⋊C₄	C₂²×D₄₃	1	6	yes	—
271	C₂×C₄²	—	43	3	no	—
272	C₂×C₄²	—	43	6	no	—
273	C₂×C₄²	—	43	12	no	—
274	C₂×C₄²	—	43	3	no	—
275	C₂×C₄²	—	43	12	no	—
276	C₂×C₄²	—	43	12	no	—
277	C₂×C₄²	—	43	24	no	—
278	C₂×C₄²	C₂²×D₄₃	1	6	yes	—
279	C₂⁴	—	43	6	no	—
280	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
281	#41	—	86	48	no	—
282	#41	—	86	48	no	—
283	#41	—	172	32	no	—
284	C₂×Q₈	—	43	12	no	—
285	C₂×Q₈	—	86	12	no	—
286	C₂×Q₈	C₂³×D₄₃	1	16	yes	—
287	C₂×D₄	—	43	24	no	—
288	C₂×D₄	—	43	12	no	—
289	C₂×D₄	—	43	48	no	—
290	C₂×D₄	—	86	24	no	—
291	C₂×D₄	—	86	12	no	—
292	C₂²×C₄	—	43	6	no	—
293	C₂²×C₄	—	43	6	no	—
294	C₂²×C₄	—	43	6	no	—
295	C₂²×C₄	—	43	24	no	—
296	C₂²×C₄	—	43	12	no	—
297	C₂²×C₄	—	43	12	no	—
298	C₂²×C₄	—	43	24	no	—
299	C₂²×C₄	—	86	24	no	—
300	C₂²×C₄	—	172	24	no	—
301	C₂²×C₄	—	172	4	no	—
302	C₂²×C₄	#10621902	1	8	yes	complemented
303	C₂²×C₄	C₂³×D₄₃	1	12	yes	—
304	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
305	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
306	C₄⋊C₄	—	43	48	no	—
307	C₄⋊C₄	—	43	48	no	—
308	C₄⋊C₄	C₂³×D₄₃	1	24	yes	—
309	C₄⋊C₄	C₂³×D₄₃	1	48	yes	—
310	C₂²⋊C₄	—	43	48	no	—
311	C₂²⋊C₄	—	43	96	no	—
312	C₂²⋊C₄	C₂³×D₄₃	1	24	yes	—
313	C₄²	—	43	6	no	—
314	C₄²	—	43	24	no	—
315	C₄²	—	43	6	no	—
316	C₄²	—	43	24	no	—
317	C₄²	—	43	1	no	—
318	C₄²	—	43	3	no	—
319	C₄²	C₂³×D₄₃	1	12	yes	—
320	C₄²	C₂³×D₄₃	1	12	yes	—
321	C₂³	—	4	2	no	—
322	C₂³	—	43	12	no	—
323	C₂³	—	86	12	no	—
324	C₂³	—	172	12	no	—
325	C₂³	C₂×#10621902	1	4	yes	—
326	C₂³	C₂⁴×D₄₃	1	3	yes	—
327	Q₈	—	4	16	no	—
328	Q₈	—	86	24	no	—
329	D₄	—	86	48	no	—
330	D₄	—	86	24	no	—
331	D₄	—	172	48	no	—
332	C₂×C₄	—	4	12	no	—
333	C₂×C₄	—	43	2	no	—
334	C₂×C₄	—	43	6	no	—
335	C₂×C₄	—	43	12	no	—
336	C₂×C₄	—	86	48	no	—
337	C₂×C₄	—	86	12	no	—
338	C₂×C₄	—	86	12	no	—
339	C₂×C₄	—	86	24	no	—
340	C₂×C₄	—	86	24	no	—
341	C₂×C₄	—	172	24	no	—
342	C₂×C₄	—	172	6	no	—
343	C₂×C₄	C₂×#10621902	1	12	yes	—
344	C₂×C₄	C₂×#10621902	1	12	yes	—
345	C₂×C₄	C₂⁴×D₄₃	1	24	yes	—
346	C₂×C₄	C₂⁴×D₄₃	1	3	yes	—
347	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
348	C₂²	—	2	6	no	—
349	C₂²	—	4	4	no	—
350	C₂²	—	86	12	no	—
351	C₂²	—	86	24	no	—
352	C₂²	—	172	24	no	—
353	C₂²	C₂²×#10621902	1	6	yes	—
354	C₂²	C₂⁵×D₄₃	1	1	yes	characteristic
355	C₄	—	2	24	no	—
356	C₄	—	43	4	no	—
357	C₄	—	86	12	no	—
358	C₄	—	86	6	no	—
359	C₄	C₂²×#10621902	1	2	yes	—
360	C₄	C₂²×#10621902	1	6	yes	—
361	C₂	—	2	6	no	—
362	C₂	—	172	12	no	—
363	C₂	C₂³×#10621902	1	1	yes	characteristic
364	C₂	#585×D₄₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂⁴×D₄₃
commutator, socle, cosocle	C₂²×C₄₃	C₂⁶
Frattini	C₂²	C₂⁵×D₄₃
Fitting	C₄₃×#3458	C₂
hypercenter	#3458	D₄₃
hypocenter	C₄₃	#62686
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ #3458

Complements

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

N	H ≅ G/N	type
1. #19148029	C₂	semidirect
2. #19148018	C₂	semidirect
3. #19148007	C₂	semidirect
4. #19147987	C₂	semidirect
5. #19147928	C₂	semidirect
6. #19147892	C₂	semidirect
7. C₂×#14583367	C₂	semidirect
8. C₄₃×#3458	C₂	semidirect
10. #14583523	C₂²	semidirect
11. #14583512	C₂²	semidirect
12. #14583505	C₂²	semidirect
13. #14583498	C₂²	semidirect
14. #14583489	C₂²	semidirect
16. #14583475	C₂²	semidirect
17. #14583468	C₂²	semidirect
19. #14583454	C₂²	semidirect
20. #14583450	C₂²	semidirect
22. #14583436	C₂²	semidirect
23. #14583435	C₂²	semidirect
24. #14583434	C₂²	semidirect
25. #14583412	C₂²	semidirect
26. #14583411	C₂²	semidirect
27. C₄×#10621902	C₂²	semidirect
28. #14583367	C₂²	semidirect
29. #14583367	C₂²	semidirect
30. #14583367	C₂²	semidirect
31. C₂×#13063290	C₂²	semidirect
32. C₂×#13063287	C₂²	semidirect
34. C₂×#13063282	C₂²	semidirect
36. C₂×C₄²×D₄₃	C₂²	semidirect
37. C₄₃×#544	C₂²	semidirect
39. C₄₃×#533	C₂²	semidirect
99. C₂²×C₄×C₄₃	#41	semidirect
182. #3458	D₄₃	semidirect
192. #544	C₂×D₄₃	semidirect
204. #533	C₂×D₄₃	semidirect
223. C₄₃	#62686	semidirect
302. C₂²×C₄	#10621902	semidirect

Maximal subgroups

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

subgroup	index
#19148029	2
#19148018	2
#19148007	2
#19147987	2
#19147928	2
#19147892	2
C₂×#14583367	2
C₄₃×#3458	2
#62686	43