Subgroups — #25964044

Order 11008.

2578 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19149382	C₂	1	2	yes	maximal, complemented
2	#19149306	C₂	1	1	yes	maximal, characteristic, complemented
3	#19144328	C₂	1	12	yes	maximal, complemented
4	#19144311	C₂	1	6	yes	maximal, complemented
5	C₂×#14583171	C₂	1	6	yes	maximal, complemented
6	#19143206	C₂	1	1	yes	maximal, characteristic, complemented
7	#19143078	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₃×#578	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583735	C₂²	1	1	yes	characteristic
10	#14583714	—	2	8	no	—
11	#14583707	—	2	8	no	—
12	#14583697	C₂²	1	12	yes	—
13	C₂×#13063356	C₂²	1	12	yes	—
14	#14583666	C₂²	1	2	yes	—
15	#14583666	C₂²	1	1	yes	characteristic
16	#14583665	C₂²	1	6	yes	—
17	#14583664	—	2	4	no	—
18	C₂×#13063331	C₂²	1	6	yes	—
19	#131×D₄₃	—	2	4	no	—
20	#14583171	C₂²	1	12	yes	complemented
21	#14583171	C₂²	1	24	yes	complemented
22	#14583166	C₂²	1	4	yes	complemented
23	#14583159	C₂²	1	24	yes	complemented
24	#14582948	C₂²	1	6	yes	complemented
25	C₂×#13063261	C₂²	1	6	yes	complemented
26	#14582883	C₂²	1	12	yes	complemented
27	C₂×#13063244	C₂²	1	12	yes	complemented
28	C₂×2₋¹⁺⁴×C₄₃	C₂²	1	2	yes	—
29	C₄₃×#578	—	2	4	no	—
30	C₄₃×#578	—	2	4	no	—
31	C₂×#137×C₄₃	C₂²	1	6	yes	complemented
32	C₂×#135×C₄₃	C₂²	1	3	yes	complemented
33	C₂²×Q₁₆×C₄₃	C₂²	1	3	yes	complemented
34	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
35	#13063373	—	2	1	no	—
36	#13063372	D₄	1	4	yes	—
37	#13063371	D₄	1	4	yes	—
38	#13063370	—	2	3	no	—
39	C₂×#10621898	C₂³	1	6	yes	—
40	#41×D₄₃	—	2	4	no	—
41	#13063356	—	2	24	no	—
42	#13063356	C₂³	1	24	yes	—
43	#13063355	—	2	12	no	—
44	#13063351	C₂³	1	8	yes	—
45	#13063350	—	2	12	no	—
46	#13063345	—	2	24	no	—
47	C₂×#10621885	—	2	12	no	—
48	C₂×#10621884	—	2	12	no	—
49	#13063333	—	2	12	no	—
50	#13063332	—	2	4	no	—
51	#13063332	—	2	8	no	—
52	#13063331	—	2	6	no	—
53	#13063331	—	2	6	no	—
54	#13063331	C₂³	1	4	yes	—
55	#13063331	C₂³	1	12	yes	—
56	C₂×#10621877	C₂³	1	6	yes	—
57	C₂×#10621877	C₂³	1	12	yes	—
58	C₂×#10621872	—	2	6	no	—
59	M₁₆×D₄₃	—	2	12	no	—
60	C₂×C₈×D₄₃	—	2	6	no	—
61	#13063261	C₂³	1	12	yes	—
62	#13063261	C₂³	1	4	yes	—
63	#13063244	C₂³	1	24	yes	—
64	#13063242	C₂³	1	8	yes	—
65	2₋¹⁺⁴×C₄₃	—	2	8	no	—
66	2₋¹⁺⁴×C₄₃	—	2	8	no	—
67	C₂×#41×C₄₃	—	2	6	no	—
68	C₂×#41×C₄₃	C₂³	1	6	yes	—
69	C₂×#41×C₄₃	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₄₃	—	2	2	no	—
71	C₂²×Q₈×C₄₃	C₂³	1	6	yes	—
72	#137×C₄₃	—	2	24	no	—
73	#135×C₄₃	—	2	6	no	—
74	#135×C₄₃	—	2	6	no	—
75	C₂×Q₁₆×C₄₃	—	2	6	no	—
76	C₂×Q₁₆×C₄₃	—	2	6	no	—
77	C₂×Q₁₆×C₄₃	C₂³	1	3	yes	—
78	C₂×Q₁₆×C₄₃	C₂³	1	6	yes	—
79	C₂×SD₁₆×C₄₃	C₂³	1	6	yes	—
80	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
81	#131×C₄₃	D₄	1	4	yes	—
82	#131×C₄₃	D₄	1	4	yes	—
83	C₂×M₁₆×C₄₃	C₂³	1	3	yes	—
84	C₂²×C₈×C₄₃	C₂³	1	3	yes	—
85	#41×C₄₃	—	2	12	no	—
86	#41×C₄₃	—	2	12	no	—
87	#41×C₄₃	—	4	24	no	—
88	#41×C₄₃	C₂×D₄	1	4	yes	—
89	#41×C₄₃	C₂×D₄	1	4	yes	—
90	C₂×Q₈×C₄₃	—	2	6	no	—
91	C₂×Q₈×C₄₃	—	2	12	no	—
92	C₂×Q₈×C₄₃	—	2	12	no	—
93	C₂×Q₈×C₄₃	—	4	8	no	—
94	C₂×Q₈×C₄₃	#41	1	1	yes	characteristic
95	C₂×Q₈×C₄₃	C₂⁴	1	6	yes	—
96	C₂×D₄×C₄₃	—	2	6	no	—
97	C₂×D₄×C₄₃	#41	1	3	yes	—
98	C₂×D₄×C₄₃	C₂⁴	1	2	yes	—
99	C₂²×C₄×C₄₃	—	2	6	no	—
100	C₂²×C₄×C₄₃	—	2	6	no	—
101	C₂²×C₄×C₄₃	C₂⁴	1	3	yes	—
102	C₂×C₂²⋊D₄₃	—	2	6	no	—
103	#10621902	—	2	4	no	—
104	Q₈×D₄₃	—	2	4	no	—
105	#10621900	—	2	12	no	—
106	D₄×D₄₃	—	2	12	no	—
107	#10621898	—	2	6	no	—
108	#10621898	—	2	6	no	—
109	#10621898	C₂×D₄	1	12	yes	—
110	#10621898	C₂⁴	1	4	yes	—
111	C₂×D₁₇₂	C₂×D₄	1	6	yes	—
112	C₂×C₄×D₄₃	—	2	6	no	—
113	C₂×Dic₈₆	C₂×D₄	1	6	yes	—
114	Q₁₆×C₄₃	—	2	3	no	—
115	Q₁₆×C₄₃	—	2	3	no	—
116	Q₁₆×C₄₃	—	2	12	no	—
117	SD₁₆×C₄₃	—	2	12	no	—
118	D₈×C₄₃	—	2	1	no	—
119	D₈×C₄₃	—	2	1	no	—
120	M₁₆×C₄₃	C₂×D₄	1	12	yes	—
121	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
122	C₂×C₈×C₄₃	C₂×D₄	1	6	yes	—
123	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
124	C₂×C₈×C₄₃	C₂⁴	1	3	yes	—
125	#10621885	—	2	24	no	—
126	#10621884	—	2	24	no	—
127	#10621883	—	2	8	no	—
128	#10621882	—	2	8	no	—
129	#10621877	—	2	6	no	—
130	#10621877	—	2	12	no	—
131	#10621877	C₂⁴	1	4	yes	—
132	#10621877	C₂⁴	1	8	yes	—
133	C₂×C₄₃⋊C₈	—	2	6	no	—
134	C₂×C₄₃⋊C₈	—	2	12	no	—
135	#10621872	—	2	12	no	—
136	#10621872	—	2	4	no	—
137	C₈×D₄₃	—	2	4	no	—
138	C₈×D₄₃	—	2	12	no	—
139	C₂³×C₄₃	—	2	2	no	—
140	C₂³×C₄₃	C₂×#41	1	3	yes	—
141	C₂²×D₄₃	—	2	6	no	—
142	Q₈×C₄₃	—	2	6	no	—
143	Q₈×C₄₃	—	2	6	no	—
144	Q₈×C₄₃	—	4	12	no	—
145	Q₈×C₄₃	#121	1	4	yes	—
146	D₄×C₄₃	—	2	2	no	—
147	D₄×C₄₃	—	2	2	no	—
148	D₄×C₄₃	—	4	12	no	—
149	D₄×C₄₃	#121	1	12	yes	—
150	C₂×C₄×C₄₃	—	2	6	no	—
151	C₂×C₄×C₄₃	—	4	12	no	—
152	C₂×C₄×C₄₃	—	4	12	no	—
153	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
154	C₂×C₄×C₄₃	C₂²×D₄	1	6	yes	—
155	C₂×C₄×C₄₃	C₂×#41	1	3	yes	—
156	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
157	C₂²⋊D₄₃	—	2	12	no	—
158	C₂²⋊D₄₃	—	2	4	no	—
159	C₂×Dic₄₃	—	2	6	no	—
160	D₁₇₂	—	2	6	no	—
161	D₁₇₂	C₂²×D₄	1	4	yes	—
162	C₄×D₄₃	—	2	12	no	—
163	C₄×D₄₃	—	2	4	no	—
164	Dic₈₆	—	2	6	no	—
165	Dic₈₆	C₂²×D₄	1	4	yes	—
166	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
167	C₈×C₄₃	C₂²×D₄	1	6	yes	—
168	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
169	C₄₃⋊C₈	—	2	4	no	—
170	C₄₃⋊C₈	—	2	8	no	—
171	#60472	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	—	4	3	no	—
173	C₂²×C₄₃	—	4	2	no	—
174	C₂²×C₄₃	—	8	2	no	—
175	C₂²×C₄₃	C₂×#121	1	6	yes	—
176	C₂²×C₄₃	C₂²×#41	1	1	yes	characteristic
177	C₂×D₄₃	—	2	4	no	—
178	C₂×D₄₃	—	8	6	no	—
179	C₄×C₄₃	—	4	6	no	—
180	C₄×C₄₃	C₂×#121	1	6	yes	—
181	C₄×C₄₃	C₂³×D₄	1	1	yes	characteristic
182	C₄×C₄₃	C₂³×D₄	1	1	yes	characteristic
183	Dic₄₃	—	2	4	no	—
184	#3586	—	43	2	no	—
185	C₂×#578	D₄₃	1	1	yes	characteristic, complemented
186	#3573	—	43	1	no	—
187	#3069	—	43	12	no	—
188	#3067	—	43	6	no	—
189	C₂×#473	—	43	6	no	—
190	#2916	—	43	1	no	—
191	#2889	—	43	2	no	—
192	C₂×C₄₃	—	2	6	no	—
193	C₂×C₄₃	—	2	1	no	—
194	C₂×C₄₃	—	8	2	no	—
195	C₂×C₄₃	C₂²×#121	1	1	yes	characteristic
196	D₄₃	—	4	4	no	—
197	#585	—	43	1	no	—
198	C₂×2₋¹⁺⁴	C₂×D₄₃	1	2	yes	—
199	#578	—	2	4	no	—
200	#578	—	2	4	no	—
201	#578	—	86	8	no	—
202	#577	—	86	8	no	—
203	#575	—	43	12	no	—
204	C₂×#137	—	43	12	no	—
205	C₂×#137	C₂×D₄₃	1	6	yes	complemented
206	C₂×#135	C₂×D₄₃	1	3	yes	complemented
207	C₂²×Q₁₆	C₂×D₄₃	1	3	yes	complemented
208	#568	—	43	2	no	—
209	#568	—	43	1	no	—
210	#568	—	43	6	no	—
211	#568	—	86	4	no	—
212	C₂×#131	—	43	6	no	—
213	C₂×#131	—	86	4	no	—
214	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
215	#473	—	43	12	no	—
216	#473	—	43	24	no	—
217	#472	—	43	4	no	—
218	#471	—	43	24	no	—
219	#430	—	43	6	no	—
220	C₂×#108	—	43	6	no	—
221	#413	—	43	12	no	—
222	C₂×#101	—	43	12	no	—
223	C₄₃	#60472	1	1	yes	characteristic, complemented, Sylow
224	2₋¹⁺⁴	—	2	8	no	—
225	2₋¹⁺⁴	—	2	8	no	—
226	2₋¹⁺⁴	—	43	4	no	—
227	2₋¹⁺⁴	—	86	1	no	—
228	H(5,𝔽₂)	—	43	4	no	—
229	H(5,𝔽₂)	—	86	3	no	—
230	C₂×#41	—	2	6	no	—
231	C₂×#41	—	43	6	no	—
232	C₂×#41	—	86	4	no	—
233	C₂×#41	C₂²×D₄₃	1	6	yes	—
234	C₂×#41	C₂²×D₄₃	1	1	yes	characteristic
235	C₂²×Q₈	—	2	2	no	—
236	C₂²×Q₈	C₂²×D₄₃	1	6	yes	—
237	#137	—	2	24	no	—
238	#137	—	43	24	no	—
239	#137	—	86	24	no	—
240	#137	—	86	12	no	—
241	AGL(1,ℤ/8)	—	43	8	no	—
242	AGL(1,ℤ/8)	—	86	12	no	—
243	#135	—	2	6	no	—
244	#135	—	2	6	no	—
245	#135	—	86	24	no	—
246	C₂×Q₁₆	—	2	6	no	—
247	C₂×Q₁₆	—	2	6	no	—
248	C₂×Q₁₆	—	86	12	no	—
249	C₂×Q₁₆	C₂²×D₄₃	1	3	yes	—
250	C₂×Q₁₆	C₂²×D₄₃	1	6	yes	—
251	C₂×SD₁₆	—	86	12	no	—
252	C₂×SD₁₆	C₂²×D₄₃	1	6	yes	—
253	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
254	#131	—	43	4	no	—
255	#131	—	43	12	no	—
256	#131	—	86	8	no	—
257	#131	—	86	4	no	—
258	#131	—	86	12	no	—
259	#131	—	86	6	no	—
260	#131	—	86	6	no	—
261	#131	C₂²⋊D₄₃	1	4	yes	—
262	#131	C₂²⋊D₄₃	1	4	yes	—
263	C₂×M₁₆	—	43	6	no	—
264	C₂×M₁₆	—	43	12	no	—
265	C₂×M₁₆	—	86	12	no	—
266	C₂×M₁₆	—	86	6	no	—
267	C₂×M₁₆	C₂²×D₄₃	1	3	yes	—
268	C₂²×C₈	—	86	6	no	—
269	C₂²×C₈	C₂²×D₄₃	1	3	yes	—
270	#108	—	43	4	no	—
271	#108	—	43	12	no	—
272	#101	—	43	24	no	—
273	#100	—	43	8	no	—
274	#41	—	2	12	no	—
275	#41	—	2	12	no	—
276	#41	—	4	24	no	—
277	#41	—	43	12	no	—
278	#41	—	43	4	no	—
279	#41	—	86	4	no	—
280	#41	—	86	12	no	—
281	#41	—	86	6	no	—
282	#41	—	86	6	no	—
283	#41	D₄×D₄₃	1	4	yes	—
284	#41	C₂×C₂²⋊D₄₃	1	4	yes	—
285	C₂×Q₈	—	2	6	no	—
286	C₂×Q₈	—	2	12	no	—
287	C₂×Q₈	—	2	12	no	—
288	C₂×Q₈	—	4	8	no	—
289	C₂×Q₈	—	43	6	no	—
290	C₂×Q₈	—	86	4	no	—
291	C₂×Q₈	#10621900	1	1	yes	characteristic
292	C₂×Q₈	C₂³×D₄₃	1	6	yes	—
293	C₂×D₄	—	2	6	no	—
294	C₂×D₄	—	43	6	no	—
295	C₂×D₄	—	86	6	no	—
296	C₂×D₄	—	86	12	no	—
297	C₂×D₄	#10621900	1	3	yes	—
298	C₂×D₄	C₂³×D₄₃	1	2	yes	—
299	C₂²×C₄	—	2	6	no	—
300	C₂²×C₄	—	2	6	no	—
301	C₂²×C₄	—	86	6	no	—
302	C₂²×C₄	C₂³×D₄₃	1	3	yes	—
303	Q₁₆	—	2	3	no	—
304	Q₁₆	—	2	3	no	—
305	Q₁₆	—	2	12	no	—
306	Q₁₆	—	86	24	no	—
307	SD₁₆	—	2	12	no	—
308	SD₁₆	—	86	8	no	—
309	SD₁₆	—	86	24	no	—
310	D₈	—	2	1	no	—
311	D₈	—	2	1	no	—
312	D₈	—	86	8	no	—
313	M₁₆	—	43	4	no	—
314	M₁₆	—	43	8	no	—
315	M₁₆	—	86	12	no	—
316	M₁₆	—	86	6	no	—
317	M₁₆	—	86	12	no	—
318	M₁₆	—	86	4	no	—
319	M₁₆	C₂×C₂²⋊D₄₃	1	12	yes	—
320	C₂×C₈	—	86	12	no	—
321	C₂×C₈	—	86	6	no	—
322	C₂×C₈	—	86	4	no	—
323	C₂×C₈	—	86	12	no	—
324	C₂×C₈	C₂×C₂²⋊D₄₃	1	6	yes	—
325	C₂×C₈	C₂×C₂²⋊D₄₃	1	6	yes	—
326	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
327	C₂×C₈	C₂³×D₄₃	1	3	yes	—
328	C₂³	—	2	2	no	—
329	C₂³	—	86	6	no	—
330	C₂³	C₂×#10621900	1	3	yes	—
331	Q₈	—	2	6	no	—
332	Q₈	—	2	6	no	—
333	Q₈	—	4	12	no	—
334	Q₈	—	43	4	no	—
335	Q₈	—	86	6	no	—
336	Q₈	#13063296	1	4	yes	—
337	D₄	—	2	2	no	—
338	D₄	—	2	2	no	—
339	D₄	—	4	12	no	—
340	D₄	—	43	4	no	—
341	D₄	—	86	12	no	—
342	D₄	—	86	4	no	—
343	D₄	—	86	6	no	—
344	D₄	#13063296	1	12	yes	—
345	C₂×C₄	—	2	6	no	—
346	C₂×C₄	—	4	12	no	—
347	C₂×C₄	—	4	12	no	—
348	C₂×C₄	—	86	6	no	—
349	C₂×C₄	—	86	12	no	—
350	C₂×C₄	—	86	4	no	—
351	C₂×C₄	C₂×D₄×D₄₃	1	6	yes	—
352	C₂×C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
353	C₂×C₄	C₂×#10621900	1	3	yes	—
354	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
355	C₈	—	86	8	no	—
356	C₈	—	86	4	no	—
357	C₈	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
358	C₈	C₂²×C₂²⋊D₄₃	1	6	yes	—
359	C₈	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
360	C₂²	—	4	3	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	8	2	no	—
363	C₂²	—	86	4	no	—
364	C₂²	—	344	6	no	—
365	C₂²	C₂×#13063296	1	6	yes	—
366	C₂²	C₂²×#10621900	1	1	yes	characteristic
367	C₄	—	4	6	no	—
368	C₄	—	86	4	no	—
369	C₄	C₂×#13063296	1	6	yes	—
370	C₄	C₂²×D₄×D₄₃	1	1	yes	characteristic
371	C₄	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
372	C₂	—	2	1	no	—
373	C₂	—	2	6	no	—
374	C₂	—	8	2	no	—
375	C₂	—	172	4	no	—
376	C₂	C₂²×#13063296	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×#13063296
commutator, cosocle	C₂×C₄×C₄₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₃
Fitting	C₂×C₄₃×#578	C₂
socle	C₂×C₄₃	C₂²×#121
hypercenter	C₂×#578	D₄₃
hypocenter	C₄₃	#60472
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#578

Complements

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

N	H ≅ G/N	type
1. #19149382	C₂	semidirect
2. #19149306	C₂	semidirect
3. #19144328	C₂	semidirect
4. #19144311	C₂	semidirect
5. C₂×#14583171	C₂	semidirect
6. #19143206	C₂	semidirect
7. #19143078	C₂	semidirect
8. C₂×C₄₃×#578	C₂	semidirect
20. #14583171	C₂²	semidirect
21. #14583171	C₂²	semidirect
22. #14583166	C₂²	semidirect
23. #14583159	C₂²	semidirect
24. #14582948	C₂²	semidirect
25. C₂×#13063261	C₂²	semidirect
26. #14582883	C₂²	semidirect
27. C₂×#13063244	C₂²	semidirect
31. C₂×#137×C₄₃	C₂²	semidirect
32. C₂×#135×C₄₃	C₂²	semidirect
33. C₂²×Q₁₆×C₄₃	C₂²	semidirect
185. C₂×#578	D₄₃	semidirect
205. C₂×#137	C₂×D₄₃	semidirect
206. C₂×#135	C₂×D₄₃	semidirect
207. C₂²×Q₁₆	C₂×D₄₃	semidirect
223. C₄₃	#60472	semidirect

Maximal subgroups

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

subgroup	index
#19149382	2
#19149306	2
#19144328	2
#19144311	2
C₂×#14583171	2
#19143206	2
#19143078	2
C₂×C₄₃×#578	2
#60472	43