Subgroups — #25107144

Order 11008.

806 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	#19137002	C₂	1	1	yes	maximal, characteristic
2	#19136993	C₂	1	1	yes	maximal, characteristic
3	#19135359	C₂	1	2	yes	maximal, complemented
4	#19135354	C₂	1	2	yes	maximal, complemented
5	#19134796	C₂	1	4	yes	maximal
6	#19134759	C₂	1	2	yes	maximal, complemented
7	#19134726	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₃×#431	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063284	C₂²	1	2	yes	—
10	C₂×#13063259	C₂²	1	4	yes	—
11	C₂×#13063257	C₂²	1	4	yes	—
12	#14582769	C₂²	1	2	yes	—
13	#14582767	—	2	1	no	—
14	#14582765	C₂²	1	1	yes	characteristic
15	#14582765	C₂²	1	2	yes	—
16	#14582765	C₂²	1	1	yes	characteristic
17	#14582762	—	2	1	no	—
18	#14582759	—	2	1	no	—
19	#14582759	—	2	1	no	—
20	#14582759	C₂²	1	2	yes	—
21	C₄×C₄⋊Dic₄₃	—	2	1	no	—
22	C₄×C₄⋊Dic₄₃	—	2	1	no	—
23	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
24	C₄×C₂²⋊Dic₄₃	—	2	1	no	—
25	#14582745	—	2	1	no	—
26	#14582662	C₂²	1	4	yes	—
27	#14582662	C₂²	1	4	yes	—
28	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
29	C₄₃×#431	C₄	1	8	yes	complemented
30	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
34	#13063284	—	4	2	no	—
35	#13063284	—	4	2	no	—
36	C₂×C₄⋊Dic₄₃	—	2	2	no	—
37	C₂×C₄⋊Dic₄₃	—	2	2	no	—
38	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
39	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
40	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
41	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
42	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
43	C₂×C₄×Dic₄₃	—	2	2	no	—
44	C₂×C₄×Dic₄₃	—	2	2	no	—
45	#13063259	—	4	4	no	—
46	#13063257	—	4	4	no	—
47	C₄₃⋊#95	—	2	2	no	—
48	C₄₃⋊#95	—	2	1	no	—
49	C₄₃⋊#95	—	2	1	no	—
50	C₄₃⋊#95	—	2	1	no	—
51	C₄₃⋊#95	—	2	1	no	—
52	C₄₃⋊#95	—	2	2	no	—
53	C₄₃⋊#95	—	2	2	no	—
54	C₄₃⋊#95	—	2	2	no	—
55	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
57	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
59	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
61	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
64	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₄₃	C₂×C₄	1	4	yes	—
67	C₄⋊C₈×C₄₃	C₂×C₄	1	4	yes	—
68	C₂²⋊C₈×C₄₃	C₂×C₄	1	8	yes	—
69	C₄×C₈×C₄₃	C₂×C₄	1	8	yes	—
70	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	—	2	4	no	—
72	C₂²×C₄×C₄₃	—	2	1	no	—
73	C₂²×C₄×C₄₃	—	2	1	no	—
74	C₂²×C₄×C₄₃	—	2	1	no	—
75	C₂²×C₄×C₄₃	—	2	1	no	—
76	C₂²×C₄×C₄₃	—	4	2	no	—
77	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
78	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
79	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₄₃	—	2	2	no	—
88	C₂²×Dic₄₃	—	2	2	no	—
89	C₂²×Dic₄₃	—	2	4	no	—
90	C₂×C₈×C₄₃	—	2	2	no	—
91	C₂×C₈×C₄₃	—	2	2	no	—
92	C₂×C₈×C₄₃	—	2	2	no	—
93	C₂×C₈×C₄₃	—	2	2	no	—
94	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
96	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
97	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
98	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
101	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
102	C₂²⋊Dic₄₃	—	4	4	no	—
103	C₂²⋊Dic₄₃	—	4	4	no	—
104	C₄⋊Dic₄₃	—	2	2	no	—
105	C₄⋊Dic₄₃	—	2	4	no	—
106	C₄⋊Dic₄₃	—	2	2	no	—
107	C₄⋊Dic₄₃	—	2	4	no	—
108	C₄⋊Dic₄₃	—	2	2	no	—
109	C₄⋊Dic₄₃	—	2	2	no	—
110	C₄⋊Dic₄₃	—	4	4	no	—
111	C₄⋊Dic₄₃	—	4	4	no	—
112	C₄×Dic₄₃	—	4	4	no	—
113	C₄×Dic₄₃	—	4	4	no	—
114	C₂³×C₄₃	—	2	1	no	—
115	C₂³×C₄₃	—	2	1	no	—
116	C₂³×C₄₃	—	2	2	no	—
117	C₂³×C₄₃	—	2	2	no	—
118	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	2	2	no	—
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	—	2	2	no	—
126	C₂×C₄×C₄₃	—	2	2	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	8	no	—
130	C₂×C₄×C₄₃	—	4	4	no	—
131	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₄₃	#135	1	4	yes	—
137	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
140	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×Dic₄₃	—	4	4	no	—
144	C₂×Dic₄₃	—	4	2	no	—
145	C₂×Dic₄₃	—	4	2	no	—
146	C₂×Dic₄₃	—	4	2	no	—
147	C₂×Dic₄₃	—	4	2	no	—
148	C₂×Dic₄₃	—	4	4	no	—
149	C₂×Dic₄₃	—	4	4	no	—
150	C₂×Dic₄₃	—	4	2	no	—
151	C₂×Dic₄₃	—	4	2	no	—
152	C₈×C₄₃	—	2	4	no	—
153	C₈×C₄₃	—	2	4	no	—
154	#21985	—	43	1	no	maximal, Sylow
155	C₂²×C₄₃	—	2	2	no	—
156	C₂²×C₄₃	—	2	2	no	—
157	C₂²×C₄₃	—	2	2	no	—
158	C₂²×C₄₃	—	4	2	no	—
159	C₂²×C₄₃	—	4	2	no	—
160	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₂²×C₄₃	#535	1	1	yes	characteristic
162	C₂²×C₄₃	#536	1	1	yes	characteristic
163	C₂²×C₄₃	C₂×#135	1	2	yes	—
164	C₂²×C₄₃	#575	1	2	yes	—
165	C₄×C₄₃	—	2	2	no	—
166	C₄×C₄₃	—	2	2	no	—
167	C₄×C₄₃	—	2	2	no	—
168	C₄×C₄₃	—	2	2	no	—
169	C₄×C₄₃	#427	1	4	yes	—
170	C₄×C₄₃	#520	1	1	yes	characteristic
171	C₄×C₄₃	#520	1	1	yes	characteristic
172	C₄×C₄₃	#544	1	1	yes	characteristic
173	C₄×C₄₃	#544	1	1	yes	characteristic
174	Dic₄₃	—	8	4	no	—
175	Dic₄₃	—	8	2	no	—
176	Dic₄₃	—	8	2	no	—
177	C₂×#431	D₄₃	1	1	yes	characteristic
178	#2285	—	43	1	no	—
179	#2283	—	43	1	no	—
180	#1948	—	43	2	no	—
181	#1947	—	43	2	no	—
182	#1854	—	43	4	no	—
183	#1846	—	43	2	no	—
184	#1835	—	43	2	no	—
185	C₂×C₄₃	—	4	2	no	—
186	C₂×C₄₃	#2317	1	1	yes	characteristic
187	C₂×C₄₃	C₂×#427	1	1	yes	characteristic
188	C₂×C₄₃	#2911	1	1	yes	characteristic
189	C₂×C₄₃	#3212	1	2	yes	—
190	C₂×C₄₃	#3233	1	2	yes	—
191	C₂×#117	—	43	2	no	—
192	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
193	#431	Dic₄₃	1	8	yes	complemented
194	C₂×#107	—	43	4	no	—
195	C₂×#106	—	43	4	no	—
196	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
197	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
198	C₂×C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
199	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
200	#384	—	43	2	no	—
201	#383	—	86	1	no	—
202	#382	—	43	1	no	—
203	#382	—	43	2	no	—
204	#382	—	43	1	no	—
205	#381	—	86	1	no	—
206	#380	—	43	2	no	—
207	#380	—	86	1	no	—
208	#380	—	86	1	no	—
209	C₄×C₄⋊C₄	—	43	2	no	—
210	C₄×C₄⋊C₄	—	86	1	no	—
211	C₄×C₄⋊C₄	—	86	1	no	—
212	C₄×C₂²⋊C₄	—	86	1	no	—
213	#376	—	86	1	no	—
214	#340	—	43	4	no	—
215	#340	—	43	4	no	—
216	C₄₃	#21985	1	1	yes	characteristic, complemented, Sylow
217	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
218	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
219	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
220	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
221	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
222	#117	—	172	2	no	—
223	#117	—	172	2	no	—
224	C₂×C₄⋊C₄	—	43	4	no	—
225	C₂×C₄⋊C₄	—	43	2	no	—
226	C₂×C₄⋊C₄	—	43	2	no	—
227	C₂×C₄⋊C₄	—	86	2	no	—
228	C₂×C₄⋊C₄	—	86	2	no	—
229	C₂×C₂²⋊C₄	—	86	2	no	—
230	C₂×C₂²⋊C₄	—	86	2	no	—
231	C₂×C₄²	—	86	2	no	—
232	C₂×C₄²	—	86	2	no	—
233	C₂×C₄²	Dic₈₆	1	4	yes	—
234	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
235	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
236	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
237	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
239	#107	—	172	4	no	—
240	#106	—	172	4	no	—
241	C₄⋊C₈	C₂×Dic₄₃	1	4	yes	—
242	C₄⋊C₈	C₂×Dic₄₃	1	4	yes	—
243	C₂²⋊C₈	C₂×Dic₄₃	1	8	yes	—
244	C₄×C₈	C₂×Dic₄₃	1	8	yes	—
245	#95	—	86	2	no	—
246	#95	—	86	1	no	—
247	#95	—	86	1	no	—
248	#95	—	86	1	no	—
249	#95	—	86	1	no	—
250	#95	—	86	2	no	—
251	#95	—	86	2	no	—
252	#95	—	86	2	no	—
253	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	—	86	2	no	—
261	C₂²×C₄	—	86	2	no	—
262	C₂²×C₄	—	86	4	no	—
263	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
264	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
265	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
266	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
271	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
272	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
278	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
279	C₄⋊C₄	—	86	2	no	—
280	C₄⋊C₄	—	86	4	no	—
281	C₄⋊C₄	—	86	2	no	—
282	C₄⋊C₄	—	86	4	no	—
283	C₄⋊C₄	—	86	2	no	—
284	C₄⋊C₄	—	86	2	no	—
285	C₄⋊C₄	—	172	4	no	—
286	C₄⋊C₄	—	172	4	no	—
287	C₂²⋊C₄	—	172	4	no	—
288	C₂²⋊C₄	—	172	4	no	—
289	C₄²	—	172	4	no	—
290	C₄²	—	172	4	no	—
291	C₄²	C₄⋊Dic₄₃	1	4	yes	—
292	C₄²	C₄⋊Dic₄₃	1	4	yes	—
293	C₄²	C₂×Dic₈₆	1	2	yes	—
294	C₄²	C₂×Dic₈₆	1	2	yes	—
295	C₄²	C₂²×Dic₄₃	1	2	yes	—
296	C₄²	C₂²×Dic₄₃	1	2	yes	—
297	C₂³	—	2	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
302	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	8	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	172	2	no	—
315	C₂×C₄	—	172	2	no	—
316	C₂×C₄	—	172	4	no	—
317	C₂×C₄	—	172	2	no	—
318	C₂×C₄	—	172	2	no	—
319	C₂×C₄	—	172	4	no	—
320	C₂×C₄	—	172	4	no	—
321	C₂×C₄	—	172	2	no	—
322	C₂×C₄	—	172	2	no	—
323	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
324	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
325	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
326	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
327	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
328	C₂×C₄	#13063347	1	4	yes	—
329	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
330	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
331	C₂×C₄	#13063371	1	1	yes	characteristic
332	C₂×C₄	#13063371	1	1	yes	characteristic
333	C₂×C₄	#13063372	1	1	yes	characteristic
334	C₂×C₄	#13063372	1	1	yes	characteristic
335	C₈	—	2	4	no	—
336	C₈	—	2	4	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
343	C₂²	#14583468	1	1	yes	characteristic
344	C₂²	#14583475	1	1	yes	characteristic
345	C₂²	C₂×#13063347	1	2	yes	—
346	C₂²	#14583699	1	2	yes	—
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	344	2	no	—
352	C₄	—	344	2	no	—
353	C₄	—	344	4	no	—
354	C₄	#14582937	1	4	yes	—
355	C₄	#14583382	1	1	yes	characteristic
356	C₄	#14583382	1	1	yes	characteristic
357	C₄	#14583527	1	1	yes	characteristic
358	C₄	#14583527	1	1	yes	characteristic
359	C₂	—	4	2	no	—
360	C₂	#19137218	1	1	yes	characteristic
361	C₂	C₂×#14582937	1	1	yes	characteristic
362	C₂	#19143185	1	1	yes	characteristic
363	C₂	#19145573	1	2	yes	—
364	C₂	#19145776	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#431	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#431	D₄₃
hypocenter	C₄₃	#21985
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #19135359	C₂	semidirect
4. #19135354	C₂	semidirect
6. #19134759	C₂	semidirect
7. #19134726	C₂	semidirect
29. C₄₃×#431	C₄	semidirect
193. #431	Dic₄₃	semidirect
216. C₄₃	#21985	semidirect

Maximal subgroups

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

subgroup	index
#19137002	2
#19136993	2
#19135359	2
#19135354	2
#19134796	2
#19134759	2
#19134726	2
C₂×C₄₃×#431	2
#21985	43