Subgroups — #25002459

Order 11008.

794 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582923	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10621877	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582664	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10621877	C₂²	1	2	yes	—
6	C₂²×#10621877	C₂²	1	2	yes	—
7	#14582923	—	2	2	no	—
8	#14582923	C₄	1	4	yes	—
9	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
11	C₄×#10621877	—	2	2	no	—
12	C₄×#10621877	C₄	1	4	yes	—
13	C₂×#13063236	—	2	1	no	—
14	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
15	#14582664	—	2	8	no	—
16	C₂³×C₈×C₄₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₄₃	—	2	2	no	—
19	C₂×C₄⋊C₈×C₄₃	—	2	2	no	—
20	C₂×C₄⋊C₈×C₄₃	C₄	1	2	yes	—
21	C₂×C₄⋊C₈×C₄₃	C₄	1	2	yes	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	—	2	2	no	—
25	C₂×#10621877	—	2	2	no	—
26	C₂×#10621877	—	2	4	no	—
27	C₂×#10621877	—	2	4	no	—
28	C₂×#10621877	C₂×C₄	1	4	yes	—
29	C₂×#10621877	C₂×C₄	1	4	yes	—
30	C₂×#10621877	D₄	1	4	yes	—
31	C₂×#10621877	Q₈	1	4	yes	—
32	C₂²×C₄₃⋊C₈	—	2	2	no	—
33	C₂²×C₄₃⋊C₈	—	2	2	no	—
34	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
35	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
36	#13063236	—	2	4	no	—
37	C₄×C₄₃⋊C₈	—	2	4	no	—
38	C₂³×C₄×C₄₃	Q₈	1	1	yes	characteristic
39	C₂³×C₄×C₄₃	Q₈	1	1	yes	characteristic
40	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₃	—	2	4	no	—
42	C₂²×C₈×C₄₃	—	2	2	no	—
43	C₂²×C₈×C₄₃	—	2	2	no	—
44	C₂²×C₈×C₄₃	—	2	4	no	—
45	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₃	D₄	1	2	yes	—
53	C₂×C₄²×C₄₃	D₄	1	2	yes	—
54	C₄⋊C₈×C₄₃	—	2	4	no	—
55	C₄⋊C₈×C₄₃	—	2	4	no	—
56	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	—	2	4	no	—
58	C₂²×C₄×C₄₃	—	4	2	no	—
59	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₄₃	#41	1	2	yes	—
76	C₂²×C₄×C₄₃	#41	1	2	yes	—
77	C₂×C₈×C₄₃	—	2	4	no	—
78	C₂×C₈×C₄₃	—	2	8	no	—
79	C₂×C₈×C₄₃	—	4	4	no	—
80	C₂×C₈×C₄₃	—	4	2	no	—
81	C₂×C₈×C₄₃	—	4	2	no	—
82	C₄²×C₄₃	C₄²	1	2	yes	—
83	C₄²×C₄₃	C₄²	1	2	yes	—
84	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
89	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
90	#10621877	—	2	4	no	—
91	#10621877	—	2	4	no	—
92	#10621877	—	2	4	no	—
93	#10621877	—	2	8	no	—
94	#10621877	—	4	2	no	—
95	#10621877	—	4	2	no	—
96	#10621877	C₄⋊C₄	1	8	yes	—
97	C₂×C₄₃⋊C₈	—	2	4	no	—
98	C₂×C₄₃⋊C₈	—	2	4	no	—
99	C₂×C₄₃⋊C₈	—	2	4	no	—
100	C₂×C₄₃⋊C₈	—	2	4	no	—
101	C₂×C₄₃⋊C₈	—	4	4	no	—
102	C₂³×C₄₃	—	2	2	no	—
103	C₂³×C₄₃	—	2	2	no	—
104	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₄₃	#125	1	2	yes	—
108	C₂³×C₄₃	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	4	no	—
113	C₂×C₄×C₄₃	—	2	4	no	—
114	C₂×C₄×C₄₃	—	4	2	no	—
115	C₂×C₄×C₄₃	—	4	2	no	—
116	C₂×C₄×C₄₃	#95	1	2	yes	—
117	C₂×C₄×C₄₃	#95	1	2	yes	—
118	C₂×C₄×C₄₃	#95	1	2	yes	—
119	C₂×C₄×C₄₃	#95	1	2	yes	—
120	C₂×C₄×C₄₃	#108	1	4	yes	—
121	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₄₃	#117	1	2	yes	—
127	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₄₃	#121	1	2	yes	—
132	C₂×C₄×C₄₃	#121	1	2	yes	—
133	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
136	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
137	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
138	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
139	C₈×C₄₃	—	4	4	no	—
140	C₄₃⋊C₈	—	2	8	no	—
141	C₄₃⋊C₈	—	4	4	no	—
142	#10853	—	43	1	no	maximal, Sylow
143	C₂²×C₄₃	—	2	4	no	—
144	C₂²×C₄₃	—	2	4	no	—
145	C₂²×C₄₃	—	2	1	no	—
146	C₂²×C₄₃	—	2	1	no	—
147	C₂²×C₄₃	—	2	1	no	—
148	C₂²×C₄₃	—	2	1	no	—
149	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
150	C₂²×C₄₃	#382	1	1	yes	characteristic
151	C₂²×C₄₃	#384	1	1	yes	characteristic
152	C₂²×C₄₃	#389	1	2	yes	—
153	C₂²×C₄₃	#389	1	2	yes	—
154	C₂²×C₄₃	C₂×#108	1	2	yes	—
155	C₂²×C₄₃	#430	1	2	yes	—
156	C₄×C₄₃	—	2	2	no	—
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	—	2	2	no	—
159	C₄×C₄₃	#341	1	4	yes	complemented
160	C₄×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
161	C₄×C₄₃	#380	1	2	yes	—
162	C₄×C₄₃	#390	1	1	yes	characteristic
163	C₄×C₄₃	#390	1	1	yes	characteristic
164	C₄×C₄₃	#391	1	1	yes	characteristic
165	C₄×C₄₃	#391	1	1	yes	characteristic
166	C₂×#423	—	43	1	no	—
167	C₂²×C₄⋊C₈	D₄₃	1	1	yes	characteristic
168	C₂×C₄×M₁₆	—	43	1	no	—
169	C₂×#341	—	43	4	no	—
170	C₂×C₄₃	—	2	1	no	—
171	C₂×C₄₃	—	2	1	no	—
172	C₂×C₄₃	—	2	2	no	—
173	C₂×C₄₃	C₄⋊#95	1	1	yes	characteristic
174	C₂×C₄₃	C₂×#341	1	1	yes	characteristic
175	C₂×C₄₃	#1738	1	1	yes	characteristic
176	C₂×C₄₃	#1950	1	4	yes	—
177	C₂²×M₁₆	—	43	2	no	—
178	C₂²×M₁₆	—	43	2	no	—
179	C₂³×C₈	C₂×D₄₃	1	2	yes	—
180	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
181	#423	—	43	4	no	—
182	#423	—	86	2	no	—
183	C₂×C₄⋊C₈	—	2	2	no	—
184	C₂×C₄⋊C₈	—	2	2	no	—
185	C₂×C₄⋊C₈	—	86	1	no	—
186	C₂×C₄⋊C₈	—	86	1	no	—
187	C₂×C₄⋊C₈	Dic₄₃	1	2	yes	—
188	C₂×C₄⋊C₈	Dic₄₃	1	2	yes	—
189	C₄×M₁₆	—	43	4	no	—
190	C₄×M₁₆	—	86	2	no	—
191	C₂×#97	—	86	1	no	—
192	C₂×C₄×C₈	—	86	1	no	—
193	#341	—	86	8	no	—
194	C₄₃	#10853	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	Dic₈₆	1	1	yes	characteristic
196	C₂³×C₄	Dic₈₆	1	1	yes	characteristic
197	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
198	C₂×M₁₆	—	43	4	no	—
199	C₂×M₁₆	—	43	4	no	—
200	C₂×M₁₆	—	43	4	no	—
201	C₂×M₁₆	—	43	4	no	—
202	C₂×M₁₆	—	86	2	no	—
203	C₂×M₁₆	—	86	2	no	—
204	C₂×M₁₆	—	86	2	no	—
205	C₂×M₁₆	—	86	2	no	—
206	C₂×M₁₆	—	86	4	no	—
207	C₂×M₁₆	—	86	4	no	—
208	C₂²×C₈	—	2	4	no	—
209	C₂²×C₈	—	2	4	no	—
210	C₂²×C₈	—	2	2	no	—
211	C₂²×C₈	—	2	2	no	—
212	C₂²×C₈	—	86	2	no	—
213	C₂²×C₈	—	86	2	no	—
214	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
215	C₂×C₄²	C₄×D₄₃	1	2	yes	—
216	C₂×C₄²	C₄×D₄₃	1	2	yes	—
217	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
218	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
219	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
220	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
221	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
222	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	2	4	no	—
225	C₄⋊C₈	—	86	4	no	—
226	C₄⋊C₈	—	86	4	no	—
227	#97	—	86	4	no	—
228	C₄×C₈	—	86	4	no	—
229	C₂⁴	C₂×Dic₈₆	1	1	yes	characteristic
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
233	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
234	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
235	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
236	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
237	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
238	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
239	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₄₃	1	2	yes	—
244	C₂²×C₄	#10621900	1	2	yes	—
245	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
246	C₂²×C₄	#10621902	1	2	yes	—
247	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
250	M₁₆	—	43	8	no	—
251	M₁₆	—	86	4	no	—
252	M₁₆	—	86	4	no	—
253	M₁₆	—	86	4	no	—
254	M₁₆	—	86	8	no	—
255	M₁₆	—	172	2	no	—
256	M₁₆	—	172	2	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	8	no	—
259	C₂×C₈	—	4	4	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	86	4	no	—
263	C₂×C₈	—	86	4	no	—
264	C₂×C₈	—	86	4	no	—
265	C₂×C₈	—	86	4	no	—
266	C₂×C₈	—	172	4	no	—
267	C₄²	C₄×Dic₄₃	1	2	yes	—
268	C₄²	C₄×Dic₄₃	1	2	yes	—
269	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
270	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
271	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
272	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
273	C₄²	C₂²⋊Dic₄₃	1	2	yes	—
274	C₄²	C₂²⋊Dic₄₃	1	2	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
278	C₂³	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
279	C₂³	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
280	C₂³	#13063316	1	2	yes	—
281	C₂³	#13063324	1	2	yes	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	C₄₃⋊#95	1	2	yes	—
290	C₂×C₄	C₄₃⋊#95	1	2	yes	—
291	C₂×C₄	C₄₃⋊#95	1	2	yes	—
292	C₂×C₄	C₄₃⋊#95	1	2	yes	—
293	C₂×C₄	#13063261	1	4	yes	—
294	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
295	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
296	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
298	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
299	C₂×C₄	#13063283	1	2	yes	—
300	C₂×C₄	#13063287	1	2	yes	—
301	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
302	C₂×C₄	#13063290	1	2	yes	—
303	C₂×C₄	Q₈×Dic₄₃	1	2	yes	—
304	C₂×C₄	#13063296	1	2	yes	—
305	C₂×C₄	#13063300	1	2	yes	—
306	C₂×C₄	#13063301	1	2	yes	—
307	C₂×C₄	#13063305	1	2	yes	—
308	C₂×C₄	#13063312	1	1	yes	characteristic
309	C₂×C₄	#13063314	1	1	yes	characteristic
310	C₂×C₄	#13063321	1	1	yes	characteristic
311	C₂×C₄	#13063323	1	1	yes	characteristic
312	C₈	—	4	4	no	—
313	C₈	—	86	8	no	—
314	C₈	—	172	4	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	2	4	no	—
321	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
322	C₂²	#14582764	1	1	yes	characteristic
323	C₂²	#14582768	1	1	yes	characteristic
324	C₂²	#14582795	1	2	yes	—
325	C₂²	#14582798	1	2	yes	—
326	C₂²	C₂×#13063261	1	2	yes	—
327	C₂²	#14582948	1	2	yes	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	#14582664	1	4	yes	complemented
332	C₄	C₄⋊C₄×Dic₄₃	1	2	yes	—
333	C₄	#14582756	1	2	yes	—
334	C₄	#14582800	1	1	yes	characteristic
335	C₄	#14582802	1	1	yes	characteristic
336	C₄	#14582804	1	1	yes	characteristic
337	C₄	#14582806	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	#19132824	1	1	yes	characteristic
342	C₂	C₂×#14582664	1	1	yes	characteristic
343	C₂	#19134322	1	1	yes	characteristic
344	C₂	#19135371	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂²×C₄⋊C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₃	C₂×Q₈
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂²×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#10853
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
159. C₄×C₄₃	#341	semidirect
194. C₄₃	#10853	semidirect
331. C₄	#14582664	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582923	2
C₂×C₄×#10621877	2
C₂×#14582664	2
C₂²×C₄⋊C₈×C₄₃	2
#10853	43