Subgroups — #25014977

Order 11008.

474 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063270	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582727	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582704	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14582701	C₂	1	1	yes	maximal, characteristic
5	#19135461	C₂	1	1	yes	maximal, characteristic
6	#19135456	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×#13063270	—	2	2	no	—
11	C₂×#13063270	—	2	2	no	—
12	C₂×#13063270	C₄	1	4	yes	complemented
13	C₂×#13063259	C₂²	1	1	yes	characteristic
14	C₂×#13063259	C₂²	1	1	yes	characteristic
15	C₂×#13063250	—	2	1	no	—
16	C₂×#13063247	—	2	1	no	—
17	#14582805	—	2	1	no	—
18	#14582801	—	2	1	no	—
19	#14582727	—	2	2	no	—
20	#14582704	—	2	2	no	—
21	#14582701	—	2	2	no	—
22	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄×C₁₆×C₄₃	—	2	2	no	—
26	C₂²×Dic₈₆	—	2	1	no	—
27	C₂²×D₁₇₂	—	2	1	no	—
28	C₂×Dic₁₇₂	—	2	1	no	—
29	C₂×Dic₁₇₂	—	2	1	no	—
30	C₂×Dic₁₇₂	—	2	1	no	—
31	C₂×Dic₁₇₂	—	2	1	no	—
32	C₂×Dic₁₇₂	—	2	1	no	—
33	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
34	C₂×D₃₄₄	—	2	1	no	—
35	C₂×D₃₄₄	—	2	1	no	—
36	C₂×D₃₄₄	—	2	1	no	—
37	C₂×D₃₄₄	—	2	1	no	—
38	C₂×D₃₄₄	—	2	1	no	—
39	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊Dic₄₃	—	2	1	no	—
41	C₂×C₄⋊Dic₄₃	—	2	1	no	—
42	C₂×(C₂×D₄₃)⋊C₄	—	4	2	no	—
43	#13063270	—	2	8	no	—
44	#13063259	—	2	1	no	—
45	#13063259	—	2	1	no	—
46	#13063259	—	2	1	no	—
47	#13063259	—	2	1	no	—
48	#13063250	—	4	2	no	—
49	#13063247	—	4	2	no	—
50	C₄₃⋊#95	—	4	2	no	—
51	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
54	C₂×C₁₆×C₄₃	—	2	1	no	—
55	C₂×C₁₆×C₄₃	—	2	1	no	—
56	C₂×C₁₆×C₄₃	C₂×C₄	1	2	yes	—
57	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
58	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
59	C₂×C₁₆×C₄₃	D₄	1	2	yes	—
60	C₄×C₈×C₄₃	—	2	1	no	—
61	C₄×C₈×C₄₃	—	2	1	no	—
62	C₂³×D₄₃	—	4	1	no	—
63	C₂²×C₄×C₄₃	D₈	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₄₃	—	4	1	no	—
66	C₂²×Dic₄₃	—	4	1	no	—
67	C₂²×Dic₄₃	—	4	1	no	—
68	C₂×D₁₇₂	—	2	2	no	—
69	C₂×D₁₇₂	—	4	1	no	—
70	C₂×D₁₇₂	—	4	1	no	—
71	C₂×D₁₇₂	—	4	1	no	—
72	C₂×D₁₇₂	—	4	1	no	—
73	C₂×D₁₇₂	—	4	1	no	—
74	C₂×Dic₈₆	—	2	2	no	—
75	C₂×Dic₈₆	—	4	1	no	—
76	C₂×Dic₈₆	—	4	1	no	—
77	C₂×Dic₈₆	—	4	1	no	—
78	C₂×Dic₈₆	—	4	1	no	—
79	C₂×Dic₈₆	—	4	1	no	—
80	C₂×C₈×C₄₃	—	2	1	no	—
81	C₂×C₈×C₄₃	—	2	1	no	—
82	C₂×C₈×C₄₃	D₈	1	2	yes	—
83	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
89	C₄²×C₄₃	—	2	1	no	—
90	C₄²×C₄₃	—	2	1	no	—
91	(C₂×D₄₃)⋊C₄	—	8	4	no	—
92	C₄⋊Dic₄₃	—	4	1	no	—
93	C₄⋊Dic₄₃	—	4	1	no	—
94	C₄⋊Dic₄₃	—	4	1	no	—
95	C₄⋊Dic₄₃	—	4	1	no	—
96	Dic₁₇₂	—	2	2	no	—
97	Dic₁₇₂	—	2	2	no	—
98	Dic₁₇₂	—	4	1	no	—
99	Dic₁₇₂	—	4	1	no	—
100	D₃₄₄	—	2	2	no	—
101	D₃₄₄	—	2	2	no	—
102	D₃₄₄	—	4	1	no	—
103	D₃₄₄	—	4	1	no	—
104	C₁₆×C₄₃	—	2	2	no	—
105	C₁₆×C₄₃	C₂²⋊C₄	1	4	yes	complemented
106	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₄₃	—	4	2	no	—
108	C₂²×D₄₃	—	8	1	no	—
109	C₂²×D₄₃	—	8	1	no	—
110	C₂²×D₄₃	—	8	2	no	—
111	C₂²×D₄₃	—	8	2	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	SD₃₂	1	4	yes	complemented
115	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
121	C₂×Dic₄₃	—	4	2	no	—
122	C₂×Dic₄₃	—	8	1	no	—
123	C₂×Dic₄₃	—	8	1	no	—
124	C₂×Dic₄₃	—	8	1	no	—
125	C₂×Dic₄₃	—	8	1	no	—
126	C₂×Dic₄₃	—	8	2	no	—
127	C₂×Dic₄₃	—	8	2	no	—
128	D₁₇₂	—	4	2	no	—
129	D₁₇₂	—	4	2	no	—
130	D₁₇₂	—	8	1	no	—
131	D₁₇₂	—	8	1	no	—
132	Dic₈₆	—	4	2	no	—
133	Dic₈₆	—	4	2	no	—
134	Dic₈₆	—	8	1	no	—
135	Dic₈₆	—	8	1	no	—
136	C₈×C₄₃	—	2	2	no	—
137	C₈×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
139	C₈×C₄₃	#121	1	1	yes	characteristic
140	C₈×C₄₃	#124	1	1	yes	characteristic
141	#13237	—	43	1	no	maximal, Sylow
142	C₂²×C₄₃	C₄×D₈	1	1	yes	characteristic
143	C₂²×C₄₃	#466	1	1	yes	characteristic
144	C₂²×C₄₃	#493	1	1	yes	characteristic
145	C₂²×C₄₃	C₂×SD₃₂	1	2	yes	—
146	C₂²×C₄₃	#508	1	2	yes	—
147	C₂×D₄₃	—	8	4	no	—
148	C₂×D₄₃	—	8	2	no	—
149	C₂×D₄₃	—	16	2	no	—
150	C₂×D₄₃	—	16	1	no	—
151	C₂×D₄₃	—	16	1	no	—
152	C₄×C₄₃	—	2	4	no	—
153	C₄×C₄₃	#390	1	1	yes	characteristic
154	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
155	C₄×C₄₃	#466	1	1	yes	characteristic
156	C₄×C₄₃	#495	1	1	yes	characteristic
157	Dic₄₃	—	8	2	no	—
158	Dic₄₃	—	16	1	no	—
159	Dic₄₃	—	16	1	no	—
160	C₂²×SD₃₂	—	43	1	no	—
161	C₂×#367	—	43	1	no	—
162	C₂×#358	—	43	1	no	—
163	C₂×#357	—	43	1	no	—
164	C₂×C₄×C₁₆	D₄₃	1	1	yes	characteristic, complemented
165	#1970	—	43	1	no	—
166	#1969	—	43	1	no	—
167	C₂×C₄₃	#1970	1	1	yes	characteristic
168	C₂×C₄₃	C₄×SD₃₂	1	2	yes	—
169	C₂×C₄₃	#2217	1	2	yes	—
170	C₂×C₄₃	#2248	1	1	yes	characteristic
171	C₂×C₄₃	#2249	1	1	yes	characteristic
172	D₄₃	—	16	2	no	—
173	C₂²×Q₁₆	—	43	1	no	—
174	C₂²×D₈	—	43	1	no	—
175	C₂×SD₃₂	—	43	4	no	—
176	C₂×SD₃₂	—	86	2	no	—
177	C₂×SD₃₂	—	86	2	no	—
178	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
179	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
180	C₂×#107	—	43	1	no	—
181	C₂×#107	—	43	1	no	—
182	C₂×#103	—	86	1	no	—
183	C₂×#102	—	86	1	no	—
184	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
185	#391	—	86	1	no	—
186	#390	—	86	1	no	—
187	#367	—	86	2	no	—
188	#358	—	86	2	no	—
189	#357	—	86	2	no	—
190	C₄×C₁₆	—	2	2	no	—
191	C₄₃	#13237	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₈	—	86	1	no	—
193	C₂²×D₄	—	86	1	no	—
194	C₂×Q₁₆	—	43	2	no	—
195	C₂×Q₁₆	—	86	1	no	—
196	C₂×Q₁₆	—	86	1	no	—
197	C₂×Q₁₆	—	86	1	no	—
198	C₂×Q₁₆	—	86	1	no	—
199	C₂×Q₁₆	—	86	1	no	—
200	C₂×D₈	—	43	2	no	—
201	C₂×D₈	—	86	1	no	—
202	C₂×D₈	—	86	1	no	—
203	C₂×D₈	—	86	1	no	—
204	C₂×D₈	—	86	1	no	—
205	C₂×D₈	—	86	1	no	—
206	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
208	C₂×C₄⋊C₄	—	86	1	no	—
209	C₂×C₄⋊C₄	—	86	1	no	—
210	C₂×C₂²⋊C₄	—	172	2	no	—
211	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
212	SD₃₂	—	86	8	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	C₄×D₄₃	1	2	yes	—
216	C₂×C₁₆	D₁₇₂	1	2	yes	—
217	C₂×C₁₆	D₁₇₂	1	2	yes	—
218	C₂×C₁₆	C₂²⋊D₄₃	1	2	yes	—
219	#107	—	86	1	no	—
220	#107	—	86	1	no	—
221	#107	—	86	1	no	—
222	#107	—	86	1	no	—
223	#103	—	172	2	no	—
224	#102	—	172	2	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	#95	—	172	2	no	—
228	C₂⁴	—	172	1	no	—
229	C₂×Q₈	—	86	2	no	—
230	C₂×Q₈	—	172	1	no	—
231	C₂×Q₈	—	172	1	no	—
232	C₂×Q₈	—	172	1	no	—
233	C₂×Q₈	—	172	1	no	—
234	C₂×Q₈	—	172	1	no	—
235	C₂×D₄	—	86	2	no	—
236	C₂×D₄	—	172	1	no	—
237	C₂×D₄	—	172	1	no	—
238	C₂×D₄	—	172	1	no	—
239	C₂×D₄	—	172	1	no	—
240	C₂×D₄	—	172	1	no	—
241	C₂²×C₄	—	172	1	no	—
242	C₂²×C₄	—	172	1	no	—
243	C₂²×C₄	—	172	1	no	—
244	C₂²×C₄	D₃₄₄	1	2	yes	—
245	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
246	Q₁₆	—	86	2	no	—
247	Q₁₆	—	86	2	no	—
248	Q₁₆	—	172	1	no	—
249	Q₁₆	—	172	1	no	—
250	D₈	—	86	2	no	—
251	D₈	—	86	2	no	—
252	D₈	—	172	1	no	—
253	D₈	—	172	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	D₃₄₄	1	2	yes	—
257	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
258	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
259	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
260	C₂×C₈	#10621898	1	1	yes	characteristic
261	C₂×C₈	#10621898	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
263	C₄⋊C₄	—	172	1	no	—
264	C₄⋊C₄	—	172	1	no	—
265	C₄⋊C₄	—	172	1	no	—
266	C₄⋊C₄	—	172	1	no	—
267	C₂²⋊C₄	—	344	4	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₁₆	—	2	2	no	—
271	C₁₆	(C₂×D₄₃)⋊C₄	1	4	yes	complemented
272	C₂³	—	344	2	no	—
273	C₂³	—	344	2	no	—
274	C₂³	—	344	1	no	—
275	C₂³	—	344	1	no	—
276	C₂³	—	172	2	no	—
277	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
278	Q₈	—	344	1	no	—
279	Q₈	—	344	1	no	—
280	Q₈	—	172	2	no	—
281	Q₈	—	172	2	no	—
282	D₄	—	344	1	no	—
283	D₄	—	344	1	no	—
284	D₄	—	172	2	no	—
285	D₄	—	172	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	344	1	no	—
289	C₂×C₄	—	344	1	no	—
290	C₂×C₄	—	344	2	no	—
291	C₂×C₄	—	344	2	no	—
292	C₂×C₄	—	344	1	no	—
293	C₂×C₄	—	344	1	no	—
294	C₂×C₄	—	172	2	no	—
295	C₂×C₄	#13063270	1	4	yes	complemented
296	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
297	C₂×C₄	#13063299	1	1	yes	characteristic
298	C₂×C₄	#13063322	1	1	yes	characteristic
299	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
300	C₂×C₄	#13063347	1	1	yes	characteristic
301	C₂×C₄	#13063347	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
304	C₈	C₄×D₁₇₂	1	1	yes	characteristic
305	C₈	#13063299	1	1	yes	characteristic
306	C₈	#13063313	1	1	yes	characteristic
307	C₂²	—	344	4	no	—
308	C₂²	—	344	2	no	—
309	C₂²	—	688	1	no	—
310	C₂²	—	688	1	no	—
311	C₂²	—	688	2	no	—
312	C₂²	C₄×D₃₄₄	1	1	yes	characteristic
313	C₂²	#14583135	1	1	yes	characteristic
314	C₂²	#14583276	1	1	yes	characteristic
315	C₂²	C₂×#13063270	1	2	yes	—
316	C₂²	#14583331	1	2	yes	—
317	C₄	—	2	4	no	—
318	C₄	—	344	2	no	—
319	C₄	—	688	1	no	—
320	C₄	—	688	1	no	—
321	C₄	#14582801	1	1	yes	characteristic
322	C₄	C₄×D₃₄₄	1	1	yes	characteristic
323	C₄	#14583135	1	1	yes	characteristic
324	C₄	#14583283	1	1	yes	characteristic
325	C₂	—	688	2	no	—
326	C₂	#19135461	1	1	yes	characteristic
327	C₂	C₄×#13063270	1	2	yes	—
328	C₂	#19136691	1	2	yes	—
329	C₂	#19136851	1	1	yes	characteristic
330	C₂	#19136856	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₄₄
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₂×C₄×C₁₆×C₄₃	C₂
socle	C₂³×C₄₃	C₂×D₈
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₄₃
hypocenter	C₄₃	#13237
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#14582727	C₂	semidirect
3. C₂×#14582704	C₂	semidirect
6. #19135456	C₂	semidirect
7. C₂×C₄×C₁₆×C₄₃	C₂	semidirect
12. C₂×#13063270	C₄	semidirect
105. C₁₆×C₄₃	C₂²⋊C₄	semidirect
114. C₂×C₄×C₄₃	SD₃₂	semidirect
164. C₂×C₄×C₁₆	D₄₃	semidirect
191. C₄₃	#13237	semidirect
271. C₁₆	(C₂×D₄₃)⋊C₄	semidirect
295. C₂×C₄	#13063270	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063270	2
C₂×#14582727	2
C₂×#14582704	2
C₂×#14582701	2
#19135461	2
#19135456	2
C₂×C₄×C₁₆×C₄₃	2
#13237	43