Subgroups — #24996113

Order 11008.

522 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583140	C₂	1	1	yes	maximal, characteristic
2	#19135580	C₂	1	2	yes	maximal
3	#19134624	C₂	1	1	yes	maximal, characteristic
4	#19133168	C₂	1	2	yes	maximal
5	C₈×C₂²⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₂	C₄	1	1	yes	characteristic, complemented
7	C₂×#13063304	C₂²	1	2	yes	—
8	#14583140	—	2	1	no	—
9	#14583140	—	2	1	no	—
10	#14583140	—	2	1	no	—
11	#14583140	—	2	1	no	—
12	C₂×#13063259	—	2	1	no	—
13	C₂×#13063259	C₄	1	1	yes	characteristic, complemented
14	C₂×#13063250	C₄	1	2	yes	complemented
15	#14582759	C₂²	1	2	yes	—
16	#14582665	—	2	4	no	—
17	#14582662	—	2	1	no	—
18	#14582637	—	2	4	no	—
19	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₃	—	2	1	no	—
22	C₄×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
23	C₄₃×#336	—	2	1	no	—
24	C₂²×Dic₈₆	C₂×C₄	1	2	yes	—
25	C₂×Dic₁₇₂	—	2	1	no	—
26	C₂×Dic₁₇₂	—	2	1	no	—
27	C₂×Dic₁₇₂	—	4	2	no	—
28	#13063304	—	2	2	no	—
29	#13063304	—	2	2	no	—
30	#13063304	—	2	2	no	—
31	#13063304	—	2	2	no	—
32	C₂×C₄⋊Dic₄₃	—	2	2	no	—
33	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
34	C₂×Dic₄₃⋊C₄	—	2	2	no	—
35	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
36	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
37	#13063259	—	2	1	no	—
38	#13063259	—	2	1	no	—
39	#13063259	—	4	2	no	—
40	#13063250	—	2	2	no	—
41	#13063250	—	2	2	no	—
42	C₄₃⋊#95	—	2	2	no	—
43	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₄₃	—	2	1	no	—
45	C₂²×C₈×C₄₃	—	4	2	no	—
46	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
53	C₂×C₄²×C₄₃	—	2	1	no	—
54	C₂²⋊C₈×C₄₃	—	2	1	no	—
55	C₂²⋊C₈×C₄₃	D₄	1	2	yes	—
56	C₄×C₈×C₄₃	—	4	2	no	—
57	#95×C₄₃	—	2	1	no	—
58	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	—	4	1	no	—
61	C₂²×C₄×C₄₃	—	4	1	no	—
62	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
68	C₂²×Dic₄₃	—	2	2	no	—
69	C₂²×Dic₄₃	—	2	2	no	—
70	C₂²×Dic₄₃	—	4	2	no	—
71	C₂×Dic₈₆	—	2	2	no	—
72	C₂×Dic₈₆	—	2	2	no	—
73	C₂×Dic₈₆	—	4	2	no	—
74	C₂×Dic₈₆	—	4	2	no	—
75	C₂×C₈×C₄₃	—	2	2	no	—
76	C₂×C₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	—	2	2	no	—
78	C₂×C₈×C₄₃	—	2	2	no	—
79	C₂×C₈×C₄₃	—	4	1	no	—
80	C₂×C₈×C₄₃	—	4	1	no	—
81	C₂×C₈×C₄₃	—	4	1	no	—
82	C₂×C₈×C₄₃	—	4	1	no	—
83	C₂×C₈×C₄₃	—	4	1	no	—
84	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²⋊C₄×C₄₃	—	2	2	no	—
89	C₂²⋊C₄×C₄₃	Q₁₆	1	4	yes	complemented
90	C₄²×C₄₃	—	4	1	no	—
91	C₄²×C₄₃	—	4	1	no	—
92	C₂²⋊Dic₄₃	—	4	2	no	—
93	C₂²⋊Dic₄₃	—	4	2	no	—
94	C₂²⋊Dic₄₃	—	4	4	no	—
95	C₄⋊Dic₄₃	—	2	2	no	—
96	C₄⋊Dic₄₃	—	2	2	no	—
97	C₄⋊Dic₄₃	—	4	2	no	—
98	C₄⋊Dic₄₃	—	4	2	no	—
99	Dic₄₃⋊C₄	—	4	2	no	—
100	Dic₄₃⋊C₄	—	4	2	no	—
101	Dic₁₇₂	—	4	2	no	—
102	Dic₁₇₂	—	4	2	no	—
103	C₂³×C₄₃	—	4	1	no	—
104	C₂³×C₄₃	—	4	1	no	—
105	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
106	C₂³×C₄₃	#121	1	1	yes	characteristic
107	C₂³×C₄₃	#127	1	1	yes	characteristic
108	C₂³×C₄₃	C₂×Q₁₆	1	2	yes	—
109	C₂³×C₄₃	#135	1	2	yes	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	4	no	—
117	C₂×C₄×C₄₃	—	4	1	no	—
118	C₂×C₄×C₄₃	—	4	1	no	—
119	C₂×C₄×C₄₃	—	4	1	no	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	4	2	no	—
122	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
126	C₂×Dic₄₃	—	4	2	no	—
127	C₂×Dic₄₃	—	4	2	no	—
128	C₂×Dic₄₃	—	8	2	no	—
129	C₂×Dic₄₃	—	8	2	no	—
130	C₂×Dic₄₃	—	8	2	no	—
131	C₂×Dic₄₃	—	8	2	no	—
132	C₂×Dic₄₃	—	8	2	no	—
133	Dic₈₆	—	4	2	no	—
134	Dic₈₆	—	4	2	no	—
135	Dic₈₆	—	8	2	no	—
136	C₈×C₄₃	—	2	1	no	—
137	C₈×C₄₃	—	2	1	no	—
138	C₈×C₄₃	—	2	1	no	—
139	C₈×C₄₃	—	4	2	no	—
140	C₈×C₄₃	#99	1	2	yes	—
141	#9782	—	43	1	no	maximal, Sylow
142	C₂²×C₄₃	—	2	1	no	—
143	C₂²×C₄₃	—	2	1	no	—
144	C₂²×C₄₃	—	2	1	no	—
145	C₂²×C₄₃	—	2	1	no	—
146	C₂²×C₄₃	—	2	1	no	—
147	C₂²×C₄₃	—	2	1	no	—
148	C₂²×C₄₃	—	4	2	no	—
149	C₂²×C₄₃	—	4	2	no	—
150	C₂²×C₄₃	#390	1	1	yes	characteristic
151	C₂²×C₄₃	C₄×Q₁₆	1	2	yes	—
152	C₂²×C₄₃	#467	1	2	yes	—
153	C₂²×C₄₃	#494	1	1	yes	characteristic
154	C₂²×C₄₃	#495	1	1	yes	characteristic
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	2	1	no	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	4	4	no	—
159	C₄×C₄₃	C₂×#99	1	1	yes	characteristic
160	C₄×C₄₃	#410	1	1	yes	characteristic
161	Dic₄₃	—	8	2	no	—
162	Dic₄₃	—	8	2	no	—
163	Dic₄₃	—	16	2	no	—
164	C₂×#467	—	43	1	no	—
165	#1996	—	43	2	no	—
166	#1808	—	43	1	no	—
167	C₈×C₂²⋊C₄	D₄₃	1	1	yes	characteristic
168	#1520	—	43	2	no	—
169	C₂×C₄₃	—	2	1	no	—
170	C₂×C₄₃	—	2	2	no	—
171	C₂×C₄₃	—	2	2	no	—
172	C₂×C₄₃	—	2	1	no	—
173	C₂×C₄₃	#1964	1	1	yes	characteristic
174	C₂×C₄₃	#1969	1	1	yes	characteristic
175	C₂×C₄₃	#1976	1	1	yes	characteristic
176	C₂²×Q₁₆	—	43	1	no	—
177	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
178	C₂×C₂²⋊Q₈	—	43	2	no	—
179	#467	—	86	1	no	—
180	#467	—	86	1	no	—
181	#467	—	86	1	no	—
182	#467	—	86	1	no	—
183	C₂×#107	—	43	1	no	—
184	C₂×#107	—	86	1	no	—
185	C₂×#103	—	43	2	no	—
186	C₂×C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
187	C₂×C₄×C₈	—	2	1	no	—
188	#380	—	43	2	no	—
189	C₄×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
190	#342	—	86	4	no	—
191	#340	—	86	1	no	—
192	#336	—	2	1	no	—
193	#328	—	86	4	no	—
194	C₄₃	#9782	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₈	—	43	2	no	—
196	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
197	C₂×Q₁₆	—	86	1	no	—
198	C₂×Q₁₆	—	86	1	no	—
199	C₂×Q₁₆	—	172	2	no	—
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	4	2	no	—
202	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
203	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
204	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
205	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
206	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
207	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
208	C₂²⋊Q₈	—	86	2	no	—
209	C₂²⋊Q₈	—	86	2	no	—
210	C₂²⋊Q₈	—	86	2	no	—
211	C₂²⋊Q₈	—	86	2	no	—
212	C₂×C₄⋊C₄	—	43	2	no	—
213	C₂×C₄⋊C₄	—	86	2	no	—
214	C₂×C₄⋊C₄	—	86	2	no	—
215	C₂×C₂²⋊C₄	—	86	2	no	—
216	C₂×C₂²⋊C₄	—	86	2	no	—
217	C₂×C₂²⋊C₄	D₁₇₂	1	2	yes	—
218	C₂×C₄²	—	2	1	no	—
219	#107	—	86	1	no	—
220	#107	—	86	1	no	—
221	#107	—	172	2	no	—
222	#103	—	86	2	no	—
223	#103	—	86	2	no	—
224	C₂²⋊C₈	—	2	1	no	—
225	C₂²⋊C₈	D₁₇₂	1	2	yes	—
226	C₄×C₈	—	4	2	no	—
227	#95	—	2	1	no	—
228	#95	—	86	2	no	—
229	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
230	C₂×Q₈	—	86	2	no	—
231	C₂×Q₈	—	86	2	no	—
232	C₂×Q₈	—	172	2	no	—
233	C₂×Q₈	—	172	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	86	2	no	—
238	C₂²×C₄	—	86	2	no	—
239	C₂²×C₄	—	172	2	no	—
240	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
243	C₂²×C₄	#10621898	1	1	yes	characteristic
244	C₂²×C₄	#10621898	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
246	Q₁₆	—	172	2	no	—
247	Q₁₆	—	172	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
258	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
259	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
260	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
261	C₄⋊C₄	—	86	2	no	—
262	C₄⋊C₄	—	86	2	no	—
263	C₄⋊C₄	—	172	2	no	—
264	C₄⋊C₄	—	172	2	no	—
265	C₄⋊C₄	—	172	2	no	—
266	C₄⋊C₄	—	172	2	no	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	—	172	4	no	—
269	C₂²⋊C₄	—	172	2	no	—
270	C₂²⋊C₄	—	172	2	no	—
271	C₂²⋊C₄	Dic₁₇₂	1	4	yes	complemented
272	C₄²	—	4	1	no	—
273	C₄²	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
277	C₂³	#13063299	1	1	yes	characteristic
278	C₂³	#13063322	1	1	yes	characteristic
279	C₂³	C₂×Dic₁₇₂	1	2	yes	—
280	C₂³	#13063347	1	2	yes	—
281	Q₈	—	344	2	no	—
282	Q₈	—	172	2	no	—
283	Q₈	—	172	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	344	2	no	—
297	C₂×C₄	—	344	2	no	—
298	C₂×C₄	—	344	2	no	—
299	C₂×C₄	—	344	2	no	—
300	C₂×C₄	—	344	2	no	—
301	C₂×C₄	—	172	2	no	—
302	C₂×C₄	—	172	2	no	—
303	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
304	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
305	C₂×C₄	#13063299	1	1	yes	characteristic
306	C₂×C₄	#13063313	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	4	2	no	—
311	C₈	#13063239	1	2	yes	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	#14582801	1	1	yes	characteristic
321	C₂²	C₄×Dic₁₇₂	1	2	yes	—
322	C₂²	#14583140	1	2	yes	—
323	C₂²	#14583279	1	1	yes	characteristic
324	C₂²	#14583283	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	4	no	—
329	C₄	—	344	2	no	—
330	C₄	—	344	2	no	—
331	C₄	—	688	2	no	—
332	C₄	C₂×#13063239	1	1	yes	characteristic
333	C₄	#14582872	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	2	1	no	—
336	C₂	—	2	2	no	—
337	C₂	—	2	1	no	—
338	C₂	#19135431	1	1	yes	characteristic
339	C₂	#19135456	1	1	yes	characteristic
340	C₂	#19135493	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14582801
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₈×C₂²⋊C₄×C₄₃	C₂
socle	C₂²×C₄₃	#390
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₈×C₂²⋊C₄	D₄₃
hypocenter	C₄₃	#9782
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
6. C₂²×Dic₁₇₂	C₄	semidirect
13. C₂×#13063259	C₄	semidirect
14. C₂×#13063250	C₄	semidirect
89. C₂²⋊C₄×C₄₃	Q₁₆	semidirect
194. C₄₃	#9782	semidirect
271. C₂²⋊C₄	Dic₁₇₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583140	2
#19135580	2
#19134624	2
#19133168	2
C₈×C₂²⋊C₄×C₄₃	2
#9782	43