Subgroups — #24996397

Order 11008.

578 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583246	C₂	1	1	yes	maximal, characteristic
2	#19135556	C₂	1	1	yes	maximal, characteristic, complemented
3	#19135401	C₂	1	1	yes	maximal, characteristic
4	#19135401	C₂	1	1	yes	maximal, characteristic
5	#19135397	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134755	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063325	C₂²	1	1	yes	characteristic
9	C₂×#13063322	C₂²	1	1	yes	characteristic
10	#14583246	—	2	2	no	—
11	#14583246	C₄	1	4	yes	complemented
12	C₂×#13063259	—	2	2	no	—
13	C₂×#13063250	—	2	2	no	—
14	C₂×#13063247	—	2	2	no	—
15	C₂×#13063247	—	2	2	no	—
16	C₂×#13063247	—	2	2	no	—
17	#14582769	C₂²	1	1	yes	characteristic
18	#14582769	C₂²	1	1	yes	characteristic
19	#14582662	—	2	2	no	—
20	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄₃×#322	—	2	2	no	—
24	C₂²×Dic₈₆	—	2	1	no	—
25	C₂²×D₁₇₂	—	2	1	no	—
26	C₂²×D₁₇₂	—	2	1	no	—
27	C₂²×D₁₇₂	—	2	1	no	—
28	#13063325	—	2	1	no	—
29	#13063325	—	2	1	no	—
30	#13063325	—	2	1	no	—
31	#13063325	C₂×C₄	1	2	yes	—
32	#13063322	—	2	1	no	—
33	#13063322	—	2	1	no	—
34	#13063322	—	2	1	no	—
35	#13063322	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₄₃	—	2	1	no	—
37	C₂×C₄⋊Dic₄₃	—	2	1	no	—
38	C₂×C₄⋊Dic₄₃	—	2	1	no	—
39	C₂×C₄⋊Dic₄₃	—	2	1	no	—
40	C₂×C₄⋊Dic₄₃	—	2	1	no	—
41	C₂×C₄⋊Dic₄₃	—	2	1	no	—
42	C₂×C₄⋊Dic₄₃	—	2	1	no	—
43	C₂×C₄⋊Dic₄₃	—	2	1	no	—
44	#13063259	—	4	4	no	—
45	#13063250	—	4	4	no	—
46	#13063247	—	2	8	no	—
47	#13063247	—	4	4	no	—
48	#13063247	—	4	4	no	—
49	C₂²×C₈×C₄₃	D₄	1	2	yes	—
50	C₂²×C₈×C₄₃	D₄	1	2	yes	—
51	C₂²×C₈×C₄₃	D₄	1	2	yes	—
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₄₃	—	2	1	no	—
54	C₄×C₈×C₄₃	—	2	1	no	—
55	C₄×C₈×C₄₃	—	2	1	no	—
56	C₄×C₈×C₄₃	—	2	1	no	—
57	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₄₃	D₄	1	2	yes	—
59	C₂³×D₄₃	—	4	1	no	—
60	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₄₃	—	4	1	no	—
64	C₂²×Dic₄₃	—	4	1	no	—
65	C₂²×Dic₄₃	—	4	1	no	—
66	C₂×D₁₇₂	—	2	2	no	—
67	C₂×D₁₇₂	—	2	2	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	2	no	—
73	C₂×D₁₇₂	—	4	2	no	—
74	C₂×D₁₇₂	—	4	2	no	—
75	C₂×D₁₇₂	—	4	2	no	—
76	C₂×D₁₇₂	—	4	2	no	—
77	C₂×D₁₇₂	—	4	2	no	—
78	C₂×Dic₈₆	—	2	2	no	—
79	C₂×Dic₈₆	—	4	2	no	—
80	C₂×Dic₈₆	—	4	2	no	—
81	C₂×Dic₈₆	—	4	1	no	—
82	C₂×C₈×C₄₃	—	2	2	no	—
83	C₂×C₈×C₄₃	—	2	2	no	—
84	C₂×C₈×C₄₃	—	2	2	no	—
85	C₂×C₈×C₄₃	—	2	2	no	—
86	C₂×C₈×C₄₃	—	2	2	no	—
87	C₂×C₈×C₄₃	—	2	2	no	—
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
90	C₂×C₈×C₄₃	D₈	1	4	yes	—
91	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₄₃	#41	1	1	yes	characteristic
94	C₄²×C₄₃	#41	1	1	yes	characteristic
95	C₄⋊Dic₄₃	—	2	2	no	—
96	C₄⋊Dic₄₃	—	2	2	no	—
97	C₄⋊Dic₄₃	—	4	1	no	—
98	C₄⋊Dic₄₃	—	4	1	no	—
99	C₄⋊Dic₄₃	—	4	2	no	—
100	C₄⋊Dic₄₃	—	4	2	no	—
101	C₄⋊Dic₄₃	—	4	2	no	—
102	C₄⋊Dic₄₃	—	4	2	no	—
103	C₄⋊Dic₄₃	—	4	2	no	—
104	C₄⋊Dic₄₃	—	4	2	no	—
105	C₂³×C₄₃	#127	1	1	yes	characteristic
106	C₂²×D₄₃	—	4	2	no	—
107	C₂²×D₄₃	—	8	1	no	—
108	C₂²×D₄₃	—	8	1	no	—
109	C₂²×D₄₃	—	8	1	no	—
110	C₂²×D₄₃	—	8	1	no	—
111	C₂²×D₄₃	—	8	1	no	—
112	C₂²×D₄₃	—	8	1	no	—
113	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₄₃	#135	1	2	yes	—
121	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₄₃	#137	1	2	yes	—
124	C₂×C₄×C₄₃	#137	1	2	yes	—
125	C₂×Dic₄₃	—	4	2	no	—
126	C₂×Dic₄₃	—	8	1	no	—
127	C₂×Dic₄₃	—	8	1	no	—
128	C₂×Dic₄₃	—	8	1	no	—
129	C₂×Dic₄₃	—	8	1	no	—
130	C₂×Dic₄₃	—	8	1	no	—
131	C₂×Dic₄₃	—	8	1	no	—
132	C₂×Dic₄₃	—	8	1	no	—
133	C₂×Dic₄₃	—	8	1	no	—
134	D₁₇₂	—	4	4	no	—
135	D₁₇₂	—	4	4	no	—
136	D₁₇₂	—	4	4	no	—
137	D₁₇₂	—	8	2	no	—
138	D₁₇₂	—	8	2	no	—
139	D₁₇₂	—	8	2	no	—
140	Dic₈₆	—	4	4	no	—
141	Dic₈₆	—	8	2	no	—
142	C₈×C₄₃	—	2	4	no	—
143	C₈×C₄₃	—	2	4	no	—
144	C₈×C₄₃	—	4	2	no	—
145	#9830	—	43	1	no	maximal, Sylow
146	C₂²×C₄₃	#390	1	1	yes	characteristic
147	C₂²×C₄₃	#493	1	1	yes	characteristic
148	C₂²×C₄₃	#495	1	1	yes	characteristic
149	C₂²×C₄₃	#496	1	1	yes	characteristic
150	C₂²×C₄₃	#496	1	1	yes	characteristic
151	C₂²×C₄₃	#497	1	1	yes	characteristic
152	C₂²×C₄₃	#497	1	1	yes	characteristic
153	C₂×D₄₃	—	8	2	no	—
154	C₂×D₄₃	—	8	2	no	—
155	C₂×D₄₃	—	8	2	no	—
156	C₂×D₄₃	—	16	1	no	—
157	C₂×D₄₃	—	16	1	no	—
158	C₂×D₄₃	—	16	1	no	—
159	C₂×D₄₃	—	16	1	no	—
160	C₄×C₄₃	#417	1	2	yes	—
161	C₄×C₄₃	C₄×D₈	1	2	yes	—
162	C₄×C₄₃	#440	1	2	yes	—
163	C₄×C₄₃	#466	1	2	yes	—
164	C₄×C₄₃	#468	1	2	yes	—
165	C₄×C₄₃	#489	1	2	yes	—
166	Dic₄₃	—	8	2	no	—
167	Dic₄₃	—	16	1	no	—
168	Dic₄₃	—	16	1	no	—
169	C₂×#486	—	43	1	no	—
170	#1990	—	43	1	no	—
171	#1957	—	43	1	no	—
172	#1957	—	43	1	no	—
173	#1956	—	43	1	no	—
174	#1845	—	43	1	no	—
175	C₂×#322	D₄₃	1	1	yes	characteristic, complemented
176	C₂×C₄₃	#1719	1	1	yes	characteristic
177	C₂×C₄₃	#1720	1	1	yes	characteristic
178	C₂×C₄₃	#1721	1	1	yes	characteristic
179	C₂×C₄₃	#1722	1	1	yes	characteristic
180	C₂×C₄₃	#1959	1	1	yes	characteristic
181	C₂×C₄₃	#1970	1	1	yes	characteristic
182	C₂×C₄₃	#1973	1	1	yes	characteristic
183	D₄₃	—	16	2	no	—
184	C₂×C₄⋊Q₈	—	43	1	no	—
185	C₂×#127	—	43	1	no	—
186	#486	—	43	4	no	—
187	#486	—	86	2	no	—
188	C₂×#107	—	86	2	no	—
189	C₂×#103	—	86	2	no	—
190	C₂×#102	—	86	2	no	—
191	C₂×#102	—	86	2	no	—
192	C₂×#102	—	86	2	no	—
193	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
194	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
195	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
196	#384	—	43	1	no	—
197	#384	—	43	1	no	—
198	#340	—	86	2	no	—
199	#322	—	2	2	no	—
200	C₄₃	#9830	1	1	yes	characteristic, complemented, Sylow
201	C₂²×Q₈	—	86	1	no	—
202	C₂²×D₄	—	86	1	no	—
203	C₂²×D₄	—	86	1	no	—
204	C₂²×D₄	—	86	1	no	—
205	C₂²×C₈	D₁₇₂	1	2	yes	—
206	C₂²×C₈	D₁₇₂	1	2	yes	—
207	C₂²×C₈	D₁₇₂	1	2	yes	—
208	C₄⋊Q₈	—	43	2	no	—
209	C₄⋊Q₈	—	86	1	no	—
210	C₄⋊Q₈	—	86	1	no	—
211	C₄⋊Q₈	—	86	1	no	—
212	#127	—	43	2	no	—
213	#127	—	86	1	no	—
214	#127	—	86	1	no	—
215	#127	—	86	1	no	—
216	C₂×C₄⋊C₄	—	86	1	no	—
217	C₂×C₄⋊C₄	—	86	1	no	—
218	C₂×C₄⋊C₄	—	86	1	no	—
219	C₂×C₄⋊C₄	—	86	1	no	—
220	C₂×C₄⋊C₄	—	86	1	no	—
221	C₂×C₄⋊C₄	—	86	1	no	—
222	C₂×C₄⋊C₄	—	86	1	no	—
223	C₂×C₄⋊C₄	—	86	1	no	—
224	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
225	#107	—	172	4	no	—
226	#103	—	172	4	no	—
227	#102	—	86	8	no	—
228	#102	—	172	4	no	—
229	#102	—	172	4	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	C₄×D₄₃	1	2	yes	—
235	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
236	C₂⁴	—	172	1	no	—
237	C₂×Q₈	—	86	2	no	—
238	C₂×Q₈	—	172	2	no	—
239	C₂×Q₈	—	172	2	no	—
240	C₂×Q₈	—	172	1	no	—
241	C₂×D₄	—	86	2	no	—
242	C₂×D₄	—	86	2	no	—
243	C₂×D₄	—	86	2	no	—
244	C₂×D₄	—	172	1	no	—
245	C₂×D₄	—	172	1	no	—
246	C₂×D₄	—	172	1	no	—
247	C₂×D₄	—	172	2	no	—
248	C₂×D₄	—	172	2	no	—
249	C₂×D₄	—	172	2	no	—
250	C₂×D₄	—	172	2	no	—
251	C₂×D₄	—	172	2	no	—
252	C₂×D₄	—	172	2	no	—
253	C₂²×C₄	—	172	1	no	—
254	C₂²×C₄	—	172	1	no	—
255	C₂²×C₄	—	172	1	no	—
256	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	D₃₄₄	1	4	yes	—
267	C₂×C₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
268	C₄⋊C₄	—	86	2	no	—
269	C₄⋊C₄	—	86	2	no	—
270	C₄⋊C₄	—	172	2	no	—
271	C₄⋊C₄	—	172	2	no	—
272	C₄⋊C₄	—	172	2	no	—
273	C₄⋊C₄	—	172	2	no	—
274	C₄⋊C₄	—	172	2	no	—
275	C₄⋊C₄	—	172	2	no	—
276	C₄⋊C₄	—	172	1	no	—
277	C₄⋊C₄	—	172	1	no	—
278	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
279	C₄²	#10621898	1	1	yes	characteristic
280	C₄²	#10621898	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
282	C₂³	—	344	1	no	—
283	C₂³	—	344	1	no	—
284	C₂³	—	344	1	no	—
285	C₂³	—	344	1	no	—
286	C₂³	—	344	1	no	—
287	C₂³	—	344	1	no	—
288	C₂³	—	172	2	no	—
289	C₂³	#13063322	1	1	yes	characteristic
290	Q₈	—	344	2	no	—
291	Q₈	—	172	4	no	—
292	D₄	—	344	2	no	—
293	D₄	—	344	2	no	—
294	D₄	—	344	2	no	—
295	D₄	—	172	4	no	—
296	D₄	—	172	4	no	—
297	D₄	—	172	4	no	—
298	C₂×C₄	—	344	1	no	—
299	C₂×C₄	—	344	1	no	—
300	C₂×C₄	—	344	1	no	—
301	C₂×C₄	—	344	1	no	—
302	C₂×C₄	—	344	1	no	—
303	C₂×C₄	—	344	1	no	—
304	C₂×C₄	—	344	1	no	—
305	C₂×C₄	—	344	1	no	—
306	C₂×C₄	—	172	2	no	—
307	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
308	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
309	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
310	C₂×C₄	#13063299	1	1	yes	characteristic
311	C₂×C₄	#13063299	1	1	yes	characteristic
312	C₂×C₄	#13063313	1	1	yes	characteristic
313	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
314	C₂×C₄	#13063347	1	2	yes	—
315	C₂×C₄	#13063353	1	2	yes	—
316	C₂×C₄	#13063353	1	2	yes	—
317	C₂×C₄	#13063358	1	2	yes	—
318	C₂×C₄	#13063358	1	2	yes	—
319	C₈	—	2	4	no	—
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₂²	—	344	2	no	—
323	C₂²	—	344	2	no	—
324	C₂²	—	344	2	no	—
325	C₂²	—	688	1	no	—
326	C₂²	—	688	1	no	—
327	C₂²	—	688	1	no	—
328	C₂²	—	688	1	no	—
329	C₂²	#14582801	1	1	yes	characteristic
330	C₂²	#14583276	1	1	yes	characteristic
331	C₂²	#14583283	1	1	yes	characteristic
332	C₂²	#14583288	1	1	yes	characteristic
333	C₂²	#14583288	1	1	yes	characteristic
334	C₂²	#14583293	1	1	yes	characteristic
335	C₂²	#14583293	1	1	yes	characteristic
336	C₄	—	344	2	no	—
337	C₄	—	688	1	no	—
338	C₄	—	688	1	no	—
339	C₄	#14582901	1	2	yes	—
340	C₄	C₄×D₃₄₄	1	2	yes	—
341	C₄	#14583001	1	2	yes	—
342	C₄	#14583135	1	2	yes	—
343	C₄	#14583145	1	2	yes	—
344	C₄	#14583259	1	2	yes	—
345	C₂	—	688	2	no	—
346	C₂	#19134248	1	1	yes	characteristic
347	C₂	#19134255	1	1	yes	characteristic
348	C₂	#19134261	1	1	yes	characteristic
349	C₂	#19134266	1	1	yes	characteristic
350	C₂	#19135411	1	1	yes	characteristic
351	C₂	#19135461	1	1	yes	characteristic
352	C₂	#19135476	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063322
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#322	C₂
socle	C₂³×C₄₃	#127
prime radical	C₂×#13063322	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#322	D₄₃
hypocenter	C₄₃	#9830
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #19135556	C₂	semidirect
5. #19135397	C₂	semidirect
6. #19134755	C₂	semidirect
7. C₂×C₄₃×#322	C₂	semidirect
11. #14583246	C₄	semidirect
175. C₂×#322	D₄₃	semidirect
200. C₄₃	#9830	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583246	2
#19135556	2
#19135401	2
#19135397	2
#19134755	2
C₂×C₄₃×#322	2
#9830	43