Subgroups — #25252130

Order 11008.

602 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145174	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136868	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136844	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136820	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136818	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136679	C₂	1	4	yes	maximal, complemented
7	C₄×#13063266	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583449	—	2	1	no	—
11	C₂×#13063266	C₂²	1	4	yes	—
12	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
13	#14583304	C₂²	1	1	yes	characteristic
14	#14583276	C₂²	1	1	yes	characteristic
15	#14583246	—	2	1	no	—
16	#14583135	—	2	1	no	—
17	#14583135	C₂²	1	2	yes	—
18	C₄×D₃₄₄	—	2	1	no	—
19	C₄×D₃₄₄	C₂²	1	2	yes	—
20	#14582724	C₂²	1	4	yes	—
21	#14582700	C₂²	1	4	yes	—
22	#14582699	C₂²	1	4	yes	—
23	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
26	C₄₃×#532	—	2	1	no	—
27	C₄₃×#500	C₂²	1	1	yes	characteristic
28	C₄₃×#493	C₂²	1	1	yes	characteristic
29	C₄₃×#486	—	2	1	no	—
30	C₄₃×#466	—	2	1	no	—
31	C₄₃×#466	C₂²	1	2	yes	—
32	C₄×D₈×C₄₃	—	2	1	no	—
33	C₄×D₈×C₄₃	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
35	C₂×#10621898	—	4	1	no	—
36	C₂×D₃₄₄	—	2	1	no	—
37	C₂×D₃₄₄	C₂³	1	2	yes	—
38	#13063325	—	2	1	no	—
39	#13063322	—	2	1	no	—
40	#13063313	—	4	1	no	—
41	#13063299	—	2	2	no	—
42	#13063299	—	4	1	no	—
43	C₄×D₁₇₂	—	2	2	no	—
44	C₄×D₁₇₂	—	4	1	no	—
45	#13063266	—	2	8	no	—
46	C₂×C₄₃⋊C₁₆	—	2	4	no	—
47	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
48	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
49	#13063259	—	2	1	no	—
50	#13063259	C₂³	1	2	yes	—
51	#13063247	—	2	2	no	—
52	#13063247	—	2	2	no	—
53	C₂×#41×C₄₃	—	4	1	no	—
54	C₂×D₈×C₄₃	—	2	1	no	—
55	C₂×D₈×C₄₃	C₂³	1	2	yes	—
56	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈×C₄₃	—	2	1	no	—
59	#127×C₄₃	—	2	1	no	—
60	#124×C₄₃	—	4	1	no	—
61	#121×C₄₃	—	2	2	no	—
62	#121×C₄₃	—	4	1	no	—
63	C₄×D₄×C₄₃	—	2	2	no	—
64	C₄×D₄×C₄₃	—	4	1	no	—
65	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
66	#107×C₄₃	—	2	1	no	—
67	#107×C₄₃	C₂³	1	2	yes	—
68	#102×C₄₃	—	2	2	no	—
69	#102×C₄₃	—	2	2	no	—
70	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
72	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	#41×C₄₃	—	8	4	no	—
75	C₂×Q₈×C₄₃	—	4	1	no	—
76	C₂×D₄×C₄₃	—	2	2	no	—
77	C₂×D₄×C₄₃	—	4	1	no	—
78	C₂×D₄×C₄₃	—	4	1	no	—
79	C₂×D₄×C₄₃	—	4	2	no	—
80	C₂²×C₄×C₄₃	—	4	2	no	—
81	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₄₃	—	4	2	no	—
85	#10621898	—	8	4	no	—
86	C₂×D₁₇₂	—	2	2	no	—
87	C₂×D₁₇₂	—	4	1	no	—
88	C₂×D₁₇₂	—	4	1	no	—
89	C₂×C₄×D₄₃	—	4	2	no	—
90	C₂×Dic₈₆	—	4	1	no	—
91	D₈×C₄₃	—	2	4	no	—
92	D₈×C₄₃	—	4	2	no	—
93	C₂×C₈×C₄₃	—	2	2	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₄₃	—	2	2	no	—
100	C₄⋊C₄×C₄₃	—	4	1	no	—
101	C₂²⋊C₄×C₄₃	—	4	2	no	—
102	C₂²⋊C₄×C₄₃	—	4	2	no	—
103	C₄²×C₄₃	D₈	1	1	yes	characteristic
104	C₄²×C₄₃	D₈	1	1	yes	characteristic
105	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
107	(C₂×D₄₃)⋊C₄	—	4	2	no	—
108	(C₂×D₄₃)⋊C₄	—	4	2	no	—
109	C₄⋊Dic₄₃	—	2	2	no	—
110	C₄⋊Dic₄₃	—	4	1	no	—
111	D₃₄₄	—	2	4	no	—
112	D₃₄₄	—	4	2	no	—
113	C₄₃⋊C₁₆	—	2	2	no	—
114	C₄₃⋊C₁₆	#41	1	4	yes	complemented
115	C₂³×C₄₃	—	4	2	no	—
116	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
117	C₂²×D₄₃	—	4	2	no	—
118	Q₈×C₄₃	—	8	2	no	—
119	D₄×C₄₃	—	4	2	no	—
120	D₄×C₄₃	—	4	2	no	—
121	D₄×C₄₃	—	8	4	no	—
122	D₄×C₄₃	—	8	2	no	—
123	D₄×C₄₃	—	8	2	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	2	1	no	—
126	C₂×C₄×C₄₃	—	2	2	no	—
127	C₂×C₄×C₄₃	—	4	2	no	—
128	C₂×C₄×C₄₃	—	8	4	no	—
129	C₂×C₄×C₄₃	D₁₆	1	4	yes	complemented
130	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
136	C₂²⋊D₄₃	—	8	4	no	—
137	C₂×Dic₄₃	—	4	2	no	—
138	D₁₇₂	—	4	2	no	—
139	D₁₇₂	—	4	2	no	—
140	D₁₇₂	—	8	2	no	—
141	D₁₇₂	—	8	2	no	—
142	C₄×D₄₃	—	8	4	no	—
143	Dic₈₆	—	8	2	no	—
144	C₈×C₄₃	—	2	1	no	—
145	C₈×C₄₃	—	2	1	no	—
146	C₈×C₄₃	#135	1	2	yes	—
147	C₈×C₄₃	C₂×#41	1	2	yes	—
148	#33733	—	43	1	no	maximal, Sylow
149	C₂²×C₄₃	—	2	1	no	—
150	C₂²×C₄₃	—	2	1	no	—
151	C₂²×C₄₃	—	8	2	no	—
152	C₂²×C₄₃	—	8	2	no	—
153	C₂²×C₄₃	—	8	2	no	—
154	C₂²×C₄₃	C₂×D₁₆	1	2	yes	—
155	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₄₃	—	8	2	no	—
157	C₂×D₄₃	—	8	2	no	—
158	C₂×D₄₃	—	8	2	no	—
159	C₄×C₄₃	—	2	1	no	—
160	C₄×C₄₃	—	2	1	no	—
161	C₄×C₄₃	—	8	2	no	—
162	C₄×C₄₃	C₂×D₁₆	1	2	yes	—
163	C₄×C₄₃	C₂×D₁₆	1	2	yes	—
164	C₄×C₄₃	#508	1	2	yes	—
165	C₄×C₄₃	#532	1	1	yes	characteristic
166	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
167	Dic₄₃	—	8	2	no	—
168	#3169	—	43	1	no	—
169	#3169	D₄₃	1	1	yes	characteristic, complemented
170	#2252	—	43	1	no	—
171	#2246	—	43	1	no	—
172	#2240	—	43	1	no	—
173	#2240	—	43	1	no	—
174	#2215	—	43	4	no	—
175	C₄×D₁₆	—	43	4	no	—
176	C₂×C₄×C₁₆	—	43	1	no	—
177	C₂×C₄₃	—	2	2	no	—
178	C₂×C₄₃	—	16	2	no	—
179	C₂×C₄₃	#3169	1	1	yes	characteristic
180	C₂×C₄₃	C₂²×D₁₆	1	1	yes	characteristic
181	C₂×C₄₃	C₂×#508	1	1	yes	characteristic
182	D₄₃	—	16	2	no	—
183	#532	—	2	1	no	—
184	#532	—	86	1	no	—
185	C₂×D₁₆	—	43	4	no	—
186	C₂²×C₁₆	—	43	2	no	—
187	#500	—	43	1	no	—
188	#500	C₂×D₄₃	1	1	yes	characteristic
189	#493	—	43	1	no	—
190	#493	C₂×D₄₃	1	1	yes	characteristic
191	#486	—	2	1	no	—
192	#486	—	86	1	no	—
193	#466	—	2	1	no	—
194	#466	—	43	2	no	—
195	#466	—	86	1	no	—
196	#466	C₂×D₄₃	1	2	yes	—
197	C₄×D₈	—	2	1	no	—
198	C₄×D₈	—	43	2	no	—
199	C₄×D₈	—	86	1	no	—
200	C₄×D₈	C₂×D₄₃	1	2	yes	—
201	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
202	#366	—	43	4	no	—
203	#357	—	43	4	no	—
204	#357	—	43	4	no	—
205	C₄×C₁₆	—	43	2	no	—
206	C₄×C₁₆	—	43	1	no	—
207	C₄×C₁₆	—	43	1	no	—
208	C₄₃	#33733	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	4	1	no	—
210	C₂×#41	—	172	1	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	—	43	2	no	—
213	C₂×D₈	—	86	1	no	—
214	C₂×D₈	C₂²×D₄₃	1	2	yes	—
215	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	86	1	no	—
219	#127	—	2	1	no	—
220	#127	—	86	1	no	—
221	#124	—	4	1	no	—
222	#124	—	172	1	no	—
223	#121	—	2	2	no	—
224	#121	—	4	1	no	—
225	#121	—	86	2	no	—
226	#121	—	172	1	no	—
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	—	4	1	no	—
229	C₄×D₄	—	86	2	no	—
230	C₄×D₄	—	172	1	no	—
231	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
232	D₁₆	—	86	8	no	—
233	C₂×C₁₆	—	43	2	no	—
234	C₂×C₁₆	—	43	2	no	—
235	C₂×C₁₆	—	86	4	no	—
236	#107	—	2	1	no	—
237	#107	—	43	2	no	—
238	#107	—	86	1	no	—
239	#107	C₂²×D₄₃	1	2	yes	—
240	#102	—	2	2	no	—
241	#102	—	2	2	no	—
242	#102	—	86	2	no	—
243	#102	—	86	2	no	—
244	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
245	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
248	#41	—	8	4	no	—
249	#41	—	344	4	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	172	1	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	86	2	no	—
257	C₂×D₄	—	172	2	no	—
258	C₂×D₄	—	172	1	no	—
259	C₂×D₄	—	172	1	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	172	2	no	—
262	C₂²×C₄	#10621882	1	1	yes	characteristic
263	C₂²×C₄	#10621882	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
265	D₈	—	2	4	no	—
266	D₈	—	4	2	no	—
267	D₈	—	86	4	no	—
268	D₈	—	172	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
273	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	86	2	no	—
278	C₄⋊C₄	—	172	1	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	172	2	no	—
282	C₂²⋊C₄	—	172	2	no	—
283	C₄²	#10621882	1	1	yes	characteristic
284	C₄²	#10621882	1	1	yes	characteristic
285	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
287	C₁₆	—	43	4	no	—
288	C₁₆	—	86	2	no	—
289	C₂³	—	4	2	no	—
290	C₂³	—	172	2	no	—
291	C₂³	C₂×#10621882	1	1	yes	characteristic
292	Q₈	—	8	2	no	—
293	Q₈	—	344	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	8	4	no	—
297	D₄	—	8	2	no	—
298	D₄	—	8	2	no	—
299	D₄	—	344	4	no	—
300	D₄	—	344	2	no	—
301	D₄	—	344	2	no	—
302	D₄	—	172	2	no	—
303	D₄	—	172	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	—	344	4	no	—
310	C₂×C₄	—	172	2	no	—
311	C₂×C₄	#13063266	1	4	yes	complemented
312	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
313	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
314	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
315	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
316	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	#13063345	1	2	yes	—
321	C₈	#41×D₄₃	1	2	yes	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	344	2	no	—
328	C₂²	—	344	2	no	—
329	C₂²	—	344	2	no	—
330	C₂²	C₂×#13063266	1	2	yes	—
331	C₂²	C₂²×#10621882	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	344	2	no	—
336	C₄	C₂×#13063266	1	2	yes	—
337	C₄	C₂×#13063266	1	2	yes	—
338	C₄	#14583329	1	2	yes	—
339	C₄	#14583445	1	1	yes	characteristic
340	C₄	C₂×#13063345	1	1	yes	characteristic
341	C₂	—	2	2	no	—
342	C₂	—	16	2	no	—
343	C₂	—	688	2	no	—
344	C₂	#19145170	1	1	yes	characteristic
345	C₂	C₂²×#13063266	1	1	yes	characteristic
346	C₂	C₂×#14583329	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3169	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#3169	D₄₃
hypocenter	C₄₃	#33733
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #19145174	C₂	semidirect
2. #19136868	C₂	semidirect
3. #19136844	C₂	semidirect
4. #19136820	C₂	semidirect
5. #19136818	C₂	semidirect
6. #19136679	C₂	semidirect
7. C₄×#13063266	C₂	semidirect
8. C₂×C₄×C₄₃⋊C₁₆	C₂	semidirect
9. C₄₃×#3169	C₂	semidirect
114. C₄₃⋊C₁₆	#41	semidirect
129. C₂×C₄×C₄₃	D₁₆	semidirect
169. #3169	D₄₃	semidirect
208. C₄₃	#33733	semidirect
311. C₂×C₄	#13063266	semidirect

Maximal subgroups

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

subgroup	index
#19145174	2
#19136868	2
#19136844	2
#19136820	2
#19136818	2
#19136679	2
C₄×#13063266	2
C₂×C₄×C₄₃⋊C₁₆	2
C₄₃×#3169	2
#33733	43