Subgroups — #24995521

Order 11008.

510 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃₄₄	C₂	1	1	yes	maximal, characteristic
2	#19132987	C₂	1	2	yes	maximal, complemented
3	#19132915	C₂	1	2	yes	maximal, complemented
4	#19132878	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₄₄	C₄	1	1	yes	characteristic
7	C₂×C₄×D₁₇₂	C₂²	1	2	yes	—
8	C₄×D₃₄₄	—	2	1	no	—
9	C₄×D₃₄₄	—	2	1	no	—
10	C₄×D₃₄₄	—	2	1	no	—
11	C₄×D₃₄₄	—	2	1	no	—
12	#14582952	C₂²	1	1	yes	characteristic
13	#14582952	C₂²	1	1	yes	characteristic
14	C₂×#13063259	C₄	1	1	yes	characteristic
15	C₂×#13063247	C₄	1	2	yes	—
16	#14582628	—	2	4	no	—
17	#14582622	—	2	4	no	—
18	#14582620	—	2	1	no	—
19	C₈×C₄₃⋊C₈	—	2	1	no	—
20	C₄₃×#431	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#335	—	2	1	no	—
23	C₄₃×#334	—	2	1	no	—
24	C₂²×D₁₇₂	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₄₃	—	2	2	no	—
26	C₂×D₃₄₄	—	2	1	no	—
27	C₂×D₃₄₄	—	2	1	no	—
28	C₂×D₃₄₄	—	2	2	no	—
29	C₂×D₃₄₄	C₈	1	4	yes	complemented
30	C₄×D₁₇₂	—	2	2	no	—
31	C₄×D₁₇₂	—	2	2	no	—
32	C₄×D₁₇₂	—	2	2	no	—
33	C₄×D₁₇₂	—	2	2	no	—
34	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
35	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
36	#13063259	—	2	1	no	—
37	#13063259	—	2	1	no	—
38	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
39	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
40	#13063247	—	2	2	no	—
41	#13063247	—	2	2	no	—
42	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
43	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
44	C₄×C₄₃⋊C₈	—	2	1	no	—
45	C₄×C₄₃⋊C₈	—	2	1	no	—
46	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₄₃	—	2	2	no	—
50	C₂²⋊C₈×C₄₃	—	2	2	no	—
51	C₄×C₈×C₄₃	—	2	2	no	—
52	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
56	C₂³×D₄₃	—	2	2	no	—
57	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
60	C₂²×Dic₄₃	—	2	2	no	—
61	C₂×D₁₇₂	—	2	2	no	—
62	C₂×D₁₇₂	—	2	2	no	—
63	C₂×D₁₇₂	—	2	2	no	—
64	C₂×D₁₇₂	—	2	2	no	—
65	C₂×D₁₇₂	—	2	2	no	—
66	C₂×D₁₇₂	C₂×C₈	1	4	yes	—
67	C₂×C₄×D₄₃	—	4	2	no	—
68	C₂×C₄×D₄₃	—	4	2	no	—
69	C₂×C₄×D₄₃	—	8	4	no	—
70	C₂×C₈×C₄₃	—	2	1	no	—
71	C₂×C₈×C₄₃	—	2	1	no	—
72	C₂×C₈×C₄₃	—	4	2	no	—
73	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₄₃	M₁₆	1	2	yes	—
79	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₄₃	#41	1	1	yes	characteristic
83	(C₂×D₄₃)⋊C₄	—	4	2	no	—
84	(C₂×D₄₃)⋊C₄	—	4	2	no	—
85	C₄⋊Dic₄₃	—	2	2	no	—
86	C₄⋊Dic₄₃	—	2	2	no	—
87	C₂×C₄₃⋊C₈	—	2	2	no	—
88	C₂×C₄₃⋊C₈	—	2	2	no	—
89	D₃₄₄	—	2	4	no	—
90	D₃₄₄	—	2	4	no	—
91	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
92	C₂²×D₄₃	—	2	4	no	—
93	C₂²×D₄₃	—	4	2	no	—
94	C₂²×D₄₃	—	4	2	no	—
95	C₂²×D₄₃	—	8	2	no	—
96	C₂²×D₄₃	—	8	2	no	—
97	C₂×C₄×C₄₃	—	2	2	no	—
98	C₂×C₄×C₄₃	—	4	1	no	—
99	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
103	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
104	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
109	C₂×Dic₄₃	—	4	2	no	—
110	C₂×Dic₄₃	—	8	2	no	—
111	D₁₇₂	—	2	4	no	—
112	D₁₇₂	—	2	4	no	—
113	D₁₇₂	—	4	2	no	—
114	D₁₇₂	—	4	2	no	—
115	C₄×D₄₃	—	8	4	no	—
116	C₄×D₄₃	—	8	4	no	—
117	C₈×C₄₃	—	2	1	no	—
118	C₈×C₄₃	—	2	1	no	—
119	C₈×C₄₃	—	8	2	no	—
120	C₈×C₄₃	C₂²⋊C₈	1	2	yes	—
121	C₈×C₄₃	C₂²⋊C₈	1	2	yes	—
122	C₄₃⋊C₈	—	4	2	no	—
123	C₄₃⋊C₈	—	4	2	no	—
124	#9668	—	43	1	no	maximal, Sylow
125	C₂²×C₄₃	—	2	1	no	—
126	C₂²×C₄₃	—	2	1	no	—
127	C₂²×C₄₃	#390	1	1	yes	characteristic
128	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
129	C₂²×C₄₃	#436	1	1	yes	characteristic
130	C₂×D₄₃	—	4	4	no	—
131	C₂×D₄₃	—	8	4	no	—
132	C₂×D₄₃	—	8	2	no	—
133	C₂×D₄₃	—	8	2	no	—
134	C₂×D₄₃	—	8	4	no	—
135	C₄×C₄₃	—	2	1	no	—
136	C₄×C₄₃	—	2	1	no	—
137	C₄×C₄₃	—	2	1	no	—
138	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₄×C₄₃	#408	1	1	yes	characteristic
140	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
141	C₄×C₄₃	#436	1	1	yes	characteristic
142	C₄×C₄₃	#472	1	2	yes	—
143	Dic₄₃	—	8	2	no	—
144	C₂×C₄×D₈	—	43	1	no	—
145	#1564	D₄₃	1	1	yes	characteristic, complemented
146	#1478	—	43	2	no	—
147	#1460	—	43	2	no	—
148	#1450	—	43	1	no	—
149	C₂×C₄₃	—	2	1	no	—
150	C₂×C₄₃	—	2	1	no	—
151	C₂×C₄₃	#1954	1	1	yes	characteristic
152	C₂×C₄₃	#1976	1	1	yes	characteristic
153	C₂×C₄₃	#1978	1	1	yes	characteristic
154	D₄₃	—	8	4	no	—
155	C₂²×D₈	—	43	1	no	—
156	C₂×C₄×D₄	—	43	2	no	—
157	C₄×D₈	—	86	1	no	—
158	C₄×D₈	—	86	1	no	—
159	C₄×D₈	—	86	1	no	—
160	C₄×D₈	—	86	1	no	—
161	#431	—	43	1	no	—
162	#431	—	43	1	no	—
163	#431	C₂×D₄₃	1	2	yes	—
164	C₂×#107	—	43	1	no	—
165	C₂×#102	—	43	2	no	—
166	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
167	#335	—	2	1	no	—
168	#334	—	2	1	no	—
169	D₄⋊C₈	—	86	4	no	—
170	#323	—	86	4	no	—
171	#322	—	86	1	no	—
172	C₈²	—	86	1	no	—
173	C₄₃	#9668	1	1	yes	characteristic, complemented, Sylow
174	C₂²×D₄	—	43	2	no	—
175	C₂³×C₄	—	86	2	no	—
176	C₂×D₈	—	43	4	no	—
177	C₂×D₈	—	86	2	no	—
178	C₂×D₈	—	86	1	no	—
179	C₂×D₈	—	86	1	no	—
180	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
181	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
182	C₄×D₄	—	86	2	no	—
183	C₄×D₄	—	86	2	no	—
184	C₄×D₄	—	86	2	no	—
185	C₄×D₄	—	86	2	no	—
186	C₂×C₄⋊C₄	—	43	2	no	—
187	C₂×C₂²⋊C₄	—	86	2	no	—
188	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
189	#107	—	86	1	no	—
190	#107	—	86	1	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₄⋊C₈	—	43	2	no	—
193	C₄⋊C₈	—	86	1	no	—
194	#102	—	86	2	no	—
195	#102	—	86	2	no	—
196	C₂²⋊C₈	—	2	2	no	—
197	C₂²⋊C₈	—	43	2	no	—
198	C₂²⋊C₈	—	86	1	no	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	—	86	1	no	—
201	C₄×C₈	—	86	1	no	—
202	C₄×C₈	D₁₇₂	1	1	yes	characteristic
203	C₄×C₈	D₁₇₂	1	1	yes	characteristic
204	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
206	C₂⁴	—	86	2	no	—
207	C₂×D₄	—	43	4	no	—
208	C₂×D₄	—	86	2	no	—
209	C₂×D₄	—	86	2	no	—
210	C₂×D₄	—	86	2	no	—
211	C₂×D₄	—	86	2	no	—
212	C₂×D₄	—	86	2	no	—
213	C₂²×C₄	—	86	2	no	—
214	C₂²×C₄	—	344	4	no	—
215	C₂²×C₄	—	172	2	no	—
216	C₂²×C₄	—	172	2	no	—
217	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
218	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
219	C₂²×C₄	#10621902	1	1	yes	characteristic
220	D₈	—	86	4	no	—
221	D₈	—	86	4	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	86	2	no	—
226	C₂×C₈	—	86	2	no	—
227	C₂×C₈	C₈×D₄₃	1	2	yes	—
228	C₂×C₈	#10621872	1	2	yes	—
229	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
230	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
231	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
232	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
233	C₄⋊C₄	—	86	2	no	—
234	C₄⋊C₄	—	86	2	no	—
235	C₂²⋊C₄	—	172	2	no	—
236	C₂²⋊C₄	—	172	2	no	—
237	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
238	C₄²	D₄×D₄₃	1	1	yes	characteristic
239	C₄²	#10621902	1	1	yes	characteristic
240	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
241	C₂³	—	86	4	no	—
242	C₂³	—	344	2	no	—
243	C₂³	—	344	2	no	—
244	C₂³	—	172	2	no	—
245	C₂³	—	172	2	no	—
246	C₂³	#13063287	1	1	yes	characteristic
247	D₄	—	86	4	no	—
248	D₄	—	86	4	no	—
249	D₄	—	172	2	no	—
250	D₄	—	172	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	344	4	no	—
254	C₂×C₄	—	344	4	no	—
255	C₂×C₄	—	344	2	no	—
256	C₂×C₄	—	172	2	no	—
257	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
258	C₂×C₄	#13063287	1	1	yes	characteristic
259	C₂×C₄	#13063300	1	1	yes	characteristic
260	C₂×C₄	#13063300	1	1	yes	characteristic
261	C₂×C₄	#13063314	1	1	yes	characteristic
262	C₂×C₄	#13063321	1	1	yes	characteristic
263	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
264	C₂×C₄	C₂×#10621872	1	1	yes	characteristic
265	C₂×C₄	#13063333	1	1	yes	characteristic
266	C₂×C₄	#13063333	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	8	2	no	—
270	C₈	—	172	2	no	—
271	C₈	—	172	2	no	—
272	C₈	(C₂×D₄₃)⋊C₈	1	2	yes	—
273	C₈	(C₂×D₄₃)⋊C₈	1	2	yes	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	344	4	no	—
277	C₂²	—	344	4	no	—
278	C₂²	—	344	2	no	—
279	C₂²	—	344	2	no	—
280	C₂²	—	172	4	no	—
281	C₂²	#14582800	1	1	yes	characteristic
282	C₂²	#14582964	1	1	yes	characteristic
283	C₂²	#14582976	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	344	2	no	—
288	C₄	C₂×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
289	C₄	#14582867	1	1	yes	characteristic
290	C₄	#14582964	1	1	yes	characteristic
291	C₄	#14582976	1	1	yes	characteristic
292	C₄	#14583170	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	344	4	no	—
296	C₂	#19135388	1	1	yes	characteristic
297	C₂	#19135491	1	1	yes	characteristic
298	C₂	#19135503	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063300
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1564	C₂
socle	C₂²×C₄₃	#390
prime radical	C₂²×D₃₄₄	C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1564	D₄₃
hypocenter	C₄₃	#9668
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. #19132987	C₂	semidirect
3. #19132915	C₂	semidirect
4. #19132878	C₂	semidirect
5. C₄₃×#1564	C₂	semidirect
29. C₂×D₃₄₄	C₈	semidirect
145. #1564	D₄₃	semidirect
173. C₄₃	#9668	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₃₄₄	2
#19132987	2
#19132915	2
#19132878	2
C₄₃×#1564	2
#9668	43