Subgroups — #25116365

Order 11008.

862 conjugacy classes of subgroups.

The subgroup lattice has 324 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	#19137566	C₂	1	2	yes	maximal, complemented
3	#19135461	C₂	1	2	yes	maximal
4	#19135223	C₂	1	4	yes	maximal, complemented
5	#19134759	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134606	C₂	1	2	yes	maximal, complemented
7	C₄×#13063259	C₂	1	2	yes	maximal, complemented
8	C₈×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₇₂	C₂²	1	2	yes	—
11	C₄×D₃₄₄	C₄	1	8	yes	complemented
12	C₂×#13063259	C₂²	1	2	yes	—
13	C₂×#13063259	C₂²	1	1	yes	characteristic
14	C₂×#13063257	C₂²	1	2	yes	—
15	C₂×#13063247	C₂²	1	4	yes	—
16	C₂×#13063247	C₂²	1	2	yes	—
17	#14582801	C₂²	1	4	yes	—
18	#14582789	—	2	4	no	—
19	#14582765	C₂²	1	2	yes	—
20	#14582761	—	2	2	no	—
21	C₄×C₄⋊Dic₄₃	C₂²	1	4	yes	—
22	#14582746	—	2	2	no	—
23	#14582662	C₂²	1	4	yes	—
24	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#336	C₂²	1	2	yes	—
29	C₂²×D₁₇₂	C₂³	1	2	yes	—
30	C₂²×C₄×D₄₃	—	2	2	no	—
31	C₂×D₃₄₄	—	2	2	no	—
32	C₂×D₃₄₄	—	2	2	no	—
33	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
34	C₄×D₁₇₂	—	2	4	no	—
35	C₄×D₁₇₂	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
38	C₂×(C₂×D₄₃)⋊C₄	—	2	2	no	—
39	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
40	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
41	C₂×C₄×Dic₄₃	—	2	4	no	—
42	#13063259	—	2	4	no	—
43	#13063259	C₂×C₄	1	4	yes	—
44	#13063257	—	4	2	no	—
45	#13063247	—	4	4	no	—
46	#13063247	C₂×C₄	1	8	yes	—
47	C₄₃⋊#95	—	2	2	no	—
48	C₄₃⋊#95	—	2	4	no	—
49	C₄₃⋊#95	—	2	2	no	—
50	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
54	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	—	2	2	no	—
57	C₄×C₈×C₄₃	—	2	4	no	—
58	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
59	#95×C₄₃	D₄	1	1	yes	characteristic
60	#95×C₄₃	D₄	1	1	yes	characteristic
61	C₂³×D₄₃	—	2	2	no	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
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	2	yes	—
66	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₃	—	2	2	no	—
68	C₂²×Dic₄₃	—	2	4	no	—
69	C₂×D₁₇₂	—	2	2	no	—
70	C₂×D₁₇₂	—	2	2	no	—
71	C₂×D₁₇₂	—	2	2	no	—
72	C₂×D₁₇₂	—	2	2	no	—
73	C₂×D₁₇₂	—	2	2	no	—
74	C₂×D₁₇₂	C₂²×C₄	1	4	yes	—
75	C₂×C₄×D₄₃	—	2	8	no	—
76	C₂×C₄×D₄₃	—	8	4	no	—
77	C₂×C₈×C₄₃	—	2	1	no	—
78	C₂×C₈×C₄₃	—	2	1	no	—
79	C₂×C₈×C₄₃	—	2	2	no	—
80	C₂×C₈×C₄₃	—	2	2	no	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₃	#41	1	2	yes	—
85	C₂×C₈×C₄₃	#41	1	2	yes	—
86	C₄⋊C₄×C₄₃	—	2	4	no	—
87	C₄²×C₄₃	—	2	2	no	—
88	C₄²×C₄₃	—	2	2	no	—
89	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₄₃	#41	1	2	yes	—
91	(C₂×D₄₃)⋊C₄	—	2	8	no	—
92	(C₂×D₄₃)⋊C₄	—	4	8	no	—
93	(C₂×D₄₃)⋊C₄	—	8	4	no	—
94	C₄⋊Dic₄₃	—	2	2	no	—
95	C₄⋊Dic₄₃	—	2	4	no	—
96	C₄⋊Dic₄₃	—	2	4	no	—
97	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
98	C₄×Dic₄₃	—	8	4	no	—
99	D₃₄₄	—	2	8	no	—
100	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
101	C₂²×D₄₃	—	2	4	no	—
102	C₂²×D₄₃	—	4	2	no	—
103	C₂²×D₄₃	—	4	2	no	—
104	C₂²×D₄₃	—	4	4	no	—
105	C₂²×D₄₃	—	4	4	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	2	2	no	—
111	C₂×C₄×C₄₃	—	2	2	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₄₃	#135	1	4	yes	—
116	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×Dic₄₃	—	2	4	no	—
123	C₂×Dic₄₃	—	4	4	no	—
124	C₂×Dic₄₃	—	4	4	no	—
125	C₂×Dic₄₃	—	4	4	no	—
126	C₂×Dic₄₃	—	4	4	no	—
127	D₁₇₂	—	2	4	no	—
128	D₁₇₂	—	2	4	no	—
129	D₁₇₂	—	4	2	no	—
130	D₁₇₂	—	4	2	no	—
131	C₄×D₄₃	—	8	8	no	—
132	C₈×C₄₃	—	2	2	no	—
133	C₈×C₄₃	—	4	2	no	—
134	C₈×C₄₃	#117	1	4	yes	—
135	#22810	—	43	1	no	maximal, Sylow
136	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂²×C₄₃	#547	1	1	yes	characteristic
138	C₂²×C₄₃	#550	1	1	yes	characteristic
139	C₂²×C₄₃	C₂×#135	1	2	yes	—
140	C₂²×C₄₃	#576	1	2	yes	—
141	C₂×D₄₃	—	4	4	no	—
142	C₂×D₄₃	—	4	4	no	—
143	C₂×D₄₃	—	4	4	no	—
144	C₂×D₄₃	—	8	4	no	—
145	C₂×D₄₃	—	8	4	no	—
146	C₄×C₄₃	—	2	4	no	—
147	C₄×C₄₃	—	2	2	no	—
148	C₄×C₄₃	—	2	2	no	—
149	C₄×C₄₃	—	4	2	no	—
150	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
151	C₄×C₄₃	#520	1	1	yes	characteristic
152	C₄×C₄₃	#555	1	1	yes	characteristic
153	C₄×C₄₃	#556	1	1	yes	characteristic
154	Dic₄₃	—	4	4	no	—
155	Dic₄₃	—	8	4	no	—
156	C₂×C₄×D₈	—	43	1	no	—
157	#2357	—	43	2	no	—
158	#1970	—	43	2	no	—
159	#1926	—	43	4	no	—
160	#1846	—	43	1	no	—
161	#1804	—	43	2	no	—
162	C₄×#107	—	43	2	no	—
163	C₈×C₄⋊C₄	D₄₃	1	1	yes	characteristic, complemented
164	C₂×C₄₃	#2354	1	1	yes	characteristic
165	C₂×C₄₃	C₄×#135	1	1	yes	characteristic
166	C₂×C₄₃	#2948	1	1	yes	characteristic
167	C₂×C₄₃	#3339	1	2	yes	—
168	C₂×C₄₃	#3396	1	2	yes	—
169	D₄₃	—	8	4	no	—
170	C₂²×D₈	—	43	1	no	—
171	C₂×C₄×D₄	—	43	2	no	—
172	C₄×D₈	—	43	8	no	—
173	C₂×#107	—	43	1	no	—
174	C₂×#107	—	43	2	no	—
175	C₂×#106	—	43	2	no	—
176	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
177	C₂×#102	—	43	2	no	—
178	C₂×#102	—	43	4	no	—
179	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
180	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
181	#390	—	43	4	no	—
182	#388	—	86	4	no	—
183	#382	—	43	2	no	—
184	#381	—	86	2	no	—
185	C₄×C₄⋊C₄	—	43	4	no	—
186	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
187	C₄×C₂²⋊C₄	—	86	2	no	—
188	#340	—	43	4	no	—
189	#336	C₂×D₄₃	1	2	yes	—
190	C₄₃	#22810	1	1	yes	characteristic, complemented, Sylow
191	C₂²×D₄	—	43	2	no	—
192	C₂³×C₄	—	86	2	no	—
193	C₂×D₈	—	43	4	no	—
194	C₂×D₈	—	86	2	no	—
195	C₂×D₈	—	86	2	no	—
196	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
197	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
199	C₄×D₄	—	43	8	no	—
200	C₄×D₄	—	86	4	no	—
201	C₂×C₄⋊C₄	—	43	2	no	—
202	C₂×C₄⋊C₄	—	43	4	no	—
203	C₂×C₄⋊C₄	D₁₇₂	1	2	yes	—
204	C₂×C₂²⋊C₄	—	86	2	no	—
205	C₂×C₂²⋊C₄	—	86	4	no	—
206	C₂×C₂²⋊C₄	—	86	4	no	—
207	C₂×C₄²	—	86	4	no	—
208	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
209	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
210	#107	—	43	4	no	—
211	#107	—	86	4	no	—
212	#106	—	172	2	no	—
213	C₄⋊C₈	—	2	2	no	—
214	#102	—	43	8	no	—
215	#102	—	172	4	no	—
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	C₄×D₄₃	1	4	yes	—
218	#95	—	86	2	no	—
219	#95	—	86	4	no	—
220	#95	—	86	2	no	—
221	#95	D₁₇₂	1	1	yes	characteristic
222	#95	D₁₇₂	1	1	yes	characteristic
223	C₂⁴	—	86	2	no	—
224	C₂×D₄	—	43	4	no	—
225	C₂×D₄	—	86	2	no	—
226	C₂×D₄	—	86	2	no	—
227	C₂×D₄	—	86	2	no	—
228	C₂×D₄	—	86	2	no	—
229	C₂×D₄	—	86	2	no	—
230	C₂²×C₄	—	86	2	no	—
231	C₂²×C₄	—	86	8	no	—
232	C₂²×C₄	—	86	4	no	—
233	C₂²×C₄	—	344	4	no	—
234	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
235	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
238	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
239	D₈	—	86	8	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
246	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
247	C₂×C₈	#10621900	1	2	yes	—
248	C₂×C₈	#10621902	1	2	yes	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	43	4	no	—
251	C₄⋊C₄	—	86	2	no	—
252	C₄⋊C₄	—	86	4	no	—
253	C₄⋊C₄	—	86	4	no	—
254	C₂²⋊C₄	—	86	8	no	—
255	C₂²⋊C₄	—	344	4	no	—
256	C₂²⋊C₄	—	172	8	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	344	4	no	—
260	C₄²	C₂×C₄×D₄₃	1	2	yes	—
261	C₄²	#10621898	1	2	yes	—
262	C₂³	—	86	4	no	—
263	C₂³	—	172	2	no	—
264	C₂³	—	172	2	no	—
265	C₂³	—	172	4	no	—
266	C₂³	—	172	4	no	—
267	C₂³	C₂²×D₁₇₂	1	1	yes	characteristic
268	D₄	—	86	4	no	—
269	D₄	—	86	4	no	—
270	D₄	—	172	2	no	—
271	D₄	—	172	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	86	4	no	—
280	C₂×C₄	—	344	8	no	—
281	C₂×C₄	—	172	4	no	—
282	C₂×C₄	—	172	4	no	—
283	C₂×C₄	—	172	4	no	—
284	C₂×C₄	—	172	4	no	—
285	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
286	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
287	C₂×C₄	#13063347	1	4	yes	—
288	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
289	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
290	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
291	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
292	C₂×C₄	#13063371	1	1	yes	characteristic
293	C₂×C₄	#13063372	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	4	2	no	—
296	C₈	#13063283	1	4	yes	—
297	C₂²	—	344	4	no	—
298	C₂²	—	344	4	no	—
299	C₂²	—	172	4	no	—
300	C₂²	—	172	4	no	—
301	C₂²	—	172	4	no	—
302	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
303	C₂²	#14583553	1	1	yes	characteristic
304	C₂²	#14583571	1	1	yes	characteristic
305	C₂²	C₂×#13063347	1	2	yes	—
306	C₂²	#14583704	1	2	yes	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	4	no	—
310	C₄	—	4	2	no	—
311	C₄	—	344	4	no	—
312	C₄	—	172	4	no	—
313	C₄	C₂×#13063283	1	1	yes	characteristic
314	C₄	#14583379	1	1	yes	characteristic
315	C₄	#14583614	1	1	yes	characteristic
316	C₄	#14583622	1	1	yes	characteristic
317	C₂	—	344	4	no	—
318	C₂	#19137542	1	1	yes	characteristic
319	C₂	C₄×#13063347	1	1	yes	characteristic
320	C₂	#19143448	1	1	yes	characteristic
321	C₂	#19146998	1	2	yes	—
322	C₂	#19147608	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₈×C₄⋊C₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂²×D₃₄₄	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#22810
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #19137566	C₂	semidirect
4. #19135223	C₂	semidirect
5. #19134759	C₂	semidirect
6. #19134606	C₂	semidirect
7. C₄×#13063259	C₂	semidirect
8. C₈×C₄⋊C₄×C₄₃	C₂	semidirect
11. C₄×D₃₄₄	C₄	semidirect
163. C₈×C₄⋊C₄	D₄₃	semidirect
190. C₄₃	#22810	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
#19137566	2
#19135461	2
#19135223	2
#19134759	2
#19134606	2
C₄×#13063259	2
C₈×C₄⋊C₄×C₄₃	2
#22810	43