Subgroups — #25115386

Order 11008.

790 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×D₁₇₂	C₂	1	2	yes	maximal
2	C₂×#14582978	C₂	1	1	yes	maximal, characteristic
3	#19135370	C₂	1	2	yes	maximal, complemented
4	#19134798	C₂	1	4	yes	maximal, complemented
5	#19134733	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134636	C₂	1	2	yes	maximal, complemented
7	#19134320	C₂	1	2	yes	maximal, complemented
8	C₄×#108×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10621872	C₂²	1	2	yes	—
10	C₂×M₁₆×D₄₃	C₂²	1	4	yes	—
11	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
12	#14582978	—	2	2	no	—
13	#14582978	C₄	1	4	yes	—
14	#14582978	C₄	1	4	yes	—
15	#14582972	—	2	4	no	—
16	C₈×D₁₇₂	—	2	2	no	—
17	C₂×#13063260	C₂²	1	4	yes	—
18	#14582923	C₂²	1	2	yes	—
19	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
20	#14582862	C₂²	1	4	yes	—
21	C₂×(C₂×D₄₃)⋊C₈	C₂²	1	2	yes	—
22	#14582664	C₂²	1	2	yes	—
23	#14582664	C₂²	1	2	yes	—
24	#14582663	C₂²	1	4	yes	—
25	C₂×#108×C₄₃	C₂²	1	2	yes	—
26	C₄×M₁₆×C₄₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#341	C₂²	1	1	yes	characteristic
29	C₄₃×#341	C₂²	1	1	yes	characteristic
30	C₂²×D₁₇₂	Q₈	1	1	yes	characteristic
31	C₂²×C₄×D₄₃	C₂³	1	2	yes	—
32	C₂×#10621877	C₂³	1	4	yes	—
33	C₂×#10621872	—	2	4	no	—
34	C₂×#10621872	—	4	4	no	—
35	C₂×#10621872	C₂×C₄	1	8	yes	—
36	M₁₆×D₄₃	—	2	8	no	—
37	M₁₆×D₄₃	—	4	4	no	—
38	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
39	C₂×C₈×D₄₃	—	2	4	no	—
40	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
41	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
42	C₄×D₁₇₂	D₄	1	2	yes	—
43	C₄×D₁₇₂	D₄	1	2	yes	—
44	C₂×C₄⋊Dic₄₃	Q₈	1	1	yes	characteristic
45	C₂×(C₂×D₄₃)⋊C₄	Q₈	1	2	yes	—
46	#13063260	—	4	4	no	—
47	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
48	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
49	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
50	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
51	(C₂×D₄₃)⋊C₈	—	2	4	no	—
52	(C₂×D₄₃)⋊C₈	—	2	4	no	—
53	(C₂×D₄₃)⋊C₈	C₂×C₄	1	8	yes	—
54	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
55	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
56	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	#108×C₄₃	—	2	4	no	—
60	#97×C₄₃	—	2	2	no	—
61	C₄×C₈×C₄₃	—	2	2	no	—
62	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
64	C₂³×D₄₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₃	C₂×Q₈	1	2	yes	—
69	C₂×D₁₇₂	—	4	2	no	—
70	C₂×D₁₇₂	C₄⋊C₄	1	2	yes	—
71	C₂×D₁₇₂	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×D₄₃	—	2	4	no	—
73	C₂×C₄×D₄₃	—	2	4	no	—
74	C₂×C₄×D₄₃	C₂²×C₄	1	4	yes	—
75	C₂×C₄×D₄₃	C₂×D₄	1	4	yes	—
76	M₁₆×C₄₃	—	2	2	no	—
77	M₁₆×C₄₃	—	2	2	no	—
78	M₁₆×C₄₃	—	2	2	no	—
79	M₁₆×C₄₃	#41	1	4	yes	—
80	C₂×C₈×C₄₃	—	2	2	no	—
81	C₂×C₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	—	2	2	no	—
83	C₂×C₈×C₄₃	—	2	2	no	—
84	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
90	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	4	yes	—
91	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	4	yes	—
92	C₄⋊Dic₄₃	C₄⋊C₄	1	2	yes	—
93	C₄⋊Dic₄₃	C₄⋊C₄	1	2	yes	—
94	#10621877	—	2	4	no	—
95	#10621877	—	2	4	no	—
96	C₂×C₄₃⋊C₈	—	2	4	no	—
97	C₂×C₄₃⋊C₈	—	2	4	no	—
98	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
99	#10621872	—	4	8	no	—
100	#10621872	—	4	8	no	—
101	C₈×D₄₃	—	4	8	no	—
102	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
103	C₂²×D₄₃	—	2	4	no	—
104	C₂²×D₄₃	—	4	2	no	—
105	C₂²×D₄₃	—	4	2	no	—
106	C₂²×D₄₃	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	—	4	1	no	—
109	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
116	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
117	C₂×Dic₄₃	—	2	4	no	—
118	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
119	D₁₇₂	—	2	4	no	—
120	D₁₇₂	—	4	2	no	—
121	C₄×D₄₃	—	2	4	no	—
122	C₄×D₄₃	—	2	4	no	—
123	C₄×D₄₃	—	2	4	no	—
124	C₄×D₄₃	—	2	4	no	—
125	C₈×C₄₃	—	2	2	no	—
126	C₈×C₄₃	—	2	2	no	—
127	C₈×C₄₃	—	2	4	no	—
128	C₈×C₄₃	—	4	2	no	—
129	C₄₃⋊C₈	—	2	4	no	—
130	C₄₃⋊C₈	—	4	4	no	—
131	#22719	—	43	1	no	maximal, Sylow
132	C₂²×C₄₃	—	2	1	no	—
133	C₂²×C₄₃	—	2	1	no	—
134	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₄₃	#554	1	2	yes	—
136	C₂×D₄₃	—	2	4	no	—
137	C₂×D₄₃	—	2	4	no	—
138	C₂×D₄₃	—	2	4	no	—
139	C₂×D₄₃	—	8	4	no	—
140	C₄×C₄₃	—	2	1	no	—
141	C₄×C₄₃	—	2	1	no	—
142	C₄×C₄₃	—	2	1	no	—
143	C₄×C₄₃	#430	1	2	yes	—
144	C₄×C₄₃	C₄×#41	1	1	yes	characteristic
145	C₄×C₄₃	#518	1	1	yes	characteristic
146	C₄×C₄₃	#546	1	2	yes	—
147	Dic₄₃	—	2	4	no	—
148	D₄×M₁₆	—	43	2	no	—
149	C₂×#436	—	43	1	no	—
150	#1950	—	43	2	no	—
151	#1855	—	43	4	no	—
152	#1837	—	43	1	no	—
153	#1811	—	43	2	no	—
154	C₄×#108	D₄₃	1	1	yes	characteristic, complemented
155	#1738	—	43	2	no	—
156	C₂×C₄₃	—	2	2	no	—
157	C₂×C₄₃	#2349	1	1	yes	characteristic
158	C₂×C₄₃	C₂×#430	1	1	yes	characteristic
159	C₂×C₄₃	#2916	1	1	yes	characteristic
160	D₄₃	—	4	4	no	—
161	C₂²×M₁₆	—	43	2	no	—
162	C₂²×M₁₆	—	43	4	no	—
163	C₂×C₄×D₄	—	43	1	no	—
164	#436	—	43	4	no	—
165	#436	—	43	4	no	—
166	#436	—	86	2	no	—
167	#435	—	86	4	no	—
168	C₈×D₄	—	86	2	no	—
169	C₂×#108	—	43	4	no	—
170	C₂×#108	C₂×D₄₃	1	2	yes	—
171	#423	—	43	2	no	—
172	C₂×C₄⋊C₈	—	43	1	no	—
173	#407	—	43	4	no	—
174	C₂×C₂²⋊C₈	—	43	2	no	—
175	C₄×M₁₆	C₂×D₄₃	1	2	yes	—
176	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
177	#341	—	43	4	no	—
178	#341	—	43	2	no	—
179	#341	—	43	2	no	—
180	#341	C₂×D₄₃	1	1	yes	characteristic
181	#341	C₂×D₄₃	1	1	yes	characteristic
182	C₄₃	#22719	1	1	yes	characteristic, complemented, Sylow
183	C₂²×D₄	—	43	1	no	—
184	C₂³×C₄	—	43	2	no	—
185	C₂×M₁₆	—	43	8	no	—
186	C₂×M₁₆	—	43	4	no	—
187	C₂×M₁₆	—	86	4	no	—
188	C₂×M₁₆	—	86	8	no	—
189	C₂×M₁₆	—	172	4	no	—
190	C₂×M₁₆	—	172	4	no	—
191	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
192	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
193	C₂²×C₈	—	43	2	no	—
194	C₂²×C₈	—	86	4	no	—
195	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
197	C₄×D₄	—	43	2	no	—
198	C₄×D₄	—	43	2	no	—
199	C₄×D₄	—	43	2	no	—
200	C₄×D₄	—	43	2	no	—
201	C₂×C₄⋊C₄	—	43	1	no	—
202	C₂×C₂²⋊C₄	—	43	2	no	—
203	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
204	#108	—	2	4	no	—
205	#108	—	172	4	no	—
206	C₄⋊C₈	—	43	2	no	—
207	C₄⋊C₈	—	43	2	no	—
208	C₄⋊C₈	—	86	2	no	—
209	C₄⋊C₈	—	86	2	no	—
210	C₂²⋊C₈	—	43	8	no	—
211	C₂²⋊C₈	—	86	4	no	—
212	C₂²⋊C₈	—	86	4	no	—
213	#97	—	2	2	no	—
214	C₄×C₈	—	2	2	no	—
215	C₄×C₈	C₄×D₄₃	1	2	yes	—
216	C₄×C₈	C₄×D₄₃	1	2	yes	—
217	C₂⁴	—	43	2	no	—
218	C₂×D₄	—	43	2	no	—
219	C₂×D₄	—	43	2	no	—
220	C₂×D₄	—	172	2	no	—
221	C₂²×C₄	—	43	2	no	—
222	C₂²×C₄	—	43	4	no	—
223	C₂²×C₄	—	43	4	no	—
224	C₂²×C₄	—	86	4	no	—
225	C₂²×C₄	—	86	4	no	—
226	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
227	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
229	M₁₆	—	2	2	no	—
230	M₁₆	—	2	2	no	—
231	M₁₆	—	2	2	no	—
232	M₁₆	—	86	4	no	—
233	M₁₆	—	86	4	no	—
234	M₁₆	—	172	8	no	—
235	M₁₆	—	172	8	no	—
236	M₁₆	#10621898	1	4	yes	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	43	4	no	—
242	C₂×C₈	—	86	4	no	—
243	C₂×C₈	—	86	4	no	—
244	C₂×C₈	—	172	8	no	—
245	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
246	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
247	C₄⋊C₄	—	43	2	no	—
248	C₄⋊C₄	—	43	2	no	—
249	C₂²⋊C₄	—	43	4	no	—
250	C₂²⋊C₄	—	43	4	no	—
251	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
252	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
253	C₄²	D₄×D₄₃	1	1	yes	characteristic
254	C₄²	D₄×D₄₃	1	1	yes	characteristic
255	C₂³	—	43	4	no	—
256	C₂³	—	86	4	no	—
257	C₂³	—	172	2	no	—
258	C₂³	—	172	2	no	—
259	C₂³	C₂×Q₈×D₄₃	1	1	yes	characteristic
260	D₄	—	86	4	no	—
261	D₄	—	172	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	43	4	no	—
265	C₂×C₄	—	86	4	no	—
266	C₂×C₄	—	86	4	no	—
267	C₂×C₄	—	86	4	no	—
268	C₂×C₄	—	86	4	no	—
269	C₂×C₄	—	86	4	no	—
270	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
271	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
272	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
273	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
276	C₂×C₄	C₂×#10621898	1	2	yes	—
277	C₂×C₄	#13063371	1	2	yes	—
278	C₈	—	2	2	no	—
279	C₈	—	2	2	no	—
280	C₈	—	2	4	no	—
281	C₈	—	4	2	no	—
282	C₈	—	86	4	no	—
283	C₈	—	172	4	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	86	4	no	—
287	C₂²	—	86	4	no	—
288	C₂²	—	86	4	no	—
289	C₂²	—	344	4	no	—
290	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
291	C₂²	#14583609	1	2	yes	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	86	4	no	—
296	C₄	#14582946	1	2	yes	—
297	C₄	C₄×#10621898	1	1	yes	characteristic
298	C₄	#14583373	1	1	yes	characteristic
299	C₄	#14583545	1	2	yes	—
300	C₂	—	2	2	no	—
301	C₂	—	172	4	no	—
302	C₂	#19137506	1	1	yes	characteristic
303	C₂	C₂×#14582946	1	1	yes	characteristic
304	C₂	#19143203	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#108×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂²×D₁₇₂	Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#108	D₄₃
hypocenter	C₄₃	#22719
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
3. #19135370	C₂	semidirect
4. #19134798	C₂	semidirect
5. #19134733	C₂	semidirect
6. #19134636	C₂	semidirect
7. #19134320	C₂	semidirect
8. C₄×#108×C₄₃	C₂	semidirect
154. C₄×#108	D₄₃	semidirect
182. C₄₃	#22719	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×D₁₇₂	2
C₂×#14582978	2
#19135370	2
#19134798	2
#19134733	2
#19134636	2
#19134320	2
C₄×#108×C₄₃	2
#22719	43