Subgroups — #25249960

Order 11008.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₆₈₈	C₂	1	1	yes	maximal, characteristic
2	#19143448	C₂	1	2	yes	maximal, complemented
3	#19136472	C₂	1	4	yes	maximal, complemented
4	C₄×D₆₈₈	C₂	1	4	yes	maximal, complemented
5	#19136379	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136312	C₂	1	2	yes	maximal, complemented
7	C₄₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₄₄	C₂²	1	2	yes	—
9	#14583373	—	2	2	no	—
10	C₂×D₆₈₈	C₄	1	8	yes	complemented
11	C₂×D₆₈₈	C₂²	1	2	yes	—
12	C₂×D₆₈₈	C₂²	1	2	yes	—
13	#14583012	—	2	4	no	—
14	C₄×D₃₄₄	C₂²	1	8	yes	complemented
15	#14582937	C₂²	1	2	yes	—
16	#14582905	—	2	2	no	—
17	#14582727	C₂²	1	2	yes	complemented
18	#14582725	C₂²	1	2	yes	complemented
19	#14582701	C₂²	1	8	yes	complemented
20	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#405	C₂²	1	1	yes	characteristic
23	C₄₃×#346	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₄₃	C₂²	1	2	yes	complemented
25	C₂²×D₁₇₂	—	2	2	no	—
26	C₂×D₃₄₄	—	2	2	no	—
27	C₂×D₃₄₄	—	2	4	no	—
28	C₂×D₃₄₄	C₂×C₄	1	8	yes	complemented
29	C₂×D₃₄₄	C₂³	1	4	yes	—
30	C₄×D₁₇₂	—	2	8	no	—
31	#13063287	—	4	4	no	—
32	#13063284	—	2	2	no	—
33	C₂²⋊C₄×D₄₃	—	4	4	no	—
34	D₆₈₈	C₂×C₄	1	8	yes	complemented
35	D₆₈₈	C₂×C₄	1	8	yes	complemented
36	#13063259	C₂³	1	4	yes	—
37	#13063257	—	2	2	no	—
38	#13063247	—	2	8	no	—
39	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
41	#117×C₄₃	D₄	1	1	yes	characteristic
42	M₃₂×C₄₃	—	2	2	no	—
43	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
44	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
45	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
47	#97×C₄₃	D₄	1	2	yes	—
48	C₄×C₈×C₄₃	C₂³	1	2	yes	—
49	C₂³×D₄₃	—	4	2	no	—
50	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
51	C₂×D₁₇₂	—	2	8	no	—
52	C₂×D₁₇₂	—	2	4	no	—
53	C₂×D₁₇₂	—	4	2	no	—
54	C₂×D₁₇₂	—	4	4	no	—
55	C₂×C₄×D₄₃	—	4	8	no	—
56	M₁₆×C₄₃	—	2	2	no	—
57	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₄₃	#41	1	2	yes	—
61	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₄₃	D₈	1	2	yes	—
63	C₂²⋊C₄×C₄₃	D₈	1	2	yes	—
64	C₄²×C₄₃	C₂×D₄	1	2	yes	—
65	C₂²⋊Dic₄₃	—	4	2	no	—
66	(C₂×D₄₃)⋊C₄	—	4	8	no	—
67	C₄⋊Dic₄₃	—	2	4	no	—
68	C₄×Dic₄₃	—	4	2	no	—
69	D₃₄₄	—	2	8	no	—
70	D₃₄₄	—	2	4	no	—
71	D₃₄₄	C₂²×C₄	1	8	yes	—
72	C₁₆×C₄₃	—	2	2	no	—
73	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
74	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
75	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
76	C₂²×D₄₃	—	4	4	no	—
77	C₂²×D₄₃	—	4	8	no	—
78	C₂²×D₄₃	—	16	4	no	—
79	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₄₃	C₂×D₈	1	4	yes	—
81	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₄₃	#135	1	2	yes	—
83	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₄₃	—	4	4	no	—
85	D₁₇₂	—	2	8	no	—
86	D₁₇₂	—	4	4	no	—
87	D₁₇₂	—	4	8	no	—
88	C₄×D₄₃	—	16	8	no	—
89	C₈×C₄₃	—	2	2	no	—
90	C₈×C₄₃	C₄×D₄	1	2	yes	—
91	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
92	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
93	#33456	—	43	1	no	maximal, Sylow
94	C₂²×C₄₃	—	2	2	no	—
95	C₂²×C₄₃	C₄×D₈	1	2	yes	—
96	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
97	C₂×D₄₃	—	4	8	no	—
98	C₂×D₄₃	—	8	8	no	—
99	C₂×D₄₃	—	16	8	no	—
100	C₄×C₄₃	—	2	4	no	—
101	C₄×C₄₃	C₄×D₈	1	2	yes	—
102	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
103	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
104	Dic₄₃	—	8	4	no	—
105	C₂²×D₁₆	—	43	1	no	—
106	#2948	—	43	2	no	—
107	#2177	—	43	4	no	—
108	C₄×D₁₆	—	43	4	no	—
109	#2158	—	43	1	no	—
110	#2140	—	43	2	no	—
111	#2108	D₄₃	1	1	yes	characteristic, complemented
112	C₂×C₄₃	—	2	2	no	—
113	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
114	C₂×C₄₃	#3415	1	2	yes	—
115	D₄₃	—	8	8	no	—
116	C₂²×D₈	—	43	2	no	—
117	#518	—	86	2	no	—
118	C₂×D₁₆	—	43	2	no	—
119	C₂×D₁₆	—	43	2	no	—
120	C₂×D₁₆	—	43	8	no	—
121	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
122	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
123	#442	—	86	4	no	—
124	C₄×D₈	—	43	8	no	—
125	#427	—	43	2	no	—
126	#418	—	86	2	no	—
127	#405	C₂×D₄₃	1	1	yes	characteristic
128	#367	—	43	2	no	—
129	#366	—	43	2	no	—
130	#357	—	43	8	no	—
131	#346	C₂×D₄₃	1	2	yes	complemented
132	C₄×C₁₆	C₂×D₄₃	1	2	yes	complemented
133	C₄₃	#33456	1	1	yes	characteristic, complemented, Sylow
134	C₂²×D₄	—	86	2	no	—
135	C₂×D₈	—	43	8	no	—
136	C₂×D₈	—	43	4	no	—
137	C₂×D₈	—	86	2	no	—
138	C₂×D₈	—	86	4	no	—
139	C₂×M₁₆	D₁₇₂	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
141	C₄×D₄	—	86	8	no	—
142	C₄×D₄	—	172	4	no	—
143	#117	—	86	2	no	—
144	#117	D₁₇₂	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	—	172	4	no	—
146	D₁₆	—	43	8	no	—
147	D₁₆	—	43	8	no	—
148	M₃₂	—	2	2	no	—
149	C₂×C₁₆	C₄×D₄₃	1	4	yes	complemented
150	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
151	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
152	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
153	#107	—	43	4	no	—
154	#106	—	86	2	no	—
155	#102	—	86	8	no	—
156	#97	D₁₇₂	1	2	yes	—
157	C₄×C₈	C₂²×D₄₃	1	2	yes	—
158	C₂⁴	—	172	2	no	—
159	C₂×D₄	—	86	8	no	—
160	C₂×D₄	—	86	4	no	—
161	C₂×D₄	—	172	2	no	—
162	C₂×D₄	—	172	4	no	—
163	C₂²×C₄	—	172	8	no	—
164	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
165	D₈	—	43	8	no	—
166	D₈	—	86	8	no	—
167	D₈	—	86	4	no	—
168	M₁₆	—	2	2	no	—
169	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
170	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
171	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
172	C₂×C₈	#10621898	1	2	yes	—
173	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
174	C₄⋊C₄	—	86	4	no	—
175	C₄⋊C₄	D₃₄₄	1	2	yes	—
176	C₂²⋊C₄	—	172	2	no	—
177	C₂²⋊C₄	—	172	8	no	—
178	C₂²⋊C₄	D₃₄₄	1	2	yes	—
179	C₄²	—	172	2	no	—
180	C₄²	C₂×D₁₇₂	1	2	yes	—
181	C₁₆	—	2	2	no	—
182	C₁₆	C₂×C₄×D₄₃	1	2	yes	—
183	C₁₆	C₂×C₄×D₄₃	1	2	yes	—
184	C₂³	—	172	4	no	—
185	C₂³	—	172	8	no	—
186	C₂³	—	688	4	no	—
187	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
188	D₄	—	86	8	no	—
189	D₄	—	172	4	no	—
190	D₄	—	172	8	no	—
191	C₂×C₄	—	172	4	no	—
192	C₂×C₄	—	688	8	no	—
193	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
194	C₂×C₄	C₂×D₃₄₄	1	4	yes	—
195	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
196	C₂×C₄	#13063347	1	2	yes	—
197	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	C₄×D₁₇₂	1	2	yes	—
200	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
201	C₈	C₂×#10621898	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	—	344	8	no	—
204	C₂²	—	172	8	no	—
205	C₂²	—	688	8	no	—
206	C₂²	C₄×D₃₄₄	1	2	yes	—
207	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	344	4	no	—
210	C₄	C₄×D₃₄₄	1	2	yes	—
211	C₄	C₂×C₄×D₁₇₂	1	1	yes	characteristic
212	C₄	C₂×#13063347	1	1	yes	characteristic
213	C₂	—	2	2	no	—
214	C₂	—	344	8	no	—
215	C₂	C₂×C₄×D₃₄₄	1	1	yes	characteristic
216	C₂	#19147736	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2108	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂²×D₆₈₈	C₂
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2108	D₄₃
hypocenter	C₄₃	#33456
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #19143448	C₂	semidirect
3. #19136472	C₂	semidirect
4. C₄×D₆₈₈	C₂	semidirect
5. #19136379	C₂	semidirect
6. #19136312	C₂	semidirect
7. C₄₃×#2108	C₂	semidirect
10. C₂×D₆₈₈	C₄	semidirect
14. C₄×D₃₄₄	C₂²	semidirect
17. #14582727	C₂²	semidirect
18. #14582725	C₂²	semidirect
19. #14582701	C₂²	semidirect
23. C₄₃×#346	C₂²	semidirect
24. C₄×C₁₆×C₄₃	C₂²	semidirect
28. C₂×D₃₄₄	C₂×C₄	semidirect
34. D₆₈₈	C₂×C₄	semidirect
35. D₆₈₈	C₂×C₄	semidirect
43. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
111. #2108	D₄₃	semidirect
131. #346	C₂×D₄₃	semidirect
132. C₄×C₁₆	C₂×D₄₃	semidirect
133. C₄₃	#33456	semidirect
149. C₂×C₁₆	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₆₈₈	2
#19143448	2
#19136472	2
C₄×D₆₈₈	2
#19136379	2
#19136312	2
C₄₃×#2108	2
#33456	43