Subgroups — #25096838

Order 11008.

818 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582923	C₂	1	2	yes	maximal
2	#19134736	C₂	1	4	yes	maximal, complemented
3	#19134733	C₂	1	4	yes	maximal, complemented
4	#19134322	C₂	1	4	yes	maximal
5	C₂×C₄×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	4	yes	—
7	#14582923	—	2	4	no	—
8	#14582923	C₂²	1	4	yes	—
9	#14582923	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
11	#14582664	C₂²	1	16	yes	—
12	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
13	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
14	C₄×M₁₆×C₄₃	C₄	1	4	yes	—
15	C₄×M₁₆×C₄₃	C₄	1	4	yes	—
16	C₂×#97×C₄₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
18	C₂×#10621877	—	2	4	no	—
19	C₂×#10621877	—	2	16	no	—
20	C₂×#10621877	C₂³	1	8	yes	—
21	C₂²×C₄₃⋊C₈	—	2	4	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	8	no	—
23	C₄⋊(C₄₃⋊C₈)	—	2	8	no	—
24	C₂³×C₄×C₄₃	Q₈	1	2	yes	—
25	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₄₃	—	2	8	no	—
27	C₂×M₁₆×C₄₃	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
29	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₃	D₄	1	4	yes	—
32	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
33	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
35	#97×C₄₃	C₂×C₄	1	8	yes	—
36	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
38	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	—	4	4	no	—
40	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	#41	1	8	yes	—
48	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
49	M₁₆×C₄₃	—	2	16	no	—
50	C₂×C₈×C₄₃	—	2	8	no	—
51	C₂×C₈×C₄₃	C₂²×C₄	1	8	yes	—
52	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
53	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
54	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
57	C₄²×C₄₃	C₂×D₄	1	2	yes	—
58	C₄²×C₄₃	C₂×D₄	1	2	yes	—
59	#10621877	—	2	16	no	—
60	#10621877	—	4	8	no	—
61	C₂×C₄₃⋊C₈	—	2	8	no	—
62	C₂×C₄₃⋊C₈	—	4	8	no	—
63	C₂³×C₄₃	—	4	2	no	—
64	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
65	C₂³×C₄₃	#125	1	4	yes	—
66	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₃	—	2	8	no	—
68	C₂×C₄×C₄₃	—	2	4	no	—
69	C₂×C₄×C₄₃	—	4	4	no	—
70	C₂×C₄×C₄₃	—	4	2	no	—
71	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
73	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₄₃	#117	1	8	yes	—
76	C₂×C₄×C₄₃	#124	1	4	yes	—
77	C₂×C₄×C₄₃	#124	1	4	yes	—
78	C₂×C₄×C₄₃	#125	1	4	yes	—
79	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
82	C₈×C₄₃	—	2	8	no	—
83	C₄₃⋊C₈	—	4	8	no	—
84	#20769	—	43	1	no	maximal, Sylow
85	C₂²×C₄₃	—	2	4	no	—
86	C₂²×C₄₃	—	2	2	no	—
87	C₂²×C₄₃	—	4	4	no	—
88	C₂²×C₄₃	#382	1	4	yes	—
89	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄₃	C₂×#125	1	2	yes	—
91	C₄×C₄₃	—	2	4	no	—
92	C₄×C₄₃	—	2	2	no	—
93	C₄×C₄₃	#382	1	4	yes	—
94	C₄×C₄₃	#430	1	4	yes	—
95	C₄×C₄₃	C₂×#117	1	2	yes	—
96	C₄×C₄₃	C₂×#124	1	2	yes	—
97	C₂×#423	—	43	2	no	—
98	C₂×C₄×M₁₆	D₄₃	1	1	yes	characteristic
99	#1838	—	43	4	no	—
100	#1837	—	43	4	no	—
101	#1738	—	43	4	no	—
102	C₂×C₄₃	—	2	2	no	—
103	C₂×C₄₃	—	2	4	no	—
104	C₂×C₄₃	C₂×#382	1	1	yes	characteristic
105	C₂×C₄₃	C₂×#430	1	2	yes	—
106	C₂²×M₁₆	—	43	4	no	—
107	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
108	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
109	#423	—	43	4	no	—
110	#423	—	43	4	no	—
111	#423	—	86	4	no	—
112	C₂×C₄⋊C₈	—	86	4	no	—
113	C₄×M₁₆	Dic₄₃	1	4	yes	—
114	C₄×M₁₆	Dic₄₃	1	4	yes	—
115	C₂×#97	C₂×D₄₃	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
117	#341	—	43	16	no	—
118	C₄₃	#20769	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₄	Dic₈₆	1	2	yes	—
120	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
121	C₂×M₁₆	—	2	8	no	—
122	C₂×M₁₆	—	43	8	no	—
123	C₂×M₁₆	—	86	4	no	—
124	C₂×M₁₆	—	86	16	no	—
125	C₂×M₁₆	C₂×Dic₄₃	1	8	yes	—
126	C₂²×C₈	—	86	4	no	—
127	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
128	C₂×C₄²	Dic₈₆	1	2	yes	—
129	C₂×C₄²	D₁₇₂	1	4	yes	—
130	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
131	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
132	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
133	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
134	C₄⋊C₈	—	86	8	no	—
135	C₄⋊C₈	—	86	8	no	—
136	#97	C₂×Dic₄₃	1	8	yes	—
137	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
138	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
139	C₂⁴	C₂×Dic₈₆	1	1	yes	characteristic
140	C₂²×C₄	—	4	4	no	—
141	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
142	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
143	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
144	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
145	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
146	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
147	C₂²×C₄	#10621898	1	8	yes	—
148	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
149	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
150	M₁₆	—	2	16	no	—
151	M₁₆	—	86	16	no	—
152	M₁₆	—	172	8	no	—
153	C₂×C₈	—	2	8	no	—
154	C₂×C₈	—	86	8	no	—
155	C₂×C₈	—	172	8	no	—
156	C₂×C₈	C₂²×Dic₄₃	1	8	yes	—
157	C₄²	C₄⋊Dic₄₃	1	4	yes	—
158	C₄²	C₄⋊Dic₄₃	1	4	yes	—
159	C₄²	C₂×D₁₇₂	1	2	yes	—
160	C₄²	C₂×D₁₇₂	1	2	yes	—
161	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
162	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
163	C₄²	C₂²×Dic₄₃	1	2	yes	—
164	C₂³	—	4	2	no	—
165	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
166	C₂³	#13063317	1	4	yes	—
167	C₂³	C₂²×Dic₈₆	1	1	yes	characteristic
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
173	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
174	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
175	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
176	C₂×C₄	#13063284	1	8	yes	—
177	C₂×C₄	#13063313	1	4	yes	—
178	C₂×C₄	#13063313	1	4	yes	—
179	C₂×C₄	#13063317	1	4	yes	—
180	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
181	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
182	C₂×C₄	C₂×#10621898	1	4	yes	—
183	C₈	—	2	8	no	—
184	C₈	—	172	8	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	4	4	no	—
188	C₂²	#14582765	1	4	yes	—
189	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
190	C₂²	C₂×#13063317	1	2	yes	—
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	#14582765	1	4	yes	—
194	C₄	#14582948	1	4	yes	—
195	C₄	C₂×#13063284	1	2	yes	—
196	C₄	C₂×#13063313	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	—	2	4	no	—
199	C₂	C₂×#14582765	1	1	yes	characteristic
200	C₂	C₂×#14582948	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄×M₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₃	C₂×Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₃
hypocenter	C₄₃	#20769
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #19134736	C₂	semidirect
3. #19134733	C₂	semidirect
118. C₄₃	#20769	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582923	2
#19134736	2
#19134733	2
#19134322	2
C₂×C₄×M₁₆×C₄₃	2
#20769	43