Subgroups — #131×#3866 #18011163

Order 4736.

572 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₇⋊#131	C₂	1	1	yes	maximal, characteristic
2	#41×#3866	C₂	1	1	yes	maximal, characteristic
3	#14558499	C₂	1	3	yes	maximal, complemented
4	#14558496	C₂	1	3	yes	maximal, complemented
5	M₁₆×#3866	C₂	1	3	yes	maximal, complemented
6	C₂×C₈×#3866	C₂	1	3	yes	maximal, complemented
7	#131×D₃₇	C₂	1	1	yes	maximal, characteristic
8	#11924624	C₄	1	2	yes	complemented
9	#11924623	C₄	1	6	yes	complemented
10	D₃₇⋊M₁₆	C₂²	1	3	yes	—
11	C₂×D₃₇⋊C₈	C₂²	1	3	yes	—
12	Q₈×#3866	C₄	1	2	yes	—
13	D₄×#3866	C₄	1	6	yes	—
14	#11924615	C₂²	1	3	yes	—
15	C₂×C₄×#3866	C₂²	1	3	yes	—
16	#11924590	C₂²	1	6	yes	—
17	C₈×#3866	C₂²	1	1	yes	characteristic
18	C₈×#3866	C₂²	1	3	yes	—
19	C₈×#3866	C₂²	1	3	yes	—
20	C₈×#3866	C₂²	1	3	yes	—
21	#41×D₃₇	C₂²	1	1	yes	characteristic
22	#11924544	C₄	1	3	yes	complemented
23	#11924543	C₄	1	1	yes	characteristic, complemented
24	#11924542	C₄	1	3	yes	complemented
25	M₁₆×D₃₇	C₂²	1	3	yes	—
26	C₂×C₈×D₃₇	C₂²	1	3	yes	—
27	#131×C₃₇	C₄	1	1	yes	characteristic, complemented
28	C₂²×#3866	C₂×C₄	1	6	yes	—
29	#41×C₃₇	C₂×C₄	1	1	yes	characteristic
30	#10596966	C₂×C₄	1	1	yes	characteristic
31	Q₈×D₃₇	C₂×C₄	1	1	yes	characteristic
32	#10596964	C₂×C₄	1	3	yes	—
33	D₄×D₃₇	C₂×C₄	1	3	yes	—
34	#10596962	C₂×C₄	1	3	yes	—
35	C₂×C₄×D₃₇	C₂³	1	3	yes	—
36	C₂²⋊#3866	C₂×C₄	1	6	yes	—
37	#10596957	C₂×C₄	1	6	yes	complemented
38	C₂×#64438	C₂×C₄	1	6	yes	complemented
39	C₄⋊#3866	C₂×C₄	1	6	yes	—
40	C₄×#3866	C₂×C₄	1	6	yes	—
41	C₄×#3866	C₂³	1	1	yes	characteristic
42	C₄×#3866	C₂³	1	3	yes	—
43	#10596953	C₂×C₄	1	6	yes	complemented
44	D₃₇⋊C₈	C₂×C₄	1	6	yes	complemented
45	D₃₇⋊C₈	C₂³	1	1	yes	characteristic
46	D₃₇⋊C₈	C₂³	1	3	yes	—
47	M₁₆×C₃₇	C₂×C₄	1	3	yes	complemented
48	C₂×C₈×C₃₇	C₂×C₄	1	3	yes	complemented
49	#10596934	C₂×C₄	1	3	yes	complemented
50	C₂×#64436	C₂×C₄	1	3	yes	complemented
51	#10596929	C₂×C₄	1	6	yes	complemented
52	C₈×D₃₇	C₂×C₄	1	3	yes	complemented
53	C₈×D₃₇	C₂×C₄	1	3	yes	complemented
54	C₈×D₃₇	C₂³	1	1	yes	characteristic
55	C₈×D₃₇	C₂³	1	3	yes	—
56	C₂²×D₃₇	C₂²×C₄	1	3	yes	—
57	C₂×#3866	—	2	12	no	—
58	C₂×#3866	C₂²×C₄	1	6	yes	—
59	C₂×#3866	C₂²×C₄	1	2	yes	—
60	Q₈×C₃₇	C₄²	1	1	yes	characteristic
61	D₄×C₃₇	C₄²	1	3	yes	—
62	C₂×C₄×C₃₇	C₂²×C₄	1	3	yes	—
63	C₂²⋊D₃₇	C₄²	1	6	yes	—
64	C₂×Dic₃₇	C₂²×C₄	1	3	yes	—
65	D₁₄₈	C₄²	1	3	yes	—
66	C₄×D₃₇	C₂²×C₄	1	3	yes	—
67	C₄×D₃₇	C₂²×C₄	1	3	yes	—
68	C₄×D₃₇	C₂²×C₄	1	3	yes	—
69	C₄×D₃₇	C₂⁴	1	1	yes	characteristic
70	Dic₇₄	C₄²	1	3	yes	—
71	#64438	C₂²×C₄	1	2	yes	—
72	#64438	C₂²×C₄	1	6	yes	—
73	C₈×C₃₇	C₂²×C₄	1	3	yes	—
74	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
75	#64436	C₂²×C₄	1	1	yes	characteristic
76	#64436	C₂²×C₄	1	3	yes	—
77	C₂²×C₃₇	C₂×C₄²	1	3	yes	—
78	C₂×D₃₇	—	2	6	no	—
79	C₂×D₃₇	C₂×C₄²	1	3	yes	—
80	C₂×D₃₇	C₂³×C₄	1	1	yes	characteristic
81	#3866	—	2	6	no	—
82	#3866	#131	1	4	yes	direct product factor
83	C₄×C₃₇	C₂×C₄²	1	3	yes	—
84	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
85	Dic₃₇	C₂×C₄²	1	3	yes	—
86	Dic₃₇	C₂³×C₄	1	1	yes	characteristic
87	C₄×#131	—	37	1	no	maximal, Sylow
88	C₂×C₃₇	—	2	3	no	—
89	C₂×C₃₇	C₂²×C₄²	1	1	yes	characteristic
90	D₃₇	—	2	3	no	—
91	D₃₇	C₂×#131	1	2	yes	—
92	C₂×#131	—	37	1	no	—
93	C₂×#131	—	37	1	no	—
94	C₄×#41	—	37	1	no	—
95	#405	—	37	3	no	—
96	#405	—	37	3	no	—
97	C₄×M₁₆	—	37	3	no	—
98	C₂×C₄×C₈	—	37	3	no	—
99	C₃₇	C₄×#131	1	1	yes	characteristic, complemented, Sylow
100	C₂×#41	—	37	1	no	—
101	#131	—	37	6	no	—
102	#131	—	37	2	no	—
103	#131	—	37	1	no	—
104	#131	—	37	3	no	—
105	#131	—	37	3	no	—
106	#131	#3866	1	1	yes	characteristic, direct product factor
107	C₂×M₁₆	—	37	3	no	—
108	C₂×M₁₆	—	37	3	no	—
109	C₂²×C₈	—	37	3	no	—
110	C₂²×C₈	—	37	3	no	—
111	C₄×Q₈	—	37	2	no	—
112	C₄×D₄	—	37	6	no	—
113	#117	—	37	3	no	—
114	C₂×C₄²	—	37	3	no	—
115	#97	—	37	6	no	—
116	C₄×C₈	—	37	1	no	—
117	C₄×C₈	—	37	3	no	—
118	C₄×C₈	—	37	3	no	—
119	C₄×C₈	—	37	3	no	—
120	#41	—	37	1	no	—
121	#41	—	37	3	no	—
122	#41	—	37	3	no	—
123	#41	C₂×#3866	1	1	yes	characteristic
124	C₂×Q₈	—	37	1	no	—
125	C₂×D₄	—	37	3	no	—
126	C₂²×C₄	—	37	3	no	—
127	C₂²×C₄	—	37	6	no	—
128	M₁₆	—	37	6	no	—
129	M₁₆	—	37	6	no	—
130	M₁₆	—	37	3	no	—
131	M₁₆	—	37	6	no	—
132	M₁₆	C₂×#3866	1	3	yes	complemented
133	C₂×C₈	—	37	6	no	—
134	C₂×C₈	—	37	1	no	—
135	C₂×C₈	—	37	3	no	—
136	C₂×C₈	—	37	6	no	—
137	C₂×C₈	—	37	3	no	—
138	C₂×C₈	—	37	3	no	—
139	C₂×C₈	—	37	3	no	—
140	C₂×C₈	—	37	3	no	—
141	C₂×C₈	—	37	1	no	—
142	C₂×C₈	C₂×#3866	1	3	yes	complemented
143	C₄⋊C₄	—	37	6	no	—
144	C₂²⋊C₄	—	37	6	no	—
145	C₄²	—	37	6	no	—
146	C₄²	—	37	1	no	—
147	C₄²	—	37	3	no	—
148	C₂³	—	37	3	no	—
149	Q₈	—	37	3	no	—
150	Q₈	C₄×#3866	1	1	yes	characteristic
151	D₄	—	37	6	no	—
152	D₄	—	37	3	no	—
153	D₄	C₄×#3866	1	3	yes	—
154	C₂×C₄	—	37	2	no	—
155	C₂×C₄	—	37	3	no	—
156	C₂×C₄	—	37	3	no	—
157	C₂×C₄	—	37	3	no	—
158	C₂×C₄	—	37	1	no	—
159	C₂×C₄	—	37	3	no	—
160	C₂×C₄	—	37	6	no	—
161	C₂×C₄	—	74	12	no	—
162	C₂×C₄	C₂²×#3866	1	3	yes	—
163	C₈	—	37	2	no	—
164	C₈	—	37	6	no	—
165	C₈	—	37	1	no	—
166	C₈	—	37	3	no	—
167	C₈	C₂²×#3866	1	3	yes	—
168	C₈	C₂²×#3866	1	1	yes	characteristic
169	C₂²	—	37	1	no	—
170	C₂²	—	37	3	no	—
171	C₂²	—	74	6	no	—
172	C₂²	C₂×C₄×#3866	1	3	yes	—
173	C₄	—	37	4	no	—
174	C₄	—	37	1	no	—
175	C₄	—	37	3	no	—
176	C₄	—	74	6	no	—
177	C₄	C₂×C₄×#3866	1	3	yes	—
178	C₄	C₂³×#3866	1	1	yes	characteristic
179	C₂	—	2	3	no	—
180	C₂	—	37	2	no	—
181	C₂	—	74	3	no	—
182	C₂	C₂²×C₄×#3866	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×#3866
commutator, socle	C₂×C₃₇	C₂²×C₄²
Frattini	C₄	C₂³×#3866
Fitting	#131×C₃₇	C₄
prime radical	#41×D₃₇	C₂²
cosocle	C₄×D₃₇	C₂⁴
hypercenter	#131	#3866
hypocenter	C₃₇	C₄×#131
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
3. #14558499	C₂	semidirect
4. #14558496	C₂	semidirect
5. M₁₆×#3866	C₂	semidirect
6. C₂×C₈×#3866	C₂	semidirect
8. #11924624	C₄	semidirect
9. #11924623	C₄	semidirect
22. #11924544	C₄	semidirect
23. #11924543	C₄	semidirect
24. #11924542	C₄	semidirect
27. #131×C₃₇	C₄	semidirect
37. #10596957	C₂×C₄	semidirect
38. C₂×#64438	C₂×C₄	semidirect
43. #10596953	C₂×C₄	semidirect
44. D₃₇⋊C₈	C₂×C₄	semidirect
47. M₁₆×C₃₇	C₂×C₄	semidirect
48. C₂×C₈×C₃₇	C₂×C₄	semidirect
49. #10596934	C₂×C₄	semidirect
50. C₂×#64436	C₂×C₄	semidirect
51. #10596929	C₂×C₄	semidirect
52. C₈×D₃₇	C₂×C₄	semidirect
53. C₈×D₃₇	C₂×C₄	semidirect
82. #3866	#131	direct (N × H)
99. C₃₇	C₄×#131	semidirect
106. #131	#3866	direct (N × H)
132. M₁₆	C₂×#3866	semidirect
142. C₂×C₈	C₂×#3866	semidirect

Maximal subgroups

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

subgroup	index
D₃₇⋊#131	2
#41×#3866	2
#14558499	2
#14558496	2
M₁₆×#3866	2
C₂×C₈×#3866	2
#131×D₃₇	2
C₄×#131	37