Subgroups — C₂×#14558009 #18007785

Order 4736.

1488 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	C₂²×Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic
2	#14558009	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×Dic₇₄	C₂	1	3	yes	maximal, complemented
5	C₂×#11924501	C₂	1	3	yes	maximal, complemented
6	C₂×#11924494	C₂	1	3	yes	maximal
7	C₂×#11924493	C₂	1	3	yes	maximal
8	C₂×C₄×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₇₄	C₂²	1	3	yes	—
10	C₂×Q₈×D₃₇	C₄	1	16	yes	complemented
11	C₂×Q₈×D₃₇	C₂²	1	8	yes	—
12	C₂²×C₄×D₃₇	C₂²	1	3	yes	—
13	Q₈×Dic₃₇	C₂²	1	8	yes	complemented
14	C₄×Dic₇₄	C₂²	1	24	yes	complemented
15	#11924501	C₂²	1	24	yes	complemented
16	#11924494	C₂²	1	24	yes	—
17	#11924493	C₂²	1	24	yes	—
18	C₂×C₄⋊Dic₃₇	C₂²	1	3	yes	—
19	C₂×#10596936	C₂²	1	3	yes	—
20	C₂×#10596936	C₂²	1	3	yes	—
21	C₂×#10596938	C₂²	1	1	yes	characteristic
22	C₂×#10596938	C₂²	1	3	yes	—
23	C₂×C₄×Dic₃₇	C₂²	1	3	yes	—
24	C₂×C₄×Dic₃₇	C₂²	1	6	yes	—
25	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₃₇	C₂²	1	8	yes	complemented
27	C₂×C₄⋊C₄×C₃₇	C₂²	1	3	yes	—
28	C₂×C₄²×C₃₇	C₂²	1	3	yes	—
29	C₂³×D₃₇	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₃₇	C₂×C₄	1	8	yes	—
31	C₂×Q₈×C₃₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₃₇	C₂³	1	3	yes	—
33	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₃₇	C₂³	1	3	yes	—
35	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
36	C₂²×Dic₃₇	C₂³	1	3	yes	—
37	C₂²×Dic₃₇	C₂³	1	3	yes	—
38	Q₈×D₃₇	C₂×C₄	1	64	yes	complemented
39	C₂×C₄×D₃₇	C₂×C₄	1	24	yes	—
40	C₂×C₄×D₃₇	C₂³	1	12	yes	—
41	C₂×Dic₇₄	C₂×C₄	1	24	yes	—
42	C₂×Dic₇₄	C₂³	1	12	yes	—
43	C₄⋊C₄×C₃₇	C₂³	1	12	yes	—
44	C₄²×C₃₇	C₂³	1	12	yes	—
45	#10596938	C₂³	1	4	yes	—
46	#10596938	C₂³	1	12	yes	—
47	C₄⋊Dic₃₇	C₂³	1	12	yes	—
48	#10596936	C₂³	1	12	yes	—
49	#10596936	C₂³	1	12	yes	—
50	C₄×Dic₃₇	C₂³	1	12	yes	—
51	C₄×Dic₃₇	C₂³	1	24	yes	—
52	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
53	C₂²×D₃₇	—	2	2	no	—
54	C₂²×D₃₇	—	2	2	no	—
55	C₂²×D₃₇	C₂²×C₄	1	4	yes	—
56	C₂²×D₃₇	C₂⁴	1	2	yes	—
57	Q₈×C₃₇	C₂²×C₄	1	16	yes	—
58	C₂×C₄×C₃₇	—	2	6	no	—
59	C₂×C₄×C₃₇	—	2	2	no	—
60	C₂×C₄×C₃₇	C₂²×C₄	1	12	yes	—
61	C₂×C₄×C₃₇	C₂⁴	1	6	yes	—
62	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
63	C₂×C₄×C₃₇	C₂⁴	1	6	yes	—
64	C₂×Dic₃₇	—	2	2	no	—
65	C₂×Dic₃₇	—	2	6	no	—
66	C₂×Dic₃₇	C₂²×C₄	1	12	yes	—
67	C₂×Dic₃₇	C₂⁴	1	2	yes	—
68	C₂×Dic₃₇	C₂⁴	1	6	yes	—
69	C₂×Dic₃₇	C₂⁴	1	6	yes	—
70	C₄×D₃₇	C₂²×C₄	1	48	yes	—
71	Dic₇₄	C₂²×C₄	1	48	yes	—
72	C₂²×C₃₇	C₂³×C₄	1	2	yes	—
73	C₂²×C₃₇	2₋¹⁺⁴	1	2	yes	—
74	C₂²×C₃₇	2₋¹⁺⁴	1	2	yes	—
75	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
76	C₂×D₃₇	—	2	8	no	—
77	C₂×D₃₇	—	2	2	no	—
78	C₂×D₃₇	—	2	2	no	—
79	C₂×D₃₇	C₂³×C₄	1	4	yes	—
80	C₄×C₃₇	—	2	6	no	—
81	C₄×C₃₇	—	2	2	no	—
82	C₄×C₃₇	C₂³×C₄	1	12	yes	—
83	Dic₃₇	—	2	2	no	—
84	Dic₃₇	—	2	6	no	—
85	Dic₃₇	C₂³×C₄	1	12	yes	—
86	C₂×#519	—	37	1	no	maximal, Sylow
87	C₂×C₃₇	#519	1	4	yes	complemented
88	C₂×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
89	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
90	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
91	D₃₇	—	2	4	no	—
92	C₂³×Q₈	—	37	1	no	—
93	#519	—	37	16	no	—
94	C₂×C₄×Q₈	—	37	1	no	—
95	C₂×C₄×Q₈	—	37	3	no	—
96	C₂×C₄×Q₈	—	37	3	no	—
97	C₂×C₄×Q₈	D₃₇	1	1	yes	characteristic, complemented
98	C₂×#117	—	37	3	no	—
99	C₂×#117	—	37	3	no	—
100	C₃₇	C₂×#519	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	—	37	16	no	—
102	C₂²×Q₈	—	37	8	no	—
103	C₂²×Q₈	—	37	3	no	—
104	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
105	C₂³×C₄	—	37	3	no	—
106	C₄×Q₈	—	37	8	no	—
107	C₄×Q₈	—	37	24	no	—
108	C₄×Q₈	—	37	24	no	—
109	C₄×Q₈	C₂×D₃₇	1	8	yes	complemented
110	#117	—	37	24	no	—
111	#117	—	37	24	no	—
112	C₂×C₄⋊C₄	—	37	3	no	—
113	C₂×C₄⋊C₄	—	37	3	no	—
114	C₂×C₄⋊C₄	—	37	3	no	—
115	C₂×C₄⋊C₄	C₂×D₃₇	1	3	yes	—
116	C₂×C₂²⋊C₄	—	37	3	no	—
117	C₂×C₂²⋊C₄	—	37	1	no	—
118	C₂×C₄²	—	37	6	no	—
119	C₂×C₄²	—	37	3	no	—
120	C₂×C₄²	C₂×D₃₇	1	3	yes	—
121	C₂⁴	—	37	1	no	—
122	C₂×Q₈	—	37	64	no	—
123	C₂×Q₈	—	37	24	no	—
124	C₂×Q₈	—	37	12	no	—
125	C₂×Q₈	C₄×D₃₇	1	8	yes	—
126	C₂×Q₈	C₂²×D₃₇	1	4	yes	—
127	C₂²×C₄	—	37	3	no	—
128	C₂²×C₄	—	37	24	no	—
129	C₂²×C₄	—	37	12	no	—
130	C₂²×C₄	—	37	1	no	—
131	C₂²×C₄	—	37	3	no	—
132	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
133	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
135	C₄⋊C₄	—	37	12	no	—
136	C₄⋊C₄	—	37	12	no	—
137	C₄⋊C₄	—	37	12	no	—
138	C₄⋊C₄	C₂²×D₃₇	1	12	yes	—
139	C₂²⋊C₄	—	37	12	no	—
140	C₂²⋊C₄	—	37	4	no	—
141	C₄²	—	37	24	no	—
142	C₄²	—	37	12	no	—
143	C₄²	C₂²×D₃₇	1	12	yes	—
144	C₂³	—	37	4	no	—
145	C₂³	—	37	2	no	—
146	C₂³	—	74	2	no	—
147	C₂³	—	74	2	no	—
148	C₂³	C₂³×D₃₇	1	1	yes	characteristic
149	Q₈	—	37	48	no	—
150	Q₈	C₂×C₄×D₃₇	1	16	yes	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	37	12	no	—
154	C₂×C₄	—	37	6	no	—
155	C₂×C₄	—	37	48	no	—
156	C₂×C₄	—	37	2	no	—
157	C₂×C₄	—	37	6	no	—
158	C₂×C₄	—	74	2	no	—
159	C₂×C₄	—	74	6	no	—
160	C₂×C₄	C₂×C₄×D₃₇	1	12	yes	—
161	C₂×C₄	C₂³×D₃₇	1	6	yes	—
162	C₂×C₄	C₂³×D₃₇	1	2	yes	—
163	C₂×C₄	C₂³×D₃₇	1	6	yes	—
164	C₂²	—	37	4	no	—
165	C₂²	—	74	8	no	—
166	C₂²	—	74	2	no	—
167	C₂²	—	74	2	no	—
168	C₂²	C₂²×C₄×D₃₇	1	2	yes	—
169	C₂²	#11924583	1	2	yes	—
170	C₂²	#11924584	1	2	yes	—
171	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
172	C₄	—	2	6	no	—
173	C₄	—	2	2	no	—
174	C₄	—	37	12	no	—
175	C₄	—	74	2	no	—
176	C₄	—	74	6	no	—
177	C₄	C₂²×C₄×D₃₇	1	12	yes	—
178	C₂	—	74	4	no	—
179	C₂	#14558009	1	4	yes	direct product factor
180	C₂	C₂³×C₄×D₃₇	1	1	yes	characteristic
181	C₂	C₂×#11924583	1	1	yes	characteristic
182	C₂	C₂×#11924584	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₃₇
commutator	C₂×C₃₇	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃₇
Fitting	C₂×C₄×Q₈×C₃₇	C₂
socle	C₂³×C₃₇	C₂⁴
prime radical	C₂³×D₃₇	C₂³
cosocle	C₂²×C₃₇	C₂⁵
hypercenter	C₂×C₄×Q₈	D₃₇
hypocenter	C₃₇	C₂×#519
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #14558009	C₂	direct (N × H)
3. C₂×Q₈×Dic₃₇	C₂	semidirect
4. C₂×C₄×Dic₇₄	C₂	semidirect
5. C₂×#11924501	C₂	semidirect
8. C₂×C₄×Q₈×C₃₇	C₂	semidirect
10. C₂×Q₈×D₃₇	C₄	semidirect
13. Q₈×Dic₃₇	C₂²	semidirect
14. C₄×Dic₇₄	C₂²	semidirect
15. #11924501	C₂²	semidirect
26. C₄×Q₈×C₃₇	C₂²	semidirect
38. Q₈×D₃₇	C₂×C₄	semidirect
87. C₂×C₃₇	#519	semidirect
97. C₂×C₄×Q₈	D₃₇	semidirect
100. C₃₇	C₂×#519	semidirect
109. C₄×Q₈	C₂×D₃₇	semidirect
179. C₂	#14558009	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₈×D₃₇	2
#14558009	2
C₂×Q₈×Dic₃₇	2
C₂×C₄×Dic₇₄	2
C₂×#11924501	2
C₂×#11924494	2
C₂×#11924493	2
C₂×C₄×Q₈×C₃₇	2
C₂×#519	37

Subgroups — C2×#14558009 #18007785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558009 #18007785