Subgroups — C₂×#14558015 #18007792

Order 4736.

1640 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10596966	C₂	1	1	yes	maximal, characteristic
2	#14558015	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₁₄₈	C₂	1	3	yes	maximal, complemented
5	C₂×#11924498	C₂	1	3	yes	maximal, complemented
6	C₂×#11924493	C₂	1	3	yes	maximal, complemented
7	C₂×C₄⋊C₄×D₃₇	C₂	1	3	yes	maximal, complemented
8	C₂×C₄×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10596966	C₄	1	16	yes	complemented
10	C₂×#10596966	C₂²	1	8	yes	—
11	C₂²×D₁₄₈	C₂²	1	3	yes	—
12	C₂²×C₄×D₃₇	C₂²	1	6	yes	—
13	C₂²×C₄×D₃₇	C₂²	1	3	yes	—
14	Q₈×Dic₃₇	C₂²	1	8	yes	complemented
15	C₄×D₁₄₈	C₂²	1	24	yes	complemented
16	#11924498	C₂²	1	24	yes	complemented
17	#11924493	C₂²	1	24	yes	complemented
18	C₂×C₄⋊Dic₃₇	C₂²	1	3	yes	—
19	C₂×#10596936	C₂²	1	3	yes	—
20	C₂×#10596936	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×D₃₇	C₂²	1	24	yes	complemented
22	C₂×#10596938	C₂²	1	3	yes	—
23	C₂×#10596938	C₂²	1	3	yes	—
24	C₂×C₄×Dic₃₇	C₂²	1	3	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	3	yes	—
30	C₂×Q₈×C₃₇	C₂×C₄	1	8	yes	complemented
31	C₂×Q₈×C₃₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₃₇	C₂³	1	3	yes	—
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	1	yes	characteristic
38	#10596966	C₂×C₄	1	64	yes	complemented
39	C₂×D₁₄₈	C₂×C₄	1	24	yes	complemented
40	C₂×D₁₄₈	C₂³	1	12	yes	—
41	C₂×C₄×D₃₇	—	4	12	no	—
42	C₂×C₄×D₃₇	C₂×C₄	1	24	yes	complemented
43	C₂×C₄×D₃₇	C₂³	1	24	yes	—
44	C₂×C₄×D₃₇	C₂³	1	12	yes	—
45	C₄⋊C₄×C₃₇	C₂³	1	12	yes	—
46	C₄²×C₃₇	C₂³	1	12	yes	—
47	#10596938	C₂³	1	12	yes	—
48	#10596938	C₂³	1	12	yes	—
49	C₄⋊Dic₃₇	C₂³	1	12	yes	—
50	#10596936	C₂³	1	12	yes	—
51	#10596936	C₂³	1	4	yes	—
52	C₄×Dic₃₇	C₂³	1	12	yes	—
53	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
54	C₂²×D₃₇	—	2	6	no	—
55	C₂²×D₃₇	—	2	6	no	—
56	C₂²×D₃₇	C₂²×C₄	1	12	yes	—
57	C₂²×D₃₇	C₂⁴	1	6	yes	—
58	Q₈×C₃₇	C₂²×C₄	1	16	yes	complemented
59	C₂×C₄×C₃₇	—	2	6	no	—
60	C₂×C₄×C₃₇	—	2	2	no	—
61	C₂×C₄×C₃₇	C₂²×C₄	1	12	yes	—
62	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
63	C₂×C₄×C₃₇	C₂⁴	1	6	yes	—
64	C₂×C₄×C₃₇	C₂⁴	1	6	yes	—
65	C₂×Dic₃₇	—	2	6	no	—
66	C₂×Dic₃₇	—	2	2	no	—
67	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
68	C₂×Dic₃₇	C₂⁴	1	2	yes	—
69	C₂×Dic₃₇	C₂⁴	1	6	yes	—
70	C₂×Dic₃₇	C₂⁴	1	2	yes	—
71	D₁₄₈	C₂²×C₄	1	48	yes	complemented
72	C₄×D₃₇	—	4	24	no	—
73	C₄×D₃₇	C₂²×C₄	1	48	yes	complemented
74	C₂²×C₃₇	C₂³×C₄	1	2	yes	—
75	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
76	C₂²×C₃₇	2₋¹⁺⁴	1	2	yes	—
77	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
78	C₂×D₃₇	—	2	24	no	—
79	C₂×D₃₇	—	2	6	no	—
80	C₂×D₃₇	—	2	6	no	—
81	C₂×D₃₇	C₂³×C₄	1	12	yes	—
82	C₄×C₃₇	—	2	6	no	—
83	C₄×C₃₇	—	2	2	no	—
84	C₄×C₃₇	C₂³×C₄	1	12	yes	—
85	Dic₃₇	—	2	6	no	—
86	Dic₃₇	—	2	2	no	—
87	Dic₃₇	C₂³×C₄	1	4	yes	—
88	C₂×#520	—	37	1	no	maximal, Sylow
89	C₂×C₃₇	#520	1	4	yes	complemented
90	C₂×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
91	C₂×C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
92	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
93	D₃₇	—	2	12	no	—
94	C₂²×#41	—	37	1	no	—
95	#520	—	37	16	no	—
96	C₂×C₄×Q₈	—	37	1	no	—
97	C₂×C₄×Q₈	D₃₇	1	1	yes	characteristic, complemented
98	C₂×C₄×D₄	—	37	3	no	—
99	C₂×C₄×D₄	—	37	3	no	—
100	C₂×#117	—	37	3	no	—
101	C₂²×C₄⋊C₄	—	37	3	no	—
102	C₃₇	C₂×#520	1	1	yes	characteristic, complemented, Sylow
103	C₂×#41	—	37	16	no	—
104	C₂×#41	—	37	8	no	—
105	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
106	C₂²×D₄	—	37	3	no	—
107	C₂³×C₄	—	37	3	no	—
108	C₂³×C₄	—	37	6	no	—
109	C₄×Q₈	—	37	8	no	—
110	C₄×Q₈	C₂×D₃₇	1	8	yes	complemented
111	C₄×D₄	—	37	24	no	—
112	C₄×D₄	—	37	24	no	—
113	#117	—	37	24	no	—
114	C₂×C₄⋊C₄	—	37	3	no	—
115	C₂×C₄⋊C₄	—	37	3	no	—
116	C₂×C₄⋊C₄	—	37	1	no	—
117	C₂×C₄⋊C₄	—	37	24	no	—
118	C₂×C₄⋊C₄	C₂×D₃₇	1	3	yes	—
119	C₂×C₂²⋊C₄	—	37	3	no	—
120	C₂×C₂²⋊C₄	—	37	3	no	—
121	C₂×C₄²	—	37	3	no	—
122	C₂×C₄²	C₂×D₃₇	1	3	yes	—
123	C₂⁴	—	37	3	no	—
124	#41	—	37	64	no	—
125	C₂×Q₈	C₄×D₃₇	1	8	yes	complemented
126	C₂×Q₈	C₂²×D₃₇	1	4	yes	—
127	C₂×D₄	—	37	24	no	—
128	C₂×D₄	—	37	12	no	—
129	C₂²×C₄	—	37	1	no	—
130	C₂²×C₄	—	37	24	no	—
131	C₂²×C₄	—	37	12	no	—
132	C₂²×C₄	—	37	1	no	—
133	C₂²×C₄	—	37	3	no	—
134	C₂²×C₄	—	37	24	no	—
135	C₂²×C₄	—	148	12	no	—
136	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
138	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
139	C₄⋊C₄	—	37	12	no	—
140	C₄⋊C₄	—	37	4	no	—
141	C₄⋊C₄	—	37	12	no	—
142	C₄⋊C₄	C₂²×D₃₇	1	12	yes	—
143	C₂²⋊C₄	—	37	12	no	—
144	C₂²⋊C₄	—	37	12	no	—
145	C₄²	—	37	12	no	—
146	C₄²	C₂²×D₃₇	1	12	yes	—
147	C₂³	—	37	12	no	—
148	C₂³	—	37	6	no	—
149	C₂³	—	74	6	no	—
150	C₂³	—	74	6	no	—
151	C₂³	C₂³×D₃₇	1	1	yes	characteristic
152	Q₈	C₂×C₄×D₃₇	1	16	yes	complemented
153	D₄	—	37	48	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	37	4	no	—
157	C₂×C₄	—	37	2	no	—
158	C₂×C₄	—	37	48	no	—
159	C₂×C₄	—	37	2	no	—
160	C₂×C₄	—	37	6	no	—
161	C₂×C₄	—	74	6	no	—
162	C₂×C₄	—	74	2	no	—
163	C₂×C₄	—	148	24	no	—
164	C₂×C₄	C₂×C₄×D₃₇	1	12	yes	—
165	C₂×C₄	C₂³×D₃₇	1	2	yes	—
166	C₂×C₄	C₂³×D₃₇	1	6	yes	—
167	C₂×C₄	C₂³×D₃₇	1	6	yes	—
168	C₂²	—	37	12	no	—
169	C₂²	—	74	24	no	—
170	C₂²	—	74	6	no	—
171	C₂²	—	74	6	no	—
172	C₂²	C₂²×C₄×D₃₇	1	2	yes	—
173	C₂²	#11924582	1	2	yes	—
174	C₂²	#11924584	1	2	yes	—
175	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
176	C₄	—	2	6	no	—
177	C₄	—	2	2	no	—
178	C₄	—	37	4	no	—
179	C₄	—	74	6	no	—
180	C₄	—	74	2	no	—
181	C₄	C₂²×C₄×D₃₇	1	12	yes	—
182	C₂	—	74	12	no	—
183	C₂	#14558015	1	4	yes	direct product factor
184	C₂	C₂³×C₄×D₃₇	1	1	yes	characteristic
185	C₂	C₂×#11924582	1	1	yes	characteristic
186	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	C₂²×#10596966	C₂
cosocle	C₂²×C₃₇	C₂⁵
hypercenter	C₂×C₄×Q₈	D₃₇
hypocenter	C₃₇	C₂×#520
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. #14558015	C₂	direct (N × H)
3. C₂×Q₈×Dic₃₇	C₂	semidirect
4. C₂×C₄×D₁₄₈	C₂	semidirect
5. C₂×#11924498	C₂	semidirect
6. C₂×#11924493	C₂	semidirect
7. C₂×C₄⋊C₄×D₃₇	C₂	semidirect
8. C₂×C₄×Q₈×C₃₇	C₂	semidirect
9. C₂×#10596966	C₄	semidirect
14. Q₈×Dic₃₇	C₂²	semidirect
15. C₄×D₁₄₈	C₂²	semidirect
16. #11924498	C₂²	semidirect
17. #11924493	C₂²	semidirect
21. C₄⋊C₄×D₃₇	C₂²	semidirect
26. C₄×Q₈×C₃₇	C₂²	semidirect
30. C₂×Q₈×C₃₇	C₂×C₄	semidirect
38. #10596966	C₂×C₄	semidirect
39. C₂×D₁₄₈	C₂×C₄	semidirect
42. C₂×C₄×D₃₇	C₂×C₄	semidirect
58. Q₈×C₃₇	C₂²×C₄	semidirect
71. D₁₄₈	C₂²×C₄	semidirect
73. C₄×D₃₇	C₂²×C₄	semidirect
89. C₂×C₃₇	#520	semidirect
97. C₂×C₄×Q₈	D₃₇	semidirect
102. C₃₇	C₂×#520	semidirect
110. C₄×Q₈	C₂×D₃₇	semidirect
125. C₂×Q₈	C₄×D₃₇	semidirect
152. Q₈	C₂×C₄×D₃₇	semidirect
183. C₂	#14558015	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×#10596966	2
#14558015	2
C₂×Q₈×Dic₃₇	2
C₂×C₄×D₁₄₈	2
C₂×#11924498	2
C₂×#11924493	2
C₂×C₄⋊C₄×D₃₇	2
C₂×C₄×Q₈×C₃₇	2
C₂×#520	37

Subgroups — C2×#14558015 #18007792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558015 #18007792