Subgroups — C₂×#14557431 #17997047

Order 4736.

800 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#10596936	C₂	1	2	yes	maximal
4	#14557431	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11924443	C₂	1	2	yes	maximal
6	C₂²×C₄²×C₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₃₇	C₂²	1	2	yes	—
8	C₂³×Dic₃₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₃₇	—	2	4	no	—
10	C₂×C₄⋊Dic₃₇	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₃₇	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₃₇	C₂²	1	4	yes	—
13	C₂×#10596936	—	2	8	no	—
14	C₂×#10596936	C₂²	1	8	yes	—
15	#11924443	C₂²	1	8	yes	—
16	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₃₇	—	2	4	no	—
20	C₂×C₄²×C₃₇	C₂²	1	4	yes	—
21	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₃₇	—	2	2	no	—
23	C₂²×C₄×C₃₇	—	2	2	no	—
24	C₂²×C₄×C₃₇	—	2	2	no	—
25	C₂²×C₄×C₃₇	—	2	2	no	—
26	C₂²×C₄×C₃₇	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₃₇	D₄	1	4	yes	—
28	C₂²×C₄×C₃₇	D₄	1	4	yes	—
29	C₂²×C₄×C₃₇	Q₈	1	4	yes	—
30	C₂²×C₄×C₃₇	Q₈	1	4	yes	—
31	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
34	C₂²×Dic₃₇	—	2	4	no	—
35	C₂²×Dic₃₇	—	2	4	no	—
36	C₂²×Dic₃₇	—	2	4	no	—
37	C₂²×Dic₃₇	—	2	4	no	—
38	C₂²×Dic₃₇	—	2	4	no	—
39	C₂²×Dic₃₇	C₂×C₄	1	8	yes	—
40	C₂²×Dic₃₇	C₂³	1	4	yes	—
41	C₂²×Dic₃₇	C₂³	1	4	yes	—
42	C₄²×C₃₇	—	2	8	no	—
43	C₄⋊Dic₃₇	—	2	8	no	—
44	C₄⋊Dic₃₇	C₂×C₄	1	16	yes	complemented
45	#10596936	—	2	16	no	—
46	C₂³×C₃₇	C₂²×C₄	1	2	yes	—
47	C₂³×C₃₇	C₂×D₄	1	2	yes	—
48	C₂³×C₃₇	C₂×D₄	1	2	yes	—
49	C₂³×C₃₇	C₂×Q₈	1	2	yes	—
50	C₂³×C₃₇	C₂×Q₈	1	2	yes	—
51	C₂³×C₃₇	#41	1	2	yes	—
52	C₂³×C₃₇	#41	1	2	yes	—
53	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
54	C₂×C₄×C₃₇	—	2	8	no	—
55	C₂×C₄×C₃₇	—	2	8	no	—
56	C₂×C₄×C₃₇	—	2	2	no	—
57	C₂×C₄×C₃₇	—	2	2	no	—
58	C₂×C₄×C₃₇	—	2	2	no	—
59	C₂×C₄×C₃₇	—	2	2	no	—
60	C₂×C₄×C₃₇	C₄⋊C₄	1	16	yes	complemented
61	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
62	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
63	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
64	C₂×C₄×C₃₇	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₃₇	C₂×Q₈	1	4	yes	—
66	C₂×Dic₃₇	—	2	16	no	—
67	C₂×Dic₃₇	—	2	4	no	—
68	C₂×Dic₃₇	—	2	4	no	—
69	C₂×Dic₃₇	—	2	4	no	—
70	C₂×Dic₃₇	—	2	4	no	—
71	C₂×Dic₃₇	—	2	4	no	—
72	C₂×Dic₃₇	—	4	8	no	—
73	C₂×Dic₃₇	C₂²×C₄	1	8	yes	—
74	C₂²×C₃₇	C₂×C₄⋊C₄	1	4	yes	—
75	C₂²×C₃₇	C₄×D₄	1	4	yes	—
76	C₂²×C₃₇	C₄×Q₈	1	4	yes	—
77	C₂²×C₃₇	#121	1	4	yes	—
78	C₂²×C₃₇	C₂²⋊Q₈	1	4	yes	—
79	C₂²×C₃₇	#125	1	4	yes	—
80	C₂²×C₃₇	C₄⋊Q₈	1	4	yes	—
81	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
82	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
83	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₃₇	C₂²×Q₈	1	1	yes	characteristic
85	C₂²×C₃₇	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
87	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
88	C₄×C₃₇	—	2	4	no	—
89	C₄×C₃₇	—	2	4	no	—
90	C₄×C₃₇	C₂×C₄⋊C₄	1	8	yes	complemented
91	Dic₃₇	—	2	8	no	—
92	Dic₃₇	—	4	4	no	—
93	C₂×#389	—	37	1	no	maximal, Sylow
94	C₂×C₃₇	#389	1	8	yes	complemented
95	C₂×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
97	C₂×C₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
98	C₂×C₃₇	C₂×#121	1	1	yes	characteristic
99	C₂×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₃₇	C₂×#125	1	1	yes	characteristic
101	C₂×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₂²×C₄⋊C₄	—	37	1	no	—
103	C₂²×C₄⋊C₄	—	37	2	no	—
104	C₂²×C₄⋊C₄	—	37	1	no	—
105	C₂²×C₄²	D₃₇	1	1	yes	characteristic
106	#389	—	37	8	no	—
107	C₂×#95	—	37	2	no	—
108	C₃₇	C₂×#389	1	1	yes	characteristic, complemented, Sylow
109	C₂³×C₄	—	37	2	no	—
110	C₂³×C₄	—	37	2	no	—
111	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
112	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
113	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	37	8	no	—
115	C₂×C₄⋊C₄	—	37	4	no	—
116	C₂×C₄⋊C₄	—	37	8	no	—
117	C₂×C₄⋊C₄	—	37	4	no	—
118	C₂×C₄⋊C₄	—	74	8	no	—
119	C₂×C₄⋊C₄	—	74	4	no	—
120	C₂×C₄²	—	2	4	no	—
121	C₂×C₄²	C₂×D₃₇	1	4	yes	—
122	#95	—	37	8	no	—
123	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	2	2	no	—
128	C₂²×C₄	—	37	8	no	—
129	C₂²×C₄	—	37	4	no	—
130	C₂²×C₄	—	37	4	no	—
131	C₂²×C₄	—	74	4	no	—
132	C₂²×C₄	—	74	4	no	—
133	C₂²×C₄	—	74	4	no	—
134	C₂²×C₄	—	74	4	no	—
135	C₂²×C₄	—	74	4	no	—
136	C₂²×C₄	Dic₇₄	1	4	yes	—
137	C₂²×C₄	Dic₇₄	1	4	yes	—
138	C₂²×C₄	C₄×D₃₇	1	4	yes	—
139	C₂²×C₄	D₁₄₈	1	4	yes	—
140	C₂²×C₄	C₂²⋊D₃₇	1	4	yes	—
141	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
142	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
143	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
144	C₄⋊C₄	—	37	16	no	—
145	C₄⋊C₄	—	74	16	no	—
146	C₄⋊C₄	—	74	8	no	—
147	C₄²	—	2	8	no	—
148	C₂³	C₂×Dic₇₄	1	2	yes	—
149	C₂³	C₂×Dic₇₄	1	2	yes	—
150	C₂³	C₂×C₄×D₃₇	1	2	yes	—
151	C₂³	C₂×D₁₄₈	1	2	yes	—
152	C₂³	#10596962	1	2	yes	—
153	C₂³	#10596962	1	2	yes	—
154	C₂³	C₂×C₂²⋊D₃₇	1	2	yes	—
155	C₂³	C₂³×D₃₇	1	1	yes	characteristic
156	C₂×C₄	—	2	8	no	—
157	C₂×C₄	—	2	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	37	8	no	—
163	C₂×C₄	—	74	16	no	—
164	C₂×C₄	—	74	4	no	—
165	C₂×C₄	—	74	4	no	—
166	C₂×C₄	—	74	4	no	—
167	C₂×C₄	—	74	4	no	—
168	C₂×C₄	—	74	4	no	—
169	C₂×C₄	—	148	8	no	—
170	C₂×C₄	#10596936	1	16	yes	complemented
171	C₂×C₄	C₂×Dic₇₄	1	4	yes	—
172	C₂×C₄	C₂×Dic₇₄	1	4	yes	—
173	C₂×C₄	C₂×C₄×D₃₇	1	4	yes	—
174	C₂×C₄	C₂×D₁₄₈	1	4	yes	—
175	C₂×C₄	C₂×C₂²⋊D₃₇	1	4	yes	—
176	C₂²	C₂×#10596936	1	4	yes	—
177	C₂²	C₄×D₁₄₈	1	4	yes	—
178	C₂²	C₄×Dic₇₄	1	4	yes	—
179	C₂²	#11924510	1	4	yes	—
180	C₂²	#11924515	1	4	yes	—
181	C₂²	#11924528	1	4	yes	—
182	C₂²	#11924536	1	4	yes	—
183	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
184	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
185	C₂²	C₂²×D₁₄₈	1	1	yes	characteristic
186	C₂²	C₂²×Dic₇₄	1	1	yes	characteristic
187	C₂²	C₂²×Dic₇₄	1	1	yes	characteristic
188	C₂²	C₂×#10596962	1	1	yes	characteristic
189	C₂²	C₂×#10596962	1	1	yes	characteristic
190	C₄	—	2	4	no	—
191	C₄	—	2	4	no	—
192	C₄	—	74	8	no	—
193	C₄	—	148	4	no	—
194	C₄	C₂×#10596936	1	8	yes	complemented
195	C₂	#14557431	1	8	yes	direct product factor
196	C₂	C₂²×#10596936	1	1	yes	characteristic
197	C₂	C₂×C₄×D₁₄₈	1	1	yes	characteristic
198	C₂	C₂×C₄×Dic₇₄	1	1	yes	characteristic
199	C₂	C₂×#11924510	1	1	yes	characteristic
200	C₂	C₂×#11924515	1	1	yes	characteristic
201	C₂	C₂×#11924528	1	1	yes	characteristic
202	C₂	C₂×#11924536	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Frattini	C₂³	C₂³×D₃₇
Fitting	C₂²×C₄²×C₃₇	C₂
socle, prime radical	C₂⁴×C₃₇	C₂³
cosocle	C₂³×C₃₇	C₂⁴
hypercenter	C₂²×C₄²	D₃₇
hypocenter	C₃₇	C₂×#389
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
4. #14557431	C₂	direct (N × H)
10. C₂×C₄⋊Dic₃₇	C₄	semidirect
44. C₄⋊Dic₃₇	C₂×C₄	semidirect
60. C₂×C₄×C₃₇	C₄⋊C₄	semidirect
90. C₄×C₃₇	C₂×C₄⋊C₄	semidirect
94. C₂×C₃₇	#389	semidirect
108. C₃₇	C₂×#389	semidirect
170. C₂×C₄	#10596936	semidirect
194. C₄	C₂×#10596936	semidirect
195. C₂	#14557431	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×C₄⋊Dic₃₇	2
C₂²×#10596936	2
#14557431	2
C₂×#11924443	2
C₂²×C₄²×C₃₇	2
C₂×#389	37

Subgroups — C2×#14557431 #17997047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557431 #17997047