Subgroups — C₂×#14558013 #18007790

Order 4736.

1640 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10596962	C₂	1	1	yes	maximal, characteristic
2	#14558013	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₇₄	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₁₄₈	C₂	1	2	yes	maximal, complemented
5	C₂×#11924496	C₂	1	4	yes	maximal, complemented
6	C₂×#11924494	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄⋊Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊C₄×D₃₇	C₂	1	2	yes	maximal, complemented
9	C₂×#117×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10596962	C₄	1	16	yes	complemented
11	C₂×#10596962	C₂²	1	8	yes	—
12	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic
13	C₂²×D₁₄₈	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₃₇	C₂²	1	2	yes	—
15	C₂³×Dic₃₇	C₂²	1	2	yes	—
16	C₂²×C₄×D₃₇	C₂²	1	4	yes	—
17	C₂²×C₄×D₃₇	C₂²	1	2	yes	—
18	C₄×Dic₇₄	C₂²	1	16	yes	complemented
19	C₄×D₁₄₈	C₂²	1	16	yes	complemented
20	#11924496	C₂²	1	32	yes	complemented
21	#11924494	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₃₇	C₂²	1	8	yes	complemented
25	C₂×#10596936	C₂²	1	4	yes	—
26	C₄⋊C₄×D₃₇	C₂²	1	16	yes	complemented
27	C₂×#10596938	C₂²	1	4	yes	—
28	C₂×C₄×Dic₃₇	C₂²	1	4	yes	—
29	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
30	#117×C₃₇	C₂²	1	8	yes	complemented
31	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
34	C₂×C₄²×C₃₇	C₂²	1	2	yes	—
35	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
36	C₂³×D₃₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₃₇	C₂×C₄	1	8	yes	complemented
38	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
43	C₂×C₂²⋊D₃₇	C₂×C₄	1	16	yes	complemented
44	C₂×C₂²⋊D₃₇	C₂³	1	8	yes	—
45	C₂²×Dic₃₇	—	4	4	no	—
46	C₂²×Dic₃₇	C₂³	1	4	yes	—
47	C₂²×Dic₃₇	C₂³	1	8	yes	—
48	C₂²×Dic₃₇	C₂³	1	2	yes	—
49	#10596962	C₂×C₄	1	64	yes	complemented
50	C₂×D₁₄₈	C₂×C₄	1	8	yes	complemented
51	C₂×D₁₄₈	C₂³	1	4	yes	—
52	C₂×C₄×D₃₇	—	4	8	no	—
53	C₂×C₄×D₃₇	C₂×C₄	1	16	yes	complemented
54	C₂×C₄×D₃₇	C₂³	1	16	yes	—
55	C₂×C₄×D₃₇	C₂³	1	8	yes	—
56	C₂×Dic₇₄	C₂×C₄	1	8	yes	complemented
57	C₂×Dic₇₄	C₂³	1	4	yes	—
58	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
59	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
60	C₂²⋊C₄×C₃₇	C₂³	1	8	yes	—
61	C₄²×C₃₇	C₂³	1	8	yes	—
62	#10596938	C₂³	1	16	yes	—
63	C₄⋊Dic₃₇	C₂³	1	8	yes	—
64	C₄⋊Dic₃₇	C₂³	1	8	yes	—
65	#10596936	C₂³	1	16	yes	—
66	C₄×Dic₃₇	C₂³	1	16	yes	—
67	C₂³×C₃₇	—	2	2	no	—
68	C₂³×C₃₇	—	2	2	no	—
69	C₂³×C₃₇	C₂²×C₄	1	4	yes	—
70	C₂³×C₃₇	C₂⁴	1	2	yes	—
71	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
72	C₂²×D₃₇	—	2	4	no	—
73	C₂²×D₃₇	—	2	4	no	—
74	C₂²×D₃₇	C₂²×C₄	1	8	yes	—
75	C₂²×D₃₇	C₂⁴	1	4	yes	—
76	C₂×C₄×C₃₇	—	2	4	no	—
77	C₂×C₄×C₃₇	—	2	4	no	—
78	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
79	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
80	C₂×C₄×C₃₇	C₂²×C₄	1	16	yes	complemented
81	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
82	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
83	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
84	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
85	C₂²⋊D₃₇	C₂²×C₄	1	32	yes	complemented
86	C₂×Dic₃₇	—	2	8	no	—
87	C₂×Dic₃₇	—	4	8	no	—
88	C₂×Dic₃₇	C₂²×C₄	1	8	yes	—
89	C₂×Dic₃₇	C₂⁴	1	8	yes	—
90	C₂×Dic₃₇	C₂⁴	1	4	yes	—
91	D₁₄₈	C₂²×C₄	1	16	yes	complemented
92	C₄×D₃₇	—	4	16	no	—
93	C₄×D₃₇	C₂²×C₄	1	32	yes	complemented
94	Dic₇₄	C₂²×C₄	1	16	yes	complemented
95	C₂²×C₃₇	—	2	8	no	—
96	C₂²×C₃₇	—	2	2	no	—
97	C₂²×C₃₇	—	2	2	no	—
98	C₂²×C₃₇	C₂³×C₄	1	4	yes	—
99	C₂²×C₃₇	C₂³×C₄	1	2	yes	—
100	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
101	C₂²×C₃₇	2₋¹⁺⁴	1	2	yes	—
102	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
103	C₂×D₃₇	—	2	16	no	—
104	C₂×D₃₇	—	2	4	no	—
105	C₂×D₃₇	—	2	4	no	—
106	C₂×D₃₇	C₂³×C₄	1	8	yes	—
107	C₄×C₃₇	—	2	4	no	—
108	C₄×C₃₇	—	2	4	no	—
109	C₄×C₃₇	C₂³×C₄	1	4	yes	—
110	C₄×C₃₇	C₂³×C₄	1	4	yes	—
111	Dic₃₇	—	2	8	no	—
112	Dic₃₇	C₂³×C₄	1	8	yes	—
113	C₂×#520	—	37	1	no	maximal, Sylow
114	C₂×C₃₇	—	2	4	no	—
115	C₂×C₃₇	#520	1	4	yes	complemented
116	C₂×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
117	C₂×C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₃₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
119	D₃₇	—	2	8	no	—
120	C₂²×#41	—	37	1	no	—
121	#520	—	37	16	no	—
122	C₂×C₄×Q₈	—	37	2	no	—
123	C₂×C₄×D₄	—	37	4	no	—
124	C₂×C₄×D₄	—	37	2	no	—
125	C₂×#117	—	37	2	no	—
126	C₂×#117	D₃₇	1	1	yes	characteristic, complemented
127	C₂²×C₄⋊C₄	—	37	1	no	—
128	C₂²×C₄⋊C₄	—	37	2	no	—
129	C₃₇	C₂×#520	1	1	yes	characteristic, complemented, Sylow
130	C₂×#41	—	37	16	no	—
131	C₂×#41	—	37	8	no	—
132	C₂²×Q₈	—	37	1	no	—
133	C₂²×D₄	—	37	2	no	—
134	C₂²×D₄	—	37	1	no	—
135	C₂³×C₄	—	37	2	no	—
136	C₂³×C₄	—	37	2	no	—
137	C₂³×C₄	—	37	4	no	—
138	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
139	C₄×Q₈	—	37	16	no	—
140	C₄×D₄	—	37	32	no	—
141	C₄×D₄	—	37	16	no	—
142	#117	—	37	16	no	—
143	#117	C₂×D₃₇	1	8	yes	complemented
144	C₂×C₄⋊C₄	—	37	2	no	—
145	C₂×C₄⋊C₄	—	37	4	no	—
146	C₂×C₄⋊C₄	—	37	2	no	—
147	C₂×C₄⋊C₄	—	37	8	no	—
148	C₂×C₄⋊C₄	—	37	16	no	—
149	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
150	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
151	C₂×C₂²⋊C₄	—	37	4	no	—
152	C₂×C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
153	C₂×C₄²	—	37	4	no	—
154	C₂×C₄²	C₂×D₃₇	1	2	yes	—
155	C₂⁴	—	37	2	no	—
156	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
157	#41	—	37	64	no	—
158	C₂×Q₈	—	37	8	no	—
159	C₂×Q₈	—	37	4	no	—
160	C₂×D₄	—	37	16	no	—
161	C₂×D₄	—	37	8	no	—
162	C₂×D₄	—	37	8	no	—
163	C₂×D₄	—	37	4	no	—
164	C₂²×C₄	—	37	2	no	—
165	C₂²×C₄	—	37	16	no	—
166	C₂²×C₄	—	37	8	no	—
167	C₂²×C₄	—	37	4	no	—
168	C₂²×C₄	—	37	8	no	—
169	C₂²×C₄	—	37	16	no	—
170	C₂²×C₄	—	148	8	no	—
171	C₂²×C₄	—	148	4	no	—
172	C₂²×C₄	C₄×D₃₇	1	8	yes	complemented
173	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
174	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
175	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
176	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₃₇	1	4	yes	—
178	C₄⋊C₄	—	37	16	no	—
179	C₄⋊C₄	—	37	8	no	—
180	C₄⋊C₄	—	37	8	no	—
181	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
182	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
183	C₂²⋊C₄	—	37	16	no	—
184	C₂²⋊C₄	C₂²×D₃₇	1	8	yes	—
185	C₄²	—	37	16	no	—
186	C₄²	C₂²×D₃₇	1	8	yes	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	37	8	no	—
190	C₂³	—	37	4	no	—
191	C₂³	—	74	4	no	—
192	C₂³	—	74	4	no	—
193	C₂³	C₂×C₄×D₃₇	1	4	yes	—
194	C₂³	C₂³×D₃₇	1	2	yes	—
195	C₂³	C₂³×D₃₇	1	1	yes	characteristic
196	Q₈	—	37	16	no	—
197	D₄	—	37	32	no	—
198	D₄	—	37	16	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	37	8	no	—
202	C₂×C₄	—	37	4	no	—
203	C₂×C₄	—	37	32	no	—
204	C₂×C₄	—	37	8	no	—
205	C₂×C₄	—	74	8	no	—
206	C₂×C₄	—	148	16	no	—
207	C₂×C₄	—	148	8	no	—
208	C₂×C₄	C₂×C₄×D₃₇	1	4	yes	—
209	C₂×C₄	C₂×C₄×D₃₇	1	4	yes	—
210	C₂×C₄	C₂×C₄×D₃₇	1	16	yes	complemented
211	C₂×C₄	C₂³×D₃₇	1	4	yes	—
212	C₂×C₄	C₂³×D₃₇	1	4	yes	—
213	C₂×C₄	C₂³×D₃₇	1	2	yes	—
214	C₂×C₄	C₂³×D₃₇	1	2	yes	—
215	C₂²	—	2	8	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	37	8	no	—
219	C₂²	—	74	16	no	—
220	C₂²	—	74	4	no	—
221	C₂²	—	74	4	no	—
222	C₂²	C₂²×C₄×D₃₇	1	4	yes	—
223	C₂²	C₂²×C₄×D₃₇	1	2	yes	—
224	C₂²	#11924582	1	2	yes	—
225	C₂²	#11924583	1	2	yes	—
226	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
227	C₄	—	2	4	no	—
228	C₄	—	2	4	no	—
229	C₄	—	37	8	no	—
230	C₄	—	74	8	no	—
231	C₄	C₂²×C₄×D₃₇	1	4	yes	—
232	C₄	C₂²×C₄×D₃₇	1	4	yes	—
233	C₂	—	2	4	no	—
234	C₂	—	74	8	no	—
235	C₂	#14558013	1	4	yes	direct product factor
236	C₂	C₂³×C₄×D₃₇	1	1	yes	characteristic
237	C₂	C₂×#11924582	1	1	yes	characteristic
238	C₂	C₂×#11924583	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₃₇
commutator	C₂×C₃₇	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃₇
Fitting	C₂×#117×C₃₇	C₂
socle	C₂³×C₃₇	C₂⁴
prime radical	C₂²×#10596962	C₂
cosocle	C₂²×C₃₇	C₂⁵
hypercenter	C₂×#117	D₃₇
hypocenter	C₃₇	C₂×#520
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #14558013	C₂	direct (N × H)
3. C₂×C₄×Dic₇₄	C₂	semidirect
4. C₂×C₄×D₁₄₈	C₂	semidirect
5. C₂×#11924496	C₂	semidirect
6. C₂×#11924494	C₂	semidirect
7. C₂²×C₄⋊Dic₃₇	C₂	semidirect
8. C₂×C₄⋊C₄×D₃₇	C₂	semidirect
9. C₂×#117×C₃₇	C₂	semidirect
10. C₂×#10596962	C₄	semidirect
18. C₄×Dic₇₄	C₂²	semidirect
19. C₄×D₁₄₈	C₂²	semidirect
20. #11924496	C₂²	semidirect
21. #11924494	C₂²	semidirect
24. C₂×C₄⋊Dic₃₇	C₂²	semidirect
26. C₄⋊C₄×D₃₇	C₂²	semidirect
30. #117×C₃₇	C₂²	semidirect
37. C₂²×C₄×C₃₇	C₂×C₄	semidirect
43. C₂×C₂²⋊D₃₇	C₂×C₄	semidirect
49. #10596962	C₂×C₄	semidirect
50. C₂×D₁₄₈	C₂×C₄	semidirect
53. C₂×C₄×D₃₇	C₂×C₄	semidirect
56. C₂×Dic₇₄	C₂×C₄	semidirect
80. C₂×C₄×C₃₇	C₂²×C₄	semidirect
85. C₂²⋊D₃₇	C₂²×C₄	semidirect
91. D₁₄₈	C₂²×C₄	semidirect
93. C₄×D₃₇	C₂²×C₄	semidirect
94. Dic₇₄	C₂²×C₄	semidirect
115. C₂×C₃₇	#520	semidirect
126. C₂×#117	D₃₇	semidirect
129. C₃₇	C₂×#520	semidirect
143. #117	C₂×D₃₇	semidirect
172. C₂²×C₄	C₄×D₃₇	semidirect
210. C₂×C₄	C₂×C₄×D₃₇	semidirect
235. C₂	#14558013	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×#10596962	2
#14558013	2
C₂×C₄×Dic₇₄	2
C₂×C₄×D₁₄₈	2
C₂×#11924496	2
C₂×#11924494	2
C₂²×C₄⋊Dic₃₇	2
C₂×C₄⋊C₄×D₃₇	2
C₂×#117×C₃₇	2
C₂×#520	37

Subgroups — C2×#14558013 #18007790

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558013 #18007790