Subgroups — C₂×#14558600 #18011573

Order 4736.

1920 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558600	C₂	1	16	yes	maximal, direct product factor
2	C₂×D₄×#3866	C₂	1	8	yes	maximal, complemented
3	C₂×#11924615	C₂	1	2	yes	maximal, complemented
4	C₂²×C₂²⋊#3866	C₂	1	4	yes	maximal, complemented
5	C₂²×D₄×D₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×#3866	C₂²	1	8	yes	complemented
7	D₄×#3866	C₂²	1	64	yes	complemented
8	#11924615	C₂²	1	16	yes	complemented
9	C₂×C₄⋊#3866	C₂²	1	4	yes	complemented
10	C₂×C₂²⋊#3866	C₂²	1	4	yes	—
11	C₂×C₂²⋊#3866	C₂²	1	32	yes	complemented
12	C₂×C₂²⋊#3866	C₂²	1	8	yes	complemented
13	C₂×C₄×#3866	C₂²	1	4	yes	complemented
14	C₂⁴×D₃₇	C₂²	1	2	yes	—
15	C₂²×D₁₄₈	C₄	1	1	yes	characteristic, complemented
16	C₂²×C₂²⋊D₃₇	C₄	1	2	yes	complemented
17	C₂×D₄×D₃₇	C₄	1	8	yes	complemented
18	C₂×D₄×D₃₇	C₄	1	4	yes	complemented
19	C₂×D₄×D₃₇	C₂²	1	4	yes	—
20	C₂×D₄×D₃₇	C₂²	1	8	yes	—
21	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
22	C₂²×D₄×C₃₇	C₄	1	1	yes	characteristic, complemented
23	C₂⁴×C₃₇	C₂×C₄	1	2	yes	complemented
24	C₂³×D₃₇	—	4	8	no	—
25	C₂³×D₃₇	C₂×C₄	1	8	yes	complemented
26	C₂³×D₃₇	C₂×C₄	1	4	yes	complemented
27	C₂³×D₃₇	C₂×C₄	1	2	yes	complemented
28	C₂³×D₃₇	C₂³	1	1	yes	characteristic
29	C₂³×D₃₇	C₂³	1	4	yes	—
30	C₂³×D₃₇	C₂³	1	8	yes	—
31	C₂²×#3866	—	4	16	no	—
32	C₂²×#3866	C₂³	1	8	yes	—
33	C₂²×#3866	C₂³	1	32	yes	complemented
34	C₂×D₄×C₃₇	C₂×C₄	1	4	yes	complemented
35	C₂×D₄×C₃₇	C₂×C₄	1	8	yes	complemented
36	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic, complemented
37	C₂×C₂²⋊D₃₇	C₂×C₄	1	8	yes	complemented
38	C₂×C₂²⋊D₃₇	C₂×C₄	1	16	yes	complemented
39	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic, complemented
40	D₄×D₃₇	C₂×C₄	1	16	yes	complemented
41	D₄×D₃₇	C₂×C₄	1	32	yes	complemented
42	D₄×D₃₇	C₂³	1	16	yes	—
43	C₂×D₁₄₈	C₂×C₄	1	4	yes	complemented
44	C₂×D₁₄₈	C₂×C₄	1	8	yes	complemented
45	C₂×C₄×D₃₇	C₂×C₄	1	4	yes	complemented
46	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	complemented
47	C₂×C₄×D₃₇	C₂³	1	2	yes	—
48	C₂×C₄×D₃₇	C₂³	1	4	yes	—
49	C₂²⋊#3866	C₂³	1	16	yes	—
50	C₂²⋊#3866	C₂³	1	32	yes	complemented
51	C₄⋊#3866	C₂³	1	16	yes	complemented
52	C₄×#3866	C₂³	1	16	yes	complemented
53	C₂³×C₃₇	—	4	4	no	—
54	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₂³×C₃₇	C₂²×C₄	1	4	yes	—
56	C₂³×C₃₇	C₂²×C₄	1	8	yes	complemented
57	C₂²×D₃₇	—	2	16	no	—
58	C₂²×D₃₇	—	2	4	no	—
59	C₂²×D₃₇	—	4	16	no	—
60	C₂²×D₃₇	—	4	16	no	—
61	C₂²×D₃₇	—	4	8	no	—
62	C₂²×D₃₇	—	4	4	no	—
63	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
64	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×D₃₇	C₂²×C₄	1	4	yes	—
66	C₂²×D₃₇	C₂²×C₄	1	8	yes	—
67	C₂²×D₃₇	C₂²×C₄	1	8	yes	complemented
68	C₂²×D₃₇	C₂²×C₄	1	16	yes	complemented
69	C₂²×D₃₇	C₂⁴	1	2	yes	—
70	C₂²×D₃₇	C₂⁴	1	8	yes	—
71	C₂²×D₃₇	C₂⁴	1	1	yes	characteristic
72	C₂×#3866	—	2	16	no	—
73	C₂×#3866	—	4	32	no	—
74	C₂×#3866	C₂⁴	1	16	yes	—
75	D₄×C₃₇	C₂²×C₄	1	16	yes	complemented
76	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	complemented
78	C₂²⋊D₃₇	C₂²×C₄	1	32	yes	complemented
79	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
80	C₂×Dic₃₇	C₂²×C₄	1	4	yes	complemented
81	D₁₄₈	C₂²×C₄	1	16	yes	complemented
82	C₄×D₃₇	C₂²×C₄	1	4	yes	—
83	C₄×D₃₇	C₂²×C₄	1	8	yes	complemented
84	C₄×D₃₇	C₂⁴	1	4	yes	—
85	C₂²×C₃₇	—	2	2	no	—
86	C₂²×C₃₇	—	2	8	no	—
87	C₂²×C₃₇	—	4	8	no	—
88	C₂²×C₃₇	C₂³×C₄	1	2	yes	—
89	C₂²×C₃₇	C₂³×C₄	1	8	yes	—
90	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×D₃₇	—	2	2	no	—
92	C₂×D₃₇	—	2	8	no	—
93	C₂×D₃₇	—	2	4	no	—
94	C₂×D₃₇	—	2	16	no	—
95	C₂×D₃₇	—	2	16	no	—
96	C₂×D₃₇	—	2	4	no	—
97	C₂×D₃₇	—	4	16	no	—
98	C₂×D₃₇	—	4	8	no	—
99	C₂×D₃₇	C₂³×C₄	1	2	yes	—
100	C₂×D₃₇	C₂³×C₄	1	8	yes	—
101	C₂×D₃₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₃₇	H(5,𝔽₂)	1	4	yes	—
103	C₂×D₃₇	C₂⁵	1	1	yes	characteristic
104	#3866	—	2	16	no	—
105	C₄×C₃₇	C₂³×C₄	1	4	yes	—
106	Dic₃₇	C₂³×C₄	1	4	yes	—
107	C₂×#518	—	37	1	no	maximal, Sylow
108	C₂×C₃₇	—	2	2	no	—
109	C₂×C₃₇	—	2	8	no	—
110	C₂×C₃₇	#518	1	2	yes	complemented
111	C₂×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
112	D₃₇	—	2	2	no	—
113	D₃₇	—	2	8	no	—
114	D₃₇	#518	1	2	yes	complemented
115	D₃₇	C₂×H(5,𝔽₂)	1	2	yes	—
116	C₂³×D₄	—	37	1	no	—
117	#518	—	37	16	no	—
118	C₂×C₄×D₄	—	37	8	no	—
119	C₂×#117	—	37	2	no	—
120	C₂²×C₂²⋊C₄	—	37	4	no	—
121	C₃₇	C₂×#518	1	1	yes	characteristic, complemented, Sylow
122	C₂⁵	—	37	2	no	—
123	C₂²×D₄	—	37	2	no	—
124	C₂²×D₄	—	37	4	no	—
125	C₂²×D₄	—	37	8	no	—
126	C₂²×D₄	—	37	4	no	—
127	C₂²×D₄	—	37	8	no	—
128	C₂²×D₄	—	37	1	no	—
129	C₂²×D₄	#3866	1	1	yes	characteristic, complemented
130	C₂³×C₄	—	37	1	no	—
131	C₂³×C₄	—	37	8	no	—
132	C₄×D₄	—	37	64	no	—
133	#117	—	37	16	no	—
134	C₂×C₄⋊C₄	—	37	4	no	—
135	C₂×C₂²⋊C₄	—	37	4	no	—
136	C₂×C₂²⋊C₄	—	37	8	no	—
137	C₂×C₂²⋊C₄	—	37	32	no	—
138	C₂×C₄²	—	37	4	no	—
139	C₂⁴	—	37	1	no	—
140	C₂⁴	—	37	4	no	—
141	C₂⁴	—	37	4	no	—
142	C₂⁴	—	37	2	no	—
143	C₂⁴	—	37	8	no	—
144	C₂⁴	—	37	8	no	—
145	C₂⁴	—	148	8	no	—
146	C₂⁴	C₂×#3866	1	2	yes	complemented
147	C₂×D₄	—	37	8	no	—
148	C₂×D₄	—	37	16	no	—
149	C₂×D₄	—	37	16	no	—
150	C₂×D₄	—	37	32	no	—
151	C₂×D₄	—	37	16	no	—
152	C₂×D₄	—	37	4	no	—
153	C₂×D₄	—	37	8	no	—
154	C₂×D₄	C₂×#3866	1	4	yes	complemented
155	C₂×D₄	C₂×#3866	1	8	yes	complemented
156	C₂²×C₄	—	37	1	no	—
157	C₂²×C₄	—	37	2	no	—
158	C₂²×C₄	—	37	2	no	—
159	C₂²×C₄	—	37	4	no	—
160	C₂²×C₄	—	37	4	no	—
161	C₂²×C₄	—	37	8	no	—
162	C₂²×C₄	—	37	32	no	—
163	C₂²×C₄	—	148	16	no	—
164	C₂²×C₄	C₂×#3866	1	1	yes	characteristic, complemented
165	C₄⋊C₄	—	37	16	no	—
166	C₂²⋊C₄	—	37	16	no	—
167	C₂²⋊C₄	—	37	32	no	—
168	C₄²	—	37	16	no	—
169	C₂³	—	4	4	no	—
170	C₂³	—	37	1	no	—
171	C₂³	—	37	2	no	—
172	C₂³	—	37	8	no	—
173	C₂³	—	37	2	no	—
174	C₂³	—	37	8	no	—
175	C₂³	—	37	1	no	—
176	C₂³	—	37	4	no	—
177	C₂³	—	37	16	no	—
178	C₂³	—	37	8	no	—
179	C₂³	—	74	16	no	—
180	C₂³	—	74	4	no	—
181	C₂³	—	148	4	no	—
182	C₂³	—	148	8	no	—
183	C₂³	—	148	16	no	—
184	C₂³	—	148	16	no	—
185	C₂³	C₂²×#3866	1	1	yes	characteristic
186	C₂³	C₂²×#3866	1	4	yes	—
187	C₂³	C₂²×#3866	1	8	yes	complemented
188	D₄	—	37	32	no	—
189	D₄	—	37	16	no	—
190	D₄	C₂²×#3866	1	16	yes	complemented
191	C₂×C₄	—	37	4	no	—
192	C₂×C₄	—	37	2	no	—
193	C₂×C₄	—	37	4	no	—
194	C₂×C₄	—	37	4	no	—
195	C₂×C₄	—	37	8	no	—
196	C₂×C₄	—	37	16	no	—
197	C₂×C₄	—	74	16	no	—
198	C₂×C₄	—	148	32	no	—
199	C₂×C₄	C₂²×#3866	1	2	yes	—
200	C₂×C₄	C₂²×#3866	1	4	yes	complemented
201	C₂²	—	2	2	no	—
202	C₂²	—	2	8	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	37	4	no	—
205	C₂²	—	37	1	no	—
206	C₂²	—	37	1	no	—
207	C₂²	—	37	2	no	—
208	C₂²	—	37	8	no	—
209	C₂²	—	74	16	no	—
210	C₂²	—	74	4	no	—
211	C₂²	—	74	4	no	—
212	C₂²	—	74	16	no	—
213	C₂²	—	74	2	no	—
214	C₂²	—	74	8	no	—
215	C₂²	—	148	8	no	—
216	C₂²	—	148	16	no	—
217	C₂²	C₂³×#3866	1	2	yes	—
218	C₂²	C₂³×#3866	1	8	yes	—
219	C₂²	C₂³×#3866	1	1	yes	characteristic
220	C₄	—	37	4	no	—
221	C₄	—	74	16	no	—
222	C₄	C₂³×#3866	1	4	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	2	8	no	—
225	C₂	—	37	2	no	—
226	C₂	—	37	2	no	—
227	C₂	—	74	2	no	—
228	C₂	—	74	8	no	—
229	C₂	#14558600	1	2	yes	direct product factor
230	C₂	C₂⁴×#3866	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×#3866
commutator	C₂×C₃₇	C₂⁴×C₄
Frattini	C₂	C₂⁴×#3866
Fitting	C₂²×D₄×C₃₇	C₄
socle	C₂²×C₃₇	C₂³×C₄
prime radical	C₂²×D₄×D₃₇	C₂
cosocle	C₂×D₃₇	C₂⁵
hypercenter	C₂²×D₄	#3866
hypocenter	C₃₇	C₂×#518
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. #14558600	C₂	direct (N × H)
2. C₂×D₄×#3866	C₂	semidirect
3. C₂×#11924615	C₂	semidirect
4. C₂²×C₂²⋊#3866	C₂	semidirect
6. C₂³×#3866	C₂²	semidirect
7. D₄×#3866	C₂²	semidirect
8. #11924615	C₂²	semidirect
9. C₂×C₄⋊#3866	C₂²	semidirect
11. C₂×C₂²⋊#3866	C₂²	semidirect
12. C₂×C₂²⋊#3866	C₂²	semidirect
13. C₂×C₄×#3866	C₂²	semidirect
15. C₂²×D₁₄₈	C₄	semidirect
16. C₂²×C₂²⋊D₃₇	C₄	semidirect
17. C₂×D₄×D₃₇	C₄	semidirect
18. C₂×D₄×D₃₇	C₄	semidirect
22. C₂²×D₄×C₃₇	C₄	semidirect
23. C₂⁴×C₃₇	C₂×C₄	semidirect
25. C₂³×D₃₇	C₂×C₄	semidirect
26. C₂³×D₃₇	C₂×C₄	semidirect
27. C₂³×D₃₇	C₂×C₄	semidirect
33. C₂²×#3866	C₂³	semidirect
34. C₂×D₄×C₃₇	C₂×C₄	semidirect
35. C₂×D₄×C₃₇	C₂×C₄	semidirect
36. C₂²×C₄×C₃₇	C₂×C₄	semidirect
37. C₂×C₂²⋊D₃₇	C₂×C₄	semidirect
38. C₂×C₂²⋊D₃₇	C₂×C₄	semidirect
39. C₂²×Dic₃₇	C₂×C₄	semidirect
40. D₄×D₃₇	C₂×C₄	semidirect
41. D₄×D₃₇	C₂×C₄	semidirect
43. C₂×D₁₄₈	C₂×C₄	semidirect
44. C₂×D₁₄₈	C₂×C₄	semidirect
45. C₂×C₄×D₃₇	C₂×C₄	semidirect
46. C₂×C₄×D₃₇	C₂×C₄	semidirect
50. C₂²⋊#3866	C₂³	semidirect
51. C₄⋊#3866	C₂³	semidirect
52. C₄×#3866	C₂³	semidirect
56. C₂³×C₃₇	C₂²×C₄	semidirect
67. C₂²×D₃₇	C₂²×C₄	semidirect
68. C₂²×D₃₇	C₂²×C₄	semidirect
75. D₄×C₃₇	C₂²×C₄	semidirect
77. C₂×C₄×C₃₇	C₂²×C₄	semidirect
78. C₂²⋊D₃₇	C₂²×C₄	semidirect
80. C₂×Dic₃₇	C₂²×C₄	semidirect
81. D₁₄₈	C₂²×C₄	semidirect
83. C₄×D₃₇	C₂²×C₄	semidirect
110. C₂×C₃₇	#518	semidirect
114. D₃₇	#518	semidirect
121. C₃₇	C₂×#518	semidirect
129. C₂²×D₄	#3866	semidirect
146. C₂⁴	C₂×#3866	semidirect
154. C₂×D₄	C₂×#3866	semidirect
155. C₂×D₄	C₂×#3866	semidirect
164. C₂²×C₄	C₂×#3866	semidirect
187. C₂³	C₂²×#3866	semidirect
190. D₄	C₂²×#3866	semidirect
200. C₂×C₄	C₂²×#3866	semidirect
229. C₂	#14558600	direct (N × H)

Maximal subgroups

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

subgroup	index
#14558600	2
C₂×D₄×#3866	2
C₂×#11924615	2
C₂²×C₂²⋊#3866	2
C₂²×D₄×D₃₇	2
C₂×#518	37

Subgroups — C2×#14558600 #18011573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558600 #18011573