Subgroups — C₂×#14557609 #18003281

Order 4736.

552 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×#10596929	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₂²⋊D₃₇	C₂	1	1	yes	maximal, characteristic
3	#14557609	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11924465	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊#64436	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924448	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₂²⋊D₃₇	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
11	C₂×#10596929	—	2	4	no	—
12	C₂×#10596929	C₂²	1	4	yes	complemented
13	C₂²×#64436	C₂²	1	1	yes	characteristic
14	C₂²×#64436	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊D₃₇	C₄	1	4	yes	—
16	C₄×C₂²⋊D₃₇	C₂²	1	4	yes	—
17	C₂×#10596936	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₃₇	C₄	1	1	yes	characteristic
19	C₂×#10596938	C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
21	#11924465	C₂²	1	4	yes	complemented
22	C₂²⋊#64436	C₂²	1	4	yes	complemented
23	#11924448	C₂²	1	4	yes	complemented
24	C₈×Dic₃₇	C₂²	1	4	yes	complemented
25	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
26	C₂×M₁₆×C₃₇	—	2	4	no	—
27	C₂×M₁₆×C₃₇	C₂²	1	4	yes	complemented
28	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
30	C₂⁴×C₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×D₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₃₇	—	2	2	no	—
33	C₂²×C₄×C₃₇	—	2	2	no	—
34	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊D₃₇	—	2	4	no	—
39	C₂×C₂²⋊D₃₇	C₂×C₄	1	4	yes	—
40	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₃₇	—	2	2	no	—
44	C₂×C₄×D₃₇	—	2	2	no	—
45	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₃₇	C₂³	1	2	yes	—
47	M₁₆×C₃₇	—	2	8	no	—
48	C₂×C₈×C₃₇	—	2	2	no	—
49	C₂×C₈×C₃₇	D₄	1	4	yes	complemented
50	C₂×C₈×C₃₇	C₂³	1	2	yes	—
51	C₂×C₈×C₃₇	C₂³	1	2	yes	—
52	C₂²⋊Dic₃₇	C₂×C₄	1	4	yes	—
53	#10596938	C₂×C₄	1	4	yes	—
54	#10596936	C₂×C₄	1	4	yes	—
55	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
56	C₄×Dic₃₇	C₂³	1	2	yes	—
57	C₂×#64436	—	2	2	no	—
58	C₂×#64436	—	2	2	no	—
59	C₂×#64436	C₂³	1	2	yes	—
60	C₂×#64436	C₂³	1	2	yes	—
61	#10596929	—	2	8	no	—
62	C₂³×C₃₇	—	2	2	no	—
63	C₂³×C₃₇	—	2	2	no	—
64	C₂³×C₃₇	—	2	2	no	—
65	C₂³×C₃₇	C₂²×C₄	1	2	yes	—
66	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₃₇	—	2	2	no	—
68	C₂²×D₃₇	—	2	2	no	—
69	C₂²×D₃₇	—	2	2	no	—
70	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₃₇	—	2	2	no	—
72	C₂×C₄×C₃₇	—	2	4	no	—
73	C₂×C₄×C₃₇	—	2	2	no	—
74	C₂×C₄×C₃₇	—	2	2	no	—
75	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₃₇	#41	1	2	yes	—
79	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₃₇	—	2	8	no	—
81	C₂×Dic₃₇	—	2	2	no	—
82	C₂×Dic₃₇	—	2	2	no	—
83	C₂×Dic₃₇	M₁₆	1	4	yes	complemented
84	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
85	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
86	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
87	C₄×D₃₇	—	2	4	no	—
88	C₄×D₃₇	—	2	2	no	—
89	C₄×D₃₇	—	2	2	no	—
90	C₈×C₃₇	—	2	2	no	—
91	C₈×C₃₇	C₂×D₄	1	4	yes	complemented
92	#64436	—	2	2	no	—
93	#64436	—	2	2	no	—
94	C₂²×C₃₇	—	2	2	no	—
95	C₂²×C₃₇	—	2	2	no	—
96	C₂²×C₃₇	—	2	2	no	—
97	C₂²×C₃₇	—	4	4	no	—
98	C₂²×C₃₇	C₄×D₄	1	2	yes	—
99	C₂²×C₃₇	C₂×M₁₆	1	2	yes	—
100	C₂²×C₃₇	#131	1	2	yes	—
101	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₃₇	—	2	2	no	—
103	C₂×D₃₇	—	2	2	no	—
104	C₂×D₃₇	—	2	2	no	—
105	C₂×D₃₇	—	4	4	no	—
106	C₄×C₃₇	—	2	2	no	—
107	C₄×C₃₇	C₄×D₄	1	2	yes	—
108	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
109	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
110	Dic₃₇	—	2	2	no	—
111	Dic₃₇	—	2	2	no	—
112	Dic₃₇	C₂×M₁₆	1	4	yes	complemented
113	C₂×#436	—	37	1	no	maximal, Sylow
114	C₂×C₃₇	—	4	2	no	—
115	C₂×C₃₇	#436	1	4	yes	complemented
116	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
117	C₂×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
118	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
119	D₃₇	—	4	2	no	—
120	C₂²×M₁₆	—	37	1	no	—
121	C₂²×M₁₆	D₃₇	1	1	yes	characteristic, complemented
122	C₂×C₄×D₄	—	37	1	no	—
123	#436	—	37	8	no	—
124	C₂×C₄⋊C₈	—	37	1	no	—
125	C₂×C₂²⋊C₈	—	37	1	no	—
126	C₂×C₂²⋊C₈	—	37	1	no	—
127	C₂×C₄×C₈	—	37	1	no	—
128	C₃₇	C₂×#436	1	1	yes	characteristic, complemented, Sylow
129	C₂²×D₄	—	37	1	no	—
130	C₂³×C₄	—	37	1	no	—
131	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
132	C₂×M₁₆	—	2	4	no	—
133	C₂×M₁₆	—	37	4	no	—
134	C₂×M₁₆	—	74	4	no	—
135	C₂×M₁₆	C₂×D₃₇	1	4	yes	complemented
136	C₂²×C₈	—	37	1	no	—
137	C₂²×C₈	—	37	1	no	—
138	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
139	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
140	C₄×D₄	—	37	4	no	—
141	C₄×D₄	—	37	4	no	—
142	C₂×C₄⋊C₄	—	37	1	no	—
143	C₂×C₂²⋊C₄	—	37	1	no	—
144	C₂×C₂²⋊C₄	—	37	1	no	—
145	C₂×C₄²	—	37	1	no	—
146	C₄⋊C₈	—	37	4	no	—
147	C₂²⋊C₈	—	37	4	no	—
148	C₂²⋊C₈	—	37	4	no	—
149	C₄×C₈	—	37	4	no	—
150	C₂⁴	—	37	1	no	—
151	C₂⁴	C₄×D₃₇	1	1	yes	characteristic
152	C₂×D₄	—	37	4	no	—
153	C₂×D₄	—	74	4	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	37	1	no	—
157	C₂²×C₄	—	37	1	no	—
158	C₂²×C₄	—	37	1	no	—
159	C₂²×C₄	—	37	2	no	—
160	C₂²×C₄	—	37	2	no	—
161	C₂²×C₄	—	74	2	no	—
162	C₂²×C₄	—	74	2	no	—
163	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
164	C₂²×C₄	C₄×D₃₇	1	2	yes	—
165	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
166	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
167	M₁₆	—	2	8	no	—
168	M₁₆	—	74	8	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	37	2	no	—
171	C₂×C₈	—	37	2	no	—
172	C₂×C₈	—	74	2	no	—
173	C₂×C₈	—	74	2	no	—
174	C₂×C₈	C₂²⋊D₃₇	1	4	yes	complemented
175	C₂×C₈	C₂²×D₃₇	1	2	yes	—
176	C₂×C₈	C₂²×D₃₇	1	2	yes	—
177	C₄⋊C₄	—	37	4	no	—
178	C₂²⋊C₄	—	37	4	no	—
179	C₂²⋊C₄	—	37	4	no	—
180	C₄²	—	37	2	no	—
181	C₄²	—	37	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	37	2	no	—
186	C₂³	—	74	2	no	—
187	C₂³	—	74	2	no	—
188	C₂³	—	74	2	no	—
189	C₂³	C₂×C₄×D₃₇	1	2	yes	—
190	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
191	D₄	—	74	8	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	37	4	no	—
197	C₂×C₄	—	37	2	no	—
198	C₂×C₄	—	37	2	no	—
199	C₂×C₄	—	37	2	no	—
200	C₂×C₄	—	74	2	no	—
201	C₂×C₄	—	74	2	no	—
202	C₂×C₄	—	74	4	no	—
203	C₂×C₄	—	74	2	no	—
204	C₂×C₄	—	74	2	no	—
205	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
207	C₂×C₄	#10596962	1	2	yes	—
208	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
209	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	74	2	no	—
212	C₈	—	74	2	no	—
213	C₈	C₂×C₂²⋊D₃₇	1	4	yes	complemented
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	—	74	2	no	—
219	C₂²	—	74	2	no	—
220	C₂²	—	74	2	no	—
221	C₂²	—	148	4	no	—
222	C₂²	C₄×C₂²⋊D₃₇	1	2	yes	—
223	C₂²	M₁₆×D₃₇	1	2	yes	—
224	C₂²	#11924544	1	2	yes	—
225	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	37	4	no	—
228	C₄	—	74	2	no	—
229	C₄	—	74	2	no	—
230	C₄	C₄×C₂²⋊D₃₇	1	2	yes	—
231	C₄	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
232	C₄	C₂×#10596962	1	1	yes	characteristic
233	C₂	—	4	2	no	—
234	C₂	—	148	2	no	—
235	C₂	#14557609	1	4	yes	direct product factor
236	C₂	C₂×C₄×C₂²⋊D₃₇	1	1	yes	characteristic
237	C₂	C₂×M₁₆×D₃₇	1	1	yes	characteristic
238	C₂	C₂×#11924544	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂²×M₁₆×C₃₇	C₂
socle	C₂³×C₃₇	C₂²×C₄
prime radical	C₂²×C₂²⋊D₃₇	C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₂²×M₁₆	D₃₇
hypocenter	C₃₇	C₂×#436
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#10596929	C₂	semidirect
3. #14557609	C₂	direct (N × H)
4. C₂×#11924465	C₂	semidirect
5. C₂×C₂²⋊#64436	C₂	semidirect
6. C₂×#11924448	C₂	semidirect
7. C₂×C₈×Dic₃₇	C₂	semidirect
8. C₂²×M₁₆×C₃₇	C₂	semidirect
12. C₂×#10596929	C₂²	semidirect
21. #11924465	C₂²	semidirect
22. C₂²⋊#64436	C₂²	semidirect
23. #11924448	C₂²	semidirect
24. C₈×Dic₃₇	C₂²	semidirect
27. C₂×M₁₆×C₃₇	C₂²	semidirect
49. C₂×C₈×C₃₇	D₄	semidirect
83. C₂×Dic₃₇	M₁₆	semidirect
91. C₈×C₃₇	C₂×D₄	semidirect
112. Dic₃₇	C₂×M₁₆	semidirect
115. C₂×C₃₇	#436	semidirect
121. C₂²×M₁₆	D₃₇	semidirect
128. C₃₇	C₂×#436	semidirect
135. C₂×M₁₆	C₂×D₃₇	semidirect
174. C₂×C₈	C₂²⋊D₃₇	semidirect
213. C₈	C₂×C₂²⋊D₃₇	semidirect
235. C₂	#14557609	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×#10596929	2
C₂×C₄×C₂²⋊D₃₇	2
#14557609	2
C₂×#11924465	2
C₂×C₂²⋊#64436	2
C₂×#11924448	2
C₂×C₈×Dic₃₇	2
C₂²×M₁₆×C₃₇	2
C₂×#436	37

Subgroups — C2×#14557609 #18003281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14557609 #18003281