Subgroups — C₂×#14558499 #18011158

Order 4736.

592 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924615	C₂	1	1	yes	maximal, characteristic
3	#14558499	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11924590	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#3866	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆×D₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂×D₃₇⋊C₈	C₄	1	4	yes	complemented
8	C₂×D₃₇⋊C₈	C₄	1	2	yes	complemented
9	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×D₃₇⋊C₈	C₂²	1	4	yes	—
12	C₂²×#64438	C₄	1	2	yes	complemented
13	#11924615	C₄	1	4	yes	—
14	#11924615	C₂²	1	4	yes	—
15	C₂×C₄⋊#3866	C₄	1	2	yes	—
16	C₂×C₂²⋊#3866	C₄	1	2	yes	—
17	C₂×C₄×#3866	C₂²	1	2	yes	—
18	#11924590	C₂²	1	8	yes	complemented
19	C₈×#3866	C₂²	1	8	yes	complemented
20	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
21	C₂×#10596934	C₄	1	1	yes	characteristic, complemented
22	C₂×#10596929	C₄	1	2	yes	complemented
23	M₁₆×D₃₇	C₄	1	4	yes	complemented
24	M₁₆×D₃₇	C₂²	1	4	yes	—
25	C₂×C₈×D₃₇	C₂²	1	2	yes	—
26	C₂×M₁₆×C₃₇	C₄	1	1	yes	characteristic, complemented
27	C₂³×D₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×#3866	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
33	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×D₃₇	C₂³	1	2	yes	—
39	C₂²⋊#3866	C₂×C₄	1	8	yes	—
40	C₂×#64438	C₂×C₄	1	2	yes	—
41	C₂×#64438	C₂×C₄	1	2	yes	—
42	C₂×#64438	C₂×C₄	1	8	yes	complemented
43	C₄⋊#3866	C₂×C₄	1	8	yes	—
44	C₄×#3866	C₂×C₄	1	4	yes	—
45	C₄×#3866	C₂³	1	4	yes	—
46	D₃₇⋊C₈	C₂×C₄	1	2	yes	—
47	D₃₇⋊C₈	C₂×C₄	1	2	yes	—
48	D₃₇⋊C₈	C₂×C₄	1	8	yes	complemented
49	D₃₇⋊C₈	C₂³	1	2	yes	—
50	D₃₇⋊C₈	C₂³	1	2	yes	—
51	M₁₆×C₃₇	C₂×C₄	1	4	yes	complemented
52	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
53	#10596934	C₂×C₄	1	4	yes	complemented
54	C₂×#64436	C₂×C₄	1	2	yes	—
55	#10596929	C₂×C₄	1	8	yes	complemented
56	C₈×D₃₇	C₂×C₄	1	4	yes	—
57	C₈×D₃₇	C₂³	1	4	yes	—
58	C₂³×C₃₇	C₄²	1	1	yes	characteristic
59	C₂²×D₃₇	—	2	4	no	—
60	C₂²×D₃₇	C₄²	1	1	yes	characteristic
61	C₂²×D₃₇	C₄²	1	2	yes	—
62	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
63	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×#3866	—	2	8	no	—
65	C₂×#3866	C₂²×C₄	1	8	yes	—
66	C₂×C₄×C₃₇	C₄²	1	2	yes	—
67	C₂×C₄×C₃₇	C₄²	1	1	yes	characteristic
68	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×Dic₃₇	C₄²	1	2	yes	—
71	C₂×Dic₃₇	C₄²	1	1	yes	characteristic
72	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
74	C₄×D₃₇	C₄²	1	2	yes	—
75	C₄×D₃₇	C₄²	1	2	yes	—
76	C₄×D₃₇	C₄²	1	2	yes	—
77	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₄×D₃₇	C₂²×C₄	1	2	yes	—
79	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
80	C₄×D₃₇	C₂²×C₄	1	2	yes	—
81	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
82	C₄×D₃₇	C₂²×C₄	1	2	yes	—
83	C₄×D₃₇	C₂⁴	1	1	yes	characteristic
84	#64438	C₂²×C₄	1	4	yes	—
85	#64438	C₂²×C₄	1	4	yes	—
86	C₈×C₃₇	C₂²×C₄	1	4	yes	—
87	#64436	C₂²×C₄	1	4	yes	—
88	C₂²×C₃₇	—	2	2	no	—
89	C₂²×C₃₇	C₂×C₄²	1	2	yes	—
90	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
91	C₂×D₃₇	—	2	2	no	—
92	C₂×D₃₇	—	2	4	no	—
93	C₂×D₃₇	—	2	4	no	—
94	C₂×D₃₇	C₂×C₄²	1	2	yes	—
95	C₂×D₃₇	C₂×C₄²	1	1	yes	characteristic
96	C₂×D₃₇	#131	1	4	yes	—
97	C₂×D₃₇	C₂³×C₄	1	1	yes	characteristic
98	#3866	—	2	8	no	—
99	C₄×C₃₇	C₂×C₄²	1	2	yes	—
100	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
101	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
102	Dic₃₇	C₂×C₄²	1	1	yes	characteristic
103	Dic₃₇	C₂×C₄²	1	2	yes	—
104	Dic₃₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×#405	—	37	1	no	maximal, Sylow
106	C₂×C₃₇	—	2	2	no	—
107	C₂×C₃₇	#405	1	2	yes	complemented
108	C₂×C₃₇	C₂²×C₄²	1	1	yes	characteristic
109	D₃₇	—	2	2	no	—
110	D₃₇	#405	1	2	yes	complemented
111	D₃₇	C₂×#131	1	2	yes	—
112	C₂²×M₁₆	—	37	1	no	—
113	C₂³×C₈	—	37	1	no	—
114	C₂×#117	—	37	1	no	—
115	#405	—	37	8	no	—
116	C₂×#97	—	37	2	no	—
117	C₂×C₄×C₈	—	37	2	no	—
118	C₃₇	C₂×#405	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₄	—	37	1	no	—
120	C₂×M₁₆	—	37	1	no	—
121	C₂×M₁₆	—	37	4	no	—
122	C₂×M₁₆	—	37	4	no	—
123	C₂×M₁₆	—	37	2	no	—
124	C₂×M₁₆	#3866	1	1	yes	characteristic, complemented
125	C₂²×C₈	—	37	2	no	—
126	C₂²×C₈	—	37	4	no	—
127	C₂²×C₈	—	37	2	no	—
128	C₂²×C₈	—	37	1	no	—
129	C₂²×C₈	—	37	1	no	—
130	C₂²×C₈	—	37	4	no	—
131	C₂²×C₈	—	37	2	no	—
132	#117	—	37	4	no	—
133	#117	—	37	4	no	—
134	C₂×C₄⋊C₄	—	37	2	no	—
135	C₂×C₂²⋊C₄	—	37	2	no	—
136	C₂×C₄²	—	37	2	no	—
137	#97	—	37	8	no	—
138	C₄×C₈	—	37	8	no	—
139	C₂⁴	—	37	1	no	—
140	C₂²×C₄	—	37	1	no	—
141	C₂²×C₄	—	37	1	no	—
142	C₂²×C₄	—	37	2	no	—
143	C₂²×C₄	—	37	2	no	—
144	C₂²×C₄	—	37	1	no	—
145	C₂²×C₄	—	37	2	no	—
146	C₂²×C₄	—	37	2	no	—
147	C₂²×C₄	—	37	1	no	—
148	C₂²×C₄	—	37	1	no	—
149	C₂²×C₄	—	37	4	no	—
150	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
151	M₁₆	—	37	4	no	—
152	M₁₆	—	37	8	no	—
153	M₁₆	C₂×#3866	1	4	yes	complemented
154	C₂×C₈	—	37	2	no	—
155	C₂×C₈	—	37	2	no	—
156	C₂×C₈	—	37	8	no	—
157	C₂×C₈	—	37	2	no	—
158	C₂×C₈	—	37	2	no	—
159	C₂×C₈	—	37	2	no	—
160	C₂×C₈	—	37	2	no	—
161	C₂×C₈	—	37	8	no	—
162	C₂×C₈	—	37	2	no	—
163	C₂×C₈	—	37	4	no	—
164	C₂×C₈	—	37	4	no	—
165	C₂×C₈	C₂×#3866	1	2	yes	—
166	C₄⋊C₄	—	37	8	no	—
167	C₂²⋊C₄	—	37	8	no	—
168	C₄²	—	37	4	no	—
169	C₄²	—	37	4	no	—
170	C₂³	—	37	1	no	—
171	C₂³	—	37	2	no	—
172	C₂³	—	37	2	no	—
173	C₂³	—	37	1	no	—
174	C₂³	—	74	4	no	—
175	C₂³	C₄×#3866	1	1	yes	characteristic
176	C₂×C₄	—	37	2	no	—
177	C₂×C₄	—	37	1	no	—
178	C₂×C₄	—	37	2	no	—
179	C₂×C₄	—	37	1	no	—
180	C₂×C₄	—	37	1	no	—
181	C₂×C₄	—	37	2	no	—
182	C₂×C₄	—	37	2	no	—
183	C₂×C₄	—	37	1	no	—
184	C₂×C₄	—	37	2	no	—
185	C₂×C₄	—	37	2	no	—
186	C₂×C₄	—	37	2	no	—
187	C₂×C₄	—	37	1	no	—
188	C₂×C₄	—	37	1	no	—
189	C₂×C₄	—	37	2	no	—
190	C₂×C₄	—	37	8	no	—
191	C₂×C₄	—	74	8	no	—
192	C₂×C₄	C₄×#3866	1	1	yes	characteristic
193	C₂×C₄	C₄×#3866	1	2	yes	—
194	C₂×C₄	C₂²×#3866	1	2	yes	—
195	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
196	C₈	—	37	4	no	—
197	C₈	—	37	4	no	—
198	C₈	—	37	4	no	—
199	C₈	C₂²×#3866	1	4	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	—	37	4	no	—
202	C₂²	—	37	1	no	—
203	C₂²	—	37	1	no	—
204	C₂²	—	37	2	no	—
205	C₂²	—	74	4	no	—
206	C₂²	—	74	4	no	—
207	C₂²	—	74	2	no	—
208	C₂²	C₂×C₄×#3866	1	2	yes	—
209	C₂²	C₂×C₄×#3866	1	1	yes	characteristic
210	C₄	—	37	1	no	—
211	C₄	—	37	1	no	—
212	C₄	—	37	2	no	—
213	C₄	—	74	8	no	—
214	C₄	C₂×C₄×#3866	1	2	yes	—
215	C₄	C₂×C₄×#3866	1	1	yes	characteristic
216	C₄	C₂³×#3866	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	37	2	no	—
219	C₂	—	37	2	no	—
220	C₂	—	74	2	no	—
221	C₂	#14558499	1	2	yes	direct product factor
222	C₂	C₂²×C₄×#3866	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#3866
commutator	C₂×C₃₇	C₂²×C₄²
Frattini	C₄	C₂³×#3866
Fitting	C₂×M₁₆×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₄²
prime radical	C₂³×D₃₇	C₂×C₄
cosocle	C₄×D₃₇	C₂⁴
hypercenter	C₂×M₁₆	#3866
hypocenter	C₃₇	C₂×#405
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14558499	C₂	direct (N × H)
4. C₂×#11924590	C₂	semidirect
5. C₂×C₈×#3866	C₂	semidirect
7. C₂×D₃₇⋊C₈	C₄	semidirect
8. C₂×D₃₇⋊C₈	C₄	semidirect
12. C₂²×#64438	C₄	semidirect
18. #11924590	C₂²	semidirect
19. C₈×#3866	C₂²	semidirect
21. C₂×#10596934	C₄	semidirect
22. C₂×#10596929	C₄	semidirect
23. M₁₆×D₃₇	C₄	semidirect
26. C₂×M₁₆×C₃₇	C₄	semidirect
42. C₂×#64438	C₂×C₄	semidirect
48. D₃₇⋊C₈	C₂×C₄	semidirect
51. M₁₆×C₃₇	C₂×C₄	semidirect
53. #10596934	C₂×C₄	semidirect
55. #10596929	C₂×C₄	semidirect
107. C₂×C₃₇	#405	semidirect
110. D₃₇	#405	semidirect
118. C₃₇	C₂×#405	semidirect
124. C₂×M₁₆	#3866	semidirect
153. M₁₆	C₂×#3866	semidirect
221. C₂	#14558499	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×D₃₇⋊C₈	2
C₂×#11924615	2
#14558499	2
C₂×#11924590	2
C₂×C₈×#3866	2
C₂×M₁₆×D₃₇	2
C₂×#405	37

Subgroups — C2×#14558499 #18011158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558499 #18011158