Subgroups — #17997506

Order 4736.

704 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	C₂×D₄×Dic₃₇	C₂	1	2	yes	maximal
2	#14557432	C₂	1	2	yes	maximal, complemented
3	#14557408	C₂	1	4	yes	maximal, complemented
4	#14557399	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557393	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11924443	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₃₇	C₂²	1	4	yes	—
10	D₄×Dic₃₇	—	2	8	no	—
11	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₃₇	C₂²	1	4	yes	—
14	C₂×C₂²⋊Dic₃₇	C₂²	1	4	yes	—
15	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
16	#11924443	C₂²	1	1	yes	characteristic
17	#11924443	C₂²	1	2	yes	—
18	#11924443	C₂²	1	4	yes	—
19	#11924443	C₂²	1	1	yes	characteristic
20	#11924443	C₂²	1	2	yes	—
21	C₂²×D₄×C₃₇	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₃₇	C₂²	1	2	yes	—
23	C₄×D₄×C₃₇	C₄	1	8	yes	complemented
24	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
27	C₂⁴×C₃₇	C₂³	1	2	yes	—
28	C₂×D₄×C₃₇	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₃₇	D₄	1	4	yes	—
30	C₂×D₄×C₃₇	Q₈	1	4	yes	—
31	C₂²×C₄×C₃₇	—	4	4	no	—
32	C₂²×C₄×C₃₇	C₂×C₄	1	8	yes	complemented
33	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
37	C₂²×Dic₃₇	—	2	8	no	—
38	C₂²×Dic₃₇	—	2	8	no	—
39	C₂²×Dic₃₇	—	4	4	no	—
40	C₂²×Dic₃₇	C₂³	1	2	yes	—
41	C₂²×Dic₃₇	C₂³	1	4	yes	—
42	C₂²×Dic₃₇	C₂³	1	2	yes	—
43	C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	complemented
44	C₂²⋊C₄×C₃₇	C₂×C₄	1	8	yes	complemented
45	C₄²×C₃₇	C₂×C₄	1	4	yes	complemented
46	C₂²⋊Dic₃₇	—	2	8	no	—
47	C₄⋊Dic₃₇	—	2	4	no	—
48	C₄⋊Dic₃₇	—	2	4	no	—
49	C₄⋊Dic₃₇	—	4	4	no	—
50	C₄×Dic₃₇	—	2	4	no	—
51	C₂³×C₃₇	—	2	4	no	—
52	C₂³×C₃₇	—	2	4	no	—
53	C₂³×C₃₇	C₂²×C₄	1	4	yes	—
54	C₂³×C₃₇	C₂×D₄	1	4	yes	—
55	C₂³×C₃₇	C₂×Q₈	1	4	yes	—
56	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
57	D₄×C₃₇	C₄⋊C₄	1	16	yes	complemented
58	C₂×C₄×C₃₇	—	2	2	no	—
59	C₂×C₄×C₃₇	—	2	4	no	—
60	C₂×C₄×C₃₇	—	2	2	no	—
61	C₂×C₄×C₃₇	—	4	8	no	—
62	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
65	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
68	C₂×Dic₃₇	—	2	2	no	—
69	C₂×Dic₃₇	—	2	2	no	—
70	C₂×Dic₃₇	—	2	2	no	—
71	C₂×Dic₃₇	—	2	4	no	—
72	C₂×Dic₃₇	—	2	2	no	—
73	C₂×Dic₃₇	—	2	4	no	—
74	C₂×Dic₃₇	—	2	2	no	—
75	C₂×Dic₃₇	—	2	4	no	—
76	C₂×Dic₃₇	—	4	16	no	—
77	C₂×Dic₃₇	#41	1	4	yes	—
78	C₂²×C₃₇	—	2	4	no	—
79	C₂²×C₃₇	—	2	4	no	—
80	C₂²×C₃₇	—	2	8	no	—
81	C₂²×C₃₇	—	2	8	no	—
82	C₂²×C₃₇	C₂×C₄⋊C₄	1	8	yes	—
83	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
84	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₃₇	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₃₇	C₂×#41	1	2	yes	—
87	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
88	C₄×C₃₇	—	2	2	no	—
89	C₄×C₃₇	—	2	2	no	—
90	C₄×C₃₇	—	2	4	no	—
91	C₄×C₃₇	C₂×C₄⋊C₄	1	4	yes	—
92	Dic₃₇	—	2	4	no	—
93	Dic₃₇	—	4	2	no	—
94	Dic₃₇	—	4	4	no	—
95	#2349	—	37	1	no	maximal, Sylow
96	C₂×C₃₇	—	2	8	no	—
97	C₂×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₃₇	C₄×#41	1	1	yes	characteristic
99	C₂×C₃₇	#518	1	1	yes	characteristic
100	C₂×C₃₇	#546	1	2	yes	—
101	C₂×C₃₇	#554	1	2	yes	—
102	C₂×C₄×D₄	—	37	2	no	—
103	C₂×C₄×D₄	D₃₇	1	1	yes	characteristic
104	#389	—	37	2	no	—
105	#385	—	37	4	no	—
106	#382	—	37	1	no	—
107	#380	—	37	2	no	—
108	C₄×C₄⋊C₄	—	37	1	no	—
109	C₂×#95	—	37	2	no	—
110	C₃₇	#2349	1	1	yes	characteristic, complemented, Sylow
111	C₂²×D₄	C₂×D₃₇	1	1	yes	characteristic
112	C₂³×C₄	—	37	4	no	—
113	C₂³×C₄	C₂×D₃₇	1	2	yes	—
114	C₄×D₄	—	74	8	no	—
115	C₄×D₄	Dic₃₇	1	8	yes	complemented
116	C₂×C₄⋊C₄	—	37	2	no	—
117	C₂×C₄⋊C₄	—	37	2	no	—
118	C₂×C₄⋊C₄	—	37	4	no	—
119	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
120	C₂×C₂²⋊C₄	—	37	4	no	—
121	C₂×C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
122	C₂×C₄²	—	37	2	no	—
123	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
124	#95	—	37	1	no	—
125	#95	—	37	2	no	—
126	#95	—	37	4	no	—
127	#95	—	37	1	no	—
128	#95	—	37	2	no	—
129	C₂⁴	C₂²×D₃₇	1	2	yes	—
130	C₂×D₄	Dic₇₄	1	4	yes	—
131	C₂×D₄	D₁₄₈	1	4	yes	—
132	C₂×D₄	C₂×Dic₃₇	1	4	yes	—
133	C₂²×C₄	—	4	4	no	—
134	C₂²×C₄	—	37	2	no	—
135	C₂²×C₄	—	37	4	no	—
136	C₂²×C₄	—	37	2	no	—
137	C₂²×C₄	—	74	8	no	—
138	C₂²×C₄	—	74	8	no	—
139	C₂²×C₄	—	148	4	no	—
140	C₂²×C₄	C₂×Dic₃₇	1	8	yes	complemented
141	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
144	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
145	C₄⋊C₄	—	74	4	no	—
146	C₄⋊C₄	—	74	4	no	—
147	C₄⋊C₄	—	148	4	no	—
148	C₄⋊C₄	C₂×Dic₃₇	1	4	yes	complemented
149	C₂²⋊C₄	—	74	8	no	—
150	C₂²⋊C₄	C₂×Dic₃₇	1	8	yes	complemented
151	C₄²	—	74	4	no	—
152	C₄²	C₂×Dic₃₇	1	4	yes	complemented
153	C₂³	—	2	4	no	—
154	C₂³	—	2	4	no	—
155	C₂³	C₂×Dic₇₄	1	4	yes	—
156	C₂³	C₂×D₁₄₈	1	4	yes	—
157	C₂³	C₂²×Dic₃₇	1	4	yes	—
158	C₂³	C₂³×D₃₇	1	1	yes	characteristic
159	D₄	C₄⋊Dic₃₇	1	16	yes	complemented
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	—	37	4	no	—
165	C₂×C₄	—	74	2	no	—
166	C₂×C₄	—	74	2	no	—
167	C₂×C₄	—	74	2	no	—
168	C₂×C₄	—	74	4	no	—
169	C₂×C₄	—	74	2	no	—
170	C₂×C₄	—	74	4	no	—
171	C₂×C₄	—	74	2	no	—
172	C₂×C₄	—	74	4	no	—
173	C₂×C₄	—	148	16	no	—
174	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
175	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
176	C₂×C₄	C₂²×Dic₃₇	1	2	yes	—
177	C₂×C₄	C₂²×Dic₃₇	1	2	yes	—
178	C₂×C₄	C₂²×Dic₃₇	1	4	yes	—
179	C₂×C₄	C₂²×Dic₃₇	1	2	yes	—
180	C₂²	—	2	8	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	8	no	—
184	C₂²	C₂×C₄⋊Dic₃₇	1	8	yes	—
185	C₂²	C₂³×Dic₃₇	1	1	yes	characteristic
186	C₂²	C₂²×D₁₄₈	1	1	yes	characteristic
187	C₂²	C₂²×Dic₇₄	1	1	yes	characteristic
188	C₂²	#41×D₃₇	1	2	yes	—
189	C₂²	#11924581	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	4	no	—
193	C₄	—	74	4	no	—
194	C₄	—	148	4	no	—
195	C₄	—	148	2	no	—
196	C₄	C₂×C₄⋊Dic₃₇	1	4	yes	—
197	C₂	—	2	8	no	—
198	C₂	C₂²×C₄⋊Dic₃₇	1	1	yes	characteristic
199	C₂	#41×Dic₃₇	1	1	yes	characteristic
200	C₂	#14558008	1	1	yes	characteristic
201	C₂	#14558178	1	2	yes	—
202	C₂	#14558242	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Fitting	C₂×C₄×D₄×C₃₇	C₂
socle, cosocle	C₂³×C₃₇	C₂⁴
prime radical	C₂²×D₄×C₃₇	C₂²
hypercenter	C₂×C₄×D₄	D₃₇
hypocenter	C₃₇	#2349
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #14557432	C₂	semidirect
3. #14557408	C₂	semidirect
4. #14557399	C₂	semidirect
5. #14557393	C₂	semidirect
6. C₄×C₄⋊Dic₃₇	C₂	semidirect
7. C₂×#11924443	C₂	semidirect
23. C₄×D₄×C₃₇	C₄	semidirect
32. C₂²×C₄×C₃₇	C₂×C₄	semidirect
43. C₄⋊C₄×C₃₇	C₂×C₄	semidirect
44. C₂²⋊C₄×C₃₇	C₂×C₄	semidirect
45. C₄²×C₃₇	C₂×C₄	semidirect
57. D₄×C₃₇	C₄⋊C₄	semidirect
110. C₃₇	#2349	semidirect
115. C₄×D₄	Dic₃₇	semidirect
140. C₂²×C₄	C₂×Dic₃₇	semidirect
148. C₄⋊C₄	C₂×Dic₃₇	semidirect
150. C₂²⋊C₄	C₂×Dic₃₇	semidirect
152. C₄²	C₂×Dic₃₇	semidirect
159. D₄	C₄⋊Dic₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Dic₃₇	2
#14557432	2
#14557408	2
#14557399	2
#14557393	2
C₄×C₄⋊Dic₃₇	2
C₂×#11924443	2
C₂×C₄×D₄×C₃₇	2
#2349	37