Subgroups — #17997219

Order 4736.

600 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924495	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₃₇	C₂	1	1	yes	maximal, characteristic
3	#14557432	C₂	1	4	yes	maximal, complemented
4	#14557408	C₂	1	4	yes	maximal
5	#14557403	C₂	1	2	yes	maximal, complemented
6	#14557399	C₂	1	2	yes	maximal, complemented
7	C₂×#117×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₃₇	C₂²	1	2	yes	—
9	#11924495	—	2	4	no	—
10	C₂×C₄⋊Dic₃₇	—	2	4	no	—
11	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₃₇	C₂²	1	8	yes	—
16	C₂×C₂²⋊Dic₃₇	C₂²	1	2	yes	—
17	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
18	#11924443	C₂²	1	8	yes	—
19	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
20	#117×C₃₇	C₄	1	8	yes	complemented
21	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₃₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₃₇	C₂²	1	2	yes	—
24	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₃₇	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₃₇	D₄	1	4	yes	—
27	C₂²×C₄×C₃₇	Q₈	1	4	yes	—
28	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
29	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
31	C₂²×Dic₃₇	—	2	4	no	—
32	C₂²×Dic₃₇	—	4	4	no	—
33	C₂²×Dic₃₇	C₂³	1	2	yes	—
34	C₂²×Dic₃₇	C₂³	1	2	yes	—
35	C₂²×Dic₃₇	C₂³	1	4	yes	—
36	C₄⋊C₄×C₃₇	C₂×C₄	1	8	yes	complemented
37	C₂²⋊C₄×C₃₇	C₂×C₄	1	8	yes	—
38	C₄²×C₃₇	C₂×C₄	1	8	yes	complemented
39	C₂²⋊Dic₃₇	—	2	4	no	—
40	C₄⋊Dic₃₇	—	2	4	no	—
41	C₄⋊Dic₃₇	—	2	4	no	—
42	C₄⋊Dic₃₇	—	2	2	no	—
43	C₄⋊Dic₃₇	—	2	2	no	—
44	C₄⋊Dic₃₇	—	4	8	no	—
45	C₄×Dic₃₇	—	2	4	no	—
46	C₂³×C₃₇	—	2	2	no	—
47	C₂³×C₃₇	—	2	2	no	—
48	C₂³×C₃₇	C₂²×C₄	1	2	yes	—
49	C₂³×C₃₇	C₂×D₄	1	2	yes	—
50	C₂³×C₃₇	C₂×Q₈	1	2	yes	—
51	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₃₇	—	2	4	no	—
53	C₂×C₄×C₃₇	—	2	4	no	—
54	C₂×C₄×C₃₇	C₄⋊C₄	1	16	yes	complemented
55	C₂×C₄×C₃₇	C₂²×C₄	1	8	yes	—
56	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
61	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
62	C₂×Dic₃₇	—	2	2	no	—
63	C₂×Dic₃₇	—	2	2	no	—
64	C₂×Dic₃₇	—	2	4	no	—
65	C₂×Dic₃₇	—	2	4	no	—
66	C₂×Dic₃₇	—	2	8	no	—
67	C₂×Dic₃₇	—	2	4	no	—
68	C₂×Dic₃₇	—	4	8	no	—
69	C₂²×C₃₇	—	2	4	no	—
70	C₂²×C₃₇	—	2	2	no	—
71	C₂²×C₃₇	—	2	2	no	—
72	C₂²×C₃₇	—	2	4	no	—
73	C₂²×C₃₇	C₂×C₄⋊C₄	1	4	yes	—
74	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
75	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
76	C₂²×C₃₇	C₂²×Q₈	1	1	yes	characteristic
77	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
78	C₂²×C₃₇	2₋¹⁺⁴	1	2	yes	—
79	C₄×C₃₇	—	2	8	no	—
80	C₄×C₃₇	C₂×C₄⋊C₄	1	4	yes	—
81	C₄×C₃₇	C₂×C₄⋊C₄	1	4	yes	—
82	Dic₃₇	—	4	2	no	—
83	Dic₃₇	—	4	4	no	—
84	Dic₃₇	—	4	2	no	—
85	#2317	—	37	1	no	maximal, Sylow
86	C₂×C₃₇	—	2	4	no	—
87	C₂×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₃₇	#520	1	2	yes	—
89	C₂×C₃₇	#535	1	1	yes	characteristic
90	C₂×C₃₇	#536	1	1	yes	characteristic
91	C₂×C₃₇	#544	1	2	yes	—
92	C₂×#117	—	37	1	no	—
93	C₂×#117	D₃₇	1	1	yes	characteristic
94	C₂²×C₄⋊C₄	—	37	1	no	—
95	#389	—	37	4	no	—
96	#385	—	37	4	no	—
97	#384	—	37	2	no	—
98	#382	—	37	2	no	—
99	C₃₇	#2317	1	1	yes	characteristic, complemented, Sylow
100	C₂³×C₄	—	37	2	no	—
101	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
102	#117	—	74	4	no	—
103	#117	Dic₃₇	1	8	yes	complemented
104	C₂×C₄⋊C₄	—	37	2	no	—
105	C₂×C₄⋊C₄	—	37	2	no	—
106	C₂×C₄⋊C₄	—	37	1	no	—
107	C₂×C₄⋊C₄	—	37	1	no	—
108	C₂×C₄⋊C₄	—	37	8	no	—
109	C₂×C₄⋊C₄	—	74	4	no	—
110	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
111	C₂×C₂²⋊C₄	—	37	2	no	—
112	C₂×C₂²⋊C₄	C₂×D₃₇	1	2	yes	—
113	C₂×C₄²	—	37	2	no	—
114	C₂×C₄²	C₂×D₃₇	1	2	yes	—
115	#95	—	37	8	no	—
116	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
117	C₂²×C₄	—	37	2	no	—
118	C₂²×C₄	—	37	4	no	—
119	C₂²×C₄	—	37	2	no	—
120	C₂²×C₄	—	74	4	no	—
121	C₂²×C₄	—	148	4	no	—
122	C₂²×C₄	Dic₇₄	1	4	yes	—
123	C₂²×C₄	D₁₄₈	1	4	yes	—
124	C₂²×C₄	C₂×Dic₃₇	1	4	yes	—
125	C₂²×C₄	C₂²×D₃₇	1	4	yes	—
126	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
128	C₄⋊C₄	—	74	4	no	—
129	C₄⋊C₄	—	74	4	no	—
130	C₄⋊C₄	—	74	2	no	—
131	C₄⋊C₄	—	74	2	no	—
132	C₄⋊C₄	—	148	8	no	—
133	C₄⋊C₄	C₂×Dic₃₇	1	8	yes	complemented
134	C₂²⋊C₄	—	74	4	no	—
135	C₂²⋊C₄	C₂×Dic₃₇	1	8	yes	—
136	C₄²	—	74	4	no	—
137	C₄²	C₂×Dic₃₇	1	8	yes	complemented
138	C₂³	—	2	2	no	—
139	C₂³	—	2	2	no	—
140	C₂³	C₂×Dic₇₄	1	2	yes	—
141	C₂³	C₂×D₁₄₈	1	2	yes	—
142	C₂³	C₂²×Dic₃₇	1	2	yes	—
143	C₂³	C₂³×D₃₇	1	1	yes	characteristic
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	74	2	no	—
147	C₂×C₄	—	74	2	no	—
148	C₂×C₄	—	74	4	no	—
149	C₂×C₄	—	74	4	no	—
150	C₂×C₄	—	74	8	no	—
151	C₂×C₄	—	74	4	no	—
152	C₂×C₄	—	148	8	no	—
153	C₂×C₄	C₄⋊Dic₃₇	1	16	yes	complemented
154	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
155	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
156	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
157	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
158	C₂×C₄	C₂²×Dic₃₇	1	8	yes	—
159	C₂×C₄	C₂²×Dic₃₇	1	2	yes	—
160	C₂×C₄	C₂²×Dic₃₇	1	2	yes	—
161	C₂²	—	2	4	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	4	no	—
165	C₂²	C₂×C₄⋊Dic₃₇	1	4	yes	—
166	C₂²	C₂³×Dic₃₇	1	1	yes	characteristic
167	C₂²	C₂²×D₁₄₈	1	1	yes	characteristic
168	C₂²	C₂²×Dic₇₄	1	1	yes	characteristic
169	C₂²	#11924582	1	2	yes	—
170	C₂²	#11924583	1	2	yes	—
171	C₄	—	2	8	no	—
172	C₄	—	148	4	no	—
173	C₄	—	148	2	no	—
174	C₄	—	148	2	no	—
175	C₄	C₂×C₄⋊Dic₃₇	1	4	yes	—
176	C₄	C₂×C₄⋊Dic₃₇	1	4	yes	—
177	C₂	—	2	4	no	—
178	C₂	C₂²×C₄⋊Dic₃₇	1	1	yes	characteristic
179	C₂	#14558016	1	2	yes	—
180	C₂	#14558102	1	1	yes	characteristic
181	C₂	#14558109	1	1	yes	characteristic
182	C₂	#14558161	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Fitting	C₂×#117×C₃₇	C₂
socle, cosocle	C₂³×C₃₇	C₂⁴
prime radical	C₂⁴×C₃₇	C₂³
hypercenter	C₂×#117	D₃₇
hypocenter	C₃₇	#2317
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #14557432	C₂	semidirect
5. #14557403	C₂	semidirect
6. #14557399	C₂	semidirect
20. #117×C₃₇	C₄	semidirect
36. C₄⋊C₄×C₃₇	C₂×C₄	semidirect
38. C₄²×C₃₇	C₂×C₄	semidirect
54. C₂×C₄×C₃₇	C₄⋊C₄	semidirect
99. C₃₇	#2317	semidirect
103. #117	Dic₃₇	semidirect
133. C₄⋊C₄	C₂×Dic₃₇	semidirect
137. C₄²	C₂×Dic₃₇	semidirect
153. C₂×C₄	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×#11924495	2
C₂²×C₄⋊Dic₃₇	2
#14557432	2
#14557408	2
#14557403	2
#14557399	2
C₂×#117×C₃₇	2
#2317	37