Subgroups — #18003157

Order 4736.

484 conjugacy classes of subgroups.

The subgroup lattice has 220 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
2	C₂×M₁₆×D₃₇	C₂	1	1	yes	maximal, characteristic
3	#117×D₃₇	C₂	1	1	yes	maximal, characteristic
4	#14557561	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557560	C₂	1	1	yes	maximal, characteristic, complemented
6	#14557559	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557558	C₂	1	4	yes	maximal, complemented
8	#14557555	C₂	1	4	yes	maximal, complemented
9	C₃₇×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
11	C₂×#10596934	C₂²	1	1	yes	characteristic
12	C₂×#10596929	—	2	4	no	—
13	C₂×#10596929	C₂²	1	2	yes	—
14	C₂×#10596929	C₂²	1	2	yes	—
15	C₂×#10596929	C₂²	1	2	yes	—
16	M₁₆×D₃₇	—	2	4	no	—
17	C₂²×#64436	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₃₇	C₂²	1	2	yes	—
19	#11924495	C₂²	1	1	yes	characteristic
20	#11924494	C₄	1	2	yes	—
21	#11924493	C₂²	1	2	yes	—
22	#11924492	C₄	1	2	yes	—
23	C₄⋊C₄×D₃₇	C₄	1	2	yes	—
24	C₂²⋊C₄×D₃₇	C₄	1	2	yes	—
25	C₄²×D₃₇	C₂²	1	2	yes	—
26	C₄⋊#64436	C₂²	1	4	yes	complemented
27	#11924465	C₂²	1	4	yes	complemented
28	#11924465	C₂²	1	4	yes	complemented
29	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
30	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
31	#117×C₃₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₃₇	C₂²	1	4	yes	complemented
33	C₂³×D₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
35	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×D₃₇	C₂×C₄	1	4	yes	—
37	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₃₇	D₄	1	4	yes	—
40	C₂×C₄×D₃₇	Q₈	1	4	yes	—
41	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
43	M₁₆×C₃₇	—	2	2	no	—
44	C₂×C₈×C₃₇	—	2	2	no	—
45	C₂×C₈×C₃₇	C₂³	1	2	yes	—
46	C₂×C₈×C₃₇	C₂³	1	2	yes	—
47	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
48	C₂²⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
49	C₄²×C₃₇	C₂³	1	2	yes	—
50	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
51	#10596938	C₂×C₄	1	4	yes	—
52	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
53	#10596936	C₂×C₄	1	4	yes	—
54	C₄×Dic₃₇	C₂×C₄	1	4	yes	—
55	C₄×Dic₃₇	C₂³	1	2	yes	—
56	#10596934	—	2	2	no	—
57	C₂×#64436	—	2	2	no	—
58	C₂×#64436	C₂³	1	2	yes	—
59	C₂×#64436	C₂³	1	2	yes	—
60	#10596929	—	2	4	no	—
61	#10596929	—	2	4	no	—
62	#10596929	—	2	4	no	—
63	C₈×D₃₇	—	2	4	no	—
64	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×D₃₇	—	4	2	no	—
66	C₂²×D₃₇	C₄⋊C₄	1	4	yes	—
67	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
70	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₃₇	C₄⋊C₄	1	4	yes	—
75	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
76	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₄×D₃₇	—	4	4	no	—
79	C₄×D₃₇	C₄⋊C₄	1	4	yes	—
80	C₄×D₃₇	C₄⋊C₄	1	4	yes	—
81	C₄×D₃₇	C₂×D₄	1	2	yes	—
82	C₄×D₃₇	C₂×D₄	1	2	yes	—
83	C₄×D₃₇	C₂×Q₈	1	2	yes	—
84	C₄×D₃₇	C₂×Q₈	1	2	yes	—
85	C₈×C₃₇	—	2	2	no	—
86	C₈×C₃₇	—	2	2	no	—
87	#64436	—	2	2	no	—
88	#64436	—	2	2	no	—
89	C₂²×C₃₇	—	2	2	no	—
90	C₂²×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×D₃₇	—	2	2	no	—
93	C₂×D₃₇	—	2	2	no	—
94	C₂×D₃₇	—	4	4	no	—
95	C₂×D₃₇	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×D₃₇	C₂×C₄⋊C₄	1	2	yes	—
97	C₄×C₃₇	—	2	4	no	—
98	C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
100	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
101	Dic₃₇	—	2	4	no	—
102	Dic₃₇	C₂×C₄⋊C₄	1	2	yes	—
103	Dic₃₇	C₂×C₄⋊C₄	1	2	yes	—
104	#2905	—	37	1	no	maximal, Sylow
105	C₂×C₃₇	—	2	2	no	—
106	C₂×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₃₇	#568	1	2	yes	—
108	D₃₇	—	2	2	no	—
109	D₃₇	—	2	2	no	—
110	C₂²×M₁₆	—	37	1	no	—
111	C₂²×M₁₆	—	37	1	no	—
112	C₂×#117	—	37	1	no	—
113	#424	—	37	1	no	—
114	#424	—	37	1	no	—
115	#424	—	37	1	no	—
116	#424	—	37	4	no	—
117	#424	D₃₇	1	1	yes	characteristic, complemented
118	#423	—	37	4	no	—
119	C₃₇	#2905	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₄	—	37	1	no	—
121	C₂×M₁₆	—	37	1	no	—
122	C₂×M₁₆	—	37	2	no	—
123	C₂×M₁₆	—	37	2	no	—
124	C₂×M₁₆	—	37	2	no	—
125	C₂×M₁₆	—	74	4	no	—
126	C₂×M₁₆	—	74	4	no	—
127	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
128	C₂²×C₈	—	37	1	no	—
129	C₂²×C₈	—	37	2	no	—
130	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
131	#117	—	37	1	no	—
132	#117	—	37	2	no	—
133	#117	—	37	2	no	—
134	#117	—	37	2	no	—
135	#117	C₂×D₃₇	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	37	2	no	—
137	C₂×C₂²⋊C₄	—	37	2	no	—
138	C₂×C₄²	—	37	2	no	—
139	C₄⋊C₈	—	37	4	no	—
140	C₄⋊C₈	—	37	4	no	—
141	C₄⋊C₈	—	37	4	no	—
142	C₄⋊C₈	C₂×D₃₇	1	4	yes	complemented
143	C₂⁴	—	37	1	no	—
144	C₂²×C₄	—	37	1	no	—
145	C₂²×C₄	—	37	1	no	—
146	C₂²×C₄	—	37	1	no	—
147	C₂²×C₄	—	37	4	no	—
148	C₂²×C₄	—	37	4	no	—
149	C₂²×C₄	—	37	1	no	—
150	C₂²×C₄	—	37	1	no	—
151	C₂²×C₄	—	37	4	no	—
152	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
153	M₁₆	—	2	2	no	—
154	M₁₆	—	74	4	no	—
155	M₁₆	—	74	4	no	—
156	M₁₆	—	74	4	no	—
157	M₁₆	—	74	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	37	2	no	—
160	C₂×C₈	—	37	2	no	—
161	C₂×C₈	—	74	4	no	—
162	C₂×C₈	—	74	2	no	—
163	C₂×C₈	C₂²×D₃₇	1	2	yes	—
164	C₂×C₈	C₂²×D₃₇	1	2	yes	—
165	C₄⋊C₄	—	37	4	no	—
166	C₄⋊C₄	—	37	2	no	—
167	C₄⋊C₄	C₄×D₃₇	1	2	yes	—
168	C₂²⋊C₄	—	37	2	no	—
169	C₂²⋊C₄	—	37	4	no	—
170	C₂²⋊C₄	C₄×D₃₇	1	2	yes	—
171	C₄²	—	37	4	no	—
172	C₄²	—	37	2	no	—
173	C₄²	C₂²×D₃₇	1	2	yes	—
174	C₂³	—	37	1	no	—
175	C₂³	—	37	1	no	—
176	C₂³	—	37	4	no	—
177	C₂³	—	148	2	no	—
178	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
179	C₂×C₄	—	37	4	no	—
180	C₂×C₄	—	37	1	no	—
181	C₂×C₄	—	37	1	no	—
182	C₂×C₄	—	37	4	no	—
183	C₂×C₄	—	37	4	no	—
184	C₂×C₄	—	37	2	no	—
185	C₂×C₄	—	37	2	no	—
186	C₂×C₄	—	37	2	no	—
187	C₂×C₄	—	37	2	no	—
188	C₂×C₄	—	37	4	no	—
189	C₂×C₄	—	148	4	no	—
190	C₂×C₄	C₂×C₄×D₃₇	1	4	yes	—
191	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
192	C₂×C₄	D₄×D₃₇	1	2	yes	—
193	C₂×C₄	Q₈×D₃₇	1	2	yes	—
194	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	2	2	no	—
197	C₈	—	74	2	no	—
198	C₈	—	74	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	37	2	no	—
201	C₂²	—	37	2	no	—
202	C₂²	—	74	2	no	—
203	C₂²	—	74	2	no	—
204	C₂²	—	148	4	no	—
205	C₂²	C₄⋊C₄×D₃₇	1	2	yes	—
206	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	37	2	no	—
209	C₄	—	37	2	no	—
210	C₄	—	74	4	no	—
211	C₄	C₄⋊C₄×D₃₇	1	2	yes	—
212	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
213	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
214	C₂	—	2	2	no	—
215	C₂	—	74	2	no	—
216	C₂	—	74	2	no	—
217	C₂	C₂×C₄⋊C₄×D₃₇	1	1	yes	characteristic
218	C₂	#14558298	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₃₇×#424	C₂
prime radical	C₂³×D₃₇	C₂×C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#424	D₃₇
hypocenter	C₃₇	#2905
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
4. #14557561	C₂	semidirect
5. #14557560	C₂	semidirect
6. #14557559	C₂	semidirect
7. #14557558	C₂	semidirect
8. #14557555	C₂	semidirect
9. C₃₇×#424	C₂	semidirect
26. C₄⋊#64436	C₂²	semidirect
27. #11924465	C₂²	semidirect
28. #11924465	C₂²	semidirect
32. C₄⋊C₈×C₃₇	C₂²	semidirect
117. #424	D₃₇	semidirect
119. C₃₇	#2905	semidirect
142. C₄⋊C₈	C₂×D₃₇	semidirect

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×M₁₆×D₃₇	2
#117×D₃₇	2
#14557561	2
#14557560	2
#14557559	2
#14557558	2
#14557555	2
C₃₇×#424	2
#2905	37