Subgroups — C₃₇×#28825 #22650078

Order 9472.

634 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3164	C₂	1	4	yes	maximal
2	C₃₇×#2060	C₂	1	4	yes	maximal
3	C₃₇×#1952	C₂	1	4	yes	maximal
4	C₃₇×#1845	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#1772	C₂	1	2	yes	maximal
6	C₃₇×#544	C₂²	1	4	yes	—
7	C₃₇×#501	—	2	8	no	—
8	C₃₇×#500	—	2	4	no	—
9	C₃₇×#499	—	2	4	no	—
10	C₂×#108×C₃₇	C₂²	1	2	yes	—
11	C₃₇×#428	C₂²	1	8	yes	—
12	C₃₇×#427	C₂²	1	2	yes	—
13	C₂×#107×C₃₇	C₂²	1	4	yes	—
14	C₃₇×#405	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
16	C₃₇×#384	C₂²	1	2	yes	—
17	C₃₇×#343	C₂²	1	4	yes	—
18	C₃₇×#339	C₂²	1	4	yes	—
19	C₂×M₁₆×C₃₇	C₂³	1	4	yes	—
20	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
21	C₄⋊Q₈×C₃₇	—	2	8	no	—
22	#125×C₃₇	—	2	8	no	—
23	C₂²⋊Q₈×C₃₇	—	4	4	no	—
24	#117×C₃₇	C₂³	1	4	yes	—
25	C₂×C₄⋊C₄×C₃₇	—	4	2	no	—
26	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
27	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
28	#108×C₃₇	Q₈	1	8	yes	complemented
29	#107×C₃₇	—	2	4	no	—
30	#107×C₃₇	—	2	4	no	—
31	#107×C₃₇	—	2	8	no	—
32	#107×C₃₇	D₄	1	2	yes	—
33	#107×C₃₇	D₄	1	2	yes	—
34	#106×C₃₇	—	2	8	no	—
35	#106×C₃₇	D₄	1	4	yes	—
36	#97×C₃₇	—	2	4	no	—
37	C₄×C₈×C₃₇	—	2	4	no	—
38	C₄×C₈×C₃₇	Q₈	1	4	yes	complemented
39	C₂×Q₈×C₃₇	—	4	4	no	—
40	C₂²×C₄×C₃₇	—	2	1	no	—
41	C₂²×C₄×C₃₇	—	2	4	no	—
42	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
43	M₁₆×C₃₇	—	2	4	no	—
44	M₁₆×C₃₇	C₂×Q₈	1	8	yes	—
45	C₂×C₈×C₃₇	—	2	4	no	—
46	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
49	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
50	C₄⋊C₄×C₃₇	—	2	8	no	—
51	C₄⋊C₄×C₃₇	—	4	4	no	—
52	C₄⋊C₄×C₃₇	—	4	8	no	—
53	C₄⋊C₄×C₃₇	—	8	4	no	—
54	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
55	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
56	C₂²⋊C₄×C₃₇	—	2	4	no	—
57	C₄²×C₃₇	—	2	4	no	—
58	C₄²×C₃₇	—	2	1	no	—
59	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
60	C₂³×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
61	Q₈×C₃₇	—	16	4	no	—
62	C₂×C₄×C₃₇	—	2	2	no	—
63	C₂×C₄×C₃₇	—	2	2	no	—
64	C₂×C₄×C₃₇	—	2	2	no	—
65	C₂×C₄×C₃₇	—	2	4	no	—
66	C₂×C₄×C₃₇	—	2	4	no	—
67	C₂×C₄×C₃₇	—	4	4	no	—
68	C₂×C₄×C₃₇	—	8	4	no	—
69	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
70	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₃₇	C₂²×Q₈	1	2	yes	—
72	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
73	C₈×C₃₇	—	2	8	no	—
74	C₈×C₃₇	C₄⋊Q₈	1	4	yes	—
75	C₈×C₃₇	C₄⋊Q₈	1	4	yes	—
76	#28825	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×C₃₇	—	2	1	no	—
78	C₂²×C₃₇	—	2	1	no	—
79	C₂²×C₃₇	#544	1	1	yes	characteristic
80	C₂²×C₃₇	D₄²	1	1	yes	characteristic
81	C₂²×C₃₇	Q₈²	1	1	yes	characteristic
82	C₄×C₃₇	—	2	4	no	—
83	C₄×C₃₇	—	4	4	no	—
84	C₄×C₃₇	—	4	4	no	—
85	C₄×C₃₇	—	8	4	no	—
86	C₄×C₃₇	C₂×C₄⋊Q₈	1	2	yes	—
87	C₄×C₃₇	D₄×Q₈	1	2	yes	—
88	#3164	C₂×C₃₇	1	4	yes	—
89	#2060	C₂×C₃₇	1	4	yes	—
90	#1952	C₂×C₃₇	1	4	yes	—
91	#1845	C₂×C₃₇	1	1	yes	characteristic
92	#1772	C₂×C₃₇	1	2	yes	—
93	C₂×C₃₇	—	2	1	no	—
94	C₂×C₃₇	—	2	1	no	—
95	C₂×C₃₇	#2551	1	1	yes	characteristic
96	C₂×C₃₇	#3292	1	1	yes	characteristic
97	C₂×C₃₇	#3293	1	1	yes	characteristic
98	#544	C₂²×C₃₇	1	4	yes	—
99	#501	—	2	8	no	—
100	#500	—	2	4	no	—
101	#499	—	2	4	no	—
102	C₂×#108	C₂²×C₃₇	1	2	yes	—
103	#428	C₂²×C₃₇	1	8	yes	—
104	#427	C₂²×C₃₇	1	2	yes	—
105	C₂×#107	C₂²×C₃₇	1	4	yes	—
106	#405	C₂²×C₃₇	1	4	yes	—
107	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
108	#384	C₂²×C₃₇	1	2	yes	—
109	#343	C₂²×C₃₇	1	4	yes	—
110	#339	C₂²×C₃₇	1	4	yes	—
111	C₃₇	#28825	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×M₁₆	C₂³×C₃₇	1	4	yes	—
113	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
114	C₄⋊Q₈	—	2	8	no	—
115	#125	—	2	8	no	—
116	C₂²⋊Q₈	—	4	4	no	—
117	#117	C₂³×C₃₇	1	4	yes	—
118	C₂×C₄⋊C₄	—	4	2	no	—
119	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
120	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
121	#108	Q₈×C₃₇	1	8	yes	complemented
122	#107	—	2	4	no	—
123	#107	—	2	4	no	—
124	#107	—	2	8	no	—
125	#107	D₄×C₃₇	1	2	yes	—
126	#107	D₄×C₃₇	1	2	yes	—
127	#106	—	2	8	no	—
128	#106	D₄×C₃₇	1	4	yes	—
129	#97	—	2	4	no	—
130	C₄×C₈	—	2	4	no	—
131	C₄×C₈	Q₈×C₃₇	1	4	yes	complemented
132	C₂×Q₈	—	4	4	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
136	M₁₆	—	2	4	no	—
137	M₁₆	C₂×Q₈×C₃₇	1	8	yes	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
140	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
141	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
142	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
143	C₄⋊C₄	—	2	8	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	8	no	—
146	C₄⋊C₄	—	8	4	no	—
147	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
148	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
149	C₂²⋊C₄	—	2	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	2	1	no	—
152	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
153	C₂³	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
154	Q₈	—	16	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
163	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
164	C₂×C₄	C₂²×Q₈×C₃₇	1	2	yes	—
165	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
166	C₈	—	2	8	no	—
167	C₈	C₄⋊Q₈×C₃₇	1	4	yes	—
168	C₈	C₄⋊Q₈×C₃₇	1	4	yes	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	C₃₇×#544	1	1	yes	characteristic
172	C₂²	D₄²×C₃₇	1	1	yes	characteristic
173	C₂²	Q₈²×C₃₇	1	1	yes	characteristic
174	C₄	—	2	4	no	—
175	C₄	—	4	4	no	—
176	C₄	—	4	4	no	—
177	C₄	—	8	4	no	—
178	C₄	C₂×C₄⋊Q₈×C₃₇	1	2	yes	—
179	C₄	D₄×Q₈×C₃₇	1	2	yes	—
180	C₂	—	2	1	no	—
181	C₂	—	2	1	no	—
182	C₂	C₃₇×#2551	1	1	yes	characteristic
183	C₂	C₃₇×#3292	1	1	yes	characteristic
184	C₂	C₃₇×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂³×C₃₇	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
28. #108×C₃₇	Q₈	semidirect
38. C₄×C₈×C₃₇	Q₈	semidirect
76. #28825	C₃₇	direct (N × H)
111. C₃₇	#28825	direct (N × H)
121. #108	Q₈×C₃₇	semidirect
131. C₄×C₈	Q₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#3164	2
C₃₇×#2060	2
C₃₇×#1952	2
C₃₇×#1845	2
C₃₇×#1772	2
#28825	37

Subgroups — C37×#28825 #22650078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28825 #22650078