Subgroups — C₃₇×#8274 #22629527

Order 9472.

438 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×#1767	C₂	1	2	yes	maximal
2	C₄²×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₇×#335	C₂	1	2	yes	maximal
4	C₃₇×#1277	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₃₇	C₂²	1	4	yes	—
6	C₂×#97×C₃₇	—	2	2	no	—
7	C₂×C₄×C₈×C₃₇	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₃₇	C₄	1	2	yes	—
9	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₄³×C₃₇	C₂²	1	1	yes	characteristic
12	C₃₇×#336	—	2	2	no	—
13	C₃₇×#335	C₄	1	8	yes	complemented
14	C₂²×C₈×C₃₇	—	2	4	no	—
15	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₃₇	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₃₇	Q₈	1	2	yes	—
22	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₈×C₃₇	—	2	4	no	—
24	C₄⋊C₈×C₃₇	C₂×C₄	1	8	yes	—
25	#97×C₃₇	—	4	4	no	—
26	C₄×C₈×C₃₇	—	2	4	no	—
27	C₄×C₈×C₃₇	—	2	4	no	—
28	C₄×C₈×C₃₇	—	2	2	no	—
29	C₄×C₈×C₃₇	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₃₇	—	2	2	no	—
36	C₂×C₈×C₃₇	—	2	2	no	—
37	C₂×C₈×C₃₇	—	2	2	no	—
38	C₂×C₈×C₃₇	—	2	8	no	—
39	C₂×C₈×C₃₇	—	4	4	no	—
40	C₂×C₈×C₃₇	—	4	4	no	—
41	C₂×C₈×C₃₇	C₄²	1	2	yes	—
42	C₂×C₈×C₃₇	C₄²	1	2	yes	—
43	C₂×C₈×C₃₇	M₁₆	1	4	yes	—
44	C₂×C₈×C₃₇	M₁₆	1	4	yes	—
45	C₄²×C₃₇	—	2	4	no	—
46	C₄²×C₃₇	—	2	2	no	—
47	C₄²×C₃₇	—	2	2	no	—
48	C₄²×C₃₇	—	2	2	no	—
49	C₄²×C₃₇	D₈	1	2	yes	—
50	C₄²×C₃₇	Q₁₆	1	2	yes	—
51	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
52	C₄²×C₃₇	#41	1	2	yes	—
53	C₂³×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₃₇	—	2	2	no	—
55	C₂×C₄×C₃₇	—	2	2	no	—
56	C₂×C₄×C₃₇	—	2	2	no	—
57	C₂×C₄×C₃₇	—	2	2	no	—
58	C₂×C₄×C₃₇	—	2	2	no	—
59	C₂×C₄×C₃₇	#107	1	4	yes	—
60	C₂×C₄×C₃₇	#108	1	4	yes	—
61	C₂×C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₃₇	#117	1	1	yes	characteristic
63	C₂×C₄×C₃₇	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₃₇	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₃₇	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₃₇	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₃₇	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₃₇	#135	1	2	yes	—
70	C₈×C₃₇	—	2	2	no	—
71	C₈×C₃₇	—	2	4	no	—
72	C₈×C₃₇	—	4	8	no	—
73	C₈×C₃₇	#97	1	4	yes	complemented
74	#8274	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂²×C₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₃₇	#423	1	2	yes	—
77	C₂²×C₃₇	C₂×#107	1	1	yes	characteristic
78	C₂²×C₃₇	#427	1	1	yes	characteristic
79	C₂²×C₃₇	C₂×#108	1	2	yes	—
80	C₄×C₃₇	—	2	4	no	—
81	C₄×C₃₇	—	2	4	no	—
82	C₄×C₃₇	—	2	2	no	—
83	C₄×C₃₇	C₂×#97	1	1	yes	characteristic
84	C₄×C₃₇	C₄×M₁₆	1	1	yes	characteristic
85	C₄×C₃₇	#432	1	2	yes	—
86	C₄×C₃₇	C₄×D₈	1	2	yes	—
87	C₄×C₃₇	C₄×Q₁₆	1	2	yes	—
88	#1767	C₂×C₃₇	1	2	yes	—
89	C₄²×C₈	C₂×C₃₇	1	1	yes	characteristic
90	C₂×#335	C₂×C₃₇	1	2	yes	—
91	#1277	C₂×C₃₇	1	2	yes	—
92	C₂×C₃₇	#1567	1	2	yes	—
93	C₂×C₃₇	#1568	1	2	yes	—
94	C₂×C₃₇	#1767	1	1	yes	characteristic
95	C₂×C₃₇	C₄×#107	1	1	yes	characteristic
96	C₂×C₃₇	C₄×#108	1	1	yes	characteristic
97	C₂×C₄⋊C₈	C₂²×C₃₇	1	4	yes	—
98	C₂×#97	—	2	2	no	—
99	C₂×C₄×C₈	C₄×C₃₇	1	2	yes	—
100	C₂×C₄×C₈	C₄×C₃₇	1	2	yes	—
101	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
103	C₄³	C₂²×C₃₇	1	1	yes	characteristic
104	#336	—	2	2	no	—
105	#335	C₄×C₃₇	1	8	yes	complemented
106	C₃₇	#8274	1	1	yes	characteristic, direct product factor, Sylow
107	C₂²×C₈	—	2	4	no	—
108	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
109	C₂²×C₈	C₂×C₄×C₃₇	1	2	yes	—
110	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
111	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
112	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
113	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
114	C₂×C₄²	Q₈×C₃₇	1	2	yes	—
115	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
116	C₄⋊C₈	—	2	4	no	—
117	C₄⋊C₈	C₂×C₄×C₃₇	1	8	yes	—
118	#97	—	4	4	no	—
119	C₄×C₈	—	2	4	no	—
120	C₄×C₈	—	2	4	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₂×C₄×C₃₇	1	4	yes	—
123	C₂²×C₄	C₄⋊C₄×C₃₇	1	2	yes	—
124	C₂²×C₄	C₄⋊C₄×C₃₇	1	2	yes	—
125	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
126	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
127	C₂²×C₄	C₂×Q₈×C₃₇	1	1	yes	characteristic
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	8	no	—
132	C₂×C₈	—	4	4	no	—
133	C₂×C₈	—	4	4	no	—
134	C₂×C₈	C₄²×C₃₇	1	2	yes	—
135	C₂×C₈	C₄²×C₃₇	1	2	yes	—
136	C₂×C₈	M₁₆×C₃₇	1	4	yes	—
137	C₂×C₈	M₁₆×C₃₇	1	4	yes	—
138	C₄²	—	2	4	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	D₈×C₃₇	1	2	yes	—
143	C₄²	Q₁₆×C₃₇	1	2	yes	—
144	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
145	C₄²	#41×C₃₇	1	2	yes	—
146	C₂³	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	#107×C₃₇	1	4	yes	—
153	C₂×C₄	#108×C₃₇	1	4	yes	—
154	C₂×C₄	C₂×C₄²×C₃₇	1	1	yes	characteristic
155	C₂×C₄	#117×C₃₇	1	1	yes	characteristic
156	C₂×C₄	C₄×D₄×C₃₇	1	2	yes	—
157	C₂×C₄	C₄×Q₈×C₃₇	1	2	yes	—
158	C₂×C₄	C₂×M₁₆×C₃₇	1	2	yes	—
159	C₂×C₄	C₂×M₁₆×C₃₇	1	2	yes	—
160	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
161	C₂×C₄	C₂×Q₁₆×C₃₇	1	1	yes	characteristic
162	C₂×C₄	#135×C₃₇	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	—	2	4	no	—
165	C₈	—	4	8	no	—
166	C₈	#97×C₃₇	1	4	yes	complemented
167	C₂²	C₄×C₄⋊C₄×C₃₇	1	1	yes	characteristic
168	C₂²	C₃₇×#423	1	2	yes	—
169	C₂²	C₂×#107×C₃₇	1	1	yes	characteristic
170	C₂²	C₃₇×#427	1	1	yes	characteristic
171	C₂²	C₂×#108×C₃₇	1	2	yes	—
172	C₄	—	2	2	no	—
173	C₄	—	2	4	no	—
174	C₄	—	2	4	no	—
175	C₄	C₂×#97×C₃₇	1	1	yes	characteristic
176	C₄	C₄×M₁₆×C₃₇	1	1	yes	characteristic
177	C₄	C₃₇×#432	1	2	yes	—
178	C₄	C₄×D₈×C₃₇	1	2	yes	—
179	C₄	C₄×Q₁₆×C₃₇	1	2	yes	—
180	C₂	C₃₇×#1567	1	2	yes	—
181	C₂	C₃₇×#1568	1	2	yes	—
182	C₂	C₃₇×#1767	1	1	yes	characteristic
183	C₂	C₄×#107×C₃₇	1	1	yes	characteristic
184	C₂	C₄×#108×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₃₇
socle, prime radical	C₂³×C₃₇	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ C₄³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₃₇×#335	C₄	semidirect
73. C₈×C₃₇	#97	semidirect
74. #8274	C₃₇	direct (N × H)
105. #335	C₄×C₃₇	semidirect
106. C₃₇	#8274	direct (N × H)
166. C₈	#97×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#1767	2
C₄²×C₈×C₃₇	2
C₂×C₃₇×#335	2
C₃₇×#1277	2
#8274	37

Subgroups — C37×#8274 #22629527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#8274 #22629527