Subgroups — C₃₇×#22277 #22643530

Order 9472.

826 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#494	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#2327	C₂	1	2	yes	maximal
3	C₃₇×#1960	C₂	1	2	yes	maximal
4	C₃₇×#1878	C₂	1	8	yes	maximal
5	C₃₇×#1835	C₂	1	1	yes	maximal, characteristic
6	C₃₇×#1746	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₃₇	C₂²	1	4	yes	—
8	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₃₇	C₂²	1	4	yes	—
10	C₃₇×#494	C₄	1	8	yes	complemented
11	C₂×#103×C₃₇	C₂²	1	8	yes	—
12	C₂×#97×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₃₇×#391	C₂²	1	4	yes	—
15	C₃₇×#387	—	2	8	no	—
16	C₃₇×#382	C₂²	1	2	yes	—
17	C₃₇×#376	C₂²	1	1	yes	characteristic
18	C₃₇×#340	C₂²	1	4	yes	—
19	C₃₇×#336	C₂²	1	4	yes	—
20	C₂²×Q₈×C₃₇	C₂³	1	4	yes	—
21	C₂×Q₁₆×C₃₇	—	4	8	no	—
22	C₂×Q₁₆×C₃₇	C₂×C₄	1	16	yes	—
23	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
24	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
25	C₄⋊Q₈×C₃₇	—	2	4	no	—
26	C₄⋊Q₈×C₃₇	C₂×C₄	1	8	yes	—
27	C₄×Q₈×C₃₇	—	4	8	no	—
28	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
29	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
30	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
31	C₂×C₄²×C₃₇	—	2	4	no	—
32	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
33	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
34	#103×C₃₇	—	4	8	no	—
35	#97×C₃₇	—	2	2	no	—
36	C₄×C₈×C₃₇	C₂×C₄	1	4	yes	—
37	#95×C₃₇	—	2	4	no	—
38	#95×C₃₇	—	2	8	no	—
39	#95×C₃₇	D₄	1	4	yes	—
40	C₂×Q₈×C₃₇	—	2	4	no	—
41	C₂×Q₈×C₃₇	—	2	8	no	—
42	C₂×Q₈×C₃₇	—	2	8	no	—
43	C₂×Q₈×C₃₇	C₂²×C₄	1	8	yes	—
44	C₂²×C₄×C₃₇	—	2	4	no	—
45	C₂²×C₄×C₃₇	—	2	4	no	—
46	C₂²×C₄×C₃₇	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
49	Q₁₆×C₃₇	—	4	16	no	—
50	C₂×C₈×C₃₇	—	2	4	no	—
51	C₂×C₈×C₃₇	—	2	2	no	—
52	C₂×C₈×C₃₇	—	2	4	no	—
53	C₂×C₈×C₃₇	C₂²×C₄	1	4	yes	—
54	C₄⋊C₄×C₃₇	—	2	16	no	—
55	C₄⋊C₄×C₃₇	—	2	8	no	—
56	C₄⋊C₄×C₃₇	—	4	8	no	—
57	C₄²×C₃₇	—	2	4	no	—
58	C₄²×C₃₇	—	4	8	no	—
59	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
60	C₄²×C₃₇	#41	1	2	yes	—
61	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
62	Q₈×C₃₇	—	2	16	no	—
63	Q₈×C₃₇	—	4	8	no	—
64	C₂×C₄×C₃₇	—	2	4	no	—
65	C₂×C₄×C₃₇	—	2	8	no	—
66	C₂×C₄×C₃₇	—	2	4	no	—
67	C₂×C₄×C₃₇	—	2	8	no	—
68	C₂×C₄×C₃₇	—	4	4	no	—
69	C₂×C₄×C₃₇	—	4	4	no	—
70	C₂×C₄×C₃₇	—	4	4	no	—
71	C₂×C₄×C₃₇	—	4	8	no	—
72	C₂×C₄×C₃₇	C₄×D₄	1	4	yes	—
73	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
75	C₂×C₄×C₃₇	H(5,𝔽₂)	1	4	yes	—
76	C₈×C₃₇	—	2	4	no	—
77	C₈×C₃₇	—	4	2	no	—
78	#22277	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
80	C₂²×C₃₇	#535	1	2	yes	—
81	C₂²×C₃₇	#578	1	4	yes	—
82	C₄×C₃₇	—	2	4	no	—
83	C₄×C₃₇	—	4	8	no	—
84	C₄×C₃₇	—	4	4	no	—
85	C₄×C₃₇	—	8	4	no	—
86	C₄×C₃₇	#518	1	2	yes	—
87	C₄×C₃₇	#540	1	2	yes	—
88	C₂×#494	C₂×C₃₇	1	1	yes	characteristic
89	#2327	C₂×C₃₇	1	2	yes	—
90	#1960	C₂×C₃₇	1	2	yes	—
91	#1878	C₂×C₃₇	1	8	yes	—
92	#1835	C₂×C₃₇	1	1	yes	characteristic
93	#1746	C₂×C₃₇	1	1	yes	characteristic
94	C₂×C₃₇	#2328	1	1	yes	characteristic
95	C₂×C₃₇	#2950	1	2	yes	—
96	C₂×C₃₇	#3223	1	4	yes	—
97	C₂²×Q₁₆	C₂²×C₃₇	1	4	yes	—
98	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
99	C₂×C₄×Q₈	C₂²×C₃₇	1	4	yes	—
100	#494	C₄×C₃₇	1	8	yes	complemented
101	C₂×#103	C₂²×C₃₇	1	8	yes	—
102	C₂×#97	C₂²×C₃₇	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
104	#391	C₂²×C₃₇	1	4	yes	—
105	#387	—	2	8	no	—
106	#382	C₂²×C₃₇	1	2	yes	—
107	#376	C₂²×C₃₇	1	1	yes	characteristic
108	#340	C₂²×C₃₇	1	4	yes	—
109	#336	C₂²×C₃₇	1	4	yes	—
110	C₃₇	#22277	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²×Q₈	C₂³×C₃₇	1	4	yes	—
112	C₂×Q₁₆	—	4	8	no	—
113	C₂×Q₁₆	C₂×C₄×C₃₇	1	16	yes	—
114	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
115	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
116	C₄⋊Q₈	—	2	4	no	—
117	C₄⋊Q₈	C₂×C₄×C₃₇	1	8	yes	—
118	C₄×Q₈	—	4	8	no	—
119	C₂×C₄⋊C₄	—	2	4	no	—
120	C₂×C₄⋊C₄	—	2	4	no	—
121	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
122	C₂×C₄²	—	2	4	no	—
123	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
124	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
125	#103	—	4	8	no	—
126	#97	—	2	2	no	—
127	C₄×C₈	C₂×C₄×C₃₇	1	4	yes	—
128	#95	—	2	4	no	—
129	#95	—	2	8	no	—
130	#95	D₄×C₃₇	1	4	yes	—
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	2	8	no	—
133	C₂×Q₈	—	2	8	no	—
134	C₂×Q₈	C₂²×C₄×C₃₇	1	8	yes	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	C₂×D₄×C₃₇	1	4	yes	—
138	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
139	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
140	Q₁₆	—	4	16	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	C₂²×C₄×C₃₇	1	4	yes	—
145	C₄⋊C₄	—	2	16	no	—
146	C₄⋊C₄	—	2	8	no	—
147	C₄⋊C₄	—	4	8	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	4	8	no	—
150	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
151	C₄²	#41×C₃₇	1	2	yes	—
152	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
153	Q₈	—	2	16	no	—
154	Q₈	—	4	8	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	8	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	8	no	—
163	C₂×C₄	C₄×D₄×C₃₇	1	4	yes	—
164	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
165	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
166	C₂×C₄	H(5,𝔽₂)×C₃₇	1	4	yes	—
167	C₈	—	2	4	no	—
168	C₈	—	4	2	no	—
169	C₂²	C₂×C₄×D₄×C₃₇	1	1	yes	characteristic
170	C₂²	C₃₇×#535	1	2	yes	—
171	C₂²	C₃₇×#578	1	4	yes	—
172	C₄	—	2	4	no	—
173	C₄	—	4	4	no	—
174	C₄	—	4	8	no	—
175	C₄	—	8	4	no	—
176	C₄	C₃₇×#518	1	2	yes	—
177	C₄	C₃₇×#540	1	2	yes	—
178	C₂	C₃₇×#2328	1	1	yes	characteristic
179	C₂	C₃₇×#2950	1	2	yes	—
180	C₂	C₃₇×#3223	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₃₇×#376 ◁ G

Complements

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

N	H ≅ G/N	type
10. C₃₇×#494	C₄	semidirect
78. #22277	C₃₇	direct (N × H)
100. #494	C₄×C₃₇	semidirect
110. C₃₇	#22277	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#494	2
C₃₇×#2327	2
C₃₇×#1960	2
C₃₇×#1878	2
C₃₇×#1835	2
C₃₇×#1746	2
#22277	37

Subgroups — C37×#22277 #22643530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#22277 #22643530