Subgroups — C₃₇×#29294 #22650547

Order 9472.

630 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#2330	C₂	1	2	yes	maximal
2	C₃₇×#2087	C₂	1	2	yes	maximal
3	C₃₇×#2086	C₂	1	2	yes	maximal
4	C₃₇×#1989	C₂	1	4	yes	maximal
5	C₃₇×#1947	C₂	1	4	yes	maximal
6	C₃₇×#1843	C₂	1	1	yes	maximal, characteristic
7	C₂×#125×C₃₇	C₂²	1	2	yes	—
8	C₂×#107×C₃₇	C₂²	1	2	yes	—
9	C₂×#107×C₃₇	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₃₇	C₂²	1	4	yes	—
11	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₃₇×#389	—	2	4	no	—
14	C₃₇×#389	C₂²	1	4	yes	—
15	C₃₇×#387	—	2	4	no	—
16	C₃₇×#384	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₃₇	C₂²	1	4	yes	—
18	C₃₇×#376	C₂²	1	1	yes	characteristic
19	C₃₇×#340	C₂²	1	4	yes	—
20	C₃₇×#340	C₂²	1	4	yes	—
21	C₃₇×#340	C₂²	1	2	yes	—
22	C₃₇×#340	C₂²	1	2	yes	—
23	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
24	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
25	#125×C₃₇	—	2	4	no	—
26	#125×C₃₇	Q₈	1	8	yes	—
27	C₂×C₄⋊C₄×C₃₇	—	2	4	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×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
32	C₂×C₄²×C₃₇	—	2	4	no	—
33	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
35	#107×C₃₇	—	4	2	no	—
36	#107×C₃₇	—	4	2	no	—
37	C₄⋊C₈×C₃₇	—	2	8	no	—
38	C₄⋊C₈×C₃₇	—	4	1	no	—
39	C₄⋊C₈×C₃₇	—	4	1	no	—
40	#95×C₃₇	—	2	4	no	—
41	#95×C₃₇	—	2	4	no	—
42	#95×C₃₇	D₄	1	2	yes	—
43	#95×C₃₇	D₄	1	2	yes	—
44	C₂²×C₄×C₃₇	—	2	4	no	—
45	C₂²×C₄×C₃₇	—	2	4	no	—
46	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₇	C₂×D₄	1	4	yes	—
49	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
50	C₂×C₈×C₃₇	—	2	4	no	—
51	C₂×C₈×C₃₇	—	2	4	no	—
52	C₂×C₈×C₃₇	—	2	2	no	—
53	C₂×C₈×C₃₇	—	2	2	no	—
54	C₄⋊C₄×C₃₇	—	2	16	no	—
55	C₄⋊C₄×C₃₇	—	2	4	no	—
56	C₄⋊C₄×C₃₇	—	2	4	no	—
57	C₄⋊C₄×C₃₇	—	2	4	no	—
58	C₄⋊C₄×C₃₇	—	8	4	no	—
59	C₄⋊C₄×C₃₇	—	8	4	no	—
60	C₄⋊C₄×C₃₇	C₂×Q₈	1	8	yes	—
61	C₄²×C₃₇	—	2	1	no	—
62	C₄²×C₃₇	—	2	1	no	—
63	C₄²×C₃₇	—	8	4	no	—
64	C₄²×C₃₇	C₂×Q₈	1	4	yes	—
65	C₄²×C₃₇	#41	1	4	yes	—
66	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
67	C₂×C₄×C₃₇	—	2	4	no	—
68	C₂×C₄×C₃₇	—	2	4	no	—
69	C₂×C₄×C₃₇	—	2	8	no	—
70	C₂×C₄×C₃₇	—	2	1	no	—
71	C₂×C₄×C₃₇	—	2	1	no	—
72	C₂×C₄×C₃₇	—	2	2	no	—
73	C₂×C₄×C₃₇	—	2	2	no	—
74	C₂×C₄×C₃₇	—	4	4	no	—
75	C₂×C₄×C₃₇	—	4	4	no	—
76	C₂×C₄×C₃₇	—	4	4	no	—
77	C₂×C₄×C₃₇	—	4	4	no	—
78	C₂×C₄×C₃₇	—	4	4	no	—
79	C₂×C₄×C₃₇	C₂²⋊Q₈	1	8	yes	—
80	C₂×C₄×C₃₇	C₂²×Q₈	1	2	yes	—
81	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
82	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
84	C₈×C₃₇	—	4	2	no	—
85	C₈×C₃₇	—	4	2	no	—
86	#29294	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₃₇	C₂×C₂²⋊Q₈	1	2	yes	—
88	C₂²×C₃₇	#537	1	1	yes	characteristic
89	C₂²×C₃₇	#576	1	2	yes	—
90	C₂²×C₃₇	#578	1	2	yes	—
91	C₄×C₃₇	—	2	8	no	—
92	C₄×C₃₇	—	4	8	no	—
93	C₄×C₃₇	—	4	1	no	—
94	C₄×C₃₇	—	4	1	no	—
95	C₄×C₃₇	—	8	4	no	—
96	C₄×C₃₇	#557	1	2	yes	—
97	C₄×C₃₇	Q₈²	1	1	yes	characteristic
98	C₄×C₃₇	#559	1	1	yes	characteristic
99	#2330	C₂×C₃₇	1	2	yes	—
100	#2087	C₂×C₃₇	1	2	yes	—
101	#2086	C₂×C₃₇	1	2	yes	—
102	#1989	C₂×C₃₇	1	4	yes	—
103	#1947	C₂×C₃₇	1	4	yes	—
104	#1843	C₂×C₃₇	1	1	yes	characteristic
105	C₂×C₃₇	#2582	1	1	yes	characteristic
106	C₂×C₃₇	#3083	1	4	yes	—
107	C₂×C₃₇	#3259	1	1	yes	characteristic
108	C₂×C₃₇	#3261	1	1	yes	characteristic
109	C₂×#125	C₂²×C₃₇	1	2	yes	—
110	C₂×#107	C₂²×C₃₇	1	2	yes	—
111	C₂×#107	C₂²×C₃₇	1	2	yes	—
112	C₂×C₄⋊C₈	C₂²×C₃₇	1	4	yes	—
113	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
114	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
115	#389	—	2	4	no	—
116	#389	C₂²×C₃₇	1	4	yes	—
117	#387	—	2	4	no	—
118	#384	C₂²×C₃₇	1	2	yes	—
119	C₄×C₄⋊C₄	C₂²×C₃₇	1	4	yes	—
120	#376	C₂²×C₃₇	1	1	yes	characteristic
121	#340	C₂²×C₃₇	1	4	yes	—
122	#340	C₂²×C₃₇	1	2	yes	—
123	#340	C₂²×C₃₇	1	4	yes	—
124	#340	C₂²×C₃₇	1	2	yes	—
125	C₃₇	#29294	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
127	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
128	#125	—	2	4	no	—
129	#125	Q₈×C₃₇	1	8	yes	—
130	C₂×C₄⋊C₄	—	2	4	no	—
131	C₂×C₄⋊C₄	—	2	4	no	—
132	C₂×C₄⋊C₄	—	2	4	no	—
133	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
134	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
135	C₂×C₄²	—	2	4	no	—
136	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
137	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
138	#107	—	4	2	no	—
139	#107	—	4	2	no	—
140	C₄⋊C₈	—	2	8	no	—
141	C₄⋊C₈	—	4	1	no	—
142	C₄⋊C₈	—	4	1	no	—
143	#95	—	2	4	no	—
144	#95	—	2	4	no	—
145	#95	D₄×C₃₇	1	2	yes	—
146	#95	D₄×C₃₇	1	2	yes	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄×C₃₇	1	4	yes	—
152	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₄⋊C₄	—	2	16	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	8	4	no	—
162	C₄⋊C₄	—	8	4	no	—
163	C₄⋊C₄	C₂×Q₈×C₃₇	1	8	yes	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	8	4	no	—
167	C₄²	C₂×Q₈×C₃₇	1	4	yes	—
168	C₄²	#41×C₃₇	1	4	yes	—
169	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	C₂²⋊Q₈×C₃₇	1	8	yes	—
183	C₂×C₄	C₂²×Q₈×C₃₇	1	2	yes	—
184	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
185	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
186	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
187	C₈	—	4	2	no	—
188	C₈	—	4	2	no	—
189	C₂²	C₂×C₂²⋊Q₈×C₃₇	1	2	yes	—
190	C₂²	C₃₇×#537	1	1	yes	characteristic
191	C₂²	C₃₇×#576	1	2	yes	—
192	C₂²	C₃₇×#578	1	2	yes	—
193	C₄	—	2	8	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	1	no	—
196	C₄	—	4	8	no	—
197	C₄	—	8	4	no	—
198	C₄	C₃₇×#557	1	2	yes	—
199	C₄	Q₈²×C₃₇	1	1	yes	characteristic
200	C₄	C₃₇×#559	1	1	yes	characteristic
201	C₂	C₃₇×#2582	1	1	yes	characteristic
202	C₂	C₃₇×#3083	1	4	yes	—
203	C₂	C₃₇×#3259	1	1	yes	characteristic
204	C₂	C₃₇×#3261	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
LCS	G ▷ C₂²×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
86. #29294	C₃₇	direct (N × H)
125. C₃₇	#29294	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#2330	2
C₃₇×#2087	2
C₃₇×#2086	2
C₃₇×#1989	2
C₃₇×#1947	2
C₃₇×#1843	2
#29294	37

Subgroups — C37×#29294 #22650547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#29294 #22650547