Subgroups — C₃₇×#33393 #22654646

Order 9472.

1102 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#578	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#3407	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#2916	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#2148	C₂	1	6	yes	maximal, complemented
5	C₂×C₃₇×#362	C₂	1	6	yes	maximal, complemented
6	C₂×2₋¹⁺⁴×C₃₇	—	2	1	no	—
7	C₃₇×#578	—	2	4	no	—
8	C₃₇×#578	C₄	1	8	yes	—
9	C₂×#137×C₃₇	—	2	3	no	—
10	C₂×#135×C₃₇	C₂²	1	3	yes	—
11	C₂²×Q₁₆×C₃₇	C₂²	1	3	yes	—
12	C₃₇×#568	—	2	1	no	—
13	C₂×#131×C₃₇	C₂²	1	1	yes	characteristic
14	C₃₇×#504	—	2	4	no	—
15	C₂×M₃₂×C₃₇	C₂²	1	6	yes	—
16	C₃₇×#430	—	2	3	no	—
17	C₂×#108×C₃₇	C₂²	1	6	yes	—
18	C₃₇×#362	C₂²	1	12	yes	—
19	C₃₇×#361	C₂²	1	4	yes	—
20	2₋¹⁺⁴×C₃₇	—	2	8	no	—
21	2₋¹⁺⁴×C₃₇	—	4	4	no	—
22	C₂×#41×C₃₇	—	2	3	no	—
23	C₂×#41×C₃₇	—	4	3	no	—
24	C₂×#41×C₃₇	D₄	1	1	yes	characteristic
25	C₂²×Q₈×C₃₇	—	2	3	no	—
26	C₂²×Q₈×C₃₇	—	4	1	no	—
27	#137×C₃₇	—	2	24	no	—
28	#135×C₃₇	—	2	6	no	—
29	#135×C₃₇	C₂×C₄	1	12	yes	—
30	C₂×Q₁₆×C₃₇	—	2	3	no	—
31	C₂×Q₁₆×C₃₇	—	2	6	no	—
32	C₂×Q₁₆×C₃₇	C₂×C₄	1	12	yes	—
33	C₂×Q₁₆×C₃₇	C₂³	1	3	yes	—
34	C₂×SD₁₆×C₃₇	—	2	3	no	—
35	C₂×D₈×C₃₇	C₂³	1	1	yes	characteristic
36	#131×C₃₇	—	4	4	no	—
37	#131×C₃₇	C₂×C₄	1	4	yes	—
38	#131×C₃₇	D₄	1	4	yes	—
39	C₂×M₁₆×C₃₇	—	2	6	no	—
40	C₂×M₁₆×C₃₇	D₄	1	3	yes	—
41	C₂²×C₈×C₃₇	C₂³	1	3	yes	—
42	M₃₂×C₃₇	—	2	6	no	—
43	M₃₂×C₃₇	C₂³	1	4	yes	—
44	C₂×C₁₆×C₃₇	—	2	6	no	—
45	#108×C₃₇	—	2	6	no	—
46	#108×C₃₇	C₂³	1	4	yes	—
47	#41×C₃₇	—	2	12	no	—
48	#41×C₃₇	—	4	24	no	—
49	#41×C₃₇	—	4	6	no	—
50	#41×C₃₇	C₂²⋊C₄	1	4	yes	—
51	#41×C₃₇	D₈	1	4	yes	—
52	C₂×Q₈×C₃₇	—	2	12	no	—
53	C₂×Q₈×C₃₇	—	2	3	no	—
54	C₂×Q₈×C₃₇	—	4	8	no	—
55	C₂×Q₈×C₃₇	—	4	6	no	—
56	C₂×Q₈×C₃₇	—	4	3	no	—
57	C₂×Q₈×C₃₇	SD₁₆	1	1	yes	characteristic
58	C₂×D₄×C₃₇	—	2	1	no	—
59	C₂×D₄×C₃₇	—	4	3	no	—
60	C₂×D₄×C₃₇	SD₁₆	1	3	yes	—
61	C₂²×C₄×C₃₇	—	4	3	no	—
62	C₂²×C₄×C₃₇	—	4	3	no	—
63	C₂²×C₄×C₃₇	C₂×D₄	1	3	yes	—
64	Q₁₆×C₃₇	—	2	12	no	—
65	Q₁₆×C₃₇	—	2	3	no	—
66	Q₁₆×C₃₇	C₂²×C₄	1	6	yes	—
67	SD₁₆×C₃₇	—	2	12	no	—
68	D₈×C₃₇	—	2	1	no	—
69	D₈×C₃₇	C₂²×C₄	1	2	yes	—
70	M₁₆×C₃₇	—	2	4	no	—
71	M₁₆×C₃₇	—	4	6	no	—
72	M₁₆×C₃₇	C₂²⋊C₄	1	12	yes	—
73	C₂×C₈×C₃₇	—	4	6	no	—
74	C₂×C₈×C₃₇	C₂²×C₄	1	6	yes	—
75	C₂×C₈×C₃₇	C₂×D₄	1	3	yes	—
76	C₂×C₈×C₃₇	C₂×D₄	1	6	yes	—
77	C₂×C₈×C₃₇	C₂⁴	1	1	yes	characteristic
78	C₁₆×C₃₇	—	2	4	no	—
79	C₂³×C₃₇	—	4	1	no	—
80	C₂³×C₃₇	C₂×SD₁₆	1	3	yes	—
81	Q₈×C₃₇	—	2	6	no	—
82	Q₈×C₃₇	—	4	3	no	—
83	Q₈×C₃₇	—	4	12	no	—
84	Q₈×C₃₇	#102	1	4	yes	—
85	D₄×C₃₇	—	2	2	no	—
86	D₄×C₃₇	—	4	12	no	—
87	D₄×C₃₇	—	4	1	no	—
88	D₄×C₃₇	#102	1	12	yes	—
89	C₂×C₄×C₃₇	—	4	3	no	—
90	C₂×C₄×C₃₇	—	4	12	no	—
91	C₂×C₄×C₃₇	—	4	12	no	—
92	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	6	yes	—
93	C₂×C₄×C₃₇	C₂×D₈	1	6	yes	—
94	C₂×C₄×C₃₇	C₂×SD₁₆	1	3	yes	—
95	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
96	C₈×C₃₇	—	4	4	no	—
97	C₈×C₃₇	C₂×C₂²⋊C₄	1	6	yes	—
98	C₈×C₃₇	C₂³×C₄	1	1	yes	characteristic
99	C₈×C₃₇	C₂²×D₄	1	1	yes	characteristic
100	#33393	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₃₇	—	4	3	no	—
102	C₂²×C₃₇	—	4	2	no	—
103	C₂²×C₃₇	—	8	2	no	—
104	C₂²×C₃₇	C₂×#102	1	6	yes	—
105	C₂²×C₃₇	C₂²×SD₁₆	1	1	yes	characteristic
106	C₄×C₃₇	—	4	6	no	—
107	C₄×C₃₇	C₂×#102	1	6	yes	—
108	C₄×C₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₃₇	C₂²×D₈	1	1	yes	characteristic
110	C₂×#578	C₂×C₃₇	1	1	yes	characteristic
111	#3407	C₂×C₃₇	1	1	yes	characteristic, complemented
112	#2916	C₂×C₃₇	1	1	yes	characteristic, complemented
113	#2148	C₂×C₃₇	1	6	yes	complemented
114	C₂×#362	C₂×C₃₇	1	6	yes	complemented
115	C₂×C₃₇	—	2	1	no	—
116	C₂×C₃₇	—	2	6	no	—
117	C₂×C₃₇	—	8	2	no	—
118	C₂×C₃₇	C₂²×#102	1	1	yes	characteristic
119	C₂×2₋¹⁺⁴	—	2	1	no	—
120	#578	—	2	4	no	—
121	#578	C₄×C₃₇	1	8	yes	—
122	C₂×#137	—	2	3	no	—
123	C₂×#135	C₂²×C₃₇	1	3	yes	—
124	C₂²×Q₁₆	C₂²×C₃₇	1	3	yes	—
125	#568	—	2	1	no	—
126	C₂×#131	C₂²×C₃₇	1	1	yes	characteristic
127	#504	—	2	4	no	—
128	C₂×M₃₂	C₂²×C₃₇	1	6	yes	—
129	#430	—	2	3	no	—
130	C₂×#108	C₂²×C₃₇	1	6	yes	—
131	#362	C₂²×C₃₇	1	12	yes	—
132	#361	C₂²×C₃₇	1	4	yes	—
133	C₃₇	#33393	1	1	yes	characteristic, direct product factor, Sylow
134	2₋¹⁺⁴	—	2	8	no	—
135	2₋¹⁺⁴	—	4	4	no	—
136	C₂×#41	—	2	3	no	—
137	C₂×#41	—	4	3	no	—
138	C₂×#41	D₄×C₃₇	1	1	yes	characteristic
139	C₂²×Q₈	—	2	3	no	—
140	C₂²×Q₈	—	4	1	no	—
141	#137	—	2	24	no	—
142	#135	—	2	6	no	—
143	#135	C₂×C₄×C₃₇	1	12	yes	—
144	C₂×Q₁₆	—	2	3	no	—
145	C₂×Q₁₆	—	2	6	no	—
146	C₂×Q₁₆	C₂×C₄×C₃₇	1	12	yes	—
147	C₂×Q₁₆	C₂³×C₃₇	1	3	yes	—
148	C₂×SD₁₆	—	2	3	no	—
149	C₂×D₈	C₂³×C₃₇	1	1	yes	characteristic
150	#131	—	4	4	no	—
151	#131	C₂×C₄×C₃₇	1	4	yes	—
152	#131	D₄×C₃₇	1	4	yes	—
153	C₂×M₁₆	—	2	6	no	—
154	C₂×M₁₆	D₄×C₃₇	1	3	yes	—
155	C₂²×C₈	C₂³×C₃₇	1	3	yes	—
156	M₃₂	—	2	6	no	—
157	M₃₂	C₂³×C₃₇	1	4	yes	—
158	C₂×C₁₆	—	2	6	no	—
159	#108	—	2	6	no	—
160	#108	C₂³×C₃₇	1	4	yes	—
161	#41	—	2	12	no	—
162	#41	—	4	24	no	—
163	#41	—	4	6	no	—
164	#41	C₂²⋊C₄×C₃₇	1	4	yes	—
165	#41	D₈×C₃₇	1	4	yes	—
166	C₂×Q₈	—	2	12	no	—
167	C₂×Q₈	—	2	3	no	—
168	C₂×Q₈	—	4	8	no	—
169	C₂×Q₈	—	4	6	no	—
170	C₂×Q₈	—	4	3	no	—
171	C₂×Q₈	SD₁₆×C₃₇	1	1	yes	characteristic
172	C₂×D₄	—	2	1	no	—
173	C₂×D₄	—	4	3	no	—
174	C₂×D₄	SD₁₆×C₃₇	1	3	yes	—
175	C₂²×C₄	—	4	3	no	—
176	C₂²×C₄	—	4	3	no	—
177	C₂²×C₄	C₂×D₄×C₃₇	1	3	yes	—
178	Q₁₆	—	2	12	no	—
179	Q₁₆	—	2	3	no	—
180	Q₁₆	C₂²×C₄×C₃₇	1	6	yes	—
181	SD₁₆	—	2	12	no	—
182	D₈	—	2	1	no	—
183	D₈	C₂²×C₄×C₃₇	1	2	yes	—
184	M₁₆	—	2	4	no	—
185	M₁₆	—	4	6	no	—
186	M₁₆	C₂²⋊C₄×C₃₇	1	12	yes	—
187	C₂×C₈	—	4	6	no	—
188	C₂×C₈	C₂²×C₄×C₃₇	1	6	yes	—
189	C₂×C₈	C₂×D₄×C₃₇	1	3	yes	—
190	C₂×C₈	C₂×D₄×C₃₇	1	6	yes	—
191	C₂×C₈	C₂⁴×C₃₇	1	1	yes	characteristic
192	C₁₆	—	2	4	no	—
193	C₂³	—	4	1	no	—
194	C₂³	C₂×SD₁₆×C₃₇	1	3	yes	—
195	Q₈	—	2	6	no	—
196	Q₈	—	4	3	no	—
197	Q₈	—	4	12	no	—
198	Q₈	#102×C₃₇	1	4	yes	—
199	D₄	—	2	2	no	—
200	D₄	—	4	12	no	—
201	D₄	—	4	1	no	—
202	D₄	#102×C₃₇	1	12	yes	—
203	C₂×C₄	—	4	3	no	—
204	C₂×C₄	—	4	12	no	—
205	C₂×C₄	—	4	12	no	—
206	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	6	yes	—
207	C₂×C₄	C₂×D₈×C₃₇	1	6	yes	—
208	C₂×C₄	C₂×SD₁₆×C₃₇	1	3	yes	—
209	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
210	C₈	—	4	4	no	—
211	C₈	C₂×C₂²⋊C₄×C₃₇	1	6	yes	—
212	C₈	C₂³×C₄×C₃₇	1	1	yes	characteristic
213	C₈	C₂²×D₄×C₃₇	1	1	yes	characteristic
214	C₂²	—	4	3	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	8	2	no	—
217	C₂²	C₂×#102×C₃₇	1	6	yes	—
218	C₂²	C₂²×SD₁₆×C₃₇	1	1	yes	characteristic
219	C₄	—	4	6	no	—
220	C₄	C₂×#102×C₃₇	1	6	yes	—
221	C₄	C₂²×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
222	C₄	C₂²×D₈×C₃₇	1	1	yes	characteristic
223	C₂	—	2	1	no	—
224	C₂	—	2	6	no	—
225	C₂	—	8	2	no	—
226	C₂	C₂²×#102×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₇	C₂²×#102
commutator	C₈	C₂³×C₄×C₃₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₇
prime radical	C₂×C₃₇×#578	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₇ ◁ C₂×#41×C₃₇ ◁ C₂×#131×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#3407	C₂	semidirect
3. C₃₇×#2916	C₂	semidirect
4. C₃₇×#2148	C₂	semidirect
5. C₂×C₃₇×#362	C₂	semidirect
100. #33393	C₃₇	direct (N × H)
111. #3407	C₂×C₃₇	semidirect
112. #2916	C₂×C₃₇	semidirect
113. #2148	C₂×C₃₇	semidirect
114. C₂×#362	C₂×C₃₇	semidirect
133. C₃₇	#33393	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×#578	2
C₃₇×#3407	2
C₃₇×#2916	2
C₃₇×#2148	2
C₂×C₃₇×#362	2
#33393	37

Subgroups — C37×#33393 #22654646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#33393 #22654646