Subgroups — C₃₇×#13279 #22634532

Order 9472.

366 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#500	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#2150	C₂	1	1	yes	maximal, characteristic
3	C₄×#108×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#1387	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈×C₃₇	—	2	1	no	—
6	C₂×M₃₂×C₃₇	C₂²	1	2	yes	—
7	C₃₇×#500	—	2	1	no	—
8	C₃₇×#500	—	2	1	no	—
9	C₃₇×#500	C₄	1	2	yes	—
10	C₃₇×#500	C₄	1	2	yes	—
11	C₂×#108×C₃₇	C₂²	1	2	yes	—
12	C₂×#107×C₃₇	—	2	1	no	—
13	C₂×#107×C₃₇	C₂²	1	2	yes	—
14	C₄×M₁₆×C₃₇	—	2	1	no	—
15	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
16	C₃₇×C₄⋊C₁₆	—	2	1	no	—
17	C₃₇×C₄⋊C₁₆	—	2	1	no	—
18	C₃₇×#341	—	2	1	no	—
19	C₂²×Q₈×C₃₇	—	4	1	no	—
20	C₂×M₁₆×C₃₇	—	2	2	no	—
21	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈×C₃₇	—	2	2	no	—
24	C₄⋊Q₈×C₃₇	—	2	2	no	—
25	C₄⋊Q₈×C₃₇	—	4	1	no	—
26	C₄⋊Q₈×C₃₇	—	4	1	no	—
27	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₃₇	—	4	1	no	—
29	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
30	M₃₂×C₃₇	—	2	4	no	—
31	C₂×C₁₆×C₃₇	—	2	2	no	—
32	#108×C₃₇	—	2	2	no	—
33	#108×C₃₇	Q₈	1	4	yes	complemented
34	#107×C₃₇	—	2	2	no	—
35	#107×C₃₇	—	2	2	no	—
36	#107×C₃₇	—	2	2	no	—
37	#107×C₃₇	C₂×C₄	1	4	yes	—
38	#97×C₃₇	—	4	1	no	—
39	C₄×C₈×C₃₇	—	4	1	no	—
40	C₄×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₃₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₃₇	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₃₇	—	4	2	no	—
45	C₂×Q₈×C₃₇	—	4	2	no	—
46	C₂×Q₈×C₃₇	—	8	4	no	—
47	C₂²×C₄×C₃₇	—	4	2	no	—
48	C₂²×C₄×C₃₇	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₃₇	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
51	M₁₆×C₃₇	—	2	4	no	—
52	M₁₆×C₃₇	—	4	2	no	—
53	C₂×C₈×C₃₇	—	2	2	no	—
54	C₂×C₈×C₃₇	—	4	2	no	—
55	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₃₇	#41	1	2	yes	—
61	C₄⋊C₄×C₃₇	—	2	4	no	—
62	C₄⋊C₄×C₃₇	—	4	2	no	—
63	C₄⋊C₄×C₃₇	—	4	2	no	—
64	C₄⋊C₄×C₃₇	—	4	2	no	—
65	C₄²×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₃₇	D₈	1	1	yes	characteristic
68	C₄²×C₃₇	D₈	1	1	yes	characteristic
69	C₁₆×C₃₇	—	4	2	no	—
70	C₂³×C₃₇	C₂×SD₁₆	1	1	yes	characteristic
71	Q₈×C₃₇	—	8	4	no	—
72	Q₈×C₃₇	—	8	4	no	—
73	C₂×C₄×C₃₇	—	2	2	no	—
74	C₂×C₄×C₃₇	—	2	2	no	—
75	C₂×C₄×C₃₇	—	4	4	no	—
76	C₂×C₄×C₃₇	—	8	4	no	—
77	C₂×C₄×C₃₇	#102	1	1	yes	characteristic
78	C₂×C₄×C₃₇	#102	1	1	yes	characteristic
79	C₂×C₄×C₃₇	#102	1	1	yes	characteristic
80	C₂×C₄×C₃₇	#102	1	1	yes	characteristic
81	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
83	C₂×C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₃₇	#137	1	2	yes	—
85	C₈×C₃₇	—	2	1	no	—
86	C₈×C₃₇	—	2	1	no	—
87	C₈×C₃₇	—	4	4	no	—
88	C₈×C₃₇	C₄×Q₈	1	2	yes	—
89	C₈×C₃₇	#124	1	1	yes	characteristic
90	C₈×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
91	#13279	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂²×C₃₇	—	2	1	no	—
93	C₂²×C₃₇	—	2	1	no	—
94	C₂²×C₃₇	C₂×#102	1	1	yes	characteristic
95	C₂²×C₃₇	#475	1	2	yes	—
96	C₄×C₃₇	—	2	1	no	—
97	C₄×C₃₇	—	2	1	no	—
98	C₄×C₃₇	—	8	4	no	—
99	C₄×C₃₇	#361	1	2	yes	—
100	C₄×C₃₇	#362	1	2	yes	—
101	C₄×C₃₇	#391	1	1	yes	characteristic
102	C₄×C₃₇	#419	1	1	yes	characteristic
103	C₄×C₃₇	#474	1	2	yes	—
104	C₂×#500	C₂×C₃₇	1	1	yes	characteristic
105	#2150	C₂×C₃₇	1	1	yes	characteristic
106	C₄×#108	C₂×C₃₇	1	1	yes	characteristic
107	#1387	C₂×C₃₇	1	4	yes	—
108	C₂×C₃₇	—	2	2	no	—
109	C₂×C₃₇	#1988	1	1	yes	characteristic
110	C₂×C₃₇	C₂×#361	1	1	yes	characteristic
111	C₂×C₃₇	C₂×#362	1	1	yes	characteristic
112	C₂×C₄⋊Q₈	—	2	1	no	—
113	C₂×M₃₂	C₂²×C₃₇	1	2	yes	—
114	#500	—	2	1	no	—
115	#500	—	2	1	no	—
116	#500	C₄×C₃₇	1	2	yes	—
117	#500	C₄×C₃₇	1	2	yes	—
118	C₂×#108	C₂²×C₃₇	1	2	yes	—
119	C₂×#107	—	2	1	no	—
120	C₂×#107	C₂²×C₃₇	1	2	yes	—
121	C₄×M₁₆	—	2	1	no	—
122	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
123	C₄⋊C₁₆	—	2	1	no	—
124	C₄⋊C₁₆	—	2	1	no	—
125	#341	—	2	1	no	—
126	C₃₇	#13279	1	1	yes	characteristic, direct product factor, Sylow
127	C₂²×Q₈	—	4	1	no	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	4	1	no	—
134	C₄⋊Q₈	—	4	1	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
138	M₃₂	—	2	4	no	—
139	C₂×C₁₆	—	2	2	no	—
140	#108	—	2	2	no	—
141	#108	Q₈×C₃₇	1	4	yes	complemented
142	#107	—	2	2	no	—
143	#107	—	2	2	no	—
144	#107	—	2	2	no	—
145	#107	C₂×C₄×C₃₇	1	4	yes	—
146	#97	—	4	1	no	—
147	C₄×C₈	—	4	1	no	—
148	C₄×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
149	C₄×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
150	C₄×C₈	D₄×C₃₇	1	1	yes	characteristic
151	C₄×C₈	D₄×C₃₇	1	1	yes	characteristic
152	C₂×Q₈	—	4	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂×Q₈	—	8	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	SD₁₆×C₃₇	1	1	yes	characteristic
157	C₂²×C₄	SD₁₆×C₃₇	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
159	M₁₆	—	2	4	no	—
160	M₁₆	—	4	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
164	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
165	C₂×C₈	C₂²×C₄×C₃₇	1	1	yes	characteristic
166	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
167	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
168	C₂×C₈	#41×C₃₇	1	2	yes	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
174	C₄²	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
175	C₄²	D₈×C₃₇	1	1	yes	characteristic
176	C₄²	D₈×C₃₇	1	1	yes	characteristic
177	C₁₆	—	4	2	no	—
178	C₂³	C₂×SD₁₆×C₃₇	1	1	yes	characteristic
179	Q₈	—	8	4	no	—
180	Q₈	—	8	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	8	4	no	—
185	C₂×C₄	#102×C₃₇	1	1	yes	characteristic
186	C₂×C₄	#102×C₃₇	1	1	yes	characteristic
187	C₂×C₄	#102×C₃₇	1	1	yes	characteristic
188	C₂×C₄	#102×C₃₇	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
190	C₂×C₄	C₂²⋊Q₈×C₃₇	1	2	yes	—
191	C₂×C₄	C₂×D₈×C₃₇	1	1	yes	characteristic
192	C₂×C₄	#137×C₃₇	1	2	yes	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	4	4	no	—
196	C₈	C₄×Q₈×C₃₇	1	2	yes	—
197	C₈	#124×C₃₇	1	1	yes	characteristic
198	C₈	C₄⋊Q₈×C₃₇	1	1	yes	characteristic
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	C₂×#102×C₃₇	1	1	yes	characteristic
202	C₂²	C₃₇×#475	1	2	yes	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	8	4	no	—
206	C₄	C₃₇×#361	1	2	yes	—
207	C₄	C₃₇×#362	1	2	yes	—
208	C₄	C₃₇×#391	1	1	yes	characteristic
209	C₄	C₃₇×#419	1	1	yes	characteristic
210	C₄	C₃₇×#474	1	2	yes	—
211	C₂	—	2	2	no	—
212	C₂	C₃₇×#1988	1	1	yes	characteristic
213	C₂	C₂×C₃₇×#361	1	1	yes	characteristic
214	C₂	C₂×C₃₇×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₃₇
prime radical	C₂³×C₃₇	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ C₂×C₄×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
33. #108×C₃₇	Q₈	semidirect
91. #13279	C₃₇	direct (N × H)
126. C₃₇	#13279	direct (N × H)
141. #108	Q₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#500	2
C₃₇×#2150	2
C₄×#108×C₃₇	2
C₃₇×#1387	2
#13279	37

Subgroups — C37×#13279 #22634532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#13279 #22634532