Subgroups — C₃₇×#28700 #22649953

Order 9472.

630 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3165	C₂	1	2	yes	maximal
2	C₂×C₃₇×#498	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2960	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2090	C₂	1	4	yes	maximal
5	C₃₇×#1997	C₂	1	2	yes	maximal
6	C₃₇×#1951	C₂	1	4	yes	maximal
7	C₄×#108×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₃₇×#533	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
10	C₃₇×#501	—	2	4	no	—
11	C₃₇×#500	—	2	2	no	—
12	C₃₇×#498	—	2	2	no	—
13	C₃₇×#498	—	2	2	no	—
14	C₃₇×#498	—	2	2	no	—
15	C₃₇×#446	—	2	2	no	—
16	C₈×Q₈×C₃₇	—	2	2	no	—
17	C₂×#108×C₃₇	C₂²	1	2	yes	—
18	C₃₇×#428	C₂²	1	8	yes	—
19	C₂×#106×C₃₇	C₂²	1	2	yes	—
20	C₂×#106×C₃₇	C₂²	1	2	yes	—
21	C₃₇×#424	C₂²	1	2	yes	—
22	C₃₇×#423	C₂²	1	1	yes	characteristic
23	C₃₇×#405	C₂²	1	2	yes	—
24	C₄×M₁₆×C₃₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
26	C₃₇×#343	C₂²	1	8	yes	—
27	C₃₇×#341	C₂²	1	2	yes	—
28	C₂²×Q₈×C₃₇	—	2	2	no	—
29	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
30	C₂×M₁₆×C₃₇	C₂³	1	4	yes	—
31	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₃₇	—	2	2	no	—
34	C₄⋊Q₈×C₃₇	—	2	2	no	—
35	C₄⋊Q₈×C₃₇	—	2	2	no	—
36	C₄⋊Q₈×C₃₇	—	2	2	no	—
37	C₄⋊Q₈×C₃₇	D₄	1	2	yes	—
38	#125×C₃₇	D₄	1	2	yes	—
39	C₂²⋊Q₈×C₃₇	—	2	2	no	—
40	C₄×Q₈×C₃₇	—	2	2	no	—
41	#117×C₃₇	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
45	#108×C₃₇	Q₈	1	8	yes	complemented
46	#107×C₃₇	—	2	8	no	—
47	#106×C₃₇	—	2	4	no	—
48	#106×C₃₇	—	2	2	no	—
49	#106×C₃₇	—	2	2	no	—
50	#106×C₃₇	—	2	8	no	—
51	C₄⋊C₈×C₃₇	—	2	4	no	—
52	C₄⋊C₈×C₃₇	—	2	2	no	—
53	#97×C₃₇	—	2	2	no	—
54	#97×C₃₇	—	2	2	no	—
55	C₄×C₈×C₃₇	—	2	2	no	—
56	C₄×C₈×C₃₇	—	2	2	no	—
57	C₄×C₈×C₃₇	D₄	1	2	yes	—
58	C₄×C₈×C₃₇	D₄	1	2	yes	—
59	C₂×Q₈×C₃₇	—	2	2	no	—
60	C₂×Q₈×C₃₇	—	4	2	no	—
61	C₂×Q₈×C₃₇	—	4	2	no	—
62	C₂×Q₈×C₃₇	—	8	4	no	—
63	C₂²×C₄×C₃₇	—	2	4	no	—
64	C₂²×C₄×C₃₇	#41	1	2	yes	—
65	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
66	M₁₆×C₃₇	—	2	4	no	—
67	M₁₆×C₃₇	—	2	4	no	—
68	M₁₆×C₃₇	C₂×Q₈	1	8	yes	—
69	C₂×C₈×C₃₇	—	2	4	no	—
70	C₂×C₈×C₃₇	—	2	2	no	—
71	C₂×C₈×C₃₇	—	2	2	no	—
72	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
76	C₄⋊C₄×C₃₇	—	2	4	no	—
77	C₄⋊C₄×C₃₇	—	2	2	no	—
78	C₄⋊C₄×C₃₇	—	2	4	no	—
79	C₄⋊C₄×C₃₇	—	2	4	no	—
80	C₄⋊C₄×C₃₇	—	4	2	no	—
81	C₄⋊C₄×C₃₇	—	4	2	no	—
82	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
83	C₂²⋊C₄×C₃₇	—	2	2	no	—
84	C₄²×C₃₇	—	2	2	no	—
85	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₃₇	C₂×D₄	1	2	yes	—
87	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
88	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
89	Q₈×C₃₇	—	4	4	no	—
90	Q₈×C₃₇	—	8	4	no	—
91	Q₈×C₃₇	—	8	4	no	—
92	C₂×C₄×C₃₇	—	2	4	no	—
93	C₂×C₄×C₃₇	—	2	4	no	—
94	C₂×C₄×C₃₇	—	4	4	no	—
95	C₂×C₄×C₃₇	—	8	4	no	—
96	C₂×C₄×C₃₇	#121	1	2	yes	—
97	C₂×C₄×C₃₇	#121	1	2	yes	—
98	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₇	C₂²×Q₈	1	2	yes	—
101	C₂×C₄×C₃₇	2₋¹⁺⁴	1	2	yes	—
102	C₈×C₃₇	—	2	2	no	—
103	C₈×C₃₇	—	2	8	no	—
104	C₈×C₃₇	—	4	2	no	—
105	C₈×C₃₇	C₄⋊Q₈	1	2	yes	—
106	C₈×C₃₇	C₄⋊Q₈	1	2	yes	—
107	#28700	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₃₇	—	2	2	no	—
109	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
110	C₂²×C₃₇	#557	1	2	yes	—
111	C₄×C₃₇	—	2	2	no	—
112	C₄×C₃₇	—	4	4	no	—
113	C₄×C₃₇	—	8	4	no	—
114	C₄×C₃₇	#471	1	4	yes	—
115	C₄×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₄×C₃₇	#536	1	1	yes	characteristic
117	C₄×C₃₇	D₄×Q₈	1	2	yes	—
118	#3165	C₂×C₃₇	1	2	yes	—
119	C₂×#498	C₂×C₃₇	1	1	yes	characteristic
120	#2960	C₂×C₃₇	1	1	yes	characteristic
121	#2090	C₂×C₃₇	1	4	yes	—
122	#1997	C₂×C₃₇	1	2	yes	—
123	#1951	C₂×C₃₇	1	4	yes	—
124	C₄×#108	C₂×C₃₇	1	1	yes	characteristic
125	C₂×C₃₇	—	2	2	no	—
126	C₂×C₃₇	#2546	1	1	yes	characteristic
127	C₂×C₃₇	C₂×#471	1	2	yes	—
128	#533	C₂²×C₃₇	1	1	yes	characteristic
129	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
130	#501	—	2	4	no	—
131	#500	—	2	2	no	—
132	#498	—	2	2	no	—
133	#498	—	2	2	no	—
134	#498	—	2	2	no	—
135	#446	—	2	2	no	—
136	C₈×Q₈	—	2	2	no	—
137	C₂×#108	C₂²×C₃₇	1	2	yes	—
138	#428	C₂²×C₃₇	1	8	yes	—
139	C₂×#106	C₂²×C₃₇	1	2	yes	—
140	C₂×#106	C₂²×C₃₇	1	2	yes	—
141	#424	C₂²×C₃₇	1	2	yes	—
142	#423	C₂²×C₃₇	1	1	yes	characteristic
143	#405	C₂²×C₃₇	1	2	yes	—
144	C₄×M₁₆	C₂²×C₃₇	1	2	yes	—
145	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
146	#343	C₂²×C₃₇	1	8	yes	—
147	#341	C₂²×C₃₇	1	2	yes	—
148	C₃₇	#28700	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×Q₈	—	2	2	no	—
150	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
151	C₂×M₁₆	C₂³×C₃₇	1	4	yes	—
152	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
153	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	—	2	2	no	—
156	C₄⋊Q₈	—	2	2	no	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	D₄×C₃₇	1	2	yes	—
159	#125	D₄×C₃₇	1	2	yes	—
160	C₂²⋊Q₈	—	2	2	no	—
161	C₄×Q₈	—	2	2	no	—
162	#117	C₂³×C₃₇	1	2	yes	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
165	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
166	#108	Q₈×C₃₇	1	8	yes	complemented
167	#107	—	2	8	no	—
168	#106	—	2	4	no	—
169	#106	—	2	2	no	—
170	#106	—	2	2	no	—
171	#106	—	2	8	no	—
172	C₄⋊C₈	—	2	4	no	—
173	C₄⋊C₈	—	2	2	no	—
174	#97	—	2	2	no	—
175	#97	—	2	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	D₄×C₃₇	1	2	yes	—
179	C₄×C₈	D₄×C₃₇	1	2	yes	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	8	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	#41×C₃₇	1	2	yes	—
186	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
187	M₁₆	—	2	4	no	—
188	M₁₆	—	2	4	no	—
189	M₁₆	C₂×Q₈×C₃₇	1	8	yes	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
194	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
195	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
196	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
207	C₄²	C₂×D₄×C₃₇	1	2	yes	—
208	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
209	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
210	Q₈	—	4	4	no	—
211	Q₈	—	8	4	no	—
212	Q₈	—	8	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	8	4	no	—
217	C₂×C₄	#121×C₃₇	1	2	yes	—
218	C₂×C₄	#121×C₃₇	1	2	yes	—
219	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
220	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
221	C₂×C₄	C₂²×Q₈×C₃₇	1	2	yes	—
222	C₂×C₄	2₋¹⁺⁴×C₃₇	1	2	yes	—
223	C₈	—	2	2	no	—
224	C₈	—	2	8	no	—
225	C₈	—	4	2	no	—
226	C₈	C₄⋊Q₈×C₃₇	1	2	yes	—
227	C₈	C₄⋊Q₈×C₃₇	1	2	yes	—
228	C₂²	—	2	2	no	—
229	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
230	C₂²	C₃₇×#557	1	2	yes	—
231	C₄	—	2	2	no	—
232	C₄	—	4	4	no	—
233	C₄	—	8	4	no	—
234	C₄	C₃₇×#471	1	4	yes	—
235	C₄	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
236	C₄	C₃₇×#536	1	1	yes	characteristic
237	C₄	D₄×Q₈×C₃₇	1	2	yes	—
238	C₂	—	2	2	no	—
239	C₂	C₃₇×#2546	1	1	yes	characteristic
240	C₂	C₂×C₃₇×#471	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂³×C₃₇	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
45. #108×C₃₇	Q₈	semidirect
107. #28700	C₃₇	direct (N × H)
148. C₃₇	#28700	direct (N × H)
166. #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
C₃₇×#3165	2
C₂×C₃₇×#498	2
C₃₇×#2960	2
C₃₇×#2090	2
C₃₇×#1997	2
C₃₇×#1951	2
C₄×#108×C₃₇	2
#28700	37

Subgroups — C37×#28700 #22649953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28700 #22649953