Subgroups — C₃₇×#17856 #22639109

Order 9472.

678 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#2982	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#1579	C₂	1	4	yes	maximal, complemented
4	C₈×Q₁₆×C₃₇	C₂	1	4	yes	maximal, complemented
5	C₃₇×#1564	C₂	1	1	yes	maximal, characteristic
6	C₃₇×#1479	C₂	1	2	yes	maximal
7	C₃₇×#1450	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₃₇	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
10	C₈×Q₈×C₃₇	—	2	4	no	—
11	C₈×Q₈×C₃₇	C₂²	1	8	yes	—
12	C₄×Q₁₆×C₃₇	C₄	1	2	yes	—
13	C₄×Q₁₆×C₃₇	C₄	1	2	yes	—
14	C₄×Q₁₆×C₃₇	C₂²	1	2	yes	—
15	C₄×Q₁₆×C₃₇	C₂²	1	2	yes	—
16	C₃₇×#431	—	2	4	no	—
17	C₃₇×#431	C₂²	1	2	yes	—
18	C₃₇×#431	C₂²	1	1	yes	characteristic
19	C₃₇×#431	C₂²	1	1	yes	characteristic
20	C₂×#107×C₃₇	C₄	1	1	yes	characteristic
21	C₂×#103×C₃₇	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₃₇×#335	C₂²	1	2	yes	—
24	C₃₇×#334	C₂²	1	2	yes	—
25	C₃₇×Q₈⋊C₈	C₂²	1	8	yes	—
26	C₃₇×#322	C₂²	1	2	yes	—
27	C₈²×C₃₇	C₂²	1	2	yes	—
28	C₂²×Q₈×C₃₇	C₂×C₄	1	2	yes	—
29	C₂×Q₁₆×C₃₇	C₈	1	8	yes	complemented
30	C₂×Q₁₆×C₃₇	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆×C₃₇	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₃₇	—	2	2	no	—
35	C₄×Q₈×C₃₇	—	2	2	no	—
36	C₄×Q₈×C₃₇	C₂×C₄	1	4	yes	—
37	C₄×Q₈×C₃₇	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₃₇	—	2	2	no	—
41	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
42	#107×C₃₇	C₂×C₄	1	2	yes	—
43	#107×C₃₇	C₂×C₄	1	2	yes	—
44	C₄⋊C₈×C₃₇	—	2	8	no	—
45	C₄⋊C₈×C₃₇	—	2	1	no	—
46	C₄⋊C₈×C₃₇	D₄	1	2	yes	—
47	C₄⋊C₈×C₃₇	C₂³	1	4	yes	—
48	#103×C₃₇	C₂×C₄	1	4	yes	—
49	#103×C₃₇	C₂×C₄	1	4	yes	—
50	C₂²⋊C₈×C₃₇	—	2	2	no	—
51	C₂²⋊C₈×C₃₇	—	2	1	no	—
52	C₂²⋊C₈×C₃₇	D₄	1	2	yes	—
53	C₄×C₈×C₃₇	—	2	8	no	—
54	C₄×C₈×C₃₇	—	2	2	no	—
55	C₄×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₃₇	C₂³	1	2	yes	—
58	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₃₇	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₃₇	C₂³	1	2	yes	—
61	C₂×Q₈×C₃₇	—	2	2	no	—
62	C₂×Q₈×C₃₇	—	2	4	no	—
63	C₂×Q₈×C₃₇	C₂×C₈	1	8	yes	—
64	C₂×Q₈×C₃₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₃₇	—	2	2	no	—
66	C₂²×C₄×C₃₇	—	2	2	no	—
67	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₇	#41	1	1	yes	characteristic
70	Q₁₆×C₃₇	C₂×C₈	1	8	yes	—
71	Q₁₆×C₃₇	C₂×C₈	1	8	yes	—
72	C₂×C₈×C₃₇	—	2	2	no	—
73	C₂×C₈×C₃₇	—	2	4	no	—
74	C₂×C₈×C₃₇	—	2	2	no	—
75	C₂×C₈×C₃₇	C₂×C₈	1	4	yes	—
76	C₂×C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
81	C₄⋊C₄×C₃₇	—	2	4	no	—
82	C₄⋊C₄×C₃₇	—	2	2	no	—
83	C₄⋊C₄×C₃₇	—	2	4	no	—
84	C₄⋊C₄×C₃₇	C₂²×C₄	1	4	yes	—
85	C₄²×C₃₇	—	2	4	no	—
86	C₄²×C₃₇	—	2	4	no	—
87	C₄²×C₃₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₃₇	#41	1	1	yes	characteristic
90	C₄²×C₃₇	C₂⁴	1	1	yes	characteristic
91	C₂³×C₃₇	C₄×D₄	1	1	yes	characteristic
92	Q₈×C₃₇	—	2	4	no	—
93	Q₈×C₃₇	—	2	8	no	—
94	Q₈×C₃₇	C₂²×C₈	1	8	yes	—
95	C₂×C₄×C₃₇	—	2	4	no	—
96	C₂×C₄×C₃₇	—	2	8	no	—
97	C₂×C₄×C₃₇	—	2	2	no	—
98	C₂×C₄×C₃₇	—	2	4	no	—
99	C₂×C₄×C₃₇	—	4	1	no	—
100	C₂×C₄×C₃₇	—	8	2	no	—
101	C₂×C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₇	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₃₇	#131	1	2	yes	—
106	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
109	C₈×C₃₇	—	2	1	no	—
110	C₈×C₃₇	—	2	1	no	—
111	C₈×C₃₇	—	2	4	no	—
112	C₈×C₃₇	—	4	2	no	—
113	C₈×C₃₇	—	4	4	no	—
114	C₈×C₃₇	C₂²×C₈	1	2	yes	—
115	C₈×C₃₇	C₂²×C₈	1	2	yes	—
116	#17856	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂²×C₃₇	—	2	1	no	—
118	C₂²×C₃₇	—	2	1	no	—
119	C₂²×C₃₇	C₈×D₄	1	2	yes	—
120	C₂²×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄×C₃₇	—	2	8	no	—
122	C₄×C₃₇	—	2	1	no	—
123	C₄×C₃₇	—	2	1	no	—
124	C₄×C₃₇	—	2	1	no	—
125	C₄×C₃₇	—	4	4	no	—
126	C₄×C₃₇	C₈×D₄	1	2	yes	—
127	C₄×C₃₇	C₂³×C₈	1	1	yes	characteristic
128	C₄×C₃₇	C₂×#131	1	1	yes	characteristic
129	C₄×C₃₇	#578	1	2	yes	—
130	#2982	C₂×C₃₇	1	2	yes	—
131	C₂×C₄×Q₁₆	C₂×C₃₇	1	1	yes	characteristic
132	#1579	C₂×C₃₇	1	4	yes	complemented
133	C₈×Q₁₆	C₂×C₃₇	1	4	yes	complemented
134	#1564	C₂×C₃₇	1	1	yes	characteristic
135	#1479	C₂×C₃₇	1	2	yes	—
136	#1450	C₂×C₃₇	1	1	yes	characteristic
137	C₂×C₃₇	—	2	2	no	—
138	C₂×C₃₇	C₂×C₈×D₄	1	1	yes	characteristic
139	C₂×C₃₇	#2950	1	1	yes	characteristic
140	C₂×C₃₇	#2957	1	1	yes	characteristic
141	C₂²×Q₁₆	C₄×C₃₇	1	1	yes	characteristic
142	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
143	C₈×Q₈	—	2	4	no	—
144	C₈×Q₈	C₂²×C₃₇	1	8	yes	—
145	C₄×Q₁₆	C₄×C₃₇	1	2	yes	—
146	C₄×Q₁₆	C₄×C₃₇	1	2	yes	—
147	C₄×Q₁₆	C₂²×C₃₇	1	2	yes	—
148	C₄×Q₁₆	C₂²×C₃₇	1	2	yes	—
149	#431	—	2	4	no	—
150	#431	C₂²×C₃₇	1	2	yes	—
151	#431	C₂²×C₃₇	1	1	yes	characteristic
152	#431	C₂²×C₃₇	1	1	yes	characteristic
153	C₂×#107	C₄×C₃₇	1	1	yes	characteristic
154	C₂×#103	C₄×C₃₇	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
156	#335	C₂²×C₃₇	1	2	yes	—
157	#334	C₂²×C₃₇	1	2	yes	—
158	Q₈⋊C₈	C₂²×C₃₇	1	8	yes	—
159	#322	C₂²×C₃₇	1	2	yes	—
160	C₈²	C₂²×C₃₇	1	2	yes	—
161	C₃₇	#17856	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²×Q₈	C₂×C₄×C₃₇	1	2	yes	—
163	C₂×Q₁₆	C₈×C₃₇	1	8	yes	complemented
164	C₂×Q₁₆	C₂×C₄×C₃₇	1	2	yes	—
165	C₂×Q₁₆	C₂×C₄×C₃₇	1	2	yes	—
166	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
167	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₄×Q₈	C₂×C₄×C₃₇	1	4	yes	—
171	C₄×Q₈	C₂³×C₃₇	1	4	yes	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	2	yes	—
174	C₂×C₄²	—	2	2	no	—
175	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
176	#107	C₂×C₄×C₃₇	1	2	yes	—
177	#107	C₂×C₄×C₃₇	1	2	yes	—
178	C₄⋊C₈	—	2	8	no	—
179	C₄⋊C₈	—	2	1	no	—
180	C₄⋊C₈	D₄×C₃₇	1	2	yes	—
181	C₄⋊C₈	C₂³×C₃₇	1	4	yes	—
182	#103	C₂×C₄×C₃₇	1	4	yes	—
183	#103	C₂×C₄×C₃₇	1	4	yes	—
184	C₂²⋊C₈	—	2	2	no	—
185	C₂²⋊C₈	—	2	1	no	—
186	C₂²⋊C₈	D₄×C₃₇	1	2	yes	—
187	C₄×C₈	—	2	8	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
190	C₄×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
191	C₄×C₈	C₂³×C₃₇	1	2	yes	—
192	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₃₇	1	1	yes	characteristic
194	C₄×C₈	C₂³×C₃₇	1	2	yes	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	C₂×C₈×C₃₇	1	8	yes	—
198	C₂×Q₈	C₂²×C₄×C₃₇	1	4	yes	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
203	C₂²×C₄	#41×C₃₇	1	1	yes	characteristic
204	Q₁₆	C₂×C₈×C₃₇	1	8	yes	—
205	Q₁₆	C₂×C₈×C₃₇	1	8	yes	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	C₂×C₈×C₃₇	1	4	yes	—
210	C₂×C₈	C₂²×C₄×C₃₇	1	1	yes	characteristic
211	C₂×C₈	C₂²×C₄×C₃₇	1	1	yes	characteristic
212	C₂×C₈	C₂²×C₄×C₃₇	1	1	yes	characteristic
213	C₂×C₈	C₂²×C₄×C₃₇	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	C₂²×C₄×C₃₇	1	4	yes	—
219	C₄²	—	2	4	no	—
220	C₄²	—	2	4	no	—
221	C₄²	C₂²×C₄×C₃₇	1	1	yes	characteristic
222	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
223	C₄²	#41×C₃₇	1	1	yes	characteristic
224	C₄²	C₂⁴×C₃₇	1	1	yes	characteristic
225	C₂³	C₄×D₄×C₃₇	1	1	yes	characteristic
226	Q₈	—	2	4	no	—
227	Q₈	—	2	8	no	—
228	Q₈	C₂²×C₈×C₃₇	1	8	yes	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	C₄×D₄×C₃₇	1	1	yes	characteristic
236	C₂×C₄	C₄×D₄×C₃₇	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₃₇	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₈×C₃₇	1	2	yes	—
239	C₂×C₄	#131×C₃₇	1	2	yes	—
240	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
242	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	4	no	—
246	C₈	—	4	2	no	—
247	C₈	—	4	4	no	—
248	C₈	C₂²×C₈×C₃₇	1	2	yes	—
249	C₈	C₂²×C₈×C₃₇	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	C₈×D₄×C₃₇	1	2	yes	—
253	C₂²	C₂×C₄×D₄×C₃₇	1	1	yes	characteristic
254	C₄	—	2	8	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	4	4	no	—
259	C₄	C₈×D₄×C₃₇	1	2	yes	—
260	C₄	C₂³×C₈×C₃₇	1	1	yes	characteristic
261	C₄	C₂×#131×C₃₇	1	1	yes	characteristic
262	C₄	C₃₇×#578	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×C₈×D₄×C₃₇	1	1	yes	characteristic
265	C₂	C₃₇×#2950	1	1	yes	characteristic
266	C₂	C₃₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂²×D₄
commutator	C₄	C₂³×C₈×C₃₇
Frattini, cosocle	C₄²	C₂⁴×C₃₇
socle	C₂²×C₃₇	C₂×C₄×D₄
prime radical	C₂³×C₃₇	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₃₇×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₇×#1579	C₂	semidirect
4. C₈×Q₁₆×C₃₇	C₂	semidirect
29. C₂×Q₁₆×C₃₇	C₈	semidirect
116. #17856	C₃₇	direct (N × H)
132. #1579	C₂×C₃₇	semidirect
133. C₈×Q₁₆	C₂×C₃₇	semidirect
161. C₃₇	#17856	direct (N × H)
163. C₂×Q₁₆	C₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#2982	2
C₂×C₄×Q₁₆×C₃₇	2
C₃₇×#1579	2
C₈×Q₁₆×C₃₇	2
C₃₇×#1564	2
C₃₇×#1479	2
C₃₇×#1450	2
#17856	37

Subgroups — C37×#17856 #22639109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#17856 #22639109