Subgroups — C₂×C₃₇×#1568 #22639045

Order 9472.

766 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#432	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#1568	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₇×#335	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₇×#334	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₃₇	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₃₇	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₃₇	C₂²	1	1	yes	characteristic
9	C₃₇×#432	—	2	4	no	—
10	C₃₇×#432	C₂²	1	8	yes	—
11	C₂×C₄⋊C₈×C₃₇	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₈×C₃₇	—	2	2	no	—
13	C₂×#97×C₃₇	—	2	2	no	—
14	C₂×C₄×C₈×C₃₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₇	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₃₇	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₇	C₂²	1	4	yes	—
20	C₃₇×#335	C₂²	1	8	yes	complemented
21	C₃₇×#334	C₂²	1	8	yes	complemented
22	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₃₇	Q₈	1	1	yes	characteristic
25	C₂²×C₈×C₃₇	—	2	4	no	—
26	C₂²×C₈×C₃₇	—	2	4	no	—
27	C₂²×C₈×C₃₇	—	2	4	no	—
28	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₇	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₃₇	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₃₇	D₄	1	2	yes	—
37	C₂×C₄²×C₃₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₃₇	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₃₇	Q₈	1	2	yes	—
40	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₃₇	—	2	4	no	—
43	C₄⋊C₈×C₃₇	C₂³	1	8	yes	—
44	C₂²⋊C₈×C₃₇	—	2	8	no	—
45	#97×C₃₇	—	2	8	no	—
46	C₄×C₈×C₃₇	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₃₇	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₃₇	C₂³	1	2	yes	—
51	C₄×C₈×C₃₇	C₂³	1	2	yes	—
52	C₂⁴×C₃₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₇	—	2	4	no	—
54	C₂²×C₄×C₃₇	—	2	2	no	—
55	C₂²×C₄×C₃₇	—	2	2	no	—
56	C₂²×C₄×C₃₇	—	2	4	no	—
57	C₂²×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₃₇	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₇	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₃₇	—	2	2	no	—
70	C₂×C₈×C₃₇	—	2	2	no	—
71	C₂×C₈×C₃₇	—	2	8	no	—
72	C₂×C₈×C₃₇	—	2	8	no	—
73	C₂×C₈×C₃₇	—	2	8	no	—
74	C₂×C₈×C₃₇	—	4	8	no	—
75	C₂×C₈×C₃₇	M₁₆	1	4	yes	—
76	C₂×C₈×C₃₇	M₁₆	1	4	yes	—
77	C₂×C₈×C₃₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₃₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₃₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₃₇	C₂²×C₄	1	2	yes	—
81	C₄²×C₃₇	C₄⋊C₄	1	2	yes	—
82	C₄²×C₃₇	C₄⋊C₄	1	2	yes	—
83	C₄²×C₃₇	C₄⋊C₄	1	2	yes	—
84	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
85	C₄²×C₃₇	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₃₇	C₂×D₄	1	2	yes	—
87	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
89	C₄²×C₃₇	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₃₇	C₂⁴	1	1	yes	characteristic
91	C₂³×C₃₇	—	2	2	no	—
92	C₂³×C₃₇	—	2	2	no	—
93	C₂³×C₃₇	#108	1	4	yes	—
94	C₂³×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	—	2	2	no	—
98	C₂×C₄×C₃₇	—	2	4	no	—
99	C₂×C₄×C₃₇	—	2	2	no	—
100	C₂×C₄×C₃₇	—	2	4	no	—
101	C₂×C₄×C₃₇	—	2	2	no	—
102	C₂×C₄×C₃₇	—	2	2	no	—
103	C₂×C₄×C₃₇	—	2	2	no	—
104	C₂×C₄×C₃₇	—	2	4	no	—
105	C₂×C₄×C₃₇	—	2	4	no	—
106	C₂×C₄×C₃₇	—	2	4	no	—
107	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₇	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₃₇	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₃₇	#135	1	4	yes	—
116	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
119	C₈×C₃₇	—	2	2	no	—
120	C₈×C₃₇	—	2	2	no	—
121	C₈×C₃₇	—	4	8	no	—
122	C₈×C₃₇	C₂×M₁₆	1	4	yes	—
123	C₈×C₃₇	C₂×M₁₆	1	4	yes	—
124	C₂×#1568	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₃₇	—	2	8	no	—
126	C₂²×C₃₇	—	2	2	no	—
127	C₂²×C₃₇	—	2	2	no	—
128	C₂²×C₃₇	#423	1	2	yes	—
129	C₂²×C₃₇	#427	1	2	yes	—
130	C₂²×C₃₇	C₂×#108	1	4	yes	—
131	C₂²×C₃₇	C₂×#108	1	2	yes	—
132	C₂²×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₃₇	—	2	2	no	—
134	C₄×C₃₇	—	2	2	no	—
135	C₄×C₃₇	—	2	1	no	—
136	C₄×C₃₇	—	2	2	no	—
137	C₄×C₃₇	—	2	1	no	—
138	C₄×C₃₇	#423	1	2	yes	—
139	C₄×C₃₇	#427	1	2	yes	—
140	C₄×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
141	C₄×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
142	C₄×C₃₇	C₂×#135	1	2	yes	—
143	C₂×#432	C₂×C₃₇	1	2	yes	—
144	C₂²×C₄×C₈	C₂×C₃₇	1	1	yes	characteristic
145	#1568	C₂×C₃₇	1	8	yes	direct product factor
146	C₂×#335	C₂×C₃₇	1	2	yes	complemented
147	C₂×#334	C₂×C₃₇	1	2	yes	complemented
148	C₂×C₃₇	—	2	4	no	—
149	C₂×C₃₇	#1568	1	4	yes	direct product factor
150	C₂×C₃₇	C₂×#423	1	1	yes	characteristic
151	C₂×C₃₇	C₂×#427	1	1	yes	characteristic
152	C₂×C₃₇	C₂²×#108	1	1	yes	characteristic
153	C₂³×C₈	C₄×C₃₇	1	1	yes	characteristic
154	C₂³×C₈	C₄×C₃₇	1	1	yes	characteristic
155	C₂²×C₄²	C₂²×C₃₇	1	1	yes	characteristic
156	#432	—	2	4	no	—
157	#432	C₂²×C₃₇	1	8	yes	—
158	C₂×C₄⋊C₈	C₂²×C₃₇	1	4	yes	—
159	C₂×C₂²⋊C₈	—	2	2	no	—
160	C₂×#97	—	2	2	no	—
161	C₂×C₄×C₈	C₄×C₃₇	1	1	yes	characteristic
162	C₂×C₄×C₈	C₄×C₃₇	1	4	yes	—
163	C₂×C₄×C₈	C₄×C₃₇	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₃₇	1	4	yes	—
167	#335	C₂²×C₃₇	1	8	yes	complemented
168	#334	C₂²×C₃₇	1	8	yes	complemented
169	C₃₇	C₂×#1568	1	1	yes	characteristic, direct product factor, Sylow
170	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
171	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
172	C₂³×C₄	Q₈×C₃₇	1	1	yes	characteristic
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	—	2	4	no	—
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
177	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
178	C₂²×C₈	C₂×C₄×C₃₇	1	4	yes	—
179	C₂²×C₈	C₂×C₄×C₃₇	1	4	yes	—
180	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
181	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
182	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
183	C₂×C₄²	C₂×C₄×C₃₇	1	1	yes	characteristic
184	C₂×C₄²	D₄×C₃₇	1	1	yes	characteristic
185	C₂×C₄²	D₄×C₃₇	1	2	yes	—
186	C₂×C₄²	Q₈×C₃₇	1	1	yes	characteristic
187	C₂×C₄²	Q₈×C₃₇	1	2	yes	—
188	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
190	C₄⋊C₈	—	2	4	no	—
191	C₄⋊C₈	C₂³×C₃₇	1	8	yes	—
192	C₂²⋊C₈	—	2	8	no	—
193	#97	—	2	8	no	—
194	C₄×C₈	C₂×C₄×C₃₇	1	4	yes	—
195	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
196	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
197	C₄×C₈	C₂×C₄×C₃₇	1	4	yes	—
198	C₄×C₈	C₂³×C₃₇	1	2	yes	—
199	C₄×C₈	C₂³×C₃₇	1	2	yes	—
200	C₂⁴	C₄⋊C₄×C₃₇	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	C₄⋊C₄×C₃₇	1	2	yes	—
206	C₂²×C₄	C₄⋊C₄×C₃₇	1	1	yes	characteristic
207	C₂²×C₄	C₄⋊C₄×C₃₇	1	1	yes	characteristic
208	C₂²×C₄	C₄⋊C₄×C₃₇	1	2	yes	—
209	C₂²×C₄	C₄⋊C₄×C₃₇	1	1	yes	characteristic
210	C₂²×C₄	C₄⋊C₄×C₃₇	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×Q₈×C₃₇	1	2	yes	—
216	C₂²×C₄	C₂×Q₈×C₃₇	1	1	yes	characteristic
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	8	no	—
220	C₂×C₈	—	2	8	no	—
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	4	8	no	—
223	C₂×C₈	M₁₆×C₃₇	1	4	yes	—
224	C₂×C₈	M₁₆×C₃₇	1	4	yes	—
225	C₂×C₈	C₂²×C₄×C₃₇	1	2	yes	—
226	C₂×C₈	C₂²×C₄×C₃₇	1	2	yes	—
227	C₂×C₈	C₂²×C₄×C₃₇	1	2	yes	—
228	C₂×C₈	C₂²×C₄×C₃₇	1	2	yes	—
229	C₄²	C₄⋊C₄×C₃₇	1	2	yes	—
230	C₄²	C₄⋊C₄×C₃₇	1	2	yes	—
231	C₄²	C₄⋊C₄×C₃₇	1	2	yes	—
232	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
233	C₄²	C₂²×C₄×C₃₇	1	1	yes	characteristic
234	C₄²	C₂×D₄×C₃₇	1	2	yes	—
235	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
236	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
237	C₄²	C₂×Q₈×C₃₇	1	1	yes	characteristic
238	C₄²	C₂⁴×C₃₇	1	1	yes	characteristic
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	#108×C₃₇	1	4	yes	—
242	C₂³	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
243	C₂³	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
256	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
258	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	2	yes	—
259	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄⋊C₄×C₃₇	1	1	yes	characteristic
261	C₂×C₄	C₂×M₁₆×C₃₇	1	2	yes	—
262	C₂×C₄	C₂×M₁₆×C₃₇	1	2	yes	—
263	C₂×C₄	#135×C₃₇	1	4	yes	—
264	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
265	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	2	no	—
269	C₈	—	4	8	no	—
270	C₈	C₂×M₁₆×C₃₇	1	4	yes	—
271	C₈	C₂×M₁₆×C₃₇	1	4	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	8	no	—
275	C₂²	C₃₇×#423	1	2	yes	—
276	C₂²	C₃₇×#427	1	2	yes	—
277	C₂²	C₂×#108×C₃₇	1	4	yes	—
278	C₂²	C₂×#108×C₃₇	1	2	yes	—
279	C₂²	C₂²×C₄⋊C₄×C₃₇	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	1	no	—
285	C₄	C₃₇×#423	1	2	yes	—
286	C₄	C₃₇×#427	1	2	yes	—
287	C₄	C₂²×M₁₆×C₃₇	1	1	yes	characteristic
288	C₄	C₂²×M₁₆×C₃₇	1	1	yes	characteristic
289	C₄	C₂×#135×C₃₇	1	2	yes	—
290	C₂	—	2	4	no	—
291	C₂	C₃₇×#1568	1	4	yes	direct product factor
292	C₂	C₂×C₃₇×#423	1	1	yes	characteristic
293	C₂	C₂×C₃₇×#427	1	1	yes	characteristic
294	C₂	C₂²×#108×C₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₃₇×#1568	C₂	direct (N × H)
4. C₂×C₃₇×#335	C₂	semidirect
5. C₂×C₃₇×#334	C₂	semidirect
20. C₃₇×#335	C₂²	semidirect
21. C₃₇×#334	C₂²	semidirect
124. C₂×#1568	C₃₇	direct (N × H)
145. #1568	C₂×C₃₇	direct (N × H)
146. C₂×#335	C₂×C₃₇	semidirect
147. C₂×#334	C₂×C₃₇	semidirect
149. C₂×C₃₇	#1568	direct (N × H)
167. #335	C₂²×C₃₇	semidirect
168. #334	C₂²×C₃₇	semidirect
169. C₃₇	C₂×#1568	direct (N × H)
291. C₂	C₃₇×#1568	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×#432	2
C₂²×C₄×C₈×C₃₇	2
C₃₇×#1568	2
C₂×C₃₇×#335	2
C₂×C₃₇×#334	2
C₂×#1568	37

Subgroups — C2×C37×#1568 #22639045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×#1568 #22639045