Subgroups — C₂²×C₃₇×#484 #22681758

Order 9472.

2662 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₇×#484	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#107×C₃₇	C₂	1	2	yes	maximal, complemented
5	C₂²×#103×C₃₇	C₂	1	2	yes	maximal, complemented
6	C₂²×C₂²⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₃₇	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₃₇	—	2	1	no	—
9	C₂⁴×C₄×C₃₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₃₇	C₂²	1	2	yes	—
11	C₂×C₂²⋊Q₈×C₃₇	—	2	6	no	—
12	C₂×C₂²⋊Q₈×C₃₇	C₂²	1	12	yes	—
13	C₂²×C₄⋊C₄×C₃₇	—	2	6	no	—
14	C₂²×C₄⋊C₄×C₃₇	—	2	1	no	—
15	C₂²×C₄⋊C₄×C₃₇	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₃₇	C₂²	1	12	yes	—
19	C₂²×C₂²⋊C₄×C₃₇	—	2	1	no	—
20	C₃₇×#484	C₂²	1	64	yes	direct product factor
21	C₂×#107×C₃₇	C₂²	1	24	yes	complemented
22	C₂×#103×C₃₇	C₂²	1	24	yes	complemented
23	C₂×C₂²⋊C₈×C₃₇	C₂²	1	12	yes	—
24	C₂⁵×C₃₇	D₄	1	1	yes	characteristic
25	C₂²×Q₈×C₃₇	—	2	3	no	—
26	C₂²×Q₈×C₃₇	—	2	8	no	—
27	C₂²×Q₈×C₃₇	C₂³	1	6	yes	—
28	C₂³×C₄×C₃₇	—	2	8	no	—
29	C₂³×C₄×C₃₇	—	2	12	no	—
30	C₂³×C₄×C₃₇	—	2	2	no	—
31	C₂³×C₄×C₃₇	—	2	1	no	—
32	C₂³×C₄×C₃₇	—	2	1	no	—
33	C₂³×C₄×C₃₇	—	4	8	no	—
34	C₂³×C₄×C₃₇	D₄	1	6	yes	—
35	C₂³×C₄×C₃₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₃₇	C₂³	1	6	yes	—
37	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₃₇	—	2	8	no	—
39	C₂²×C₈×C₃₇	C₂³	1	12	yes	—
40	C₂²⋊Q₈×C₃₇	—	2	24	no	—
41	C₂²⋊Q₈×C₃₇	C₂³	1	16	yes	—
42	C₂×C₄⋊C₄×C₃₇	—	2	3	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	12	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	12	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	24	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	6	no	—
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	6	yes	—
48	C₂×C₄⋊C₄×C₃₇	C₂³	1	12	yes	—
49	C₂×C₄⋊C₄×C₃₇	C₂³	1	16	yes	—
50	C₂×C₂²⋊C₄×C₃₇	—	2	12	no	—
51	#107×C₃₇	C₂³	1	32	yes	complemented
52	#103×C₃₇	C₂³	1	32	yes	complemented
53	C₂²⋊C₈×C₃₇	C₂³	1	16	yes	—
54	C₂⁴×C₃₇	—	2	4	no	—
55	C₂⁴×C₃₇	—	2	4	no	—
56	C₂⁴×C₃₇	Q₁₆	1	8	yes	—
57	C₂⁴×C₃₇	C₂×D₄	1	6	yes	—
58	C₂⁴×C₃₇	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₃₇	—	2	6	no	—
60	C₂×Q₈×C₃₇	—	2	24	no	—
61	C₂×Q₈×C₃₇	—	4	12	no	—
62	C₂×Q₈×C₃₇	C₂⁴	1	4	yes	—
63	C₂²×C₄×C₃₇	—	2	4	no	—
64	C₂²×C₄×C₃₇	—	2	24	no	—
65	C₂²×C₄×C₃₇	—	2	24	no	—
66	C₂²×C₄×C₃₇	—	2	6	no	—
67	C₂²×C₄×C₃₇	—	2	6	no	—
68	C₂²×C₄×C₃₇	—	2	16	no	—
69	C₂²×C₄×C₃₇	—	2	12	no	—
70	C₂²×C₄×C₃₇	—	4	48	no	—
71	C₂²×C₄×C₃₇	—	4	4	no	—
72	C₂²×C₄×C₃₇	—	4	4	no	—
73	C₂²×C₄×C₃₇	—	4	8	no	—
74	C₂²×C₄×C₃₇	C₂×D₄	1	6	yes	—
75	C₂²×C₄×C₃₇	C₂×D₄	1	12	yes	—
76	C₂²×C₄×C₃₇	C₂×D₄	1	3	yes	—
77	C₂²×C₄×C₃₇	#41	1	8	yes	—
78	C₂²×C₄×C₃₇	C₂⁴	1	4	yes	—
79	C₂²×C₄×C₃₇	C₂⁴	1	3	yes	—
80	C₂×C₈×C₃₇	—	2	24	no	—
81	C₂×C₈×C₃₇	C₂⁴	1	8	yes	—
82	C₄⋊C₄×C₃₇	—	2	16	no	—
83	C₄⋊C₄×C₃₇	—	2	12	no	—
84	C₄⋊C₄×C₃₇	—	2	16	no	—
85	C₄⋊C₄×C₃₇	—	2	6	no	—
86	C₄⋊C₄×C₃₇	C₂⁴	1	8	yes	—
87	C₄⋊C₄×C₃₇	C₂⁴	1	4	yes	—
88	C₂²⋊C₄×C₃₇	—	2	16	no	—
89	C₂³×C₃₇	—	2	12	no	—
90	C₂³×C₃₇	—	2	12	no	—
91	C₂³×C₃₇	—	2	32	no	—
92	C₂³×C₃₇	#123	1	4	yes	—
93	C₂³×C₃₇	C₂×Q₁₆	1	24	yes	—
94	C₂³×C₃₇	C₂×Q₁₆	1	4	yes	—
95	C₂³×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
96	C₂³×C₃₇	C₂²×D₄	1	4	yes	—
97	C₂³×C₃₇	C₂²×D₄	1	3	yes	—
98	Q₈×C₃₇	—	2	8	no	—
99	Q₈×C₃₇	—	4	12	no	—
100	C₂×C₄×C₃₇	—	2	12	no	—
101	C₂×C₄×C₃₇	—	2	24	no	—
102	C₂×C₄×C₃₇	—	2	8	no	—
103	C₂×C₄×C₃₇	—	2	4	no	—
104	C₂×C₄×C₃₇	—	2	4	no	—
105	C₂×C₄×C₃₇	—	2	8	no	—
106	C₂×C₄×C₃₇	—	4	32	no	—
107	C₂×C₄×C₃₇	—	4	12	no	—
108	C₂×C₄×C₃₇	—	4	12	no	—
109	C₂×C₄×C₃₇	—	4	24	no	—
110	C₂×C₄×C₃₇	#123	1	12	yes	—
111	C₂×C₄×C₃₇	C₂²×D₄	1	4	yes	—
112	C₂×C₄×C₃₇	C₂²×D₄	1	3	yes	—
113	C₂×C₄×C₃₇	C₂×#41	1	12	yes	—
114	C₂×C₄×C₃₇	C₂⁵	1	1	yes	characteristic
115	C₈×C₃₇	—	2	8	no	—
116	C₂²×#484	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂²×C₃₇	—	2	48	no	—
118	C₂²×C₃₇	—	2	4	no	—
119	C₂²×C₃₇	—	2	4	no	—
120	C₂²×C₃₇	#484	1	16	yes	direct product factor
121	C₂²×C₃₇	C₂×#123	1	6	yes	—
122	C₂²×C₃₇	C₂²×Q₁₆	1	8	yes	—
123	C₂²×C₃₇	C₂²×Q₁₆	1	6	yes	—
124	C₂²×C₃₇	C₂×AGL(1,ℤ/8)	1	6	yes	—
125	C₂²×C₃₇	C₂³×D₄	1	1	yes	characteristic
126	C₄×C₃₇	—	2	4	no	—
127	C₄×C₃₇	—	4	4	no	—
128	C₄×C₃₇	—	4	4	no	—
129	C₄×C₃₇	—	4	8	no	—
130	C₄×C₃₇	C₂×#123	1	6	yes	—
131	C₄×C₃₇	C₂²×#41	1	2	yes	—
132	C₂²×C₂²⋊Q₈	C₂×C₃₇	1	1	yes	characteristic
133	C₂³×C₄⋊C₄	C₂×C₃₇	1	1	yes	characteristic
134	C₂×#484	C₂×C₃₇	1	24	yes	direct product factor
135	C₂²×#107	C₂×C₃₇	1	2	yes	complemented
136	C₂²×#103	C₂×C₃₇	1	2	yes	complemented
137	C₂²×C₂²⋊C₈	C₂×C₃₇	1	1	yes	characteristic
138	C₂×C₃₇	—	2	8	no	—
139	C₂×C₃₇	C₂×#484	1	12	yes	direct product factor
140	C₂×C₃₇	C₂²×#123	1	1	yes	characteristic
141	C₂×C₃₇	C₂³×Q₁₆	1	1	yes	characteristic
142	C₂×C₃₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂³×Q₈	C₂²×C₃₇	1	1	yes	characteristic
144	C₂⁴×C₄	—	2	1	no	—
145	C₂⁴×C₄	C₂²×C₃₇	1	1	yes	characteristic
146	C₂³×C₈	C₂²×C₃₇	1	2	yes	—
147	C₂×C₂²⋊Q₈	—	2	6	no	—
148	C₂×C₂²⋊Q₈	C₂²×C₃₇	1	12	yes	—
149	C₂²×C₄⋊C₄	—	2	1	no	—
150	C₂²×C₄⋊C₄	—	2	6	no	—
151	C₂²×C₄⋊C₄	—	2	1	no	—
152	C₂²×C₄⋊C₄	C₂²×C₃₇	1	12	yes	—
153	C₂²×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
154	C₂²×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
155	C₂²×C₂²⋊C₄	—	2	1	no	—
156	#484	C₂²×C₃₇	1	64	yes	direct product factor
157	C₂×#107	C₂²×C₃₇	1	24	yes	complemented
158	C₂×#103	C₂²×C₃₇	1	24	yes	complemented
159	C₂×C₂²⋊C₈	C₂²×C₃₇	1	12	yes	—
160	C₃₇	C₂²×#484	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁵	D₄×C₃₇	1	1	yes	characteristic
162	C₂²×Q₈	—	2	3	no	—
163	C₂²×Q₈	—	2	8	no	—
164	C₂²×Q₈	C₂³×C₃₇	1	6	yes	—
165	C₂³×C₄	—	2	8	no	—
166	C₂³×C₄	—	2	1	no	—
167	C₂³×C₄	—	2	1	no	—
168	C₂³×C₄	—	2	12	no	—
169	C₂³×C₄	—	2	2	no	—
170	C₂³×C₄	—	4	8	no	—
171	C₂³×C₄	D₄×C₃₇	1	1	yes	characteristic
172	C₂³×C₄	D₄×C₃₇	1	6	yes	—
173	C₂³×C₄	C₂³×C₃₇	1	6	yes	—
174	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
175	C₂²×C₈	—	2	8	no	—
176	C₂²×C₈	C₂³×C₃₇	1	12	yes	—
177	C₂²⋊Q₈	—	2	24	no	—
178	C₂²⋊Q₈	C₂³×C₃₇	1	16	yes	—
179	C₂×C₄⋊C₄	—	2	12	no	—
180	C₂×C₄⋊C₄	—	2	24	no	—
181	C₂×C₄⋊C₄	—	2	6	no	—
182	C₂×C₄⋊C₄	—	2	12	no	—
183	C₂×C₄⋊C₄	—	2	3	no	—
184	C₂×C₄⋊C₄	C₂³×C₃₇	1	12	yes	—
185	C₂×C₄⋊C₄	C₂³×C₃₇	1	16	yes	—
186	C₂×C₄⋊C₄	C₂³×C₃₇	1	6	yes	—
187	C₂×C₂²⋊C₄	—	2	12	no	—
188	#107	C₂³×C₃₇	1	32	yes	complemented
189	#103	C₂³×C₃₇	1	32	yes	complemented
190	C₂²⋊C₈	C₂³×C₃₇	1	16	yes	—
191	C₂⁴	—	2	4	no	—
192	C₂⁴	—	2	4	no	—
193	C₂⁴	Q₁₆×C₃₇	1	8	yes	—
194	C₂⁴	C₂×D₄×C₃₇	1	6	yes	—
195	C₂⁴	C₂×D₄×C₃₇	1	1	yes	characteristic
196	C₂×Q₈	—	2	6	no	—
197	C₂×Q₈	—	2	24	no	—
198	C₂×Q₈	—	4	12	no	—
199	C₂×Q₈	C₂⁴×C₃₇	1	4	yes	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	24	no	—
202	C₂²×C₄	—	2	24	no	—
203	C₂²×C₄	—	2	6	no	—
204	C₂²×C₄	—	2	6	no	—
205	C₂²×C₄	—	2	16	no	—
206	C₂²×C₄	—	2	12	no	—
207	C₂²×C₄	—	4	48	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	8	no	—
211	C₂²×C₄	C₂×D₄×C₃₇	1	6	yes	—
212	C₂²×C₄	C₂×D₄×C₃₇	1	12	yes	—
213	C₂²×C₄	C₂×D₄×C₃₇	1	3	yes	—
214	C₂²×C₄	#41×C₃₇	1	8	yes	—
215	C₂²×C₄	C₂⁴×C₃₇	1	4	yes	—
216	C₂²×C₄	C₂⁴×C₃₇	1	3	yes	—
217	C₂×C₈	—	2	24	no	—
218	C₂×C₈	C₂⁴×C₃₇	1	8	yes	—
219	C₄⋊C₄	—	2	16	no	—
220	C₄⋊C₄	—	2	12	no	—
221	C₄⋊C₄	—	2	16	no	—
222	C₄⋊C₄	—	2	6	no	—
223	C₄⋊C₄	C₂⁴×C₃₇	1	8	yes	—
224	C₄⋊C₄	C₂⁴×C₃₇	1	4	yes	—
225	C₂²⋊C₄	—	2	16	no	—
226	C₂³	—	2	12	no	—
227	C₂³	—	2	12	no	—
228	C₂³	—	2	32	no	—
229	C₂³	#123×C₃₇	1	4	yes	—
230	C₂³	C₂×Q₁₆×C₃₇	1	24	yes	—
231	C₂³	C₂×Q₁₆×C₃₇	1	4	yes	—
232	C₂³	AGL(1,ℤ/8)×C₃₇	1	4	yes	—
233	C₂³	C₂²×D₄×C₃₇	1	4	yes	—
234	C₂³	C₂²×D₄×C₃₇	1	3	yes	—
235	Q₈	—	2	8	no	—
236	Q₈	—	4	12	no	—
237	C₂×C₄	—	2	12	no	—
238	C₂×C₄	—	2	24	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	4	32	no	—
244	C₂×C₄	—	4	12	no	—
245	C₂×C₄	—	4	24	no	—
246	C₂×C₄	—	4	12	no	—
247	C₂×C₄	#123×C₃₇	1	12	yes	—
248	C₂×C₄	C₂²×D₄×C₃₇	1	4	yes	—
249	C₂×C₄	C₂²×D₄×C₃₇	1	3	yes	—
250	C₂×C₄	C₂×#41×C₃₇	1	12	yes	—
251	C₂×C₄	C₂⁵×C₃₇	1	1	yes	characteristic
252	C₈	—	2	8	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	48	no	—
256	C₂²	C₃₇×#484	1	16	yes	direct product factor
257	C₂²	C₂×#123×C₃₇	1	6	yes	—
258	C₂²	C₂²×Q₁₆×C₃₇	1	8	yes	—
259	C₂²	C₂²×Q₁₆×C₃₇	1	6	yes	—
260	C₂²	C₂×AGL(1,ℤ/8)×C₃₇	1	6	yes	—
261	C₂²	C₂³×D₄×C₃₇	1	1	yes	characteristic
262	C₄	—	2	4	no	—
263	C₄	—	4	4	no	—
264	C₄	—	4	8	no	—
265	C₄	—	4	4	no	—
266	C₄	C₂×#123×C₃₇	1	6	yes	—
267	C₄	C₂²×#41×C₃₇	1	2	yes	—
268	C₂	—	2	8	no	—
269	C₂	C₂×C₃₇×#484	1	12	yes	direct product factor
270	C₂	C₂²×#123×C₃₇	1	1	yes	characteristic
271	C₂	C₂³×Q₁₆×C₃₇	1	1	yes	characteristic
272	C₂	C₂²×AGL(1,ℤ/8)×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₇
prime radical	C₂⁵×C₃₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₇ ◁ C₂⁴×C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃₇×#484	C₂	direct (N × H)
4. C₂²×#107×C₃₇	C₂	semidirect
5. C₂²×#103×C₃₇	C₂	semidirect
20. C₃₇×#484	C₂²	direct (N × H)
21. C₂×#107×C₃₇	C₂²	semidirect
22. C₂×#103×C₃₇	C₂²	semidirect
51. #107×C₃₇	C₂³	semidirect
52. #103×C₃₇	C₂³	semidirect
116. C₂²×#484	C₃₇	direct (N × H)
120. C₂²×C₃₇	#484	direct (N × H)
134. C₂×#484	C₂×C₃₇	direct (N × H)
135. C₂²×#107	C₂×C₃₇	semidirect
136. C₂²×#103	C₂×C₃₇	semidirect
139. C₂×C₃₇	C₂×#484	direct (N × H)
156. #484	C₂²×C₃₇	direct (N × H)
157. C₂×#107	C₂²×C₃₇	semidirect
158. C₂×#103	C₂²×C₃₇	semidirect
160. C₃₇	C₂²×#484	direct (N × H)
188. #107	C₂³×C₃₇	semidirect
189. #103	C₂³×C₃₇	semidirect
256. C₂²	C₃₇×#484	direct (N × H)
269. C₂	C₂×C₃₇×#484	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊Q₈×C₃₇	2
C₂³×C₄⋊C₄×C₃₇	2
C₂×C₃₇×#484	2
C₂²×#107×C₃₇	2
C₂²×#103×C₃₇	2
C₂²×C₂²⋊C₈×C₃₇	2
C₂²×#484	37

Subgroups — C22×C37×#484 #22681758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₇×#484 #22681758