Subgroups — #476×D₃₇ #18004343

Order 4736.

480 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	C₄⋊Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₃₇	C₂	1	1	yes	maximal, characteristic
3	#14557824	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557823	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557822	C₂	1	2	yes	maximal, complemented
6	#14557821	C₂	1	2	yes	maximal, complemented
7	#14557820	C₂	1	1	yes	maximal, characteristic, complemented
8	#106×D₃₇	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₈×D₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#102×D₃₇	C₂	1	2	yes	maximal
11	C₃₇×#476	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₃₇	—	2	1	no	—
13	C₂×D₄×D₃₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₃₇	—	2	1	no	—
15	C₂×C₈×D₃₇	C₂²	1	2	yes	—
16	#11924536	C₂²	1	1	yes	characteristic, complemented
17	#11924535	—	2	1	no	—
18	#11924535	C₂²	1	2	yes	complemented
19	#11924534	—	2	1	no	—
20	C₄×D₁₄₈	C₂²	1	1	yes	characteristic
21	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
22	#11924498	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₃₇	—	2	1	no	—
24	#11924496	—	2	1	no	—
25	C₄⋊C₄×D₃₇	—	2	1	no	—
26	C₄⋊C₄×D₃₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₃₇	C₂²	1	2	yes	—
28	C₂²⋊C₄×D₃₇	—	2	1	no	—
29	C₄²×D₃₇	C₂²	1	1	yes	characteristic
30	#11924468	C₂²	1	2	yes	complemented
31	#11924467	C₂²	1	2	yes	complemented
32	#11924467	C₂²	1	2	yes	complemented
33	C₄⋊#64436	C₂²	1	1	yes	characteristic, complemented
34	#11924465	C₂²	1	2	yes	complemented
35	#11924458	C₂²	1	2	yes	—
36	#11924457	C₂²	1	2	yes	—
37	#11924456	C₂²	1	2	yes	—
38	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
39	C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
40	#106×C₃₇	C₂²	1	2	yes	complemented
41	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
42	#102×C₃₇	C₂²	1	2	yes	—
43	C₂³×D₃₇	—	2	1	no	—
44	C₂×Q₈×C₃₇	—	2	1	no	—
45	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₃₇	—	2	1	no	—
47	C₂×C₂²⋊D₃₇	—	2	1	no	—
48	C₂²×Dic₃₇	—	2	1	no	—
49	Q₈×D₃₇	—	4	4	no	—
50	D₄×D₃₇	—	2	2	no	—
51	D₄×D₃₇	Q₈	1	4	yes	complemented
52	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₃₇	—	2	1	no	—
54	C₂×C₄×D₃₇	—	2	2	no	—
55	C₂×C₄×D₃₇	—	2	1	no	—
56	C₂×C₄×D₃₇	—	2	1	no	—
57	C₂×C₄×D₃₇	—	4	4	no	—
58	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
59	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
60	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
61	C₂×Dic₇₄	—	2	1	no	—
62	C₂×Dic₇₄	—	2	2	no	—
63	C₂×C₈×C₃₇	C₂³	1	2	yes	—
64	C₄⋊C₄×C₃₇	—	2	1	no	—
65	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
67	C₂²⋊C₄×C₃₇	—	2	1	no	—
68	C₄²×C₃₇	C₂³	1	1	yes	characteristic
69	C₂²⋊Dic₃₇	—	2	1	no	—
70	#10596938	—	2	1	no	—
71	#10596938	—	2	1	no	—
72	C₄⋊Dic₃₇	—	2	1	no	—
73	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₃₇	C₂³	1	2	yes	—
75	#10596936	—	2	2	no	—
76	#10596936	—	2	1	no	—
77	#10596936	—	2	2	no	—
78	C₄×Dic₃₇	D₄	1	1	yes	characteristic
79	C₄×Dic₃₇	D₄	1	1	yes	characteristic
80	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
81	C₂×#64436	C₂³	1	2	yes	—
82	C₈×D₃₇	—	2	4	no	—
83	C₂³×C₃₇	—	2	1	no	—
84	C₂²×D₃₇	—	2	1	no	—
85	C₂²×D₃₇	—	2	4	no	—
86	C₂²×D₃₇	—	4	2	no	—
87	C₂²×D₃₇	—	4	2	no	—
88	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₃₇	—	4	2	no	—
90	D₄×C₃₇	—	2	1	no	—
91	D₄×C₃₇	C₂×Q₈	1	2	yes	complemented
92	C₂×C₄×C₃₇	—	2	2	no	—
93	C₂×C₄×C₃₇	—	2	1	no	—
94	C₂×C₄×C₃₇	—	4	2	no	—
95	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₃₇	—	2	4	no	—
99	C₂×Dic₃₇	—	2	2	no	—
100	C₂×Dic₃₇	—	2	1	no	—
101	C₂×Dic₃₇	—	4	2	no	—
102	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
103	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
105	D₁₄₈	—	2	1	no	—
106	D₁₄₈	C₂×Q₈	1	2	yes	complemented
107	C₄×D₃₇	—	2	2	no	—
108	C₄×D₃₇	—	4	2	no	—
109	C₄×D₃₇	—	4	4	no	—
110	C₄×D₃₇	—	4	2	no	—
111	C₄×D₃₇	—	4	2	no	—
112	C₄×D₃₇	SD₁₆	1	4	yes	—
113	C₄×D₃₇	C₂×Q₈	1	2	yes	—
114	C₄×D₃₇	#41	1	2	yes	—
115	Dic₇₄	—	4	4	no	—
116	Dic₇₄	—	4	2	no	—
117	C₈×C₃₇	—	2	2	no	—
118	#64436	—	2	2	no	—
119	C₂²×C₃₇	—	2	2	no	—
120	C₂²×C₃₇	—	4	1	no	—
121	C₂²×C₃₇	—	4	1	no	—
122	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₃₇	—	2	2	no	—
124	C₂×D₃₇	—	4	1	no	—
125	C₂×D₃₇	—	4	1	no	—
126	C₂×D₃₇	—	4	8	no	—
127	C₂×D₃₇	C₂²⋊Q₈	1	2	yes	—
128	C₂×D₃₇	C₂×SD₁₆	1	2	yes	—
129	C₂×D₃₇	AGL(1,ℤ/8)	1	2	yes	—
130	C₄×C₃₇	—	2	1	no	—
131	C₄×C₃₇	—	4	1	no	—
132	C₄×C₃₇	—	4	2	no	—
133	C₄×C₃₇	C₂×SD₁₆	1	2	yes	—
134	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
135	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
136	Dic₃₇	—	2	1	no	—
137	Dic₃₇	—	4	1	no	—
138	Dic₃₇	—	4	2	no	—
139	Dic₃₇	C₂²⋊Q₈	1	2	yes	—
140	Dic₃₇	C₂×SD₁₆	1	2	yes	—
141	C₂×#476	—	37	1	no	maximal, Sylow
142	C₂×C₃₇	—	4	2	no	—
143	C₂×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
144	C₂×C₃₇	C₂²×SD₁₆	1	1	yes	characteristic
145	C₂×C₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
146	D₃₇	—	4	2	no	—
147	D₃₇	#476	1	4	yes	direct product factor
148	C₂×C₄⋊Q₈	—	37	1	no	—
149	C₂×C₄×D₄	—	37	1	no	—
150	#476	—	37	1	no	—
151	#476	—	37	1	no	—
152	#476	—	37	2	no	—
153	#476	—	37	1	no	—
154	#476	—	37	2	no	—
155	#476	D₃₇	1	1	yes	characteristic, direct product factor
156	C₂×#106	—	37	2	no	—
157	C₂×C₄⋊C₈	—	37	1	no	—
158	C₂×#102	—	37	2	no	—
159	C₃₇	C₂×#476	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	74	1	no	—
161	C₂²×D₄	—	37	1	no	—
162	C₂³×C₄	—	74	1	no	—
163	C₂²×C₈	—	37	2	no	—
164	C₄⋊Q₈	—	37	2	no	—
165	C₄⋊Q₈	—	37	1	no	—
166	C₄⋊Q₈	—	74	1	no	—
167	C₄⋊Q₈	—	74	1	no	—
168	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic, complemented
169	C₄×D₄	—	37	1	no	—
170	C₄×D₄	—	37	1	no	—
171	C₄×D₄	—	37	1	no	—
172	C₄×D₄	—	74	1	no	—
173	C₄×D₄	—	74	1	no	—
174	C₄×D₄	C₂×D₃₇	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	37	1	no	—
176	C₂×C₄⋊C₄	—	37	2	no	—
177	C₂×C₄⋊C₄	—	74	1	no	—
178	C₂×C₂²⋊C₄	—	74	1	no	—
179	C₂×C₄²	—	37	1	no	—
180	#106	—	37	2	no	—
181	#106	—	37	2	no	—
182	#106	—	37	2	no	—
183	#106	C₂×D₃₇	1	2	yes	complemented
184	C₄⋊C₈	—	37	2	no	—
185	C₄⋊C₈	—	37	1	no	—
186	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic, complemented
187	#102	—	37	2	no	—
188	#102	—	37	2	no	—
189	#102	—	37	2	no	—
190	#102	C₂×D₃₇	1	2	yes	—
191	C₂⁴	—	74	1	no	—
192	C₂×Q₈	—	2	1	no	—
193	C₂×Q₈	—	74	2	no	—
194	C₂×Q₈	—	74	1	no	—
195	C₂×Q₈	—	148	4	no	—
196	C₂×D₄	—	37	4	no	—
197	C₂×D₄	—	37	1	no	—
198	C₂×D₄	—	74	2	no	—
199	C₂×D₄	—	74	1	no	—
200	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	37	1	no	—
203	C₂²×C₄	—	37	1	no	—
204	C₂²×C₄	—	37	1	no	—
205	C₂²×C₄	—	74	1	no	—
206	C₂²×C₄	—	74	1	no	—
207	C₂²×C₄	—	74	2	no	—
208	C₂²×C₄	—	74	1	no	—
209	C₂²×C₄	—	74	1	no	—
210	C₂²×C₄	—	148	4	no	—
211	C₂×C₈	—	37	2	no	—
212	C₂×C₈	—	74	4	no	—
213	C₂×C₈	C₂²×D₃₇	1	2	yes	—
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	—	37	2	no	—
216	C₄⋊C₄	—	37	1	no	—
217	C₄⋊C₄	—	74	1	no	—
218	C₄⋊C₄	—	74	2	no	—
219	C₄⋊C₄	—	74	2	no	—
220	C₄⋊C₄	—	74	1	no	—
221	C₄⋊C₄	C₂²×D₃₇	1	2	yes	—
222	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
223	C₂²⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	74	1	no	—
225	C₂²⋊C₄	—	74	1	no	—
226	C₂²⋊C₄	—	74	1	no	—
227	C₄²	—	37	1	no	—
228	C₄²	—	37	1	no	—
229	C₄²	—	37	1	no	—
230	C₄²	C₂²×D₃₇	1	1	yes	characteristic
231	C₂³	—	2	1	no	—
232	C₂³	—	37	1	no	—
233	C₂³	—	74	4	no	—
234	C₂³	—	74	1	no	—
235	C₂³	—	148	2	no	—
236	C₂³	—	148	2	no	—
237	Q₈	—	4	2	no	—
238	Q₈	—	148	2	no	—
239	Q₈	—	148	4	no	—
240	D₄	—	2	1	no	—
241	D₄	—	37	2	no	—
242	D₄	—	74	4	no	—
243	D₄	—	74	1	no	—
244	D₄	Q₈×D₃₇	1	2	yes	complemented
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	37	1	no	—
249	C₂×C₄	—	37	1	no	—
250	C₂×C₄	—	37	4	no	—
251	C₂×C₄	—	37	2	no	—
252	C₂×C₄	—	37	2	no	—
253	C₂×C₄	—	37	1	no	—
254	C₂×C₄	—	74	2	no	—
255	C₂×C₄	—	74	1	no	—
256	C₂×C₄	—	74	2	no	—
257	C₂×C₄	—	148	2	no	—
258	C₂×C₄	—	148	2	no	—
259	C₂×C₄	—	148	4	no	—
260	C₂×C₄	—	148	2	no	—
261	C₂×C₄	—	148	2	no	—
262	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
263	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
264	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	74	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	37	2	no	—
271	C₂²	—	37	2	no	—
272	C₂²	—	37	2	no	—
273	C₂²	—	74	2	no	—
274	C₂²	—	148	8	no	—
275	C₂²	—	148	1	no	—
276	C₂²	—	148	1	no	—
277	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	2	no	—
281	C₄	—	37	2	no	—
282	C₄	—	37	2	no	—
283	C₄	—	74	1	no	—
284	C₄	—	148	1	no	—
285	C₄	—	148	2	no	—
286	C₄	SD₁₆×D₃₇	1	2	yes	—
287	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
288	C₄	#41×D₃₇	1	1	yes	characteristic
289	C₂	—	4	2	no	—
290	C₂	—	37	4	no	—
291	C₂	—	148	2	no	—
292	C₂	C₂²⋊Q₈×D₃₇	1	1	yes	characteristic
293	C₂	C₂×SD₁₆×D₃₇	1	1	yes	characteristic
294	C₂	AGL(1,ℤ/8)×D₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#476	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×D₄×D₃₇	C₂²
hypercenter	#476	D₃₇
hypocenter	C₃₇	C₂×#476
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. C₄⋊Q₈×D₃₇	C₂	semidirect
3. #14557824	C₂	semidirect
4. #14557823	C₂	semidirect
5. #14557822	C₂	semidirect
6. #14557821	C₂	semidirect
7. #14557820	C₂	semidirect
8. #106×D₃₇	C₂	semidirect
9. C₄⋊C₈×D₃₇	C₂	semidirect
11. C₃₇×#476	C₂	semidirect
16. #11924536	C₂²	semidirect
18. #11924535	C₂²	semidirect
30. #11924468	C₂²	semidirect
31. #11924467	C₂²	semidirect
32. #11924467	C₂²	semidirect
33. C₄⋊#64436	C₂²	semidirect
34. #11924465	C₂²	semidirect
38. C₄⋊Q₈×C₃₇	C₂²	semidirect
40. #106×C₃₇	C₂²	semidirect
41. C₄⋊C₈×C₃₇	C₂²	semidirect
51. D₄×D₃₇	Q₈	semidirect
91. D₄×C₃₇	C₂×Q₈	semidirect
106. D₁₄₈	C₂×Q₈	semidirect
147. D₃₇	#476	direct (N × H)
155. #476	D₃₇	direct (N × H)
159. C₃₇	C₂×#476	semidirect
168. C₄⋊Q₈	C₂×D₃₇	semidirect
183. #106	C₂×D₃₇	semidirect
186. C₄⋊C₈	C₂×D₃₇	semidirect
244. D₄	Q₈×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×D₃₇	2
C₄×D₄×D₃₇	2
#14557824	2
#14557823	2
#14557822	2
#14557821	2
#14557820	2
#106×D₃₇	2
C₄⋊C₈×D₃₇	2
#102×D₃₇	2
C₃₇×#476	2
C₂×#476	37

Subgroups — #476×D37 #18004343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #476×D₃₇ #18004343