Subgroups — #18003429

Order 4736.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558016	C₂	1	1	yes	maximal, characteristic, complemented
2	#14558008	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557645	C₂	1	2	yes	maximal, complemented
4	#14557634	C₂	1	2	yes	maximal, complemented
5	SD₁₆×Dic₃₇	C₂	1	2	yes	maximal, complemented
6	D₈×Dic₃₇	C₂	1	2	yes	maximal, complemented
7	#14557575	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557529	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11924457	C₂	1	1	yes	maximal, characteristic, complemented
10	#14557491	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×AGL(1,ℤ/8)×C₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×#64436	C₂²	1	1	yes	characteristic
13	Q₈×Dic₃₇	C₂²	1	2	yes	complemented
14	D₄×Dic₃₇	—	2	2	no	—
15	D₄×Dic₃₇	—	2	1	no	—
16	D₄×Dic₃₇	—	2	1	no	—
17	D₄×Dic₃₇	C₂²	1	2	yes	—
18	D₄×Dic₃₇	C₂²	1	2	yes	—
19	D₄×Dic₃₇	C₂²	1	2	yes	complemented
20	#11924495	—	2	1	no	—
21	#11924495	C₂²	1	1	yes	characteristic
22	#11924495	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₃₇	—	2	1	no	—
24	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Dic₃₇	—	2	2	no	—
26	#11924470	C₂²	1	2	yes	complemented
27	#11924468	C₂²	1	2	yes	complemented
28	#11924460	C₂²	1	1	yes	characteristic, complemented
29	#11924460	C₂²	1	1	yes	characteristic, complemented
30	#11924457	C₂²	1	2	yes	—
31	#11924457	C₂²	1	2	yes	—
32	#11924457	C₂²	1	1	yes	characteristic, complemented
33	#11924457	C₂²	1	1	yes	characteristic, complemented
34	#11924446	C₂²	1	2	yes	complemented
35	C₈×Dic₃₇	C₂²	1	2	yes	complemented
36	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
37	C₂²×D₄×C₃₇	C₂²	1	1	yes	characteristic
38	AGL(1,ℤ/8)×C₃₇	C₄	1	8	yes	complemented
39	C₂×SD₁₆×C₃₇	C₂²	1	2	yes	—
40	C₂×D₈×C₃₇	C₂²	1	2	yes	—
41	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
42	C₂⁴×C₃₇	—	2	1	no	—
43	#41×C₃₇	—	2	2	no	—
44	#41×C₃₇	C₂×C₄	1	4	yes	complemented
45	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₃₇	—	2	1	no	—
47	C₂×D₄×C₃₇	—	2	1	no	—
48	C₂×D₄×C₃₇	—	2	2	no	—
49	C₂×D₄×C₃₇	C₂×C₄	1	4	yes	complemented
50	C₂×D₄×C₃₇	C₂³	1	2	yes	—
51	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₃₇	—	2	1	no	—
53	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
54	C₂²×Dic₃₇	—	2	1	no	—
55	C₂²×Dic₃₇	—	2	2	no	—
56	C₂²×Dic₃₇	—	2	2	no	—
57	C₂²×Dic₃₇	—	2	2	no	—
58	SD₁₆×C₃₇	C₂×C₄	1	8	yes	complemented
59	D₈×C₃₇	C₂×C₄	1	8	yes	complemented
60	M₁₆×C₃₇	C₂×C₄	1	4	yes	complemented
61	C₂×C₈×C₃₇	C₂³	1	2	yes	—
62	C₂²⋊Dic₃₇	—	2	1	no	—
63	C₂²⋊Dic₃₇	—	2	2	no	—
64	C₂²⋊Dic₃₇	—	2	2	no	—
65	C₂²⋊Dic₃₇	—	2	2	no	—
66	C₂²⋊Dic₃₇	D₄	1	2	yes	—
67	C₄⋊Dic₃₇	—	2	1	no	—
68	C₄⋊Dic₃₇	—	2	2	no	—
69	C₄⋊Dic₃₇	D₄	1	1	yes	characteristic
70	C₄⋊Dic₃₇	D₄	1	1	yes	characteristic
71	C₄⋊Dic₃₇	C₂³	1	2	yes	—
72	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
74	C₄×Dic₃₇	—	2	1	no	—
75	C₄×Dic₃₇	—	2	2	no	—
76	C₄×Dic₃₇	C₂³	1	2	yes	—
77	C₂×#64436	—	2	2	no	—
78	C₂×#64436	C₂³	1	1	yes	characteristic
79	C₂×#64436	C₂³	1	1	yes	characteristic
80	C₂³×C₃₇	—	2	1	no	—
81	C₂³×C₃₇	—	2	2	no	—
82	C₂³×C₃₇	—	2	4	no	—
83	C₂³×C₃₇	—	8	1	no	—
84	C₂³×C₃₇	—	8	1	no	—
85	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
86	Q₈×C₃₇	—	2	1	no	—
87	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
88	D₄×C₃₇	—	2	4	no	—
89	D₄×C₃₇	—	2	4	no	—
90	D₄×C₃₇	—	2	2	no	—
91	D₄×C₃₇	—	2	1	no	—
92	D₄×C₃₇	C₂²×C₄	1	2	yes	—
93	D₄×C₃₇	C₂²×C₄	1	4	yes	—
94	C₂×C₄×C₃₇	—	2	1	no	—
95	C₂×C₄×C₃₇	—	2	4	no	—
96	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₃₇	#41	1	2	yes	—
99	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
100	C₂×Dic₃₇	—	2	2	no	—
101	C₂×Dic₃₇	—	2	1	no	—
102	C₂×Dic₃₇	—	2	1	no	—
103	C₂×Dic₃₇	—	8	2	no	—
104	C₂×Dic₃₇	—	8	2	no	—
105	C₂×Dic₃₇	—	8	2	no	—
106	C₂×Dic₃₇	—	8	1	no	—
107	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
108	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
109	C₈×C₃₇	C₂²×C₄	1	4	yes	—
110	#64436	—	2	1	no	—
111	#64436	—	2	1	no	—
112	C₂²×C₃₇	—	2	1	no	—
113	C₂²×C₃₇	—	2	1	no	—
114	C₂²×C₃₇	—	2	2	no	—
115	C₂²×C₃₇	—	2	4	no	—
116	C₂²×C₃₇	—	4	1	no	—
117	C₂²×C₃₇	—	4	2	no	—
118	C₂²×C₃₇	—	4	2	no	—
119	C₂²×C₃₇	—	4	1	no	—
120	C₂²×C₃₇	—	8	4	no	—
121	C₂²×C₃₇	C₄×D₄	1	2	yes	—
122	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
123	C₄×C₃₇	—	2	2	no	—
124	C₄×C₃₇	C₄×D₄	1	2	yes	—
125	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
127	Dic₃₇	—	2	2	no	—
128	Dic₃₇	—	2	2	no	—
129	Dic₃₇	—	4	2	no	—
130	Dic₃₇	—	4	1	no	—
131	Dic₃₇	—	4	1	no	—
132	#2946	—	37	1	no	maximal, Sylow
133	C₂×C₃₇	—	2	2	no	—
134	C₂×C₃₇	—	4	2	no	—
135	C₂×C₃₇	—	4	4	no	—
136	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×C₃₇	#576	1	1	yes	characteristic
138	C₂×C₃₇	#577	1	1	yes	characteristic
139	C₂×AGL(1,ℤ/8)	D₃₇	1	1	yes	characteristic
140	#520	—	37	1	no	—
141	#518	—	37	1	no	—
142	#442	—	37	2	no	—
143	#440	—	37	2	no	—
144	C₄×SD₁₆	—	37	2	no	—
145	C₄×D₈	—	37	2	no	—
146	#428	—	37	1	no	—
147	#416	—	37	1	no	—
148	C₂×#102	—	37	1	no	—
149	#405	—	37	1	no	—
150	C₃₇	#2946	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
152	C₂²×D₄	C₂×D₃₇	1	1	yes	characteristic
153	AGL(1,ℤ/8)	Dic₃₇	1	8	yes	complemented
154	C₂×SD₁₆	C₂×D₃₇	1	2	yes	—
155	C₂×D₈	C₂×D₃₇	1	2	yes	—
156	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
157	C₂²×C₈	—	37	1	no	—
158	C₄×Q₈	—	37	2	no	—
159	C₄×D₄	—	37	2	no	—
160	C₄×D₄	—	37	2	no	—
161	C₄×D₄	—	37	2	no	—
162	C₄×D₄	—	74	2	no	—
163	C₄×D₄	—	74	1	no	—
164	C₄×D₄	—	74	1	no	—
165	#117	—	37	1	no	—
166	#117	—	37	1	no	—
167	#117	—	74	1	no	—
168	C₂×C₄⋊C₄	—	37	1	no	—
169	C₂×C₄⋊C₄	—	74	1	no	—
170	C₂×C₂²⋊C₄	—	74	2	no	—
171	#107	—	37	2	no	—
172	#106	—	37	2	no	—
173	#103	—	37	1	no	—
174	#103	—	37	1	no	—
175	#102	—	37	2	no	—
176	#102	—	37	2	no	—
177	#102	—	37	1	no	—
178	#102	—	37	1	no	—
179	#97	—	37	2	no	—
180	C₄×C₈	—	37	2	no	—
181	C₂⁴	—	2	1	no	—
182	#41	—	2	2	no	—
183	#41	C₂×Dic₃₇	1	4	yes	complemented
184	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	C₂×Dic₃₇	1	4	yes	complemented
189	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
190	C₂×D₄	C₂²×D₃₇	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	74	2	no	—
193	C₂²×C₄	—	74	2	no	—
194	C₂²×C₄	—	74	2	no	—
195	C₂²×C₄	—	74	1	no	—
196	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
197	SD₁₆	C₂×Dic₃₇	1	8	yes	complemented
198	D₈	C₂×Dic₃₇	1	8	yes	complemented
199	M₁₆	C₂×Dic₃₇	1	4	yes	complemented
200	C₂×C₈	—	37	1	no	—
201	C₂×C₈	—	37	1	no	—
202	C₂×C₈	—	74	2	no	—
203	C₂×C₈	C₂²×D₃₇	1	2	yes	—
204	C₄⋊C₄	—	37	1	no	—
205	C₄⋊C₄	—	37	1	no	—
206	C₄⋊C₄	—	37	2	no	—
207	C₄⋊C₄	—	37	1	no	—
208	C₄⋊C₄	—	37	1	no	—
209	C₄⋊C₄	—	74	2	no	—
210	C₄⋊C₄	—	74	1	no	—
211	C₂²⋊C₄	—	37	2	no	—
212	C₂²⋊C₄	—	74	2	no	—
213	C₂²⋊C₄	—	74	2	no	—
214	C₂²⋊C₄	—	74	2	no	—
215	C₂²⋊C₄	—	74	1	no	—
216	C₄²	—	37	2	no	—
217	C₄²	—	74	2	no	—
218	C₄²	—	74	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	8	1	no	—
224	C₂³	D₄×D₃₇	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	C₂²×Dic₃₇	1	2	yes	—
227	D₄	—	2	4	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	2	no	—
230	D₄	—	2	1	no	—
231	D₄	C₂²×Dic₃₇	1	2	yes	—
232	D₄	C₂²×Dic₃₇	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	37	2	no	—
236	C₂×C₄	—	37	2	no	—
237	C₂×C₄	—	296	2	no	—
238	C₂×C₄	—	296	2	no	—
239	C₂×C₄	—	296	2	no	—
240	C₂×C₄	—	296	1	no	—
241	C₂×C₄	—	74	1	no	—
242	C₂×C₄	—	74	1	no	—
243	C₂×C₄	—	74	2	no	—
244	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
245	C₂×C₄	#10596964	1	2	yes	—
246	C₂×C₄	C₂²×Dic₃₇	1	2	yes	—
247	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
248	C₈	—	74	1	no	—
249	C₈	—	74	1	no	—
250	C₈	C₂²×Dic₃₇	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	8	4	no	—
260	C₂²	D₄×Dic₃₇	1	2	yes	—
261	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	74	2	no	—
264	C₄	—	74	2	no	—
265	C₄	—	148	1	no	—
266	C₄	—	148	2	no	—
267	C₄	—	148	1	no	—
268	C₄	D₄×Dic₃₇	1	2	yes	—
269	C₄	C₂³×Dic₃₇	1	1	yes	characteristic
270	C₄	C₂×#10596964	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	4	no	—
273	C₂	—	4	2	no	—
274	C₂	C₂×D₄×Dic₃₇	1	1	yes	characteristic
275	C₂	#14558337	1	1	yes	characteristic
276	C₂	#14558343	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting, prime radical	C₂×AGL(1,ℤ/8)×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₂×AGL(1,ℤ/8)	D₃₇
hypocenter	C₃₇	#2946
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #14558016	C₂	semidirect
2. #14558008	C₂	semidirect
3. #14557645	C₂	semidirect
4. #14557634	C₂	semidirect
5. SD₁₆×Dic₃₇	C₂	semidirect
6. D₈×Dic₃₇	C₂	semidirect
7. #14557575	C₂	semidirect
8. #14557529	C₂	semidirect
9. C₂×#11924457	C₂	semidirect
10. #14557491	C₂	semidirect
13. Q₈×Dic₃₇	C₂²	semidirect
19. D₄×Dic₃₇	C₂²	semidirect
26. #11924470	C₂²	semidirect
27. #11924468	C₂²	semidirect
28. #11924460	C₂²	semidirect
29. #11924460	C₂²	semidirect
32. #11924457	C₂²	semidirect
33. #11924457	C₂²	semidirect
34. #11924446	C₂²	semidirect
35. C₈×Dic₃₇	C₂²	semidirect
38. AGL(1,ℤ/8)×C₃₇	C₄	semidirect
44. #41×C₃₇	C₂×C₄	semidirect
49. C₂×D₄×C₃₇	C₂×C₄	semidirect
58. SD₁₆×C₃₇	C₂×C₄	semidirect
59. D₈×C₃₇	C₂×C₄	semidirect
60. M₁₆×C₃₇	C₂×C₄	semidirect
150. C₃₇	#2946	semidirect
153. AGL(1,ℤ/8)	Dic₃₇	semidirect
183. #41	C₂×Dic₃₇	semidirect
188. C₂×D₄	C₂×Dic₃₇	semidirect
197. SD₁₆	C₂×Dic₃₇	semidirect
198. D₈	C₂×Dic₃₇	semidirect
199. M₁₆	C₂×Dic₃₇	semidirect

Maximal subgroups

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

subgroup	index
#14558016	2
#14558008	2
#14557645	2
#14557634	2
SD₁₆×Dic₃₇	2
D₈×Dic₃₇	2
#14557575	2
#14557529	2
C₂×#11924457	2
#14557491	2
C₂×AGL(1,ℤ/8)×C₃₇	2
#2946	37