Subgroups — #477×D₃₇ #18004363

Order 4736.

416 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	C₄⋊Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic
3	#14557829	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557828	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557827	C₂	1	2	yes	maximal, complemented
6	#14557826	C₂	1	2	yes	maximal, complemented
7	#14557825	C₂	1	1	yes	maximal, characteristic, complemented
8	#107×D₃₇	C₂	1	2	yes	maximal
9	C₄⋊C₈×D₃₇	C₂	1	1	yes	maximal, characteristic
10	#103×D₃₇	C₂	1	2	yes	maximal
11	C₃₇×#477	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₃₇	—	2	1	no	—
13	C₂×Q₈×D₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₃₇	C₂²	1	2	yes	—
15	#11924536	C₂²	1	1	yes	characteristic
16	#11924535	—	2	1	no	—
17	#11924535	C₂²	1	2	yes	—
18	#11924534	—	2	1	no	—
19	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
20	C₄×Dic₇₄	—	2	1	no	—
21	C₄×Dic₇₄	C₂²	1	1	yes	characteristic
22	#11924501	—	2	1	no	—
23	#11924501	C₂²	1	1	yes	characteristic
24	C₄⋊C₄×D₃₇	—	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	C₄²×D₃₇	—	2	1	no	—
29	C₄²×D₃₇	C₂²	1	1	yes	characteristic
30	#11924470	C₂²	1	2	yes	—
31	#11924469	C₂²	1	2	yes	—
32	#11924469	C₂²	1	2	yes	—
33	C₄⋊#64436	C₂²	1	1	yes	characteristic
34	#11924465	C₂²	1	2	yes	—
35	#11924461	C₂²	1	2	yes	—
36	#11924460	C₂²	1	2	yes	—
37	#11924459	C₂²	1	2	yes	—
38	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
39	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
40	#107×C₃₇	C₂²	1	2	yes	—
41	C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
42	#103×C₃₇	C₂²	1	2	yes	—
43	C₂×Q₈×C₃₇	—	2	1	no	—
44	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
45	Q₈×D₃₇	—	2	2	no	—
46	Q₈×D₃₇	—	4	4	no	—
47	Q₈×D₃₇	Q₈	1	4	yes	complemented
48	C₂×C₄×D₃₇	—	2	1	no	—
49	C₂×C₄×D₃₇	—	2	1	no	—
50	C₂×C₄×D₃₇	—	2	2	no	—
51	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
52	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
53	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
54	C₂×Dic₇₄	—	2	1	no	—
55	C₂×Dic₇₄	—	2	1	no	—
56	C₂×Dic₇₄	—	2	2	no	—
57	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₃₇	C₂³	1	2	yes	—
59	C₄⋊C₄×C₃₇	—	2	1	no	—
60	C₄⋊C₄×C₃₇	—	2	1	no	—
61	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
63	C₄²×C₃₇	—	2	1	no	—
64	C₄²×C₃₇	C₂³	1	1	yes	characteristic
65	C₄⋊Dic₃₇	—	2	1	no	—
66	C₄⋊Dic₃₇	—	2	1	no	—
67	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₃₇	C₂³	1	2	yes	—
69	#10596936	—	2	1	no	—
70	#10596936	—	2	1	no	—
71	#10596936	—	2	2	no	—
72	#10596936	—	2	1	no	—
73	#10596936	—	2	2	no	—
74	C₄×Dic₃₇	—	2	1	no	—
75	C₄×Dic₃₇	—	2	1	no	—
76	C₄×Dic₃₇	—	2	1	no	—
77	C₄×Dic₃₇	D₄	1	1	yes	characteristic
78	C₄×Dic₃₇	D₄	1	1	yes	characteristic
79	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
80	C₂×#64436	C₂³	1	2	yes	—
81	C₈×D₃₇	—	2	4	no	—
82	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₃₇	—	2	1	no	—
84	Q₈×C₃₇	—	4	2	no	—
85	Q₈×C₃₇	C₂×Q₈	1	2	yes	complemented
86	C₂×C₄×C₃₇	—	2	1	no	—
87	C₂×C₄×C₃₇	—	2	1	no	—
88	C₂×C₄×C₃₇	—	2	2	no	—
89	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
92	C₂×Dic₃₇	—	2	1	no	—
93	C₂×Dic₃₇	—	2	1	no	—
94	C₂×Dic₃₇	—	2	2	no	—
95	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
96	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
97	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
98	C₄×D₃₇	—	2	4	no	—
99	C₄×D₃₇	—	2	2	no	—
100	C₄×D₃₇	—	4	2	no	—
101	C₄×D₃₇	—	4	4	no	—
102	C₄×D₃₇	Q₁₆	1	4	yes	—
103	C₄×D₃₇	C₂×Q₈	1	2	yes	—
104	C₄×D₃₇	#41	1	2	yes	—
105	Dic₇₄	—	2	1	no	—
106	Dic₇₄	—	2	4	no	—
107	Dic₇₄	—	4	4	no	—
108	Dic₇₄	—	4	2	no	—
109	Dic₇₄	C₂×Q₈	1	2	yes	complemented
110	C₈×C₃₇	—	2	2	no	—
111	#64436	—	2	2	no	—
112	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×D₃₇	C₂²⋊Q₈	1	2	yes	—
114	C₂×D₃₇	C₂×Q₁₆	1	2	yes	—
115	C₂×D₃₇	AGL(1,ℤ/8)	1	2	yes	—
116	C₄×C₃₇	—	2	2	no	—
117	C₄×C₃₇	—	2	1	no	—
118	C₄×C₃₇	—	4	2	no	—
119	C₄×C₃₇	—	4	1	no	—
120	C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
121	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
122	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
123	Dic₃₇	—	2	2	no	—
124	Dic₃₇	—	2	1	no	—
125	Dic₃₇	—	4	2	no	—
126	Dic₃₇	—	4	1	no	—
127	Dic₃₇	C₂²⋊Q₈	1	2	yes	—
128	Dic₃₇	C₂×Q₁₆	1	2	yes	—
129	C₂×#477	—	37	1	no	maximal, Sylow
130	C₂×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₃₇	C₂²×Q₁₆	1	1	yes	characteristic
132	C₂×C₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
133	D₃₇	#477	1	4	yes	direct product factor
134	C₂×C₄⋊Q₈	—	37	1	no	—
135	C₂×C₄×Q₈	—	37	1	no	—
136	#477	—	37	1	no	—
137	#477	—	37	1	no	—
138	#477	—	37	2	no	—
139	#477	—	37	1	no	—
140	#477	—	37	2	no	—
141	#477	D₃₇	1	1	yes	characteristic, direct product factor
142	C₂×#107	—	37	2	no	—
143	C₂×C₄⋊C₈	—	37	1	no	—
144	C₂×#103	—	37	2	no	—
145	C₃₇	C₂×#477	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₈	—	37	1	no	—
147	C₂²×Q₈	—	74	1	no	—
148	C₂²×C₈	—	37	2	no	—
149	C₄⋊Q₈	—	37	2	no	—
150	C₄⋊Q₈	—	37	1	no	—
151	C₄⋊Q₈	—	74	1	no	—
152	C₄⋊Q₈	—	74	1	no	—
153	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
154	C₄×Q₈	—	37	1	no	—
155	C₄×Q₈	—	37	1	no	—
156	C₄×Q₈	—	37	1	no	—
157	C₄×Q₈	—	74	1	no	—
158	C₄×Q₈	—	74	1	no	—
159	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
160	C₂×C₄⋊C₄	—	37	1	no	—
161	C₂×C₄⋊C₄	—	37	2	no	—
162	C₂×C₄⋊C₄	—	74	1	no	—
163	C₂×C₄⋊C₄	—	74	1	no	—
164	C₂×C₄²	—	37	1	no	—
165	C₂×C₄²	—	74	1	no	—
166	#107	—	37	2	no	—
167	#107	—	37	2	no	—
168	#107	—	37	2	no	—
169	#107	C₂×D₃₇	1	2	yes	—
170	C₄⋊C₈	—	37	2	no	—
171	C₄⋊C₈	—	37	1	no	—
172	C₄⋊C₈	C₂×D₃₇	1	1	yes	characteristic
173	#103	—	37	2	no	—
174	#103	—	37	2	no	—
175	#103	—	37	2	no	—
176	#103	C₂×D₃₇	1	2	yes	—
177	C₂×Q₈	—	2	1	no	—
178	C₂×Q₈	—	37	4	no	—
179	C₂×Q₈	—	37	1	no	—
180	C₂×Q₈	—	74	1	no	—
181	C₂×Q₈	—	74	2	no	—
182	C₂×Q₈	—	74	2	no	—
183	C₂×Q₈	—	74	1	no	—
184	C₂×Q₈	—	148	4	no	—
185	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
186	C₂²×C₄	—	37	1	no	—
187	C₂²×C₄	—	37	1	no	—
188	C₂²×C₄	—	37	1	no	—
189	C₂²×C₄	—	74	1	no	—
190	C₂²×C₄	—	74	1	no	—
191	C₂²×C₄	—	74	2	no	—
192	C₂×C₈	—	37	2	no	—
193	C₂×C₈	—	74	4	no	—
194	C₂×C₈	C₂²×D₃₇	1	2	yes	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	37	2	no	—
198	C₄⋊C₄	—	37	1	no	—
199	C₄⋊C₄	—	74	1	no	—
200	C₄⋊C₄	—	74	2	no	—
201	C₄⋊C₄	—	74	1	no	—
202	C₄⋊C₄	—	74	2	no	—
203	C₄⋊C₄	—	74	1	no	—
204	C₄⋊C₄	—	74	1	no	—
205	C₄⋊C₄	—	74	1	no	—
206	C₄⋊C₄	C₂²×D₃₇	1	2	yes	—
207	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
208	C₄²	—	2	1	no	—
209	C₄²	—	37	1	no	—
210	C₄²	—	37	1	no	—
211	C₄²	—	37	1	no	—
212	C₄²	—	74	1	no	—
213	C₄²	—	74	1	no	—
214	C₄²	—	74	1	no	—
215	C₄²	C₂²×D₃₇	1	1	yes	characteristic
216	C₂³	—	37	1	no	—
217	Q₈	—	2	1	no	—
218	Q₈	—	4	2	no	—
219	Q₈	—	37	2	no	—
220	Q₈	—	74	4	no	—
221	Q₈	—	74	1	no	—
222	Q₈	—	148	2	no	—
223	Q₈	—	148	4	no	—
224	Q₈	Q₈×D₃₇	1	2	yes	complemented
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	37	1	no	—
229	C₂×C₄	—	37	1	no	—
230	C₂×C₄	—	37	4	no	—
231	C₂×C₄	—	37	2	no	—
232	C₂×C₄	—	37	2	no	—
233	C₂×C₄	—	37	1	no	—
234	C₂×C₄	—	74	2	no	—
235	C₂×C₄	—	74	4	no	—
236	C₂×C₄	—	74	1	no	—
237	C₂×C₄	—	74	1	no	—
238	C₂×C₄	—	74	2	no	—
239	C₂×C₄	—	148	4	no	—
240	C₂×C₄	—	148	2	no	—
241	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
242	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	74	2	no	—
246	C₂²	—	37	2	no	—
247	C₂²	—	37	2	no	—
248	C₂²	—	37	2	no	—
249	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	37	2	no	—
255	C₄	—	37	2	no	—
256	C₄	—	74	2	no	—
257	C₄	—	74	1	no	—
258	C₄	—	148	2	no	—
259	C₄	—	148	1	no	—
260	C₄	Q₁₆×D₃₇	1	2	yes	—
261	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
262	C₄	#41×D₃₇	1	1	yes	characteristic
263	C₂	—	37	4	no	—
264	C₂	C₂²⋊Q₈×D₃₇	1	1	yes	characteristic
265	C₂	C₂×Q₁₆×D₃₇	1	1	yes	characteristic
266	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×#477	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂²×D₃₇	C₂×D₄
hypercenter	#477	D₃₇
hypocenter	C₃₇	C₂×#477
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

Complements

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

N	H ≅ G/N	type
3. #14557829	C₂	semidirect
4. #14557828	C₂	semidirect
5. #14557827	C₂	semidirect
6. #14557826	C₂	semidirect
7. #14557825	C₂	semidirect
11. C₃₇×#477	C₂	semidirect
47. Q₈×D₃₇	Q₈	semidirect
85. Q₈×C₃₇	C₂×Q₈	semidirect
109. Dic₇₄	C₂×Q₈	semidirect
133. D₃₇	#477	direct (N × H)
141. #477	D₃₇	direct (N × H)
145. C₃₇	C₂×#477	semidirect
224. Q₈	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×Q₈×D₃₇	2
#14557829	2
#14557828	2
#14557827	2
#14557826	2
#14557825	2
#107×D₃₇	2
C₄⋊C₈×D₃₇	2
#103×D₃₇	2
C₃₇×#477	2
C₂×#477	37

Subgroups — #477×D37 #18004363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #477×D₃₇ #18004363