Subgroups — #137×Dic₃₇ #18003417

Order 4736.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×Dic₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557640	C₂	1	2	yes	maximal, complemented
4	#14557634	C₂	1	2	yes	maximal, complemented
5	Q₁₆×Dic₃₇	C₂	1	2	yes	maximal, complemented
6	SD₁₆×Dic₃₇	C₂	1	2	yes	maximal, complemented
7	#14557575	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557542	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557534	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#137×C₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#10596934	C₂²	1	1	yes	characteristic
13	Q₈×Dic₃₇	—	2	1	no	—
14	Q₈×Dic₃₇	—	2	1	no	—
15	Q₈×Dic₃₇	C₂²	1	2	yes	—
16	Q₈×Dic₃₇	C₂²	1	2	yes	—
17	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
18	Q₈×Dic₃₇	C₂²	1	1	yes	characteristic
19	D₄×Dic₃₇	—	2	1	no	—
20	D₄×Dic₃₇	—	2	1	no	—
21	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
22	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
23	#11924495	—	2	1	no	—
24	#11924495	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₃₇	—	2	1	no	—
26	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₃₇	—	2	1	no	—
28	C₂×C₄×Dic₃₇	—	2	1	no	—
29	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
30	#11924470	C₂²	1	2	yes	—
31	#11924468	C₂²	1	2	yes	—
32	#11924460	C₂²	1	2	yes	—
33	#11924460	C₂²	1	2	yes	—
34	#11924460	C₂²	1	2	yes	—
35	#11924457	C₂²	1	2	yes	—
36	#11924446	C₂²	1	2	yes	—
37	C₈×Dic₃₇	C₂²	1	2	yes	—
38	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic
40	#137×C₃₇	C₄	1	8	yes	complemented
41	C₂×Q₁₆×C₃₇	C₂²	1	2	yes	—
42	C₂×SD₁₆×C₃₇	C₂²	1	2	yes	—
43	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
44	#41×C₃₇	—	2	2	no	—
45	#41×C₃₇	C₂×C₄	1	4	yes	—
46	C₂×Q₈×C₃₇	—	2	1	no	—
47	C₂×Q₈×C₃₇	—	2	2	no	—
48	C₂×Q₈×C₃₇	C₂×C₄	1	4	yes	complemented
49	C₂×Q₈×C₃₇	C₂³	1	2	yes	—
50	C₂×Q₈×C₃₇	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₃₇	—	2	1	no	—
52	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₃₇	—	2	1	no	—
54	C₂²×C₄×C₃₇	—	2	1	no	—
55	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
56	C₂²×Dic₃₇	—	2	1	no	—
57	C₂²×Dic₃₇	—	2	1	no	—
58	C₂²×Dic₃₇	—	2	1	no	—
59	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
60	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
61	Q₁₆×C₃₇	C₂×C₄	1	8	yes	complemented
62	SD₁₆×C₃₇	C₂×C₄	1	8	yes	complemented
63	M₁₆×C₃₇	C₂×C₄	1	4	yes	complemented
64	C₂×C₈×C₃₇	C₂³	1	2	yes	—
65	C₂²⋊Dic₃₇	—	2	1	no	—
66	C₂²⋊Dic₃₇	—	2	1	no	—
67	C₂²⋊Dic₃₇	—	2	1	no	—
68	C₄⋊Dic₃₇	—	2	1	no	—
69	C₄⋊Dic₃₇	—	2	2	no	—
70	C₄⋊Dic₃₇	—	2	2	no	—
71	C₄⋊Dic₃₇	—	2	1	no	—
72	C₄⋊Dic₃₇	—	2	1	no	—
73	C₄⋊Dic₃₇	C₂³	1	2	yes	—
74	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
76	C₄×Dic₃₇	—	2	1	no	—
77	C₄×Dic₃₇	—	2	1	no	—
78	C₄×Dic₃₇	—	2	1	no	—
79	C₄×Dic₃₇	—	2	2	no	—
80	C₄×Dic₃₇	—	2	2	no	—
81	C₄×Dic₃₇	D₄	1	1	yes	characteristic
82	C₄×Dic₃₇	D₄	1	1	yes	characteristic
83	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
84	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
85	#10596934	—	2	2	no	—
86	C₂×#64436	C₂³	1	2	yes	—
87	C₂³×C₃₇	—	2	1	no	—
88	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₃₇	—	2	4	no	—
90	Q₈×C₃₇	—	2	2	no	—
91	Q₈×C₃₇	—	2	1	no	—
92	Q₈×C₃₇	C₂²×C₄	1	4	yes	—
93	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
94	D₄×C₃₇	—	2	4	no	—
95	D₄×C₃₇	—	2	1	no	—
96	D₄×C₃₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₃₇	—	2	4	no	—
98	C₂×C₄×C₃₇	—	2	1	no	—
99	C₂×C₄×C₃₇	—	2	2	no	—
100	C₂×C₄×C₃₇	—	2	4	no	—
101	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₇	#41	1	2	yes	—
104	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
105	C₂×Dic₃₇	—	2	2	no	—
106	C₂×Dic₃₇	—	2	2	no	—
107	C₂×Dic₃₇	—	2	1	no	—
108	C₂×Dic₃₇	—	2	1	no	—
109	C₂×Dic₃₇	—	4	1	no	—
110	C₂×Dic₃₇	—	4	2	no	—
111	C₂×Dic₃₇	—	4	2	no	—
112	C₂×Dic₃₇	—	4	2	no	—
113	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
114	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
117	C₈×C₃₇	C₂²×C₄	1	4	yes	—
118	#64436	—	2	2	no	—
119	C₂²×C₃₇	—	2	1	no	—
120	C₂²×C₃₇	—	2	1	no	—
121	C₂²×C₃₇	—	2	2	no	—
122	C₂²×C₃₇	—	4	1	no	—
123	C₂²×C₃₇	—	4	1	no	—
124	C₂²×C₃₇	C₄×D₄	1	2	yes	—
125	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₃₇	—	2	2	no	—
127	C₄×C₃₇	—	2	4	no	—
128	C₄×C₃₇	C₄×D₄	1	2	yes	—
129	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
131	Dic₃₇	—	2	1	no	—
132	Dic₃₇	—	2	1	no	—
133	Dic₃₇	—	2	1	no	—
134	Dic₃₇	—	4	1	no	—
135	Dic₃₇	—	4	2	no	—
136	Dic₃₇	—	4	1	no	—
137	Dic₃₇	#137	1	2	yes	direct product factor
138	C₄×#137	—	37	1	no	maximal, Sylow
139	C₂×C₃₇	—	2	2	no	—
140	C₂×C₃₇	—	4	2	no	—
141	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂×C₃₇	C₂×#137	1	1	yes	characteristic
143	C₂×C₃₇	#575	1	1	yes	characteristic
144	C₂×#137	D₃₇	1	1	yes	characteristic
145	C₄×#41	—	37	1	no	—
146	C₂×C₄×Q₈	—	37	1	no	—
147	#441	—	37	2	no	—
148	#440	—	37	2	no	—
149	C₄×Q₁₆	—	37	2	no	—
150	C₄×SD₁₆	—	37	2	no	—
151	#428	—	37	1	no	—
152	#419	—	37	1	no	—
153	#417	—	37	1	no	—
154	C₄×M₁₆	—	37	1	no	—
155	C₃₇	C₄×#137	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
157	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
158	#137	Dic₃₇	1	8	yes	direct product factor
159	C₂×Q₁₆	C₂×D₃₇	1	2	yes	—
160	C₂×SD₁₆	C₂×D₃₇	1	2	yes	—
161	C₂×M₁₆	—	37	1	no	—
162	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
163	C₄×Q₈	—	37	2	no	—
164	C₄×Q₈	—	37	2	no	—
165	C₄×Q₈	—	37	1	no	—
166	C₄×Q₈	—	37	1	no	—
167	C₄×Q₈	—	74	1	no	—
168	C₄×Q₈	—	74	1	no	—
169	C₄×D₄	—	37	1	no	—
170	C₄×D₄	—	37	1	no	—
171	C₄×D₄	—	74	1	no	—
172	C₄×D₄	—	74	1	no	—
173	#117	—	37	1	no	—
174	#117	—	74	1	no	—
175	C₂×C₄⋊C₄	—	37	1	no	—
176	C₂×C₄⋊C₄	—	74	1	no	—
177	C₂×C₄²	—	37	1	no	—
178	C₂×C₄²	—	74	1	no	—
179	C₂×C₄²	—	74	1	no	—
180	#107	—	37	2	no	—
181	#106	—	37	2	no	—
182	#103	—	37	2	no	—
183	#103	—	37	2	no	—
184	#103	—	37	2	no	—
185	#102	—	37	2	no	—
186	#97	—	37	2	no	—
187	C₄×C₈	—	37	2	no	—
188	#41	—	2	2	no	—
189	#41	C₂×Dic₃₇	1	4	yes	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	C₂×Dic₃₇	1	4	yes	complemented
193	C₂×Q₈	C₂²×D₃₇	1	2	yes	—
194	C₂×Q₈	C₂²×D₃₇	1	1	yes	characteristic
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	37	1	no	—
200	C₂²×C₄	—	37	1	no	—
201	C₂²×C₄	—	74	1	no	—
202	C₂²×C₄	—	74	1	no	—
203	C₂²×C₄	—	74	1	no	—
204	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
205	Q₁₆	C₂×Dic₃₇	1	8	yes	complemented
206	SD₁₆	C₂×Dic₃₇	1	8	yes	complemented
207	M₁₆	—	74	2	no	—
208	M₁₆	C₂×Dic₃₇	1	4	yes	complemented
209	C₂×C₈	—	37	2	no	—
210	C₂×C₈	C₂²×D₃₇	1	2	yes	—
211	C₄⋊C₄	—	37	2	no	—
212	C₄⋊C₄	—	37	1	no	—
213	C₄⋊C₄	—	37	1	no	—
214	C₄⋊C₄	—	74	2	no	—
215	C₄⋊C₄	—	74	2	no	—
216	C₄⋊C₄	—	74	1	no	—
217	C₄⋊C₄	—	74	1	no	—
218	C₄⋊C₄	—	74	1	no	—
219	C₂²⋊C₄	—	74	1	no	—
220	C₂²⋊C₄	—	74	1	no	—
221	C₂²⋊C₄	—	74	1	no	—
222	C₄²	—	37	1	no	—
223	C₄²	—	37	1	no	—
224	C₄²	—	37	1	no	—
225	C₄²	—	37	1	no	—
226	C₄²	—	74	1	no	—
227	C₄²	—	74	1	no	—
228	C₄²	—	74	2	no	—
229	C₄²	—	74	2	no	—
230	C₄²	—	74	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	D₄×D₃₇	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	2	1	no	—
236	Q₈	C₂²×Dic₃₇	1	4	yes	—
237	Q₈	C₂²×Dic₃₇	1	2	yes	—
238	D₄	—	2	4	no	—
239	D₄	—	2	1	no	—
240	D₄	C₂²×Dic₃₇	1	2	yes	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	37	1	no	—
246	C₂×C₄	—	37	1	no	—
247	C₂×C₄	—	37	1	no	—
248	C₂×C₄	—	37	1	no	—
249	C₂×C₄	—	74	2	no	—
250	C₂×C₄	—	74	1	no	—
251	C₂×C₄	—	74	1	no	—
252	C₂×C₄	—	74	2	no	—
253	C₂×C₄	—	148	2	no	—
254	C₂×C₄	—	148	1	no	—
255	C₂×C₄	—	148	2	no	—
256	C₂×C₄	—	148	2	no	—
257	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
258	C₂×C₄	#10596964	1	2	yes	—
259	C₂×C₄	C₂²×Dic₃₇	1	2	yes	—
260	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
261	C₈	—	74	2	no	—
262	C₈	C₂²×Dic₃₇	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	D₄×Dic₃₇	1	2	yes	—
269	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	4	no	—
272	C₄	—	37	2	no	—
273	C₄	—	74	1	no	—
274	C₄	—	74	1	no	—
275	C₄	—	74	1	no	—
276	C₄	—	148	1	no	—
277	C₄	—	148	1	no	—
278	C₄	—	148	2	no	—
279	C₄	D₄×Dic₃₇	1	2	yes	—
280	C₄	C₂³×Dic₃₇	1	1	yes	characteristic
281	C₄	C₂×#10596964	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	4	2	no	—
284	C₂	C₂×D₄×Dic₃₇	1	1	yes	characteristic
285	C₂	#137×D₃₇	1	1	yes	characteristic
286	C₂	#14558332	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×#137×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#41×C₃₇	C₂²
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₂×#137	D₃₇
hypocenter	C₃₇	C₄×#137
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#137

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×Dic₃₇	C₂	semidirect
3. #14557640	C₂	semidirect
4. #14557634	C₂	semidirect
5. Q₁₆×Dic₃₇	C₂	semidirect
6. SD₁₆×Dic₃₇	C₂	semidirect
7. #14557575	C₂	semidirect
8. #14557542	C₂	semidirect
10. M₁₆×Dic₃₇	C₂	semidirect
40. #137×C₃₇	C₄	semidirect
48. C₂×Q₈×C₃₇	C₂×C₄	semidirect
61. Q₁₆×C₃₇	C₂×C₄	semidirect
62. SD₁₆×C₃₇	C₂×C₄	semidirect
63. M₁₆×C₃₇	C₂×C₄	semidirect
137. Dic₃₇	#137	direct (N × H)
155. C₃₇	C₄×#137	semidirect
158. #137	Dic₃₇	direct (N × H)
192. C₂×Q₈	C₂×Dic₃₇	semidirect
205. Q₁₆	C₂×Dic₃₇	semidirect
206. SD₁₆	C₂×Dic₃₇	semidirect
208. 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
#41×Dic₃₇	2
C₂×Q₈×Dic₃₇	2
#14557640	2
#14557634	2
Q₁₆×Dic₃₇	2
SD₁₆×Dic₃₇	2
#14557575	2
#14557542	2
#14557534	2
M₁₆×Dic₃₇	2
C₂×#137×C₃₇	2
C₄×#137	37

Subgroups — #137×Dic37 #18003417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #137×Dic₃₇ #18003417