Subgroups — #18003331

Order 4736.

544 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₃₇	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₁₄₈	C₂	1	1	yes	maximal, characteristic
3	#14557610	C₂	1	2	yes	maximal, complemented
4	#14557603	C₂	1	4	yes	maximal, complemented
5	#14557598	C₂	1	2	yes	maximal, complemented
6	#14557496	C₂	1	2	yes	maximal, complemented
7	C₄×#10596934	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#423	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₁₄₈	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₃₇	C₂²	1	2	yes	—
11	C₂×#10596934	C₂²	1	1	yes	characteristic
12	C₂×#10596934	C₂²	1	1	yes	characteristic
13	C₂×#10596929	C₂²	1	4	yes	complemented
14	M₁₆×D₃₇	—	2	8	no	—
15	C₂×C₈×D₃₇	C₂²	1	4	yes	complemented
16	C₄×D₁₄₈	C₄	1	2	yes	—
17	C₄×D₁₄₈	C₄	1	1	yes	characteristic
18	C₄×D₁₄₈	C₄	1	1	yes	characteristic
19	C₄×D₁₄₈	C₂²	1	1	yes	characteristic
20	C₄×D₁₄₈	C₂²	1	2	yes	—
21	C₄×D₁₄₈	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₃₇	C₄	1	1	yes	characteristic
23	C₂×#10596938	C₄	1	2	yes	—
24	#11924448	C₂²	1	4	yes	complemented
25	#11924448	C₂²	1	4	yes	complemented
26	#11924447	C₂²	1	2	yes	complemented
27	C₄×#64436	C₂²	1	2	yes	complemented
28	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
29	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×C₃₇	C₂²	1	2	yes	complemented
31	C₄⋊C₈×C₃₇	C₂²	1	2	yes	complemented
32	C₂³×D₃₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
36	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
37	C₂×D₁₄₈	—	2	4	no	—
38	C₂×D₁₄₈	C₂×C₄	1	2	yes	—
39	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
40	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₃₇	—	2	4	no	—
42	C₂×C₄×D₃₇	—	2	4	no	—
43	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₃₇	C₂³	1	2	yes	—
46	C₂×C₄×D₃₇	C₂³	1	2	yes	—
47	M₁₆×C₃₇	—	2	4	no	—
48	C₂×C₈×C₃₇	C₂³	1	4	yes	—
49	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
50	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₃₇	C₂³	1	1	yes	characteristic
52	C₄²×C₃₇	C₂³	1	1	yes	characteristic
53	#10596938	C₂×C₄	1	2	yes	—
54	#10596938	C₂×C₄	1	2	yes	—
55	#10596938	C₂×C₄	1	2	yes	—
56	#10596938	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
59	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
60	#10596934	—	2	2	no	—
61	#10596934	D₄	1	4	yes	complemented
62	C₂×#64436	C₂³	1	2	yes	—
63	C₂×#64436	C₂³	1	2	yes	—
64	#10596929	—	2	8	no	—
65	C₈×D₃₇	—	2	8	no	—
66	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₃₇	—	2	4	no	—
68	C₂²×D₃₇	—	2	4	no	—
69	C₂²×D₃₇	—	4	2	no	—
70	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
71	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₃₇	—	2	2	no	—
73	C₂×C₄×C₃₇	—	4	1	no	—
74	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₃₇	#41	1	2	yes	—
81	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₃₇	—	2	4	no	—
83	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
84	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
85	D₁₄₈	—	2	4	no	—
86	D₁₄₈	—	2	2	no	—
87	D₁₄₈	M₁₆	1	4	yes	complemented
88	C₄×D₃₇	—	2	4	no	—
89	C₄×D₃₇	—	2	4	no	—
90	C₄×D₃₇	—	2	2	no	—
91	C₄×D₃₇	—	2	2	no	—
92	C₄×D₃₇	—	2	2	no	—
93	C₄×D₃₇	—	2	2	no	—
94	C₈×C₃₇	—	2	4	no	—
95	#64436	—	2	2	no	—
96	#64436	C₂×D₄	1	4	yes	complemented
97	C₂²×C₃₇	—	2	1	no	—
98	C₂²×C₃₇	—	2	1	no	—
99	C₂²×C₃₇	C₄×D₄	1	2	yes	—
100	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₃₇	—	2	2	no	—
102	C₂×D₃₇	—	2	2	no	—
103	C₂×D₃₇	—	2	2	no	—
104	C₂×D₃₇	—	2	2	no	—
105	C₂×D₃₇	—	2	2	no	—
106	C₂×D₃₇	—	4	8	no	—
107	C₂×D₃₇	C₂×M₁₆	1	4	yes	complemented
108	C₄×C₃₇	—	2	1	no	—
109	C₄×C₃₇	—	2	1	no	—
110	C₄×C₃₇	—	2	1	no	—
111	C₄×C₃₇	C₄×D₄	1	2	yes	—
112	C₄×C₃₇	C₂×M₁₆	1	2	yes	complemented
113	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
115	Dic₃₇	—	2	2	no	—
116	Dic₃₇	—	2	2	no	—
117	D₄×M₁₆	—	37	1	no	maximal, Sylow
118	C₂×C₃₇	—	2	2	no	—
119	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
121	C₂×C₃₇	#568	1	1	yes	characteristic
122	D₃₇	—	2	4	no	—
123	D₃₇	—	4	2	no	—
124	C₂²×M₁₆	—	37	2	no	—
125	C₂×C₄×D₄	—	37	1	no	—
126	#436	—	37	2	no	—
127	#435	—	37	4	no	—
128	C₈×D₄	—	37	2	no	—
129	#423	D₃₇	1	1	yes	characteristic, complemented
130	#407	—	37	2	no	—
131	C₄×M₁₆	—	37	1	no	—
132	C₃₇	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
133	C₂²×D₄	—	37	1	no	—
134	C₂³×C₄	—	37	2	no	—
135	C₂×M₁₆	—	37	1	no	—
136	C₂×M₁₆	—	37	1	no	—
137	C₂×M₁₆	—	37	4	no	—
138	C₂×M₁₆	—	74	8	no	—
139	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
140	C₂²×C₈	—	37	4	no	—
141	C₄×D₄	—	37	1	no	—
142	C₄×D₄	—	37	2	no	—
143	C₄×D₄	—	37	1	no	—
144	C₄×D₄	—	37	1	no	—
145	C₄×D₄	—	37	2	no	—
146	C₄×D₄	—	37	1	no	—
147	C₂×C₄⋊C₄	—	37	1	no	—
148	C₂×C₂²⋊C₄	—	37	2	no	—
149	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
150	C₄⋊C₈	C₂×D₃₇	1	2	yes	complemented
151	C₄⋊C₈	C₂×D₃₇	1	2	yes	complemented
152	C₂²⋊C₈	—	37	4	no	—
153	C₂²⋊C₈	—	37	4	no	—
154	#97	—	37	2	no	—
155	C₄×C₈	—	37	2	no	—
156	C₂⁴	—	37	2	no	—
157	C₂×D₄	—	37	1	no	—
158	C₂×D₄	—	37	2	no	—
159	C₂×D₄	—	37	1	no	—
160	C₂×D₄	—	74	4	no	—
161	C₂²×C₄	—	37	2	no	—
162	C₂²×C₄	—	37	2	no	—
163	C₂²×C₄	—	37	2	no	—
164	C₂²×C₄	—	37	2	no	—
165	C₂²×C₄	—	37	2	no	—
166	C₂²×C₄	—	74	4	no	—
167	C₂²×C₄	—	74	4	no	—
168	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
171	M₁₆	—	2	4	no	—
172	M₁₆	—	37	4	no	—
173	M₁₆	—	74	8	no	—
174	M₁₆	—	74	2	no	—
175	C₂×C₈	—	37	2	no	—
176	C₂×C₈	—	37	2	no	—
177	C₂×C₈	—	74	8	no	—
178	C₂×C₈	C₂²×D₃₇	1	4	yes	—
179	C₄⋊C₄	—	37	2	no	—
180	C₄⋊C₄	—	37	1	no	—
181	C₄⋊C₄	—	37	1	no	—
182	C₂²⋊C₄	—	37	2	no	—
183	C₂²⋊C₄	—	37	2	no	—
184	C₂²⋊C₄	—	37	2	no	—
185	C₂²⋊C₄	—	37	2	no	—
186	C₄²	C₄×D₃₇	1	1	yes	characteristic
187	C₄²	C₄×D₃₇	1	1	yes	characteristic
188	C₄²	C₂²×D₃₇	1	1	yes	characteristic
189	C₄²	C₂²×D₃₇	1	1	yes	characteristic
190	C₂³	—	37	2	no	—
191	C₂³	—	37	2	no	—
192	C₂³	—	74	4	no	—
193	C₂³	—	74	4	no	—
194	C₂³	—	148	2	no	—
195	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
196	D₄	—	37	4	no	—
197	D₄	—	74	4	no	—
198	D₄	—	74	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	37	2	no	—
202	C₂×C₄	—	37	2	no	—
203	C₂×C₄	—	74	4	no	—
204	C₂×C₄	—	74	4	no	—
205	C₂×C₄	—	74	4	no	—
206	C₂×C₄	—	74	2	no	—
207	C₂×C₄	—	74	2	no	—
208	C₂×C₄	—	74	2	no	—
209	C₂×C₄	—	74	2	no	—
210	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
215	C₂×C₄	D₄×D₃₇	1	2	yes	—
216	C₂×C₄	#10596966	1	2	yes	—
217	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
218	C₈	—	2	4	no	—
219	C₈	—	37	4	no	—
220	C₈	—	74	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	37	4	no	—
224	C₂²	—	74	2	no	—
225	C₂²	—	74	2	no	—
226	C₂²	—	74	2	no	—
227	C₂²	—	74	2	no	—
228	C₂²	—	74	2	no	—
229	C₂²	—	148	8	no	—
230	C₂²	#11924498	1	2	yes	—
231	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	74	2	no	—
236	C₄	—	74	2	no	—
237	C₄	#11924498	1	2	yes	—
238	C₄	M₁₆×D₃₇	1	2	yes	complemented
239	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
240	C₄	C₂×#10596966	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	74	4	no	—
243	C₂	—	148	2	no	—
244	C₂	C₂×#11924498	1	1	yes	characteristic
245	C₂	C₂×M₁₆×D₃₇	1	1	yes	characteristic
246	C₂	#14558299	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₃₇×#423	C₂
prime radical	C₂²×D₁₄₈	C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#423	D₃₇
hypocenter	C₃₇	D₄×M₁₆
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×D₃₇	C₂	semidirect
3. #14557610	C₂	semidirect
4. #14557603	C₂	semidirect
5. #14557598	C₂	semidirect
6. #14557496	C₂	semidirect
7. C₄×#10596934	C₂	semidirect
8. C₃₇×#423	C₂	semidirect
13. C₂×#10596929	C₂²	semidirect
15. C₂×C₈×D₃₇	C₂²	semidirect
24. #11924448	C₂²	semidirect
25. #11924448	C₂²	semidirect
26. #11924447	C₂²	semidirect
27. C₄×#64436	C₂²	semidirect
30. C₄⋊C₈×C₃₇	C₂²	semidirect
31. C₄⋊C₈×C₃₇	C₂²	semidirect
61. #10596934	D₄	semidirect
87. D₁₄₈	M₁₆	semidirect
96. #64436	C₂×D₄	semidirect
107. C₂×D₃₇	C₂×M₁₆	semidirect
112. C₄×C₃₇	C₂×M₁₆	semidirect
129. #423	D₃₇	semidirect
132. C₃₇	D₄×M₁₆	semidirect
150. C₄⋊C₈	C₂×D₃₇	semidirect
151. C₄⋊C₈	C₂×D₃₇	semidirect
238. C₄	M₁₆×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₃₇	2
C₂×C₄×D₁₄₈	2
#14557610	2
#14557603	2
#14557598	2
#14557496	2
C₄×#10596934	2
C₃₇×#423	2
D₄×M₁₆	37