Subgroups — #18003503

Order 4736.

458 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558331	C₂	1	1	yes	maximal, characteristic
2	#14558299	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557653	C₂	1	4	yes	maximal, complemented
4	#14557648	C₂	1	4	yes	maximal, complemented
5	#14557581	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924463	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11924462	C₂	1	1	yes	maximal, characteristic, complemented
8	M₁₆×Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10596962	C₂²	1	1	yes	characteristic
11	#11924567	C₄	1	2	yes	complemented
12	#11924566	C₄	1	2	yes	complemented
13	#11924562	C₄	1	2	yes	complemented
14	#11924561	C₄	1	2	yes	complemented
15	#11924556	C₂²	1	4	yes	—
16	#11924544	—	2	2	no	—
17	#11924542	C₂²	1	4	yes	complemented
18	C₂×#10596934	C₂²	1	1	yes	characteristic
19	C₂×#10596934	C₂²	1	1	yes	characteristic
20	M₁₆×D₃₇	—	2	2	no	—
21	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
22	#11924471	C₂²	1	4	yes	complemented
23	#11924463	C₂²	1	2	yes	complemented
24	#11924463	C₂²	1	2	yes	complemented
25	#11924462	C₂²	1	2	yes	complemented
26	#11924462	C₂²	1	2	yes	complemented
27	#11924446	C₂²	1	2	yes	complemented
28	C₈×Dic₃₇	C₂²	1	2	yes	complemented
29	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
30	#131×C₃₇	—	2	2	no	—
31	#131×C₃₇	C₂²	1	4	yes	complemented
32	C₂×M₁₆×C₃₇	—	2	2	no	—
33	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
34	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
35	#41×C₃₇	—	2	1	no	—
36	#41×C₃₇	—	2	1	no	—
37	#41×C₃₇	C₂×C₄	1	2	yes	—
38	#41×C₃₇	C₂³	1	2	yes	—
39	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
40	C₂×D₄×C₃₇	—	2	1	no	—
41	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₇	—	2	1	no	—
43	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₃₇	—	2	1	no	—
45	C₂²×Dic₃₇	—	2	1	no	—
46	#10596962	—	2	1	no	—
47	#10596962	—	2	1	no	—
48	#10596962	C₂×C₄	1	2	yes	—
49	#10596962	C₂³	1	2	yes	—
50	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₃₇	—	2	1	no	—
52	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
53	M₁₆×C₃₇	—	2	1	no	—
54	M₁₆×C₃₇	—	2	4	no	—
55	M₁₆×C₃₇	D₄	1	2	yes	—
56	M₁₆×C₃₇	D₄	1	2	yes	—
57	M₁₆×C₃₇	C₂³	1	2	yes	—
58	C₂×C₈×C₃₇	—	2	1	no	—
59	C₂×C₈×C₃₇	—	2	4	no	—
60	C₂×C₈×C₃₇	C₂³	1	2	yes	—
61	#10596942	C₂×C₄	1	4	yes	complemented
62	#10596941	C₂×C₄	1	4	yes	complemented
63	#10596940	C₂×C₄	1	4	yes	complemented
64	#10596939	C₂×C₄	1	4	yes	complemented
65	C₄×Dic₃₇	—	2	1	no	—
66	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
67	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
68	#10596934	—	2	1	no	—
69	#10596934	C₂×C₄	1	2	yes	—
70	#10596934	C₂×C₄	1	2	yes	—
71	#10596934	C₂³	1	2	yes	—
72	C₂×#64436	—	2	1	no	—
73	C₂×#64436	C₂³	1	2	yes	—
74	#10596929	—	2	4	no	—
75	C₈×D₃₇	—	2	4	no	—
76	C₂³×C₃₇	—	2	1	no	—
77	C₂³×C₃₇	#41	1	1	yes	characteristic
78	C₂²×D₃₇	—	2	1	no	—
79	Q₈×C₃₇	—	2	1	no	—
80	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
81	D₄×C₃₇	—	2	2	no	—
82	D₄×C₃₇	—	2	2	no	—
83	D₄×C₃₇	—	2	1	no	—
84	D₄×C₃₇	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₃₇	—	2	1	no	—
86	C₂×C₄×C₃₇	—	2	2	no	—
87	C₂×C₄×C₃₇	—	2	2	no	—
88	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
93	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₃₇	—	2	2	no	—
95	C₂²⋊D₃₇	—	2	2	no	—
96	C₂×Dic₃₇	—	2	1	no	—
97	C₂×Dic₃₇	—	2	1	no	—
98	C₂×Dic₃₇	—	2	1	no	—
99	C₂×Dic₃₇	—	4	2	no	—
100	D₁₄₈	—	2	1	no	—
101	D₁₄₈	C₂²×C₄	1	2	yes	—
102	C₄×D₃₇	—	2	2	no	—
103	C₄×D₃₇	—	2	2	no	—
104	Dic₇₄	—	2	1	no	—
105	Dic₇₄	C₂²×C₄	1	2	yes	—
106	C₈×C₃₇	—	2	2	no	—
107	C₈×C₃₇	C₂×D₄	1	4	yes	—
108	#64436	—	2	2	no	—
109	#64436	C₂²×C₄	1	4	yes	—
110	C₂²×C₃₇	—	2	2	no	—
111	C₂²×C₃₇	—	4	1	no	—
112	C₂²×C₃₇	—	8	1	no	—
113	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
114	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
116	C₂×D₃₇	—	2	2	no	—
117	C₂×D₃₇	—	8	1	no	—
118	C₄×C₃₇	—	2	2	no	—
119	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
120	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
121	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
123	Dic₃₇	—	2	2	no	—
124	Dic₃₇	—	2	2	no	—
125	Dic₃₇	—	4	1	no	—
126	#2956	—	37	1	no	maximal, Sylow
127	C₂×C₃₇	—	2	1	no	—
128	C₂×C₃₇	—	2	1	no	—
129	C₂×C₃₇	—	2	1	no	—
130	C₂×C₃₇	—	4	2	no	—
131	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
132	D₃₇	—	4	2	no	—
133	#575	—	37	1	no	—
134	#568	—	37	1	no	—
135	#568	D₃₇	1	1	yes	characteristic, complemented
136	#444	—	37	4	no	—
137	#443	—	37	4	no	—
138	#430	—	37	1	no	—
139	C₂×#104	—	37	1	no	—
140	C₂×#104	—	37	1	no	—
141	C₄×M₁₆	—	37	1	no	—
142	C₃₇	#2956	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	—	37	1	no	—
144	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
145	#137	—	37	2	no	—
146	#137	—	37	2	no	—
147	AGL(1,ℤ/8)	—	37	2	no	—
148	AGL(1,ℤ/8)	—	37	2	no	—
149	#135	—	37	4	no	—
150	#131	—	2	2	no	—
151	#131	—	37	4	no	—
152	#131	—	74	2	no	—
153	#131	C₂×D₃₇	1	4	yes	complemented
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	37	1	no	—
156	C₂×M₁₆	—	37	1	no	—
157	C₂×M₁₆	—	74	2	no	—
158	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
159	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
160	C₂×C₄²	—	37	1	no	—
161	#108	—	37	4	no	—
162	#104	—	37	2	no	—
163	#104	—	37	2	no	—
164	#104	—	37	2	no	—
165	#104	—	37	2	no	—
166	#97	—	37	2	no	—
167	C₄×C₈	—	37	2	no	—
168	#41	—	2	1	no	—
169	#41	—	2	1	no	—
170	#41	—	37	2	no	—
171	#41	—	37	2	no	—
172	#41	—	74	1	no	—
173	#41	—	74	1	no	—
174	#41	C₄×D₃₇	1	2	yes	—
175	#41	C₂²×D₃₇	1	2	yes	—
176	C₂×Q₈	—	37	1	no	—
177	C₂×Q₈	C₄×D₃₇	1	1	yes	characteristic
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	37	1	no	—
180	C₂×D₄	—	74	1	no	—
181	C₂×D₄	C₄×D₃₇	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	74	1	no	—
184	C₂²×C₄	—	74	1	no	—
185	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
186	Q₁₆	—	37	4	no	—
187	SD₁₆	—	37	4	no	—
188	SD₁₆	—	37	4	no	—
189	D₈	—	37	4	no	—
190	M₁₆	—	2	1	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	37	2	no	—
193	M₁₆	—	37	2	no	—
194	M₁₆	—	37	2	no	—
195	M₁₆	—	74	1	no	—
196	M₁₆	—	74	4	no	—
197	M₁₆	C₂²⋊D₃₇	1	2	yes	—
198	M₁₆	C₂²⋊D₃₇	1	2	yes	—
199	M₁₆	C₂²×D₃₇	1	2	yes	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	37	2	no	—
203	C₂×C₈	—	74	1	no	—
204	C₂×C₈	—	74	4	no	—
205	C₂×C₈	C₂²×D₃₇	1	2	yes	—
206	C₄²	—	37	1	no	—
207	C₄²	—	37	1	no	—
208	C₄²	—	74	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	74	1	no	—
211	C₂³	#10596962	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	37	2	no	—
214	Q₈	—	74	1	no	—
215	Q₈	C₂×C₄×D₃₇	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	37	2	no	—
220	D₄	—	74	2	no	—
221	D₄	—	74	2	no	—
222	D₄	—	74	1	no	—
223	D₄	C₂×C₄×D₃₇	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	74	1	no	—
228	C₂×C₄	—	74	1	no	—
229	C₂×C₄	—	74	2	no	—
230	C₂×C₄	—	74	2	no	—
231	C₂×C₄	—	74	1	no	—
232	C₂×C₄	—	148	2	no	—
233	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
235	C₂×C₄	#10596962	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	37	4	no	—
241	C₈	—	74	2	no	—
242	C₈	C₂×C₂²⋊D₃₇	1	4	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	296	1	no	—
247	C₂²	—	74	2	no	—
248	C₂²	C₄×C₂²⋊D₃₇	1	1	yes	characteristic
249	C₂²	C₄×C₂²⋊D₃₇	1	1	yes	characteristic
250	C₂²	C₂×#10596962	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	74	2	no	—
253	C₄	—	74	2	no	—
254	C₄	—	148	1	no	—
255	C₄	C₄×C₂²⋊D₃₇	1	1	yes	characteristic
256	C₄	C₄×C₂²⋊D₃₇	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
258	C₄	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	148	2	no	—
264	C₂	C₂×C₄×C₂²⋊D₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#568	C₂
socle	C₂×C₃₇	C₂×C₄×D₄
prime radical	#14558331	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#568	D₃₇
hypocenter	C₃₇	#2956
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #568

Complements

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

N	H ≅ G/N	type
2. #14558299	C₂	semidirect
3. #14557653	C₂	semidirect
4. #14557648	C₂	semidirect
5. #14557581	C₂	semidirect
6. C₂×#11924463	C₂	semidirect
7. C₂×#11924462	C₂	semidirect
8. M₁₆×Dic₃₇	C₂	semidirect
9. C₃₇×#568	C₂	semidirect
11. #11924567	C₄	semidirect
12. #11924566	C₄	semidirect
13. #11924562	C₄	semidirect
14. #11924561	C₄	semidirect
17. #11924542	C₂²	semidirect
22. #11924471	C₂²	semidirect
23. #11924463	C₂²	semidirect
24. #11924463	C₂²	semidirect
25. #11924462	C₂²	semidirect
26. #11924462	C₂²	semidirect
27. #11924446	C₂²	semidirect
28. C₈×Dic₃₇	C₂²	semidirect
31. #131×C₃₇	C₂²	semidirect
61. #10596942	C₂×C₄	semidirect
62. #10596941	C₂×C₄	semidirect
63. #10596940	C₂×C₄	semidirect
64. #10596939	C₂×C₄	semidirect
135. #568	D₃₇	semidirect
142. C₃₇	#2956	semidirect
153. #131	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#14558331	2
#14558299	2
#14557653	2
#14557648	2
#14557581	2
C₂×#11924463	2
C₂×#11924462	2
M₁₆×Dic₃₇	2
C₃₇×#568	2
#2956	37