Subgroups — #18003430

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	C₂×#11924564	C₂	1	1	yes	maximal, characteristic
2	#14558013	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558007	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557646	C₂	1	2	yes	maximal, complemented
5	#14557635	C₂	1	2	yes	maximal, complemented
6	C₄×#10596930	C₂	1	2	yes	maximal, complemented
7	C₄×D₂₉₆	C₂	1	2	yes	maximal, complemented
8	#14557575	C₂	1	1	yes	maximal, characteristic, complemented
9	#14557530	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11924458	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₇×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596962	C₂²	1	1	yes	characteristic
13	C₂²×D₁₄₈	C₂²	1	1	yes	characteristic
14	#11924564	C₄	1	8	yes	complemented
15	C₂×#10596930	C₂²	1	2	yes	—
16	C₂×D₂₉₆	C₂²	1	2	yes	—
17	C₄×Dic₇₄	C₂²	1	2	yes	complemented
18	C₄×D₁₄₈	C₂²	1	2	yes	—
19	C₄×D₁₄₈	C₂²	1	2	yes	—
20	C₄×D₁₄₈	C₂²	1	2	yes	complemented
21	#11924498	—	2	1	no	—
22	#11924498	—	2	1	no	—
23	#11924496	—	2	2	no	—
24	#11924495	C₂²	1	1	yes	characteristic
25	#11924494	—	2	1	no	—
26	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₃₇	—	2	1	no	—
28	C₂²⋊C₄×D₃₇	—	2	2	no	—
29	#11924470	C₂²	1	2	yes	complemented
30	#11924468	C₂²	1	2	yes	complemented
31	#11924461	C₂²	1	1	yes	characteristic, complemented
32	#11924461	C₂²	1	1	yes	characteristic, complemented
33	#11924458	C₂²	1	2	yes	—
34	#11924458	C₂²	1	2	yes	—
35	#11924458	C₂²	1	1	yes	characteristic, complemented
36	#11924458	C₂²	1	1	yes	characteristic, complemented
37	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
39	#117×C₃₇	C₂²	1	1	yes	characteristic
40	#97×C₃₇	C₂²	1	2	yes	complemented
41	C₄×C₈×C₃₇	C₂²	1	2	yes	complemented
42	C₂³×D₃₇	—	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	2	no	—
47	#10596962	C₂×C₄	1	4	yes	complemented
48	C₂×D₁₄₈	—	2	1	no	—
49	C₂×D₁₄₈	—	2	2	no	—
50	C₂×D₁₄₈	C₂×C₄	1	4	yes	complemented
51	C₂×D₁₄₈	C₂³	1	2	yes	—
52	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₃₇	—	2	1	no	—
54	C₂×C₄×D₃₇	—	2	2	no	—
55	C₂×C₄×D₃₇	—	2	2	no	—
56	C₂×C₄×D₃₇	—	2	2	no	—
57	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
58	M₁₆×C₃₇	C₂×C₄	1	4	yes	complemented
59	C₂×C₈×C₃₇	—	2	2	no	—
60	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₃₇	C₂³	1	2	yes	—
63	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
65	C₂²⋊C₄×C₃₇	D₄	1	2	yes	—
66	C₄²×C₃₇	C₂³	1	2	yes	—
67	C₂²⋊Dic₃₇	—	2	1	no	—
68	#10596938	—	2	2	no	—
69	#10596938	—	2	2	no	—
70	#10596938	—	2	2	no	—
71	C₄⋊Dic₃₇	—	2	1	no	—
72	C₄⋊Dic₃₇	C₂³	1	2	yes	—
73	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
75	#10596936	—	2	2	no	—
76	C₄×Dic₃₇	—	2	2	no	—
77	C₄×Dic₃₇	—	2	1	no	—
78	D₂₉₆	C₂×C₄	1	8	yes	complemented
79	#10596930	C₂×C₄	1	8	yes	complemented
80	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₃₇	—	2	1	no	—
82	C₂²×D₃₇	—	2	2	no	—
83	C₂²×D₃₇	—	2	4	no	—
84	C₂²×D₃₇	—	8	1	no	—
85	C₂²×D₃₇	—	8	1	no	—
86	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₃₇	#41	1	2	yes	—
91	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
92	C₂²⋊D₃₇	—	2	4	no	—
93	C₂×Dic₃₇	—	2	1	no	—
94	C₂×Dic₃₇	—	2	2	no	—
95	C₂×Dic₃₇	—	2	1	no	—
96	C₂×Dic₃₇	—	2	1	no	—
97	C₂×Dic₃₇	—	8	1	no	—
98	D₁₄₈	—	2	4	no	—
99	D₁₄₈	—	2	2	no	—
100	D₁₄₈	—	2	1	no	—
101	D₁₄₈	C₂²×C₄	1	4	yes	—
102	D₁₄₈	C₂²×C₄	1	2	yes	—
103	C₄×D₃₇	—	2	4	no	—
104	C₄×D₃₇	—	8	2	no	—
105	C₄×D₃₇	—	8	2	no	—
106	C₄×D₃₇	—	8	2	no	—
107	Dic₇₄	—	2	1	no	—
108	Dic₇₄	C₂²×C₄	1	2	yes	—
109	C₈×C₃₇	—	2	1	no	—
110	C₈×C₃₇	—	2	1	no	—
111	C₈×C₃₇	C₂²×C₄	1	4	yes	—
112	C₂²×C₃₇	—	2	1	no	—
113	C₂²×C₃₇	—	2	1	no	—
114	C₂²×C₃₇	C₄×D₄	1	2	yes	—
115	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×D₃₇	—	2	2	no	—
117	C₂×D₃₇	—	2	4	no	—
118	C₂×D₃₇	—	4	1	no	—
119	C₂×D₃₇	—	4	2	no	—
120	C₂×D₃₇	—	4	2	no	—
121	C₂×D₃₇	—	4	1	no	—
122	C₂×D₃₇	—	8	4	no	—
123	C₄×C₃₇	—	2	2	no	—
124	C₄×C₃₇	—	2	2	no	—
125	C₄×C₃₇	C₄×D₄	1	2	yes	—
126	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
127	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
128	Dic₃₇	—	2	2	no	—
129	Dic₃₇	—	4	1	no	—
130	Dic₃₇	—	4	2	no	—
131	Dic₃₇	—	4	1	no	—
132	#2946	—	37	1	no	maximal, Sylow
133	C₂×C₃₇	—	2	2	no	—
134	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×C₃₇	#576	1	1	yes	characteristic
136	C₂×C₃₇	#577	1	1	yes	characteristic
137	D₃₇	—	4	2	no	—
138	D₃₇	—	4	4	no	—
139	C₂×AGL(1,ℤ/8)	—	37	1	no	—
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	D₃₇	1	1	yes	characteristic, complemented
150	C₃₇	#2946	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	—	37	1	no	—
152	C₂²×D₄	—	37	1	no	—
153	AGL(1,ℤ/8)	—	37	8	no	—
154	C₂×SD₁₆	—	37	2	no	—
155	C₂×D₈	—	37	2	no	—
156	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
157	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
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	1	no	—
163	C₄×D₄	—	74	1	no	—
164	C₄×D₄	—	74	2	no	—
165	#117	—	37	1	no	—
166	#117	—	74	1	no	—
167	#117	C₂×D₃₇	1	1	yes	characteristic
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	C₂×D₃₇	1	2	yes	complemented
180	C₄×C₈	C₂×D₃₇	1	2	yes	complemented
181	C₂⁴	—	74	1	no	—
182	#41	—	37	4	no	—
183	#41	—	74	2	no	—
184	C₂×Q₈	—	37	1	no	—
185	C₂×D₄	—	37	4	no	—
186	C₂×D₄	—	37	1	no	—
187	C₂×D₄	—	37	2	no	—
188	C₂×D₄	—	74	1	no	—
189	C₂×D₄	—	74	1	no	—
190	C₂×D₄	—	74	2	no	—
191	C₂²×C₄	—	74	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	—	37	8	no	—
198	D₈	—	37	8	no	—
199	M₁₆	C₄×D₃₇	1	4	yes	complemented
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₃₇	1	2	yes	—
204	C₄⋊C₄	—	37	2	no	—
205	C₄⋊C₄	—	37	1	no	—
206	C₄⋊C₄	—	37	1	no	—
207	C₄⋊C₄	—	74	2	no	—
208	C₄⋊C₄	—	74	1	no	—
209	C₄⋊C₄	D₁₄₈	1	1	yes	characteristic
210	C₄⋊C₄	D₁₄₈	1	1	yes	characteristic
211	C₂²⋊C₄	—	74	1	no	—
212	C₂²⋊C₄	—	74	2	no	—
213	C₂²⋊C₄	—	74	2	no	—
214	C₂²⋊C₄	—	74	2	no	—
215	C₂²⋊C₄	D₁₄₈	1	2	yes	—
216	C₄²	—	74	1	no	—
217	C₄²	—	74	2	no	—
218	C₄²	C₂²×D₃₇	1	2	yes	—
219	C₂³	—	296	1	no	—
220	C₂³	—	296	1	no	—
221	C₂³	—	74	4	no	—
222	C₂³	—	74	1	no	—
223	C₂³	—	74	2	no	—
224	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
225	Q₈	—	37	2	no	—
226	Q₈	—	74	1	no	—
227	D₄	—	37	2	no	—
228	D₄	—	37	4	no	—
229	D₄	—	74	4	no	—
230	D₄	—	74	4	no	—
231	D₄	—	74	2	no	—
232	D₄	—	74	1	no	—
233	C₂×C₄	—	296	2	no	—
234	C₂×C₄	—	296	2	no	—
235	C₂×C₄	—	296	2	no	—
236	C₂×C₄	—	296	1	no	—
237	C₂×C₄	—	74	1	no	—
238	C₂×C₄	—	74	4	no	—
239	C₂×C₄	—	74	1	no	—
240	C₂×C₄	—	74	1	no	—
241	C₂×C₄	—	74	2	no	—
242	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
243	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
244	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
245	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
246	C₂×C₄	#10596962	1	2	yes	—
247	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	C₂×C₄×D₃₇	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	296	4	no	—
254	C₂²	—	74	2	no	—
255	C₂²	—	74	4	no	—
256	C₂²	—	148	1	no	—
257	C₂²	—	148	2	no	—
258	C₂²	—	148	2	no	—
259	C₂²	—	148	1	no	—
260	C₂²	C₄×D₁₄₈	1	2	yes	—
261	C₂²	C₂²×D₁₄₈	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	74	2	no	—
265	C₄	—	148	1	no	—
266	C₄	—	148	2	no	—
267	C₄	—	148	1	no	—
268	C₄	C₄×D₁₄₈	1	2	yes	—
269	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
270	C₄	C₂×#10596962	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	148	4	no	—
273	C₂	—	148	2	no	—
274	C₂	C₂×C₄×D₁₄₈	1	1	yes	characteristic
275	C₂	#14558338	1	1	yes	characteristic
276	C₂	#14558345	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₄₈
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#405	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#11924564	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#405	D₃₇
hypocenter	C₃₇	#2946
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #14558013	C₂	semidirect
3. #14558007	C₂	semidirect
4. #14557646	C₂	semidirect
5. #14557635	C₂	semidirect
6. C₄×#10596930	C₂	semidirect
7. C₄×D₂₉₆	C₂	semidirect
8. #14557575	C₂	semidirect
9. #14557530	C₂	semidirect
10. C₂×#11924458	C₂	semidirect
11. C₃₇×#405	C₂	semidirect
14. #11924564	C₄	semidirect
17. C₄×Dic₇₄	C₂²	semidirect
20. C₄×D₁₄₈	C₂²	semidirect
29. #11924470	C₂²	semidirect
30. #11924468	C₂²	semidirect
31. #11924461	C₂²	semidirect
32. #11924461	C₂²	semidirect
35. #11924458	C₂²	semidirect
36. #11924458	C₂²	semidirect
40. #97×C₃₇	C₂²	semidirect
41. C₄×C₈×C₃₇	C₂²	semidirect
47. #10596962	C₂×C₄	semidirect
50. C₂×D₁₄₈	C₂×C₄	semidirect
58. M₁₆×C₃₇	C₂×C₄	semidirect
78. D₂₉₆	C₂×C₄	semidirect
79. #10596930	C₂×C₄	semidirect
149. #405	D₃₇	semidirect
150. C₃₇	#2946	semidirect
179. #97	C₂×D₃₇	semidirect
180. C₄×C₈	C₂×D₃₇	semidirect
199. M₁₆	C₄×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924564	2
#14558013	2
#14558007	2
#14557646	2
#14557635	2
C₄×#10596930	2
C₄×D₂₉₆	2
#14557575	2
#14557530	2
C₂×#11924458	2
C₃₇×#405	2
#2946	37