Subgroups — #18003428

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×#11924562	C₂	1	1	yes	maximal, characteristic
2	#14558013	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557643	C₂	1	2	yes	maximal, complemented
4	#14557633	C₂	1	2	yes	maximal, complemented
5	C₄×#10596940	C₂	1	2	yes	maximal, complemented
6	C₄×#10596939	C₂	1	2	yes	maximal, complemented
7	#14557576	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557527	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11924457	C₂	1	1	yes	maximal, characteristic, complemented
10	#14557489	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₇×#518	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10596962	C₂²	1	1	yes	characteristic
13	#11924562	C₄	1	8	yes	complemented
14	C₂×#10596940	C₂²	1	2	yes	—
15	C₂×#10596939	C₂²	1	2	yes	—
16	C₂×#10596934	C₂²	1	1	yes	characteristic
17	C₂²×#64436	C₂²	1	1	yes	characteristic
18	C₄×Dic₇₄	C₂²	1	2	yes	complemented
19	C₄×D₁₄₈	C₂²	1	2	yes	complemented
20	#11924496	—	2	2	no	—
21	#11924494	—	2	1	no	—
22	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₃₇	—	2	1	no	—
24	#11924469	C₂²	1	2	yes	complemented
25	#11924467	C₂²	1	2	yes	complemented
26	#11924459	C₂²	1	1	yes	characteristic, complemented
27	#11924459	C₂²	1	1	yes	characteristic, complemented
28	#11924457	C₂²	1	2	yes	—
29	#11924457	C₂²	1	2	yes	—
30	#11924456	C₂²	1	1	yes	characteristic, complemented
31	#11924456	C₂²	1	1	yes	characteristic, complemented
32	#11924447	C₂²	1	2	yes	complemented
33	C₄×#64436	C₂²	1	2	yes	complemented
34	C₂²×D₄×C₃₇	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₃₇	—	2	1	no	—
36	C₄×D₄×C₃₇	—	2	1	no	—
37	C₄×D₄×C₃₇	C₂²	1	2	yes	—
38	C₄×D₄×C₃₇	C₂²	1	2	yes	—
39	#117×C₃₇	C₂²	1	1	yes	characteristic
40	#117×C₃₇	C₂²	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
42	C₂⁴×C₃₇	—	2	1	no	—
43	C₂×D₄×C₃₇	—	2	1	no	—
44	C₂×D₄×C₃₇	—	2	2	no	—
45	C₂×D₄×C₃₇	C₂×C₄	1	4	yes	complemented
46	C₂×D₄×C₃₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₃₇	—	2	2	no	—
48	C₂²×C₄×C₃₇	—	2	2	no	—
49	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₃₇	—	2	1	no	—
51	C₂²×Dic₃₇	—	2	1	no	—
52	#10596962	—	2	2	no	—
53	#10596962	C₂×C₄	1	4	yes	complemented
54	C₂×D₁₄₈	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₃₇	—	2	1	no	—
56	C₂×C₄×D₃₇	—	2	2	no	—
57	C₂×Dic₇₄	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₃₇	C₂³	1	1	yes	characteristic
62	C₂²⋊C₄×C₃₇	—	2	1	no	—
63	C₂²⋊C₄×C₃₇	—	2	2	no	—
64	C₂²⋊C₄×C₃₇	—	2	2	no	—
65	C₂²⋊C₄×C₃₇	D₄	1	2	yes	—
66	C₄²×C₃₇	—	2	1	no	—
67	C₄²×C₃₇	C₂³	1	2	yes	—
68	#10596940	C₂×C₄	1	8	yes	complemented
69	#10596939	C₂×C₄	1	8	yes	complemented
70	#10596938	—	2	2	no	—
71	C₄⋊Dic₃₇	—	2	1	no	—
72	C₄⋊Dic₃₇	C₂³	1	2	yes	—
73	#10596936	—	2	2	no	—
74	C₄×Dic₃₇	—	2	2	no	—
75	#10596934	C₂×C₄	1	4	yes	complemented
76	C₂×#64436	—	2	2	no	—
77	C₂×#64436	C₂³	1	1	yes	characteristic
78	C₂×#64436	C₂³	1	1	yes	characteristic
79	C₂×#64436	C₂³	1	2	yes	—
80	C₂³×C₃₇	—	2	2	no	—
81	C₂³×C₃₇	—	2	4	no	—
82	C₂³×C₃₇	—	8	1	no	—
83	C₂³×C₃₇	—	8	1	no	—
84	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×D₃₇	—	2	1	no	—
86	D₄×C₃₇	—	2	4	no	—
87	D₄×C₃₇	—	2	2	no	—
88	D₄×C₃₇	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₃₇	—	2	1	no	—
90	C₂×C₄×C₃₇	—	2	1	no	—
91	C₂×C₄×C₃₇	—	8	2	no	—
92	C₂×C₄×C₃₇	—	8	2	no	—
93	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₇	#41	1	2	yes	—
98	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₃₇	—	2	4	no	—
100	C₂×Dic₃₇	—	2	1	no	—
101	C₂×Dic₃₇	—	2	2	no	—
102	C₂×Dic₃₇	—	8	1	no	—
103	D₁₄₈	—	2	1	no	—
104	D₁₄₈	C₂²×C₄	1	2	yes	—
105	C₄×D₃₇	—	2	4	no	—
106	C₄×D₃₇	—	8	2	no	—
107	Dic₇₄	—	2	1	no	—
108	Dic₇₄	C₂²×C₄	1	2	yes	—
109	#64436	—	2	1	no	—
110	#64436	—	2	1	no	—
111	#64436	C₂²×C₄	1	4	yes	—
112	C₂²×C₃₇	—	2	1	no	—
113	C₂²×C₃₇	—	2	1	no	—
114	C₂²×C₃₇	—	2	4	no	—
115	C₂²×C₃₇	—	4	2	no	—
116	C₂²×C₃₇	—	4	2	no	—
117	C₂²×C₃₇	—	8	4	no	—
118	C₂²×C₃₇	C₄×D₄	1	2	yes	—
119	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₃₇	—	2	2	no	—
121	C₂×D₃₇	—	4	1	no	—
122	C₂×D₃₇	—	4	1	no	—
123	C₄×C₃₇	—	2	2	no	—
124	C₄×C₃₇	—	2	2	no	—
125	C₄×C₃₇	—	4	1	no	—
126	C₄×C₃₇	—	4	1	no	—
127	C₄×C₃₇	C₄×D₄	1	2	yes	—
128	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
130	Dic₃₇	—	2	2	no	—
131	Dic₃₇	—	4	2	no	—
132	#2946	—	37	1	no	maximal, Sylow
133	C₂×C₃₇	—	2	2	no	—
134	C₂×C₃₇	—	4	4	no	—
135	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₃₇	#576	1	1	yes	characteristic
137	C₂×C₃₇	#577	1	1	yes	characteristic
138	D₃₇	—	4	2	no	—
139	C₂×AGL(1,ℤ/8)	—	37	1	no	—
140	#520	—	37	1	no	—
141	#518	D₃₇	1	1	yes	characteristic, complemented
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	—	37	1	no	—
150	C₃₇	#2946	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	—	37	1	no	—
152	C₂²×D₄	C₂×D₃₇	1	1	yes	characteristic
153	AGL(1,ℤ/8)	—	37	8	no	—
154	C₂×SD₁₆	—	37	2	no	—
155	C₂×D₈	—	37	2	no	—
156	C₂×M₁₆	—	37	1	no	—
157	C₂²×C₈	—	37	1	no	—
158	C₄×Q₈	—	37	2	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	—	37	2	no	—
162	C₄×D₄	—	74	2	no	—
163	C₄×D₄	C₂×D₃₇	1	2	yes	—
164	C₄×D₄	C₂×D₃₇	1	2	yes	—
165	#117	—	74	1	no	—
166	#117	C₂×D₃₇	1	1	yes	characteristic
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	—	2	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	—	37	2	no	—
180	C₄×C₈	—	37	2	no	—
181	C₂⁴	—	2	1	no	—
182	#41	—	37	4	no	—
183	#41	—	74	2	no	—
184	C₂×Q₈	—	37	1	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	37	1	no	—
188	C₂×D₄	—	74	1	no	—
189	C₂×D₄	C₄×D₃₇	1	4	yes	complemented
190	C₂×D₄	C₂²×D₃₇	1	2	yes	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	74	1	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	—	37	4	no	—
200	C₂×C₈	—	37	2	no	—
201	C₂×C₈	—	37	1	no	—
202	C₂×C₈	—	37	1	no	—
203	C₂×C₈	—	74	2	no	—
204	C₄⋊C₄	—	37	2	no	—
205	C₄⋊C₄	—	74	2	no	—
206	C₄⋊C₄	—	74	1	no	—
207	C₄⋊C₄	C₂²⋊D₃₇	1	1	yes	characteristic
208	C₄⋊C₄	C₂²⋊D₃₇	1	1	yes	characteristic
209	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₃₇	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	74	2	no	—
215	C₂²⋊C₄	C₂²⋊D₃₇	1	2	yes	—
216	C₄²	—	2	1	no	—
217	C₄²	—	74	2	no	—
218	C₄²	C₂²×D₃₇	1	2	yes	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	8	1	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	74	1	no	—
224	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
225	Q₈	—	37	2	no	—
226	Q₈	—	74	1	no	—
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	—	37	2	no	—
230	D₄	—	74	4	no	—
231	D₄	—	74	1	no	—
232	D₄	C₂×C₄×D₃₇	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	296	2	no	—
238	C₂×C₄	—	296	1	no	—
239	C₂×C₄	—	74	1	no	—
240	C₂×C₄	—	74	4	no	—
241	C₂×C₄	—	74	2	no	—
242	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
243	C₂×C₄	#10596962	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
246	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
247	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
248	C₈	—	37	4	no	—
249	C₈	—	74	1	no	—
250	C₈	—	74	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	74	2	no	—
258	C₂²	—	148	1	no	—
259	C₂²	—	148	1	no	—
260	C₂²	C₄×C₂²⋊D₃₇	1	2	yes	—
261	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	74	2	no	—
267	C₄	—	148	2	no	—
268	C₄	C₄×C₂²⋊D₃₇	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	—	4	4	no	—
273	C₂	—	148	2	no	—
274	C₂	C₂×C₄×C₂²⋊D₃₇	1	1	yes	characteristic
275	C₂	#14558336	1	1	yes	characteristic
276	C₂	#14558344	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#518	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×#11924562	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#518	D₃₇
hypocenter	C₃₇	#2946
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #518

Complements

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

N	H ≅ G/N	type
2. #14558013	C₂	semidirect
3. #14557643	C₂	semidirect
4. #14557633	C₂	semidirect
5. C₄×#10596940	C₂	semidirect
6. C₄×#10596939	C₂	semidirect
7. #14557576	C₂	semidirect
8. #14557527	C₂	semidirect
9. C₂×#11924457	C₂	semidirect
10. #14557489	C₂	semidirect
11. C₃₇×#518	C₂	semidirect
13. #11924562	C₄	semidirect
18. C₄×Dic₇₄	C₂²	semidirect
19. C₄×D₁₄₈	C₂²	semidirect
24. #11924469	C₂²	semidirect
25. #11924467	C₂²	semidirect
26. #11924459	C₂²	semidirect
27. #11924459	C₂²	semidirect
30. #11924456	C₂²	semidirect
31. #11924456	C₂²	semidirect
32. #11924447	C₂²	semidirect
33. C₄×#64436	C₂²	semidirect
45. C₂×D₄×C₃₇	C₂×C₄	semidirect
53. #10596962	C₂×C₄	semidirect
68. #10596940	C₂×C₄	semidirect
69. #10596939	C₂×C₄	semidirect
75. #10596934	C₂×C₄	semidirect
141. #518	D₃₇	semidirect
150. C₃₇	#2946	semidirect
189. C₂×D₄	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×#11924562	2
#14558013	2
#14557643	2
#14557633	2
C₄×#10596940	2
C₄×#10596939	2
#14557576	2
#14557527	2
C₂×#11924457	2
#14557489	2
C₃₇×#518	2
#2946	37