Subgroups — #18003509

Order 4736.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924556	C₂	1	1	yes	maximal, characteristic
2	#14558300	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558299	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557652	C₂	1	4	yes	maximal, complemented
5	#14557647	C₂	1	4	yes	maximal, complemented
6	C₂×#11924471	C₂	1	1	yes	maximal, characteristic, complemented
7	#14557551	C₂	1	1	yes	maximal, characteristic, complemented
8	#14557489	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#421	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10596962	C₂²	1	1	yes	characteristic
11	#11924556	C₄	1	2	yes	complemented
12	#11924556	C₄	1	2	yes	complemented
13	#11924556	C₂²	1	2	yes	—
14	#11924556	C₂²	1	2	yes	—
15	C₂×#10596942	C₄	1	1	yes	characteristic, complemented
16	C₂×#10596940	C₄	1	1	yes	characteristic, complemented
17	C₂×#10596941	C₄	1	1	yes	characteristic, complemented
18	C₂×#10596939	C₄	1	1	yes	characteristic, complemented
19	#11924544	C₂²	1	4	yes	complemented
20	#11924543	—	2	2	no	—
21	#11924543	C₂²	1	4	yes	complemented
22	#11924542	—	2	2	no	—
23	C₂×#10596934	—	2	2	no	—
24	C₂×#10596934	C₂²	1	1	yes	characteristic
25	C₂×#10596934	C₂²	1	1	yes	characteristic
26	M₁₆×D₃₇	—	2	2	no	—
27	C₂²×#64436	C₂²	1	1	yes	characteristic
28	#11924471	C₂²	1	2	yes	complemented
29	#11924471	C₂²	1	2	yes	complemented
30	#11924464	C₂²	1	4	yes	complemented
31	#11924447	C₂²	1	2	yes	complemented
32	C₄×#64436	C₂²	1	2	yes	complemented
33	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
34	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
35	#117×C₃₇	C₂²	1	1	yes	characteristic
36	#104×C₃₇	C₂²	1	4	yes	complemented
37	#41×C₃₇	—	2	1	no	—
38	#41×C₃₇	—	2	1	no	—
39	#41×C₃₇	C₂×C₄	1	2	yes	—
40	#41×C₃₇	C₂³	1	2	yes	—
41	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₃₇	—	2	1	no	—
43	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₇	—	2	1	no	—
45	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₃₇	—	2	1	no	—
47	#10596962	—	2	1	no	—
48	#10596962	—	2	1	no	—
49	#10596962	C₂×C₄	1	2	yes	—
50	#10596962	C₂³	1	2	yes	—
51	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₃₇	—	2	1	no	—
53	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
54	M₁₆×C₃₇	—	2	1	no	—
55	M₁₆×C₃₇	C₂³	1	2	yes	—
56	C₂×C₈×C₃₇	—	2	1	no	—
57	C₄⋊C₄×C₃₇	—	2	1	no	—
58	C₂²⋊C₄×C₃₇	—	2	1	no	—
59	C₄²×C₃₇	C₂³	1	2	yes	—
60	#10596942	C₂×C₄	1	2	yes	—
61	#10596942	C₂×C₄	1	2	yes	—
62	#10596941	C₂×C₄	1	2	yes	—
63	#10596941	C₂×C₄	1	2	yes	—
64	#10596940	C₂×C₄	1	2	yes	—
65	#10596940	C₂×C₄	1	2	yes	—
66	#10596939	C₂×C₄	1	2	yes	—
67	#10596939	C₂×C₄	1	2	yes	—
68	#10596934	—	2	1	no	—
69	#10596934	—	2	4	no	—
70	#10596934	D₄	1	2	yes	—
71	#10596934	D₄	1	2	yes	—
72	#10596934	C₂³	1	2	yes	—
73	C₂×#64436	—	2	1	no	—
74	C₂×#64436	—	2	4	no	—
75	C₂×#64436	C₂×C₄	1	1	yes	characteristic
76	C₂×#64436	C₂×C₄	1	1	yes	characteristic
77	C₂×#64436	C₂×C₄	1	1	yes	characteristic
78	C₂×#64436	C₂×C₄	1	1	yes	characteristic
79	C₂×#64436	C₂³	1	1	yes	characteristic
80	C₂×#64436	C₂³	1	1	yes	characteristic
81	C₂×#64436	C₂³	1	2	yes	—
82	#10596929	—	2	4	no	—
83	C₈×D₃₇	—	2	4	no	—
84	C₂³×C₃₇	—	2	1	no	—
85	C₂³×C₃₇	#41	1	1	yes	characteristic
86	C₂²×D₃₇	—	2	1	no	—
87	Q₈×C₃₇	—	2	1	no	—
88	Q₈×C₃₇	C₂²×C₄	1	2	yes	—
89	D₄×C₃₇	—	2	2	no	—
90	D₄×C₃₇	—	2	2	no	—
91	D₄×C₃₇	—	2	1	no	—
92	D₄×C₃₇	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₃₇	—	2	1	no	—
94	C₂×C₄×C₃₇	—	2	2	no	—
95	C₂×C₄×C₃₇	—	2	2	no	—
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
102	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
103	C₂²⋊D₃₇	—	2	2	no	—
104	C₂²⋊D₃₇	—	2	2	no	—
105	C₂×Dic₃₇	—	2	1	no	—
106	D₁₄₈	—	2	1	no	—
107	D₁₄₈	C₂²×C₄	1	2	yes	—
108	C₄×D₃₇	—	2	2	no	—
109	C₄×D₃₇	—	2	2	no	—
110	Dic₇₄	—	2	1	no	—
111	Dic₇₄	C₂²×C₄	1	2	yes	—
112	C₈×C₃₇	—	2	2	no	—
113	#64436	—	2	2	no	—
114	#64436	C₂²×C₄	1	1	yes	characteristic
115	#64436	C₂²×C₄	1	1	yes	characteristic
116	#64436	C₂²×C₄	1	1	yes	characteristic
117	#64436	C₂²×C₄	1	1	yes	characteristic
118	#64436	C₂×D₄	1	4	yes	—
119	C₂²×C₃₇	—	2	2	no	—
120	C₂²×C₃₇	—	4	1	no	—
121	C₂²×C₃₇	—	8	1	no	—
122	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
123	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
125	C₂×D₃₇	—	2	2	no	—
126	C₂×D₃₇	—	8	1	no	—
127	C₄×C₃₇	—	2	2	no	—
128	C₄×C₃₇	—	4	2	no	—
129	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
130	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
131	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
132	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
133	Dic₃₇	—	2	2	no	—
134	#2957	—	37	1	no	maximal, Sylow
135	C₂×C₃₇	—	2	1	no	—
136	C₂×C₃₇	—	2	1	no	—
137	C₂×C₃₇	—	2	1	no	—
138	C₂×C₃₇	—	4	2	no	—
139	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
140	D₃₇	—	4	2	no	—
141	C₂×#135	—	37	1	no	—
142	#568	—	37	1	no	—
143	#568	—	37	1	no	—
144	#444	—	37	4	no	—
145	#443	—	37	4	no	—
146	C₂×#108	—	37	1	no	—
147	#421	—	37	1	no	—
148	#421	D₃₇	1	1	yes	characteristic, complemented
149	#405	—	37	1	no	—
150	C₃₇	#2957	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	—	37	1	no	—
152	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
153	#135	—	37	2	no	—
154	#135	—	37	2	no	—
155	#135	—	37	2	no	—
156	#135	—	37	2	no	—
157	C₂×Q₁₆	—	37	1	no	—
158	C₂×SD₁₆	—	37	1	no	—
159	C₂×SD₁₆	—	37	1	no	—
160	C₂×D₈	—	37	1	no	—
161	#131	—	37	4	no	—
162	#131	—	37	4	no	—
163	#131	—	74	2	no	—
164	#131	—	74	2	no	—
165	C₂×M₁₆	—	37	1	no	—
166	C₂×M₁₆	—	37	1	no	—
167	C₂×M₁₆	—	74	2	no	—
168	C₂×M₁₆	—	74	2	no	—
169	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
170	C₂²×C₈	—	37	1	no	—
171	#117	C₂×D₃₇	1	1	yes	characteristic
172	#108	—	37	2	no	—
173	#108	—	37	2	no	—
174	#104	—	37	4	no	—
175	#104	C₂×D₃₇	1	4	yes	complemented
176	#97	—	37	2	no	—
177	C₄×C₈	—	37	2	no	—
178	#41	—	2	1	no	—
179	#41	—	2	1	no	—
180	#41	—	37	2	no	—
181	#41	—	37	2	no	—
182	#41	—	74	1	no	—
183	#41	—	74	1	no	—
184	#41	C₄×D₃₇	1	2	yes	—
185	#41	C₂²×D₃₇	1	2	yes	—
186	C₂×Q₈	—	37	1	no	—
187	C₂×Q₈	C₄×D₃₇	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	37	1	no	—
190	C₂×D₄	—	74	1	no	—
191	C₂×D₄	C₄×D₃₇	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	74	1	no	—
194	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
195	Q₁₆	—	37	2	no	—
196	Q₁₆	—	37	2	no	—
197	SD₁₆	—	37	2	no	—
198	SD₁₆	—	37	2	no	—
199	SD₁₆	—	37	2	no	—
200	SD₁₆	—	37	2	no	—
201	D₈	—	37	2	no	—
202	D₈	—	37	2	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	37	2	no	—
205	M₁₆	—	37	2	no	—
206	M₁₆	—	37	2	no	—
207	M₁₆	—	74	4	no	—
208	M₁₆	—	74	4	no	—
209	M₁₆	—	74	1	no	—
210	M₁₆	C₂²×D₃₇	1	2	yes	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	37	1	no	—
213	C₂×C₈	—	37	1	no	—
214	C₂×C₈	—	37	1	no	—
215	C₂×C₈	—	37	1	no	—
216	C₂×C₈	—	37	1	no	—
217	C₂×C₈	—	37	2	no	—
218	C₂×C₈	—	37	1	no	—
219	C₂×C₈	—	74	4	no	—
220	C₂×C₈	—	74	4	no	—
221	C₂×C₈	—	74	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₄²	C₂²×D₃₇	1	2	yes	—
225	C₂³	—	2	1	no	—
226	C₂³	—	74	1	no	—
227	C₂³	#10596964	1	1	yes	characteristic
228	Q₈	—	2	1	no	—
229	Q₈	—	37	2	no	—
230	Q₈	—	74	1	no	—
231	Q₈	C₂×C₄×D₃₇	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	1	no	—
235	D₄	—	37	2	no	—
236	D₄	—	74	2	no	—
237	D₄	—	74	2	no	—
238	D₄	—	74	1	no	—
239	D₄	C₂×C₄×D₃₇	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	74	1	no	—
245	C₂×C₄	—	74	2	no	—
246	C₂×C₄	—	74	2	no	—
247	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
249	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
250	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
251	C₂×C₄	#10596964	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	37	1	no	—
255	C₈	—	37	1	no	—
256	C₈	—	37	1	no	—
257	C₈	—	37	1	no	—
258	C₈	—	37	4	no	—
259	C₈	—	74	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	—	296	1	no	—
264	C₂²	—	74	2	no	—
265	C₂²	#11924496	1	1	yes	characteristic
266	C₂²	#11924496	1	1	yes	characteristic
267	C₂²	C₂×#10596964	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	74	2	no	—
271	C₄	#11924496	1	1	yes	characteristic
272	C₄	#11924496	1	1	yes	characteristic
273	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
274	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	4	2	no	—
279	C₂	—	148	2	no	—
280	C₂	C₂×#11924496	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#421	C₂
socle	C₂×C₃₇	C₂×C₄×D₄
prime radical	C₂×#11924556	C₂
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#421	D₃₇
hypocenter	C₃₇	#2957
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
2. #14558300	C₂	semidirect
3. #14558299	C₂	semidirect
4. #14557652	C₂	semidirect
5. #14557647	C₂	semidirect
6. C₂×#11924471	C₂	semidirect
7. #14557551	C₂	semidirect
8. #14557489	C₂	semidirect
9. C₃₇×#421	C₂	semidirect
11. #11924556	C₄	semidirect
12. #11924556	C₄	semidirect
15. C₂×#10596942	C₄	semidirect
16. C₂×#10596940	C₄	semidirect
17. C₂×#10596941	C₄	semidirect
18. C₂×#10596939	C₄	semidirect
19. #11924544	C₂²	semidirect
21. #11924543	C₂²	semidirect
28. #11924471	C₂²	semidirect
29. #11924471	C₂²	semidirect
30. #11924464	C₂²	semidirect
31. #11924447	C₂²	semidirect
32. C₄×#64436	C₂²	semidirect
36. #104×C₃₇	C₂²	semidirect
148. #421	D₃₇	semidirect
150. C₃₇	#2957	semidirect
175. #104	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924556	2
#14558300	2
#14558299	2
#14557652	2
#14557647	2
C₂×#11924471	2
#14557551	2
#14557489	2
C₃₇×#421	2
#2957	37