Subgroups — #17997547

Order 4736.

632 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₄₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924494	C₂	1	2	yes	maximal, complemented
3	#14557434	C₂	1	2	yes	maximal
4	#14557423	C₂	1	4	yes	maximal, complemented
5	#14557390	C₂	1	2	yes	maximal, complemented
6	C₄⋊C₄×Dic₃₇	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₁₄₈	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₃₇	C₂²	1	2	yes	—
11	C₄×D₁₄₈	C₄	1	8	yes	complemented
12	#11924494	—	2	8	no	—
13	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
16	C₂×#10596938	C₂²	1	4	yes	—
17	C₂×#10596938	C₂²	1	4	yes	—
18	C₂×#10596938	C₂²	1	2	yes	—
19	C₂×C₄×Dic₃₇	C₂²	1	4	yes	—
20	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
21	#11924443	C₂²	1	2	yes	—
22	#11924443	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
27	C₂³×D₃₇	C₂³	1	2	yes	—
28	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
29	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
32	C₂²×Dic₃₇	C₂³	1	2	yes	—
33	C₂²×Dic₃₇	C₂³	1	2	yes	—
34	C₂²×Dic₃₇	C₂³	1	2	yes	—
35	C₂×D₁₄₈	—	2	2	no	—
36	C₂×D₁₄₈	—	2	2	no	—
37	C₂×D₁₄₈	C₂×C₄	1	4	yes	—
38	C₂×C₄×D₃₇	—	2	4	no	—
39	C₂×C₄×D₃₇	—	2	4	no	—
40	C₂×C₄×D₃₇	C₂×C₄	1	8	yes	—
41	C₄⋊C₄×C₃₇	—	2	4	no	—
42	C₄⋊C₄×C₃₇	—	2	4	no	—
43	C₄⋊C₄×C₃₇	—	4	2	no	—
44	C₄²×C₃₇	C₂×C₄	1	4	yes	—
45	#10596938	—	2	8	no	—
46	#10596938	—	4	4	no	—
47	#10596938	C₂×C₄	1	8	yes	—
48	C₄⋊Dic₃₇	—	2	2	no	—
49	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
50	C₄⋊Dic₃₇	D₄	1	4	yes	complemented
51	C₄×Dic₃₇	—	2	4	no	—
52	C₄×Dic₃₇	—	2	8	no	—
53	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
54	C₂²×D₃₇	—	2	2	no	—
55	C₂²×D₃₇	—	2	2	no	—
56	C₂²×D₃₇	—	2	2	no	—
57	C₂²×D₃₇	—	2	2	no	—
58	C₂²×D₃₇	—	2	2	no	—
59	C₂²×D₃₇	—	2	2	no	—
60	C₂²×D₃₇	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₃₇	—	2	1	no	—
62	C₂×C₄×C₃₇	—	2	1	no	—
63	C₂×C₄×C₃₇	—	2	4	no	—
64	C₂×C₄×C₃₇	—	2	4	no	—
65	C₂×C₄×C₃₇	—	2	4	no	—
66	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₃₇	#41	1	2	yes	—
71	C₂×C₄×C₃₇	#41	1	2	yes	—
72	C₂×C₄×C₃₇	#41	1	2	yes	—
73	C₂×Dic₃₇	—	2	2	no	—
74	C₂×Dic₃₇	—	2	2	no	—
75	C₂×Dic₃₇	—	2	2	no	—
76	C₂×Dic₃₇	—	2	2	no	—
77	C₂×Dic₃₇	—	2	2	no	—
78	C₂×Dic₃₇	—	2	2	no	—
79	C₂×Dic₃₇	—	2	2	no	—
80	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
81	C₂×Dic₃₇	C₂×D₄	1	4	yes	—
82	D₁₄₈	—	2	8	no	—
83	C₄×D₃₇	—	2	16	no	—
84	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
85	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
87	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
88	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
89	C₂²×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
90	C₂²×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
91	C₂×D₃₇	—	2	4	no	—
92	C₂×D₃₇	—	2	4	no	—
93	C₂×D₃₇	—	2	4	no	—
94	C₂×D₃₇	—	4	2	no	—
95	C₂×D₃₇	—	4	2	no	—
96	C₂×D₃₇	—	4	2	no	—
97	C₂×D₃₇	—	4	2	no	—
98	C₄×C₃₇	—	2	2	no	—
99	C₄×C₃₇	—	4	4	no	—
100	C₄×C₃₇	#117	1	4	yes	complemented
101	C₄×C₃₇	C₄×D₄	1	4	yes	—
102	Dic₃₇	—	2	4	no	—
103	Dic₃₇	—	2	4	no	—
104	Dic₃₇	—	4	2	no	—
105	#2354	—	37	1	no	maximal, Sylow
106	C₂×C₃₇	C₂×#117	1	1	yes	characteristic
107	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
108	C₂×C₃₇	#520	1	1	yes	characteristic
109	C₂×C₃₇	#547	1	1	yes	characteristic
110	C₂×C₃₇	#550	1	1	yes	characteristic
111	C₂×C₃₇	#555	1	1	yes	characteristic
112	C₂×C₃₇	#556	1	1	yes	characteristic
113	D₃₇	—	4	4	no	—
114	C₂×C₄×D₄	—	37	1	no	—
115	C₂×#117	—	37	2	no	—
116	#390	—	37	2	no	—
117	#388	—	37	4	no	—
118	#384	D₃₇	1	1	yes	characteristic, complemented
119	#380	—	37	2	no	—
120	C₄×C₄⋊C₄	—	37	2	no	—
121	C₄×C₄⋊C₄	—	37	1	no	—
122	C₃₇	#2354	1	1	yes	characteristic, complemented, Sylow
123	C₂²×D₄	—	37	1	no	—
124	C₂³×C₄	—	37	2	no	—
125	C₄×D₄	—	37	8	no	—
126	#117	—	74	8	no	—
127	C₂×C₄⋊C₄	—	37	1	no	—
128	C₂×C₄⋊C₄	—	37	1	no	—
129	C₂×C₄⋊C₄	—	37	1	no	—
130	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
131	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
132	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
133	C₂×C₂²⋊C₄	—	37	2	no	—
134	C₂×C₂²⋊C₄	—	37	4	no	—
135	C₂×C₂²⋊C₄	—	37	4	no	—
136	C₂×C₄²	—	37	4	no	—
137	C₂×C₄²	—	37	2	no	—
138	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
139	#95	—	37	4	no	—
140	#95	—	37	2	no	—
141	C₂⁴	—	37	2	no	—
142	C₂×D₄	—	37	4	no	—
143	C₂×D₄	—	74	2	no	—
144	C₂×D₄	—	74	2	no	—
145	C₂²×C₄	—	37	2	no	—
146	C₂²×C₄	—	37	2	no	—
147	C₂²×C₄	—	37	8	no	—
148	C₂²×C₄	—	37	2	no	—
149	C₂²×C₄	—	74	4	no	—
150	C₂²×C₄	—	74	4	no	—
151	C₂²×C₄	C₂²×D₃₇	1	4	yes	—
152	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
153	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	37	4	no	—
159	C₄⋊C₄	—	37	4	no	—
160	C₄⋊C₄	—	74	2	no	—
161	C₂²⋊C₄	—	37	8	no	—
162	C₂²⋊C₄	—	74	8	no	—
163	C₂²⋊C₄	—	148	4	no	—
164	C₄²	—	74	8	no	—
165	C₄²	—	74	4	no	—
166	C₄²	C₄×D₃₇	1	4	yes	—
167	C₂³	—	37	4	no	—
168	C₂³	—	74	2	no	—
169	C₂³	—	74	2	no	—
170	C₂³	—	74	2	no	—
171	C₂³	—	74	2	no	—
172	C₂³	—	74	2	no	—
173	C₂³	—	74	2	no	—
174	C₂³	C₂³×D₃₇	1	1	yes	characteristic
175	D₄	—	74	8	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	37	4	no	—
182	C₂×C₄	—	37	4	no	—
183	C₂×C₄	—	74	2	no	—
184	C₂×C₄	—	74	2	no	—
185	C₂×C₄	—	74	2	no	—
186	C₂×C₄	—	74	2	no	—
187	C₂×C₄	—	74	16	no	—
188	C₂×C₄	—	74	2	no	—
189	C₂×C₄	—	74	2	no	—
190	C₂×C₄	—	74	2	no	—
191	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
192	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
193	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
194	C₂×C₄	D₄×D₃₇	1	2	yes	—
195	C₂×C₄	#10596964	1	2	yes	—
196	C₂×C₄	#10596966	1	2	yes	—
197	C₂×C₄	#10596966	1	2	yes	—
198	C₂²	—	74	4	no	—
199	C₂²	—	74	4	no	—
200	C₂²	—	74	4	no	—
201	C₂²	—	148	2	no	—
202	C₂²	—	148	2	no	—
203	C₂²	—	148	2	no	—
204	C₂²	—	148	2	no	—
205	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
206	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
207	C₂²	C₂×#10596964	1	1	yes	characteristic
208	C₂²	C₂×#10596966	1	1	yes	characteristic
209	C₂²	C₂×#10596966	1	1	yes	characteristic
210	C₂²	#11924581	1	1	yes	characteristic
211	C₂²	#11924584	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	—	4	4	no	—
214	C₄	—	74	4	no	—
215	C₄	—	74	4	no	—
216	C₄	—	148	2	no	—
217	C₄	#11924494	1	4	yes	complemented
218	C₄	#11924498	1	4	yes	—
219	C₂	—	148	4	no	—
220	C₂	C₂×#11924494	1	1	yes	characteristic
221	C₂	C₂×#11924498	1	1	yes	characteristic
222	C₂	#14558014	1	1	yes	characteristic
223	C₂	#14558186	1	1	yes	characteristic
224	C₂	#14558203	1	1	yes	characteristic
225	C₂	#14558249	1	1	yes	characteristic
226	C₂	#14558257	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Fitting	C₃₇×#384	C₂
socle, cosocle	C₂³×C₃₇	C₂⁴
prime radical	C₂²×D₁₄₈	C₂²
hypercenter	#384	D₃₇
hypocenter	C₃₇	#2354
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
2. C₂×#11924494	C₂	semidirect
4. #14557423	C₂	semidirect
5. #14557390	C₂	semidirect
6. C₄⋊C₄×Dic₃₇	C₂	semidirect
7. C₄×C₄⋊Dic₃₇	C₂	semidirect
8. C₃₇×#384	C₂	semidirect
11. C₄×D₁₄₈	C₄	semidirect
50. C₄⋊Dic₃₇	D₄	semidirect
100. C₄×C₃₇	#117	semidirect
118. #384	D₃₇	semidirect
122. C₃₇	#2354	semidirect
217. C₄	#11924494	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₄₈	2
C₂×#11924494	2
#14557434	2
#14557423	2
#14557390	2
C₄⋊C₄×Dic₃₇	2
C₄×C₄⋊Dic₃₇	2
C₃₇×#384	2
#2354	37