Subgroups — C₂²×C₃₇×#407 #22681313

Order 9472.

7214 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	C₂⁵×C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×M₁₆×C₃₇	C₂	1	2	yes	maximal, complemented
3	C₂×C₃₇×#407	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₂²⋊C₈×C₃₇	C₂	1	4	yes	maximal, complemented
5	C₂⁶×C₃₇	C₄	1	1	yes	characteristic
6	C₂⁴×C₄×C₃₇	—	2	16	no	—
7	C₂⁴×C₄×C₃₇	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₃₇	C₄	1	12	yes	—
9	C₂⁴×C₄×C₃₇	C₄	1	2	yes	—
10	C₂⁴×C₄×C₃₇	C₂²	1	12	yes	—
11	C₂⁴×C₄×C₃₇	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₃₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₇	—	2	16	no	—
14	C₂²×M₁₆×C₃₇	C₂²	1	24	yes	complemented
15	C₂³×C₈×C₃₇	C₂²	1	4	yes	complemented
16	C₃₇×#407	C₂²	1	64	yes	direct product factor
17	C₂×C₂²⋊C₈×C₃₇	C₂²	1	48	yes	complemented
18	C₂⁵×C₃₇	—	2	8	no	—
19	C₂⁵×C₃₇	—	2	16	no	—
20	C₂⁵×C₃₇	C₂×C₄	1	12	yes	—
21	C₂⁵×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂⁵×C₃₇	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₃₇	—	2	8	no	—
24	C₂³×C₄×C₃₇	—	2	96	no	—
25	C₂³×C₄×C₃₇	—	2	16	no	—
26	C₂³×C₄×C₃₇	—	2	16	no	—
27	C₂³×C₄×C₃₇	—	2	96	no	—
28	C₂³×C₄×C₃₇	C₂×C₄	1	12	yes	—
29	C₂³×C₄×C₃₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₃₇	C₂×C₄	1	48	yes	—
31	C₂³×C₄×C₃₇	C₂×C₄	1	24	yes	—
32	C₂³×C₄×C₃₇	C₂×C₄	1	12	yes	—
33	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₃₇	C₂×C₄	1	6	yes	—
35	C₂³×C₄×C₃₇	D₄	1	16	yes	—
36	C₂³×C₄×C₃₇	C₂³	1	12	yes	—
37	C₂³×C₄×C₃₇	C₂³	1	16	yes	—
38	C₂³×C₄×C₃₇	C₂³	1	6	yes	—
39	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₃₇	—	2	96	no	—
41	C₂×M₁₆×C₃₇	C₂³	1	32	yes	complemented
42	C₂²×C₈×C₃₇	—	2	16	no	—
43	C₂²×C₈×C₃₇	C₂³	1	24	yes	complemented
44	C₂²⋊C₈×C₃₇	C₂³	1	64	yes	complemented
45	C₂⁴×C₃₇	—	2	128	no	—
46	C₂⁴×C₃₇	—	2	48	no	—
47	C₂⁴×C₃₇	—	2	8	no	—
48	C₂⁴×C₃₇	—	2	8	no	—
49	C₂⁴×C₃₇	—	2	8	no	—
50	C₂⁴×C₃₇	—	2	96	no	—
51	C₂⁴×C₃₇	C₂²⋊C₄	1	8	yes	—
52	C₂⁴×C₃₇	M₁₆	1	16	yes	complemented
53	C₂⁴×C₃₇	C₂²×C₄	1	16	yes	—
54	C₂⁴×C₃₇	C₂²×C₄	1	6	yes	—
55	C₂⁴×C₃₇	C₂²×C₄	1	12	yes	—
56	C₂⁴×C₃₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₇	—	2	48	no	—
58	C₂²×C₄×C₃₇	—	2	8	no	—
59	C₂²×C₄×C₃₇	—	2	192	no	—
60	C₂²×C₄×C₃₇	—	2	96	no	—
61	C₂²×C₄×C₃₇	—	2	48	no	—
62	C₂²×C₄×C₃₇	—	2	48	no	—
63	C₂²×C₄×C₃₇	—	2	64	no	—
64	C₂²×C₄×C₃₇	C₂²⋊C₄	1	48	yes	—
65	C₂²×C₄×C₃₇	C₂²⋊C₄	1	8	yes	—
66	C₂²×C₄×C₃₇	C₂²×C₄	1	16	yes	—
67	C₂²×C₄×C₃₇	C₂²×C₄	1	12	yes	—
68	C₂²×C₄×C₃₇	C₂²×C₄	1	24	yes	—
69	C₂²×C₄×C₃₇	C₂²×C₄	1	32	yes	—
70	C₂²×C₄×C₃₇	C₂²×C₄	1	12	yes	—
71	C₂²×C₄×C₃₇	C₂²×C₄	1	6	yes	—
72	C₂²×C₄×C₃₇	C₂²×C₄	1	3	yes	—
73	C₂²×C₄×C₃₇	C₂×D₄	1	8	yes	—
74	C₂²×C₄×C₃₇	C₂×D₄	1	48	yes	—
75	C₂²×C₄×C₃₇	C₂⁴	1	8	yes	—
76	C₂²×C₄×C₃₇	C₂⁴	1	3	yes	—
77	C₂²×C₄×C₃₇	C₂⁴	1	4	yes	—
78	M₁₆×C₃₇	—	2	64	no	—
79	C₂×C₈×C₃₇	—	2	48	no	—
80	C₂×C₈×C₃₇	C₂⁴	1	16	yes	complemented
81	C₂³×C₃₇	—	2	384	no	—
82	C₂³×C₃₇	—	2	32	no	—
83	C₂³×C₃₇	—	2	24	no	—
84	C₂³×C₃₇	—	2	24	no	—
85	C₂³×C₃₇	—	2	24	no	—
86	C₂³×C₃₇	—	2	64	no	—
87	C₂³×C₃₇	—	2	32	no	—
88	C₂³×C₃₇	—	2	64	no	—
89	C₂³×C₃₇	C₂×C₂²⋊C₄	1	24	yes	—
90	C₂³×C₃₇	C₂×C₂²⋊C₄	1	4	yes	—
91	C₂³×C₃₇	C₂×M₁₆	1	8	yes	—
92	C₂³×C₃₇	C₂×M₁₆	1	48	yes	complemented
93	C₂³×C₃₇	C₂³×C₄	1	4	yes	—
94	C₂³×C₃₇	C₂³×C₄	1	8	yes	—
95	C₂³×C₃₇	C₂³×C₄	1	3	yes	—
96	C₂×C₄×C₃₇	—	2	32	no	—
97	C₂×C₄×C₃₇	—	2	24	no	—
98	C₂×C₄×C₃₇	—	2	48	no	—
99	C₂×C₄×C₃₇	—	2	64	no	—
100	C₂×C₄×C₃₇	—	2	16	no	—
101	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	48	yes	—
102	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	24	yes	—
103	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	12	yes	—
104	C₂×C₄×C₃₇	C₂³×C₄	1	8	yes	—
105	C₂×C₄×C₃₇	C₂³×C₄	1	4	yes	—
106	C₂×C₄×C₃₇	C₂³×C₄	1	3	yes	—
107	C₂×C₄×C₃₇	C₂²×D₄	1	12	yes	—
108	C₂×C₄×C₃₇	C₂²×D₄	1	16	yes	—
109	C₂×C₄×C₃₇	C₂⁵	1	1	yes	characteristic
110	C₈×C₃₇	—	2	16	no	—
111	C₂²×#407	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂²×C₃₇	—	2	128	no	—
113	C₂²×C₃₇	—	2	48	no	—
114	C₂²×C₃₇	—	2	96	no	—
115	C₂²×C₃₇	—	2	8	no	—
116	C₂²×C₃₇	—	2	8	no	—
117	C₂²×C₃₇	—	2	8	no	—
118	C₂²×C₃₇	#407	1	16	yes	direct product factor
119	C₂²×C₃₇	C₂²×C₂²⋊C₄	1	8	yes	—
120	C₂²×C₃₇	C₂²×C₂²⋊C₄	1	6	yes	—
121	C₂²×C₃₇	C₂²×M₁₆	1	12	yes	—
122	C₂²×C₃₇	C₂²×M₁₆	1	16	yes	complemented
123	C₂²×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
124	C₄×C₃₇	—	2	8	no	—
125	C₄×C₃₇	C₂²×C₂²⋊C₄	1	6	yes	—
126	C₄×C₃₇	C₂²×C₂²⋊C₄	1	8	yes	—
127	C₄×C₃₇	C₂³×D₄	1	2	yes	—
128	C₂⁵×C₄	C₂×C₃₇	1	1	yes	characteristic
129	C₂³×M₁₆	C₂×C₃₇	1	2	yes	complemented
130	C₂×#407	C₂×C₃₇	1	24	yes	direct product factor
131	C₂²×C₂²⋊C₈	C₂×C₃₇	1	4	yes	complemented
132	C₂×C₃₇	—	2	8	no	—
133	C₂×C₃₇	—	2	16	no	—
134	C₂×C₃₇	C₂×#407	1	12	yes	direct product factor
135	C₂×C₃₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₃₇	C₂³×M₁₆	1	2	yes	—
137	C₂⁶	C₄×C₃₇	1	1	yes	characteristic
138	C₂⁴×C₄	—	2	16	no	—
139	C₂⁴×C₄	C₄×C₃₇	1	1	yes	characteristic
140	C₂⁴×C₄	C₄×C₃₇	1	12	yes	—
141	C₂⁴×C₄	C₄×C₃₇	1	2	yes	—
142	C₂⁴×C₄	C₂²×C₃₇	1	2	yes	—
143	C₂⁴×C₄	C₂²×C₃₇	1	12	yes	—
144	C₂⁴×C₄	C₂²×C₃₇	1	1	yes	characteristic
145	C₂²×M₁₆	—	2	16	no	—
146	C₂²×M₁₆	C₂²×C₃₇	1	24	yes	complemented
147	C₂³×C₈	C₂²×C₃₇	1	4	yes	complemented
148	#407	C₂²×C₃₇	1	64	yes	direct product factor
149	C₂×C₂²⋊C₈	C₂²×C₃₇	1	48	yes	complemented
150	C₃₇	C₂²×#407	1	1	yes	characteristic, direct product factor, Sylow
151	C₂⁵	—	2	8	no	—
152	C₂⁵	—	2	16	no	—
153	C₂⁵	C₂×C₄×C₃₇	1	12	yes	—
154	C₂⁵	C₂×C₄×C₃₇	1	1	yes	characteristic
155	C₂⁵	C₂×C₄×C₃₇	1	2	yes	—
156	C₂³×C₄	—	2	8	no	—
157	C₂³×C₄	—	2	96	no	—
158	C₂³×C₄	—	2	16	no	—
159	C₂³×C₄	—	2	16	no	—
160	C₂³×C₄	—	2	96	no	—
161	C₂³×C₄	C₂×C₄×C₃₇	1	12	yes	—
162	C₂³×C₄	C₂×C₄×C₃₇	1	2	yes	—
163	C₂³×C₄	C₂×C₄×C₃₇	1	48	yes	—
164	C₂³×C₄	C₂×C₄×C₃₇	1	12	yes	—
165	C₂³×C₄	C₂×C₄×C₃₇	1	24	yes	—
166	C₂³×C₄	C₂×C₄×C₃₇	1	6	yes	—
167	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
168	C₂³×C₄	D₄×C₃₇	1	16	yes	—
169	C₂³×C₄	C₂³×C₃₇	1	12	yes	—
170	C₂³×C₄	C₂³×C₃₇	1	16	yes	—
171	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
172	C₂³×C₄	C₂³×C₃₇	1	6	yes	—
173	C₂×M₁₆	—	2	96	no	—
174	C₂×M₁₆	C₂³×C₃₇	1	32	yes	complemented
175	C₂²×C₈	—	2	16	no	—
176	C₂²×C₈	C₂³×C₃₇	1	24	yes	complemented
177	C₂²⋊C₈	C₂³×C₃₇	1	64	yes	complemented
178	C₂⁴	—	2	128	no	—
179	C₂⁴	—	2	48	no	—
180	C₂⁴	—	2	8	no	—
181	C₂⁴	—	2	8	no	—
182	C₂⁴	—	2	8	no	—
183	C₂⁴	—	2	96	no	—
184	C₂⁴	C₂²⋊C₄×C₃₇	1	8	yes	—
185	C₂⁴	M₁₆×C₃₇	1	16	yes	complemented
186	C₂⁴	C₂²×C₄×C₃₇	1	16	yes	—
187	C₂⁴	C₂²×C₄×C₃₇	1	6	yes	—
188	C₂⁴	C₂²×C₄×C₃₇	1	12	yes	—
189	C₂⁴	C₂²×C₄×C₃₇	1	1	yes	characteristic
190	C₂²×C₄	—	2	48	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	192	no	—
193	C₂²×C₄	—	2	48	no	—
194	C₂²×C₄	—	2	96	no	—
195	C₂²×C₄	—	2	48	no	—
196	C₂²×C₄	—	2	64	no	—
197	C₂²×C₄	C₂²⋊C₄×C₃₇	1	48	yes	—
198	C₂²×C₄	C₂²⋊C₄×C₃₇	1	8	yes	—
199	C₂²×C₄	C₂²×C₄×C₃₇	1	16	yes	—
200	C₂²×C₄	C₂²×C₄×C₃₇	1	12	yes	—
201	C₂²×C₄	C₂²×C₄×C₃₇	1	24	yes	—
202	C₂²×C₄	C₂²×C₄×C₃₇	1	32	yes	—
203	C₂²×C₄	C₂²×C₄×C₃₇	1	12	yes	—
204	C₂²×C₄	C₂²×C₄×C₃₇	1	6	yes	—
205	C₂²×C₄	C₂²×C₄×C₃₇	1	3	yes	—
206	C₂²×C₄	C₂×D₄×C₃₇	1	8	yes	—
207	C₂²×C₄	C₂×D₄×C₃₇	1	48	yes	—
208	C₂²×C₄	C₂⁴×C₃₇	1	8	yes	—
209	C₂²×C₄	C₂⁴×C₃₇	1	3	yes	—
210	C₂²×C₄	C₂⁴×C₃₇	1	4	yes	—
211	M₁₆	—	2	64	no	—
212	C₂×C₈	—	2	48	no	—
213	C₂×C₈	C₂⁴×C₃₇	1	16	yes	complemented
214	C₂³	—	2	384	no	—
215	C₂³	—	2	24	no	—
216	C₂³	—	2	24	no	—
217	C₂³	—	2	64	no	—
218	C₂³	—	2	32	no	—
219	C₂³	—	2	64	no	—
220	C₂³	—	2	32	no	—
221	C₂³	—	2	24	no	—
222	C₂³	C₂×C₂²⋊C₄×C₃₇	1	24	yes	—
223	C₂³	C₂×C₂²⋊C₄×C₃₇	1	4	yes	—
224	C₂³	C₂×M₁₆×C₃₇	1	8	yes	—
225	C₂³	C₂×M₁₆×C₃₇	1	48	yes	complemented
226	C₂³	C₂³×C₄×C₃₇	1	8	yes	—
227	C₂³	C₂³×C₄×C₃₇	1	4	yes	—
228	C₂³	C₂³×C₄×C₃₇	1	3	yes	—
229	C₂×C₄	—	2	32	no	—
230	C₂×C₄	—	2	24	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	48	no	—
233	C₂×C₄	—	2	64	no	—
234	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	48	yes	—
235	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	12	yes	—
236	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	24	yes	—
237	C₂×C₄	C₂³×C₄×C₃₇	1	8	yes	—
238	C₂×C₄	C₂³×C₄×C₃₇	1	3	yes	—
239	C₂×C₄	C₂³×C₄×C₃₇	1	4	yes	—
240	C₂×C₄	C₂²×D₄×C₃₇	1	12	yes	—
241	C₂×C₄	C₂²×D₄×C₃₇	1	16	yes	—
242	C₂×C₄	C₂⁵×C₃₇	1	1	yes	characteristic
243	C₈	—	2	16	no	—
244	C₂²	—	2	128	no	—
245	C₂²	—	2	48	no	—
246	C₂²	—	2	96	no	—
247	C₂²	—	2	8	no	—
248	C₂²	—	2	8	no	—
249	C₂²	—	2	8	no	—
250	C₂²	C₃₇×#407	1	16	yes	direct product factor
251	C₂²	C₂²×C₂²⋊C₄×C₃₇	1	8	yes	—
252	C₂²	C₂²×C₂²⋊C₄×C₃₇	1	6	yes	—
253	C₂²	C₂²×M₁₆×C₃₇	1	12	yes	—
254	C₂²	C₂²×M₁₆×C₃₇	1	16	yes	complemented
255	C₂²	C₂⁴×C₄×C₃₇	1	1	yes	characteristic
256	C₄	—	2	8	no	—
257	C₄	C₂²×C₂²⋊C₄×C₃₇	1	6	yes	—
258	C₄	C₂²×C₂²⋊C₄×C₃₇	1	8	yes	—
259	C₄	C₂³×D₄×C₃₇	1	2	yes	—
260	C₂	—	2	16	no	—
261	C₂	—	2	8	no	—
262	C₂	C₂×C₃₇×#407	1	12	yes	direct product factor
263	C₂	C₂³×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
264	C₂	C₂³×M₁₆×C₃₇	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₃₇	C₂³
commutator	C₂²	C₂⁴×C₄×C₃₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₃₇
socle	C₂⁴×C₃₇	C₂²×C₄
prime radical	C₂⁶×C₃₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×M₁₆×C₃₇	C₂	semidirect
3. C₂×C₃₇×#407	C₂	direct (N × H)
4. C₂²×C₂²⋊C₈×C₃₇	C₂	semidirect
14. C₂²×M₁₆×C₃₇	C₂²	semidirect
15. C₂³×C₈×C₃₇	C₂²	semidirect
16. C₃₇×#407	C₂²	direct (N × H)
17. C₂×C₂²⋊C₈×C₃₇	C₂²	semidirect
41. C₂×M₁₆×C₃₇	C₂³	semidirect
43. C₂²×C₈×C₃₇	C₂³	semidirect
44. C₂²⋊C₈×C₃₇	C₂³	semidirect
52. C₂⁴×C₃₇	M₁₆	semidirect
80. C₂×C₈×C₃₇	C₂⁴	semidirect
92. C₂³×C₃₇	C₂×M₁₆	semidirect
111. C₂²×#407	C₃₇	direct (N × H)
118. C₂²×C₃₇	#407	direct (N × H)
122. C₂²×C₃₇	C₂²×M₁₆	semidirect
129. C₂³×M₁₆	C₂×C₃₇	semidirect
130. C₂×#407	C₂×C₃₇	direct (N × H)
131. C₂²×C₂²⋊C₈	C₂×C₃₇	semidirect
134. C₂×C₃₇	C₂×#407	direct (N × H)
146. C₂²×M₁₆	C₂²×C₃₇	semidirect
147. C₂³×C₈	C₂²×C₃₇	semidirect
148. #407	C₂²×C₃₇	direct (N × H)
149. C₂×C₂²⋊C₈	C₂²×C₃₇	semidirect
150. C₃₇	C₂²×#407	direct (N × H)
174. C₂×M₁₆	C₂³×C₃₇	semidirect
176. C₂²×C₈	C₂³×C₃₇	semidirect
177. C₂²⋊C₈	C₂³×C₃₇	semidirect
185. C₂⁴	M₁₆×C₃₇	semidirect
213. C₂×C₈	C₂⁴×C₃₇	semidirect
225. C₂³	C₂×M₁₆×C₃₇	semidirect
250. C₂²	C₃₇×#407	direct (N × H)
254. C₂²	C₂²×M₁₆×C₃₇	semidirect
262. C₂	C₂×C₃₇×#407	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₃₇	2
C₂³×M₁₆×C₃₇	2
C₂×C₃₇×#407	2
C₂²×C₂²⋊C₈×C₃₇	2
C₂²×#407	37

Subgroups — C22×C37×#407 #22681313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₇×#407 #22681313