Subgroups — C₂×C₃₇×#2383 #22655602

Order 9472.

2502 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#126×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#2383	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃₇×#389	C₂	1	3	yes	maximal, complemented
4	C₂×C₃₇×#388	C₂	1	3	yes	maximal
5	C₂×C₃₇×#387	C₂	1	3	yes	maximal, complemented
6	C₂×C₃₇×#385	C₂	1	3	yes	maximal
7	C₂×C₃₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₇×#383	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×C₄×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×#126×C₃₇	C₄	1	16	yes	complemented
11	C₂×#126×C₃₇	C₂²	1	8	yes	—
12	C₂²×C₄⋊C₄×C₃₇	C₂²	1	3	yes	—
13	C₂²×C₄⋊C₄×C₃₇	C₂²	1	3	yes	—
14	C₂²×C₄⋊C₄×C₃₇	C₂²	1	3	yes	—
15	C₂²×C₄⋊C₄×C₃₇	C₂²	1	3	yes	—
16	C₂²×C₂²⋊C₄×C₃₇	C₂²	1	3	yes	—
17	C₂²×C₂²⋊C₄×C₃₇	C₂²	1	3	yes	—
18	C₂²×C₄²×C₃₇	C₂²	1	3	yes	—
19	C₂²×C₄²×C₃₇	C₂²	1	1	yes	characteristic
20	C₃₇×#389	C₂²	1	24	yes	complemented
21	C₃₇×#388	C₂²	1	24	yes	—
22	C₃₇×#387	C₂²	1	24	yes	complemented
23	C₃₇×#385	C₂²	1	24	yes	—
24	C₃₇×#384	C₂²	1	8	yes	complemented
25	C₃₇×#383	C₂²	1	8	yes	complemented
26	C₂×#95×C₃₇	C₂²	1	3	yes	—
27	C₂×#95×C₃₇	C₂²	1	3	yes	—
28	C₂×#95×C₃₇	C₂²	1	3	yes	—
29	C₂×#95×C₃₇	C₂²	1	3	yes	—
30	C₂⁵×C₃₇	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₃₇	—	4	6	no	—
32	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₃₇	C₂³	1	3	yes	—
35	C₂³×C₄×C₃₇	C₂³	1	3	yes	—
36	C₂³×C₄×C₃₇	C₂³	1	3	yes	—
37	C₂³×C₄×C₃₇	C₂³	1	3	yes	—
38	C₂³×C₄×C₃₇	C₂³	1	4	yes	—
39	#126×C₃₇	C₂×C₄	1	64	yes	complemented
40	C₂×C₄⋊C₄×C₃₇	—	4	6	no	—
41	C₂×C₄⋊C₄×C₃₇	—	4	6	no	—
42	C₂×C₄⋊C₄×C₃₇	—	4	6	no	—
43	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	24	yes	complemented
44	C₂×C₄⋊C₄×C₃₇	C₂³	1	12	yes	—
45	C₂×C₄⋊C₄×C₃₇	C₂³	1	12	yes	—
46	C₂×C₄⋊C₄×C₃₇	C₂³	1	12	yes	—
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	12	yes	—
48	C₂×C₂²⋊C₄×C₃₇	—	4	6	no	—
49	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	24	yes	—
50	C₂×C₂²⋊C₄×C₃₇	C₂³	1	12	yes	—
51	C₂×C₂²⋊C₄×C₃₇	C₂³	1	12	yes	—
52	C₂×C₄²×C₃₇	—	4	6	no	—
53	C₂×C₄²×C₃₇	C₂×C₄	1	8	yes	complemented
54	C₂×C₄²×C₃₇	C₂³	1	12	yes	—
55	C₂×C₄²×C₃₇	C₂³	1	4	yes	—
56	#95×C₃₇	C₂³	1	12	yes	—
57	#95×C₃₇	C₂³	1	12	yes	—
58	#95×C₃₇	C₂³	1	12	yes	—
59	#95×C₃₇	C₂³	1	12	yes	—
60	C₂⁴×C₃₇	—	2	6	no	—
61	C₂⁴×C₃₇	—	2	6	no	—
62	C₂⁴×C₃₇	C₂²×C₄	1	4	yes	—
63	C₂⁴×C₃₇	C₂⁴	1	2	yes	—
64	C₂⁴×C₃₇	C₂⁴	1	1	yes	characteristic
65	C₂²×C₄×C₃₇	—	2	6	no	—
66	C₂²×C₄×C₃₇	—	2	6	no	—
67	C₂²×C₄×C₃₇	—	2	6	no	—
68	C₂²×C₄×C₃₇	—	2	6	no	—
69	C₂²×C₄×C₃₇	—	2	6	no	—
70	C₂²×C₄×C₃₇	—	2	6	no	—
71	C₂²×C₄×C₃₇	—	2	6	no	—
72	C₂²×C₄×C₃₇	—	2	6	no	—
73	C₂²×C₄×C₃₇	—	2	6	no	—
74	C₂²×C₄×C₃₇	—	2	6	no	—
75	C₂²×C₄×C₃₇	—	2	12	no	—
76	C₂²×C₄×C₃₇	—	4	12	no	—
77	C₂²×C₄×C₃₇	—	4	16	no	—
78	C₂²×C₄×C₃₇	C₂²×C₄	1	12	yes	—
79	C₂²×C₄×C₃₇	C₂²×C₄	1	12	yes	—
80	C₂²×C₄×C₃₇	C₂⁴	1	6	yes	—
81	C₂²×C₄×C₃₇	C₂⁴	1	2	yes	—
82	C₂²×C₄×C₃₇	C₂⁴	1	2	yes	—
83	C₂²×C₄×C₃₇	C₂⁴	1	6	yes	—
84	C₂²×C₄×C₃₇	C₂⁴	1	6	yes	—
85	C₂²×C₄×C₃₇	C₂⁴	1	6	yes	—
86	C₄⋊C₄×C₃₇	—	4	12	no	—
87	C₄⋊C₄×C₃₇	—	4	12	no	—
88	C₄⋊C₄×C₃₇	—	4	12	no	—
89	C₄⋊C₄×C₃₇	C₂²×C₄	1	48	yes	complemented
90	C₂²⋊C₄×C₃₇	—	4	12	no	—
91	C₂²⋊C₄×C₃₇	C₂²×C₄	1	48	yes	—
92	C₄²×C₃₇	—	4	12	no	—
93	C₄²×C₃₇	C₂²×C₄	1	16	yes	complemented
94	C₂³×C₃₇	—	2	24	no	—
95	C₂³×C₃₇	—	2	6	no	—
96	C₂³×C₃₇	—	2	6	no	—
97	C₂³×C₃₇	—	4	4	no	—
98	C₂³×C₃₇	—	4	12	no	—
99	C₂³×C₃₇	C₂³×C₄	1	4	yes	—
100	C₂³×C₃₇	C₂³×C₄	1	2	yes	—
101	C₂³×C₃₇	H(5,𝔽₂)	1	6	yes	—
102	C₂³×C₃₇	2₋¹⁺⁴	1	6	yes	—
103	C₂³×C₃₇	C₂⁵	1	1	yes	characteristic
104	C₂×C₄×C₃₇	—	2	24	no	—
105	C₂×C₄×C₃₇	—	2	24	no	—
106	C₂×C₄×C₃₇	—	2	6	no	—
107	C₂×C₄×C₃₇	—	2	6	no	—
108	C₂×C₄×C₃₇	—	2	6	no	—
109	C₂×C₄×C₃₇	—	2	6	no	—
110	C₂×C₄×C₃₇	—	2	6	no	—
111	C₂×C₄×C₃₇	—	2	6	no	—
112	C₂×C₄×C₃₇	—	2	6	no	—
113	C₂×C₄×C₃₇	—	2	6	no	—
114	C₂×C₄×C₃₇	—	2	6	no	—
115	C₂×C₄×C₃₇	—	2	6	no	—
116	C₂×C₄×C₃₇	—	2	12	no	—
117	C₂×C₄×C₃₇	—	4	16	no	—
118	C₂×C₄×C₃₇	—	4	12	no	—
119	C₂×C₄×C₃₇	—	4	4	no	—
120	C₂×C₄×C₃₇	—	4	4	no	—
121	C₂×C₄×C₃₇	—	4	12	no	—
122	C₂×C₄×C₃₇	C₂³×C₄	1	12	yes	—
123	C₂×C₄×C₃₇	C₂³×C₄	1	12	yes	—
124	C₂×#2383	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂²×C₃₇	—	2	12	no	—
126	C₂²×C₃₇	—	4	16	no	—
127	C₂²×C₃₇	—	4	2	no	—
128	C₂²×C₃₇	—	4	6	no	—
129	C₂²×C₃₇	#520	1	12	yes	—
130	C₂²×C₃₇	#560	1	4	yes	—
131	C₂²×C₃₇	#563	1	12	yes	—
132	C₂²×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
133	C₂²×C₃₇	C₂×H(5,𝔽₂)	1	3	yes	—
134	C₂²×C₃₇	C₂×2₋¹⁺⁴	1	3	yes	—
135	C₄×C₃₇	—	2	12	no	—
136	C₄×C₃₇	—	2	12	no	—
137	C₄×C₃₇	—	4	6	no	—
138	C₄×C₃₇	—	4	2	no	—
139	C₄×C₃₇	—	4	2	no	—
140	C₄×C₃₇	—	4	6	no	—
141	C₂²×#126	C₂×C₃₇	1	1	yes	characteristic
142	#2383	C₂×C₃₇	1	16	yes	direct product factor
143	C₂×#389	C₂×C₃₇	1	3	yes	complemented
144	C₂×#388	C₂×C₃₇	1	3	yes	—
145	C₂×#387	C₂×C₃₇	1	3	yes	complemented
146	C₂×#385	C₂×C₃₇	1	3	yes	—
147	C₂×#384	C₂×C₃₇	1	1	yes	characteristic, complemented
148	C₂×#383	C₂×C₃₇	1	1	yes	characteristic, complemented
149	C₂×C₃₇	—	4	4	no	—
150	C₂×C₃₇	#2383	1	8	yes	direct product factor
151	C₂×C₃₇	C₂×#520	1	3	yes	—
152	C₂×C₃₇	C₂×#560	1	1	yes	characteristic
153	C₂×C₃₇	C₂×#563	1	3	yes	—
154	C₂⁴×C₄	C₂²×C₃₇	1	1	yes	characteristic
155	C₂×#126	C₄×C₃₇	1	16	yes	complemented
156	C₂×#126	C₂²×C₃₇	1	8	yes	—
157	C₂²×C₄⋊C₄	C₂²×C₃₇	1	3	yes	—
158	C₂²×C₄⋊C₄	C₂²×C₃₇	1	3	yes	—
159	C₂²×C₄⋊C₄	C₂²×C₃₇	1	3	yes	—
160	C₂²×C₄⋊C₄	C₂²×C₃₇	1	3	yes	—
161	C₂²×C₂²⋊C₄	C₂²×C₃₇	1	3	yes	—
162	C₂²×C₂²⋊C₄	C₂²×C₃₇	1	3	yes	—
163	C₂²×C₄²	C₂²×C₃₇	1	3	yes	—
164	C₂²×C₄²	C₂²×C₃₇	1	1	yes	characteristic
165	#389	C₂²×C₃₇	1	24	yes	complemented
166	#388	C₂²×C₃₇	1	24	yes	—
167	#387	C₂²×C₃₇	1	24	yes	complemented
168	#385	C₂²×C₃₇	1	24	yes	—
169	#384	C₂²×C₃₇	1	8	yes	complemented
170	#383	C₂²×C₃₇	1	8	yes	complemented
171	C₂×#95	C₂²×C₃₇	1	3	yes	—
172	C₂×#95	C₂²×C₃₇	1	3	yes	—
173	C₂×#95	C₂²×C₃₇	1	3	yes	—
174	C₂×#95	C₂²×C₃₇	1	3	yes	—
175	C₃₇	C₂×#2383	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁵	C₂³×C₃₇	1	1	yes	characteristic
177	C₂³×C₄	—	4	6	no	—
178	C₂³×C₄	C₂³×C₃₇	1	3	yes	—
179	C₂³×C₄	C₂³×C₃₇	1	3	yes	—
180	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
181	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
182	C₂³×C₄	C₂³×C₃₇	1	3	yes	—
183	C₂³×C₄	C₂³×C₃₇	1	3	yes	—
184	C₂³×C₄	C₂³×C₃₇	1	4	yes	—
185	#126	C₂×C₄×C₃₇	1	64	yes	complemented
186	C₂×C₄⋊C₄	—	4	6	no	—
187	C₂×C₄⋊C₄	—	4	6	no	—
188	C₂×C₄⋊C₄	—	4	6	no	—
189	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	24	yes	complemented
190	C₂×C₄⋊C₄	C₂³×C₃₇	1	12	yes	—
191	C₂×C₄⋊C₄	C₂³×C₃₇	1	12	yes	—
192	C₂×C₄⋊C₄	C₂³×C₃₇	1	12	yes	—
193	C₂×C₄⋊C₄	C₂³×C₃₇	1	12	yes	—
194	C₂×C₂²⋊C₄	—	4	6	no	—
195	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	24	yes	—
196	C₂×C₂²⋊C₄	C₂³×C₃₇	1	12	yes	—
197	C₂×C₂²⋊C₄	C₂³×C₃₇	1	12	yes	—
198	C₂×C₄²	—	4	6	no	—
199	C₂×C₄²	C₂×C₄×C₃₇	1	8	yes	complemented
200	C₂×C₄²	C₂³×C₃₇	1	12	yes	—
201	C₂×C₄²	C₂³×C₃₇	1	4	yes	—
202	#95	C₂³×C₃₇	1	12	yes	—
203	#95	C₂³×C₃₇	1	12	yes	—
204	#95	C₂³×C₃₇	1	12	yes	—
205	#95	C₂³×C₃₇	1	12	yes	—
206	C₂⁴	—	2	6	no	—
207	C₂⁴	—	2	6	no	—
208	C₂⁴	C₂²×C₄×C₃₇	1	4	yes	—
209	C₂⁴	C₂⁴×C₃₇	1	2	yes	—
210	C₂⁴	C₂⁴×C₃₇	1	1	yes	characteristic
211	C₂²×C₄	—	2	6	no	—
212	C₂²×C₄	—	2	6	no	—
213	C₂²×C₄	—	2	6	no	—
214	C₂²×C₄	—	2	6	no	—
215	C₂²×C₄	—	2	6	no	—
216	C₂²×C₄	—	2	6	no	—
217	C₂²×C₄	—	2	6	no	—
218	C₂²×C₄	—	2	6	no	—
219	C₂²×C₄	—	2	6	no	—
220	C₂²×C₄	—	2	6	no	—
221	C₂²×C₄	—	2	12	no	—
222	C₂²×C₄	—	4	12	no	—
223	C₂²×C₄	—	4	16	no	—
224	C₂²×C₄	C₂²×C₄×C₃₇	1	12	yes	—
225	C₂²×C₄	C₂²×C₄×C₃₇	1	12	yes	—
226	C₂²×C₄	C₂⁴×C₃₇	1	6	yes	—
227	C₂²×C₄	C₂⁴×C₃₇	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₃₇	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₃₇	1	6	yes	—
230	C₂²×C₄	C₂⁴×C₃₇	1	6	yes	—
231	C₂²×C₄	C₂⁴×C₃₇	1	6	yes	—
232	C₄⋊C₄	—	4	12	no	—
233	C₄⋊C₄	—	4	12	no	—
234	C₄⋊C₄	—	4	12	no	—
235	C₄⋊C₄	C₂²×C₄×C₃₇	1	48	yes	complemented
236	C₂²⋊C₄	—	4	12	no	—
237	C₂²⋊C₄	C₂²×C₄×C₃₇	1	48	yes	—
238	C₄²	—	4	12	no	—
239	C₄²	C₂²×C₄×C₃₇	1	16	yes	complemented
240	C₂³	—	2	24	no	—
241	C₂³	—	2	6	no	—
242	C₂³	—	2	6	no	—
243	C₂³	—	4	4	no	—
244	C₂³	—	4	12	no	—
245	C₂³	C₂³×C₄×C₃₇	1	4	yes	—
246	C₂³	C₂³×C₄×C₃₇	1	2	yes	—
247	C₂³	H(5,𝔽₂)×C₃₇	1	6	yes	—
248	C₂³	2₋¹⁺⁴×C₃₇	1	6	yes	—
249	C₂³	C₂⁵×C₃₇	1	1	yes	characteristic
250	C₂×C₄	—	2	24	no	—
251	C₂×C₄	—	2	24	no	—
252	C₂×C₄	—	2	6	no	—
253	C₂×C₄	—	2	6	no	—
254	C₂×C₄	—	2	6	no	—
255	C₂×C₄	—	2	6	no	—
256	C₂×C₄	—	2	6	no	—
257	C₂×C₄	—	2	6	no	—
258	C₂×C₄	—	2	6	no	—
259	C₂×C₄	—	2	6	no	—
260	C₂×C₄	—	2	6	no	—
261	C₂×C₄	—	2	6	no	—
262	C₂×C₄	—	2	12	no	—
263	C₂×C₄	—	4	16	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	12	no	—
267	C₂×C₄	—	4	12	no	—
268	C₂×C₄	C₂³×C₄×C₃₇	1	12	yes	—
269	C₂×C₄	C₂³×C₄×C₃₇	1	12	yes	—
270	C₂²	—	2	12	no	—
271	C₂²	—	4	16	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	6	no	—
274	C₂²	C₃₇×#520	1	12	yes	—
275	C₂²	C₃₇×#560	1	4	yes	—
276	C₂²	C₃₇×#563	1	12	yes	—
277	C₂²	C₂⁴×C₄×C₃₇	1	1	yes	characteristic
278	C₂²	C₂×H(5,𝔽₂)×C₃₇	1	3	yes	—
279	C₂²	C₂×2₋¹⁺⁴×C₃₇	1	3	yes	—
280	C₄	—	2	12	no	—
281	C₄	—	2	12	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	6	no	—
285	C₄	—	4	6	no	—
286	C₂	—	4	4	no	—
287	C₂	C₃₇×#2383	1	8	yes	direct product factor
288	C₂	C₂×C₃₇×#520	1	3	yes	—
289	C₂	C₂×C₃₇×#560	1	1	yes	characteristic
290	C₂	C₂×C₃₇×#563	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃₇×#2383	C₂	direct (N × H)
3. C₂×C₃₇×#389	C₂	semidirect
5. C₂×C₃₇×#387	C₂	semidirect
7. C₂×C₃₇×#384	C₂	semidirect
8. C₂×C₃₇×#383	C₂	semidirect
10. C₂×#126×C₃₇	C₄	semidirect
20. C₃₇×#389	C₂²	semidirect
22. C₃₇×#387	C₂²	semidirect
24. C₃₇×#384	C₂²	semidirect
25. C₃₇×#383	C₂²	semidirect
39. #126×C₃₇	C₂×C₄	semidirect
43. C₂×C₄⋊C₄×C₃₇	C₂×C₄	semidirect
53. C₂×C₄²×C₃₇	C₂×C₄	semidirect
89. C₄⋊C₄×C₃₇	C₂²×C₄	semidirect
93. C₄²×C₃₇	C₂²×C₄	semidirect
124. C₂×#2383	C₃₇	direct (N × H)
142. #2383	C₂×C₃₇	direct (N × H)
143. C₂×#389	C₂×C₃₇	semidirect
145. C₂×#387	C₂×C₃₇	semidirect
147. C₂×#384	C₂×C₃₇	semidirect
148. C₂×#383	C₂×C₃₇	semidirect
150. C₂×C₃₇	#2383	direct (N × H)
155. C₂×#126	C₄×C₃₇	semidirect
165. #389	C₂²×C₃₇	semidirect
167. #387	C₂²×C₃₇	semidirect
169. #384	C₂²×C₃₇	semidirect
170. #383	C₂²×C₃₇	semidirect
175. C₃₇	C₂×#2383	direct (N × H)
185. #126	C₂×C₄×C₃₇	semidirect
189. C₂×C₄⋊C₄	C₂×C₄×C₃₇	semidirect
199. C₂×C₄²	C₂×C₄×C₃₇	semidirect
235. C₄⋊C₄	C₂²×C₄×C₃₇	semidirect
239. C₄²	C₂²×C₄×C₃₇	semidirect
287. C₂	C₃₇×#2383	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#126×C₃₇	2
C₃₇×#2383	2
C₂×C₃₇×#389	2
C₂×C₃₇×#388	2
C₂×C₃₇×#387	2
C₂×C₃₇×#385	2
C₂×C₃₇×#384	2
C₂×C₃₇×#383	2
C₂×#2383	37

Subgroups — C2×C37×#2383 #22655602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×#2383 #22655602