Subgroups — C₃₇×#28438 #22649691

Order 9472.

1258 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#492	C₂	1	2	yes	maximal
2	C₂×C₃₇×#461	C₂	1	4	yes	maximal, complemented
3	C₃₇×#2868	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#127×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#2013	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1956	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₃₇	C₂²	1	8	yes	—
9	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
12	C₂×#127×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₇	—	2	2	no	—
14	C₂×C₄×D₄×C₃₇	C₂²	1	2	yes	—
15	C₃₇×#492	—	2	8	no	—
16	C₃₇×#461	—	2	16	no	—
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
18	C₂×#103×C₃₇	C₂²	1	8	yes	—
19	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
20	C₃₇×#390	—	2	2	no	—
21	C₃₇×#390	C₂²	1	2	yes	—
22	C₃₇×#384	—	2	2	no	—
23	C₃₇×#384	—	2	2	no	—
24	C₃₇×#384	C₂²	1	1	yes	characteristic
25	C₄³×C₃₇	C₂²	1	1	yes	characteristic
26	C₃₇×#336	C₂²	1	2	yes	—
27	C₂²×Q₈×C₃₇	C₂³	1	4	yes	—
28	C₂²×D₄×C₃₇	—	2	2	no	—
29	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
30	C₂³×C₄×C₃₇	—	2	2	no	—
31	C₂×SD₁₆×C₃₇	—	2	16	no	—
32	C₂×SD₁₆×C₃₇	—	2	16	no	—
33	C₂×SD₁₆×C₃₇	—	4	8	no	—
34	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
35	C₄⋊Q₈×C₃₇	—	2	8	no	—
36	C₄⋊Q₈×C₃₇	—	2	4	no	—
37	C₄⋊Q₈×C₃₇	—	2	4	no	—
38	C₄⋊Q₈×C₃₇	—	4	4	no	—
39	#127×C₃₇	—	2	2	no	—
40	#127×C₃₇	D₄	1	4	yes	—
41	C₄×D₄×C₃₇	—	2	8	no	—
42	C₄×D₄×C₃₇	—	4	8	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
51	C₂×C₄²×C₃₇	D₄	1	2	yes	—
52	C₂×C₄²×C₃₇	D₄	1	2	yes	—
53	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₃₇	D₄	1	8	yes	complemented
57	#103×C₃₇	—	2	16	no	—
58	C₄×C₈×C₃₇	—	2	4	no	—
59	C₂⁴×C₃₇	—	2	2	no	—
60	C₂×Q₈×C₃₇	—	2	8	no	—
61	C₂×Q₈×C₃₇	—	2	4	no	—
62	C₂×Q₈×C₃₇	—	2	4	no	—
63	C₂×Q₈×C₃₇	—	2	8	no	—
64	C₂×D₄×C₃₇	—	2	8	no	—
65	C₂×D₄×C₃₇	—	2	4	no	—
66	C₂×D₄×C₃₇	—	2	2	no	—
67	C₂×D₄×C₃₇	—	2	4	no	—
68	C₂×D₄×C₃₇	—	4	4	no	—
69	C₂×D₄×C₃₇	—	4	4	no	—
70	C₂×D₄×C₃₇	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₃₇	—	2	2	no	—
72	C₂²×C₄×C₃₇	—	2	4	no	—
73	C₂²×C₄×C₃₇	—	4	4	no	—
74	C₂²×C₄×C₃₇	—	4	8	no	—
75	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
80	SD₁₆×C₃₇	—	4	32	no	—
81	C₂×C₈×C₃₇	—	2	4	no	—
82	C₂×C₈×C₃₇	—	2	4	no	—
83	C₂×C₈×C₃₇	C₂×D₄	1	8	yes	—
84	C₄⋊C₄×C₃₇	—	2	4	no	—
85	C₄⋊C₄×C₃₇	—	4	8	no	—
86	C₄⋊C₄×C₃₇	—	4	8	no	—
87	C₄⋊C₄×C₃₇	—	4	4	no	—
88	C₄⋊C₄×C₃₇	—	4	4	no	—
89	C₄⋊C₄×C₃₇	—	4	4	no	—
90	C₂²⋊C₄×C₃₇	—	4	4	no	—
91	C₂²⋊C₄×C₃₇	—	4	8	no	—
92	C₄²×C₃₇	—	2	4	no	—
93	C₄²×C₃₇	—	2	2	no	—
94	C₄²×C₃₇	SD₁₆	1	8	yes	complemented
95	C₄²×C₃₇	C₂×D₄	1	2	yes	—
96	C₄²×C₃₇	C₂×D₄	1	4	yes	—
97	C₄²×C₃₇	#41	1	2	yes	—
98	C₂³×C₃₇	—	2	4	no	—
99	C₂³×C₃₇	—	4	2	no	—
100	C₂³×C₃₇	—	4	2	no	—
101	C₂³×C₃₇	—	4	4	no	—
102	C₂³×C₃₇	—	4	4	no	—
103	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₃₇	—	4	8	no	—
105	Q₈×C₃₇	—	4	8	no	—
106	D₄×C₃₇	—	2	8	no	—
107	D₄×C₃₇	—	4	4	no	—
108	D₄×C₃₇	—	4	16	no	—
109	C₂×C₄×C₃₇	—	2	2	no	—
110	C₂×C₄×C₃₇	—	2	2	no	—
111	C₂×C₄×C₃₇	—	2	2	no	—
112	C₂×C₄×C₃₇	—	4	4	no	—
113	C₂×C₄×C₃₇	—	4	2	no	—
114	C₂×C₄×C₃₇	—	4	4	no	—
115	C₂×C₄×C₃₇	—	4	4	no	—
116	C₂×C₄×C₃₇	—	4	4	no	—
117	C₂×C₄×C₃₇	—	8	8	no	—
118	C₂×C₄×C₃₇	#121	1	4	yes	—
119	C₂×C₄×C₃₇	#127	1	2	yes	—
120	C₂×C₄×C₃₇	#127	1	2	yes	—
121	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
122	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
123	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
124	C₂×C₄×C₃₇	#137	1	4	yes	—
125	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
127	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
129	C₈×C₃₇	—	2	8	no	—
130	#28438	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₃₇	—	4	8	no	—
132	C₂²×C₃₇	—	4	4	no	—
133	C₂²×C₃₇	—	8	2	no	—
134	C₂²×C₃₇	—	8	2	no	—
135	C₂²×C₃₇	—	8	4	no	—
136	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
137	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
138	C₂²×C₃₇	#535	1	1	yes	characteristic
139	C₂²×C₃₇	C₂²×SD₁₆	1	2	yes	—
140	C₂²×C₃₇	C₂×#137	1	2	yes	—
141	C₄×C₃₇	—	2	2	no	—
142	C₄×C₃₇	—	2	4	no	—
143	C₄×C₃₇	—	8	2	no	—
144	C₄×C₃₇	—	8	4	no	—
145	C₄×C₃₇	#461	1	8	yes	complemented
146	C₄×C₃₇	#492	1	4	yes	complemented
147	C₄×C₃₇	D₄²	1	2	yes	—
148	C₄×C₃₇	#550	1	2	yes	—
149	C₂×#492	C₂×C₃₇	1	2	yes	—
150	C₂×#461	C₂×C₃₇	1	4	yes	complemented
151	#2868	C₂×C₃₇	1	1	yes	characteristic, complemented
152	C₄×#127	C₂×C₃₇	1	1	yes	characteristic
153	#2013	C₂×C₃₇	1	4	yes	complemented
154	#1956	C₂×C₃₇	1	2	yes	complemented
155	C₄×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic, complemented
156	C₂×C₃₇	—	8	4	no	—
157	C₂×C₃₇	#2541	1	1	yes	characteristic
158	C₂×C₃₇	C₂×#461	1	2	yes	—
159	C₂×C₃₇	C₂×#492	1	1	yes	characteristic
160	C₂×C₃₇	#3152	1	1	yes	characteristic
161	C₂×C₃₇	#3206	1	2	yes	—
162	C₂²×SD₁₆	C₂²×C₃₇	1	8	yes	—
163	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
164	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
165	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
166	C₂×#127	C₂²×C₃₇	1	1	yes	characteristic
167	C₂×C₄×D₄	—	2	2	no	—
168	C₂×C₄×D₄	C₂²×C₃₇	1	2	yes	—
169	#492	—	2	8	no	—
170	#461	—	2	16	no	—
171	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
172	C₂×#103	C₂²×C₃₇	1	8	yes	—
173	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
174	#390	—	2	2	no	—
175	#390	C₂²×C₃₇	1	2	yes	—
176	#384	—	2	2	no	—
177	#384	—	2	2	no	—
178	#384	C₂²×C₃₇	1	1	yes	characteristic
179	C₄³	C₂²×C₃₇	1	1	yes	characteristic
180	#336	C₂²×C₃₇	1	2	yes	—
181	C₃₇	#28438	1	1	yes	characteristic, direct product factor, Sylow
182	C₂²×Q₈	C₂³×C₃₇	1	4	yes	—
183	C₂²×D₄	—	2	2	no	—
184	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
185	C₂³×C₄	—	2	2	no	—
186	C₂×SD₁₆	—	2	16	no	—
187	C₂×SD₁₆	—	2	16	no	—
188	C₂×SD₁₆	—	4	8	no	—
189	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
190	C₄⋊Q₈	—	2	8	no	—
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	—	2	4	no	—
193	C₄⋊Q₈	—	4	4	no	—
194	#127	—	2	2	no	—
195	#127	D₄×C₃₇	1	4	yes	—
196	C₄×D₄	—	2	8	no	—
197	C₄×D₄	—	4	8	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
204	C₂×C₂²⋊C₄	—	2	4	no	—
205	C₂×C₂²⋊C₄	—	2	2	no	—
206	C₂×C₄²	D₄×C₃₇	1	2	yes	—
207	C₂×C₄²	D₄×C₃₇	1	2	yes	—
208	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
210	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
211	C₄⋊C₈	D₄×C₃₇	1	8	yes	complemented
212	#103	—	2	16	no	—
213	C₄×C₈	—	2	4	no	—
214	C₂⁴	—	2	2	no	—
215	C₂×Q₈	—	2	8	no	—
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	—	2	8	no	—
219	C₂×D₄	—	2	8	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	4	4	no	—
224	C₂×D₄	—	4	4	no	—
225	C₂×D₄	C₂×D₄×C₃₇	1	4	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	8	no	—
230	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
234	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
235	SD₁₆	—	4	32	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	C₂×D₄×C₃₇	1	8	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	8	no	—
241	C₄⋊C₄	—	4	8	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₂²⋊C₄	—	4	8	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	2	no	—
249	C₄²	SD₁₆×C₃₇	1	8	yes	complemented
250	C₄²	C₂×D₄×C₃₇	1	2	yes	—
251	C₄²	C₂×D₄×C₃₇	1	4	yes	—
252	C₄²	#41×C₃₇	1	2	yes	—
253	C₂³	—	2	4	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	4	no	—
257	C₂³	—	4	4	no	—
258	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
259	Q₈	—	4	8	no	—
260	Q₈	—	4	8	no	—
261	D₄	—	2	8	no	—
262	D₄	—	4	4	no	—
263	D₄	—	4	16	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	8	8	no	—
273	C₂×C₄	#121×C₃₇	1	4	yes	—
274	C₂×C₄	#127×C₃₇	1	2	yes	—
275	C₂×C₄	#127×C₃₇	1	2	yes	—
276	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
277	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
278	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
279	C₂×C₄	#137×C₃₇	1	4	yes	—
280	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
282	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
283	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
284	C₈	—	2	8	no	—
285	C₂²	—	4	8	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	8	2	no	—
288	C₂²	—	8	2	no	—
289	C₂²	—	8	4	no	—
290	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
291	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
292	C₂²	C₃₇×#535	1	1	yes	characteristic
293	C₂²	C₂²×SD₁₆×C₃₇	1	2	yes	—
294	C₂²	C₂×#137×C₃₇	1	2	yes	—
295	C₄	—	2	2	no	—
296	C₄	—	2	4	no	—
297	C₄	—	8	2	no	—
298	C₄	—	8	4	no	—
299	C₄	C₃₇×#461	1	8	yes	complemented
300	C₄	C₃₇×#492	1	4	yes	complemented
301	C₄	D₄²×C₃₇	1	2	yes	—
302	C₄	C₃₇×#550	1	2	yes	—
303	C₂	—	8	4	no	—
304	C₂	C₃₇×#2541	1	1	yes	characteristic
305	C₂	C₂×C₃₇×#461	1	2	yes	—
306	C₂	C₂×C₃₇×#492	1	1	yes	characteristic
307	C₂	C₃₇×#3152	1	1	yes	characteristic
308	C₂	C₃₇×#3206	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂×#127×C₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₄³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₇×#461	C₂	semidirect
3. C₃₇×#2868	C₂	semidirect
5. C₃₇×#2013	C₂	semidirect
6. C₃₇×#1956	C₂	semidirect
7. C₄×C₄⋊C₈×C₃₇	C₂	semidirect
56. C₄⋊C₈×C₃₇	D₄	semidirect
94. C₄²×C₃₇	SD₁₆	semidirect
130. #28438	C₃₇	direct (N × H)
145. C₄×C₃₇	#461	semidirect
146. C₄×C₃₇	#492	semidirect
150. C₂×#461	C₂×C₃₇	semidirect
151. #2868	C₂×C₃₇	semidirect
153. #2013	C₂×C₃₇	semidirect
154. #1956	C₂×C₃₇	semidirect
155. C₄×C₄⋊C₈	C₂×C₃₇	semidirect
181. C₃₇	#28438	direct (N × H)
211. C₄⋊C₈	D₄×C₃₇	semidirect
249. C₄²	SD₁₆×C₃₇	semidirect
299. C₄	C₃₇×#461	semidirect
300. C₄	C₃₇×#492	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#492	2
C₂×C₃₇×#461	2
C₃₇×#2868	2
C₄×#127×C₃₇	2
C₃₇×#2013	2
C₃₇×#1956	2
C₄×C₄⋊C₈×C₃₇	2
#28438	37

Subgroups — C37×#28438 #22649691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28438 #22649691