Subgroups — C₃₇×#28462 #22649715

Order 9472.

1062 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#494	C₂	1	2	yes	maximal
3	C₂×C₃₇×#462	C₂	1	2	yes	maximal
4	C₂×C₄×Q₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#2546	C₂	1	2	yes	maximal
6	C₃₇×#2016	C₂	1	4	yes	maximal
7	C₃₇×#1969	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₃₇	C₂²	1	2	yes	—
10	C₂²×Q₁₆×C₃₇	C₂²	1	4	yes	—
11	C₂²×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
12	C₂²×Q₁₆×C₃₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₃₇	C₂²	1	4	yes	—
15	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
16	C₃₇×#494	—	2	8	no	—
17	C₃₇×#494	—	2	4	no	—
18	C₃₇×#462	—	2	8	no	—
19	C₄×Q₁₆×C₃₇	—	2	4	no	—
20	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
22	C₂×#103×C₃₇	C₂²	1	2	yes	—
23	C₂×#103×C₃₇	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
26	C₃₇×#398	—	2	4	no	—
27	C₃₇×#391	C₂²	1	4	yes	—
28	C₃₇×#389	—	2	4	no	—
29	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
30	C₃₇×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₃₇	C₂³	1	4	yes	—
32	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
33	C₂×Q₁₆×C₃₇	—	2	8	no	—
34	C₂×Q₁₆×C₃₇	—	2	2	no	—
35	C₂×Q₁₆×C₃₇	—	2	8	no	—
36	C₂×Q₁₆×C₃₇	—	2	4	no	—
37	C₂×Q₁₆×C₃₇	—	2	4	no	—
38	C₂×Q₁₆×C₃₇	—	2	4	no	—
39	C₂×Q₁₆×C₃₇	—	2	2	no	—
40	C₂×Q₁₆×C₃₇	—	2	4	no	—
41	C₂×Q₁₆×C₃₇	—	4	4	no	—
42	C₂×Q₁₆×C₃₇	—	4	2	no	—
43	C₂×Q₁₆×C₃₇	—	4	2	no	—
44	C₂×Q₁₆×C₃₇	D₄	1	4	yes	—
45	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₃₇	—	2	8	no	—
49	C₄⋊Q₈×C₃₇	—	2	4	no	—
50	C₄⋊Q₈×C₃₇	—	2	4	no	—
51	C₄⋊Q₈×C₃₇	—	4	4	no	—
52	C₄×Q₈×C₃₇	—	2	4	no	—
53	C₄×Q₈×C₃₇	—	2	4	no	—
54	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₇	—	2	8	no	—
56	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
60	C₂×C₄²×C₃₇	—	2	2	no	—
61	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
62	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
63	#107×C₃₇	—	2	2	no	—
64	C₄⋊C₈×C₃₇	D₄	1	4	yes	—
65	#103×C₃₇	—	2	4	no	—
66	#103×C₃₇	—	2	8	no	—
67	C₄×C₈×C₃₇	—	2	4	no	—
68	C₄×C₈×C₃₇	D₄	1	4	yes	—
69	#95×C₃₇	—	2	8	no	—
70	#95×C₃₇	D₄	1	2	yes	—
71	C₂×Q₈×C₃₇	—	2	2	no	—
72	C₂×Q₈×C₃₇	—	2	2	no	—
73	C₂×Q₈×C₃₇	—	2	2	no	—
74	C₂×Q₈×C₃₇	—	2	4	no	—
75	C₂×Q₈×C₃₇	—	2	4	no	—
76	C₂×Q₈×C₃₇	—	2	4	no	—
77	C₂×Q₈×C₃₇	—	2	2	no	—
78	C₂×Q₈×C₃₇	—	2	4	no	—
79	C₂×Q₈×C₃₇	—	2	2	no	—
80	C₂×Q₈×C₃₇	—	2	4	no	—
81	C₂×Q₈×C₃₇	—	2	4	no	—
82	C₂×Q₈×C₃₇	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₃₇	—	2	2	no	—
84	C₂²×C₄×C₃₇	—	2	2	no	—
85	C₂²×C₄×C₃₇	—	2	4	no	—
86	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₇	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
90	Q₁₆×C₃₇	—	2	8	no	—
91	Q₁₆×C₃₇	—	4	16	no	—
92	Q₁₆×C₃₇	—	4	8	no	—
93	Q₁₆×C₃₇	—	4	8	no	—
94	C₂×C₈×C₃₇	—	2	2	no	—
95	C₂×C₈×C₃₇	—	2	1	no	—
96	C₂×C₈×C₃₇	—	2	1	no	—
97	C₂×C₈×C₃₇	—	2	2	no	—
98	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₃₇	#41	1	2	yes	—
103	C₄⋊C₄×C₃₇	—	2	2	no	—
104	C₄⋊C₄×C₃₇	—	2	2	no	—
105	C₄⋊C₄×C₃₇	—	4	16	no	—
106	C₄⋊C₄×C₃₇	—	4	4	no	—
107	C₄⋊C₄×C₃₇	—	4	4	no	—
108	C₄⋊C₄×C₃₇	—	4	4	no	—
109	C₄⋊C₄×C₃₇	Q₁₆	1	8	yes	complemented
110	C₄²×C₃₇	—	2	2	no	—
111	C₄²×C₃₇	—	2	2	no	—
112	C₄²×C₃₇	—	4	4	no	—
113	C₄²×C₃₇	C₂×D₄	1	2	yes	—
114	C₄²×C₃₇	C₂×D₄	1	2	yes	—
115	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
116	Q₈×C₃₇	—	2	4	no	—
117	Q₈×C₃₇	—	2	4	no	—
118	Q₈×C₃₇	—	4	2	no	—
119	Q₈×C₃₇	—	4	2	no	—
120	Q₈×C₃₇	—	4	4	no	—
121	Q₈×C₃₇	—	4	4	no	—
122	Q₈×C₃₇	—	4	4	no	—
123	Q₈×C₃₇	—	4	4	no	—
124	C₂×C₄×C₃₇	—	2	4	no	—
125	C₂×C₄×C₃₇	—	2	2	no	—
126	C₂×C₄×C₃₇	—	2	4	no	—
127	C₂×C₄×C₃₇	—	2	4	no	—
128	C₂×C₄×C₃₇	—	4	2	no	—
129	C₂×C₄×C₃₇	—	4	2	no	—
130	C₂×C₄×C₃₇	—	4	2	no	—
131	C₂×C₄×C₃₇	—	4	4	no	—
132	C₂×C₄×C₃₇	—	4	4	no	—
133	C₂×C₄×C₃₇	—	4	4	no	—
134	C₂×C₄×C₃₇	—	4	4	no	—
135	C₂×C₄×C₃₇	#127	1	2	yes	—
136	C₂×C₄×C₃₇	#127	1	2	yes	—
137	C₂×C₄×C₃₇	C₂×Q₁₆	1	8	yes	—
138	C₂×C₄×C₃₇	C₂×Q₁₆	1	4	yes	—
139	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
145	C₈×C₃₇	—	2	2	no	—
146	C₈×C₃₇	—	2	4	no	—
147	C₈×C₃₇	#121	1	4	yes	—
148	#28462	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
150	C₂²×C₃₇	D₄²	1	1	yes	characteristic
151	C₂²×C₃₇	#550	1	1	yes	characteristic
152	C₂²×C₃₇	C₂²×Q₁₆	1	2	yes	—
153	C₂²×C₃₇	#578	1	2	yes	—
154	C₄×C₃₇	—	2	2	no	—
155	C₄×C₃₇	—	2	8	no	—
156	C₄×C₃₇	—	4	4	no	—
157	C₄×C₃₇	—	8	2	no	—
158	C₄×C₃₇	—	8	4	no	—
159	C₄×C₃₇	#494	1	4	yes	complemented
160	C₄×C₃₇	C₂×#121	1	1	yes	characteristic
161	C₄×C₃₇	#535	1	1	yes	characteristic
162	C₄×C₃₇	D₄²	1	1	yes	characteristic
163	C₄×C₃₇	#550	1	1	yes	characteristic
164	C₂×#494	C₂×C₃₇	1	1	yes	characteristic
165	C₂×#494	C₂×C₃₇	1	2	yes	—
166	C₂×#462	C₂×C₃₇	1	2	yes	—
167	C₂×C₄×Q₁₆	C₂×C₃₇	1	1	yes	characteristic
168	#2546	C₂×C₃₇	1	2	yes	—
169	#2016	C₂×C₃₇	1	4	yes	—
170	#1969	C₂×C₃₇	1	2	yes	—
171	C₈×C₄⋊C₄	C₂×C₃₇	1	1	yes	characteristic
172	C₂×C₃₇	#2541	1	1	yes	characteristic
173	C₂×C₃₇	C₂×#494	1	1	yes	characteristic
174	C₂×C₃₇	#3156	1	1	yes	characteristic
175	C₂×C₃₇	D₄×Q₁₆	1	2	yes	—
176	C₂×C₃₇	#3395	1	2	yes	—
177	C₂²×Q₁₆	C₂²×C₃₇	1	2	yes	—
178	C₂²×Q₁₆	C₂²×C₃₇	1	4	yes	—
179	C₂²×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic
180	C₂²×Q₁₆	C₂²×C₃₇	1	2	yes	—
181	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
182	C₂×C₄⋊Q₈	C₂²×C₃₇	1	4	yes	—
183	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
184	#494	—	2	8	no	—
185	#494	—	2	4	no	—
186	#462	—	2	8	no	—
187	C₄×Q₁₆	—	2	4	no	—
188	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
189	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
190	C₂×#103	C₂²×C₃₇	1	2	yes	—
191	C₂×#103	C₂²×C₃₇	1	4	yes	—
192	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
193	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
194	#398	—	2	4	no	—
195	#391	C₂²×C₃₇	1	4	yes	—
196	#389	—	2	4	no	—
197	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
198	#336	C₂²×C₃₇	1	2	yes	—
199	C₃₇	#28462	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²×Q₈	C₂³×C₃₇	1	4	yes	—
201	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
202	C₂×Q₁₆	—	2	8	no	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	—	2	8	no	—
205	C₂×Q₁₆	—	2	4	no	—
206	C₂×Q₁₆	—	2	4	no	—
207	C₂×Q₁₆	—	2	4	no	—
208	C₂×Q₁₆	—	2	2	no	—
209	C₂×Q₁₆	—	2	4	no	—
210	C₂×Q₁₆	—	4	4	no	—
211	C₂×Q₁₆	—	4	2	no	—
212	C₂×Q₁₆	—	4	2	no	—
213	C₂×Q₁₆	D₄×C₃₇	1	4	yes	—
214	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
215	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
216	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	8	no	—
218	C₄⋊Q₈	—	2	4	no	—
219	C₄⋊Q₈	—	2	4	no	—
220	C₄⋊Q₈	—	4	4	no	—
221	C₄×Q₈	—	2	4	no	—
222	C₄×Q₈	—	2	4	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	8	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
228	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
229	C₂×C₄²	—	2	2	no	—
230	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
231	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
232	#107	—	2	2	no	—
233	C₄⋊C₈	D₄×C₃₇	1	4	yes	—
234	#103	—	2	4	no	—
235	#103	—	2	8	no	—
236	C₄×C₈	—	2	4	no	—
237	C₄×C₈	D₄×C₃₇	1	4	yes	—
238	#95	—	2	8	no	—
239	#95	D₄×C₃₇	1	2	yes	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	—	2	4	no	—
251	C₂×Q₈	C₂×D₄×C₃₇	1	4	yes	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₃₇	1	4	yes	—
257	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
258	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
259	Q₁₆	—	2	8	no	—
260	Q₁₆	—	4	16	no	—
261	Q₁₆	—	4	8	no	—
262	Q₁₆	—	4	8	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
269	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
270	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
271	C₂×C₈	#41×C₃₇	1	2	yes	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	16	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	Q₁₆×C₃₇	1	8	yes	complemented
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	4	4	no	—
282	C₄²	C₂×D₄×C₃₇	1	2	yes	—
283	C₄²	C₂×D₄×C₃₇	1	2	yes	—
284	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	4	no	—
290	Q₈	—	4	4	no	—
291	Q₈	—	4	4	no	—
292	Q₈	—	4	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	#127×C₃₇	1	2	yes	—
305	C₂×C₄	#127×C₃₇	1	2	yes	—
306	C₂×C₄	C₂×Q₁₆×C₃₇	1	8	yes	—
307	C₂×C₄	C₂×Q₁₆×C₃₇	1	4	yes	—
308	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
311	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
312	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
313	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	2	4	no	—
316	C₈	#121×C₃₇	1	4	yes	—
317	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
318	C₂²	D₄²×C₃₇	1	1	yes	characteristic
319	C₂²	C₃₇×#550	1	1	yes	characteristic
320	C₂²	C₂²×Q₁₆×C₃₇	1	2	yes	—
321	C₂²	C₃₇×#578	1	2	yes	—
322	C₄	—	2	2	no	—
323	C₄	—	2	8	no	—
324	C₄	—	4	4	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	4	no	—
327	C₄	C₃₇×#494	1	4	yes	complemented
328	C₄	C₂×#121×C₃₇	1	1	yes	characteristic
329	C₄	C₃₇×#535	1	1	yes	characteristic
330	C₄	D₄²×C₃₇	1	1	yes	characteristic
331	C₄	C₃₇×#550	1	1	yes	characteristic
332	C₂	C₃₇×#2541	1	1	yes	characteristic
333	C₂	C₂×C₃₇×#494	1	1	yes	characteristic
334	C₂	C₃₇×#3156	1	1	yes	characteristic
335	C₂	D₄×Q₁₆×C₃₇	1	2	yes	—
336	C₂	C₃₇×#3395	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
109. C₄⋊C₄×C₃₇	Q₁₆	semidirect
148. #28462	C₃₇	direct (N × H)
159. C₄×C₃₇	#494	semidirect
199. C₃₇	#28462	direct (N × H)
278. C₄⋊C₄	Q₁₆×C₃₇	semidirect
327. C₄	C₃₇×#494	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#494	2
C₂×C₃₇×#462	2
C₂×C₄×Q₁₆×C₃₇	2
C₃₇×#2546	2
C₃₇×#2016	2
C₃₇×#1969	2
C₈×C₄⋊C₄×C₃₇	2
#28462	37

Subgroups — C37×#28462 #22649715

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28462 #22649715