Subgroups — C₃₇×#28374 #22649627

Order 9472.

1038 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#475	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₇×#460	C₂	1	2	yes	maximal
3	C₃₇×#2858	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#2094	C₂	1	2	yes	maximal, complemented
6	C₃₇×#2013	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1968	C₂	1	4	yes	maximal, complemented
8	C₄×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₃₇	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×#127×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×#124×C₃₇	—	2	2	no	—
14	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
15	C₃₇×#475	—	2	8	no	—
16	C₃₇×#460	—	2	8	no	—
17	C₂×#106×C₃₇	C₂²	1	4	yes	—
18	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
19	C₂×#103×C₃₇	C₂²	1	4	yes	—
20	C₂×#102×C₃₇	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
22	C₃₇×#391	C₂²	1	2	yes	—
23	C₃₇×#390	—	2	2	no	—
24	C₃₇×#390	C₂²	1	2	yes	—
25	C₃₇×#390	C₂²	1	2	yes	—
26	C₃₇×#389	—	2	2	no	—
27	C₃₇×#384	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₃₇	—	2	2	no	—
29	C₄³×C₃₇	C₂²	1	1	yes	characteristic
30	C₃₇×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₃₇	—	2	1	no	—
32	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
33	C₂²×D₄×C₃₇	—	2	1	no	—
34	C₂²×D₄×C₃₇	—	2	1	no	—
35	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
36	C₂×SD₁₆×C₃₇	—	2	8	no	—
37	C₂×SD₁₆×C₃₇	—	2	8	no	—
38	C₂×SD₁₆×C₃₇	—	4	4	no	—
39	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
40	C₄⋊Q₈×C₃₇	—	2	4	no	—
41	C₄⋊Q₈×C₃₇	—	2	4	no	—
42	#127×C₃₇	—	2	4	no	—
43	#124×C₃₇	—	4	8	no	—
44	C₄×Q₈×C₃₇	—	2	4	no	—
45	C₄×Q₈×C₃₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
48	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
52	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
53	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
54	C₂×C₄²×C₃₇	—	2	2	no	—
55	C₂×C₄²×C₃₇	D₄	1	2	yes	—
56	C₂×C₄²×C₃₇	D₄	1	2	yes	—
57	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
60	#106×C₃₇	—	2	8	no	—
61	C₄⋊C₈×C₃₇	D₄	1	8	yes	—
62	#103×C₃₇	—	2	8	no	—
63	#102×C₃₇	—	2	8	no	—
64	C₄×C₈×C₃₇	—	4	2	no	—
65	#95×C₃₇	—	2	4	no	—
66	C₂⁴×C₃₇	—	2	2	no	—
67	C₂×Q₈×C₃₇	—	2	2	no	—
68	C₂×Q₈×C₃₇	—	2	2	no	—
69	C₂×Q₈×C₃₇	—	2	2	no	—
70	C₂×Q₈×C₃₇	—	2	2	no	—
71	C₂×Q₈×C₃₇	—	2	2	no	—
72	C₂×Q₈×C₃₇	—	4	2	no	—
73	C₂×Q₈×C₃₇	—	4	4	no	—
74	C₂×Q₈×C₃₇	#41	1	4	yes	—
75	C₂×D₄×C₃₇	—	2	4	no	—
76	C₂×D₄×C₃₇	—	2	4	no	—
77	C₂×D₄×C₃₇	—	2	2	no	—
78	C₂×D₄×C₃₇	—	2	2	no	—
79	C₂×D₄×C₃₇	—	4	4	no	—
80	C₂×D₄×C₃₇	—	4	2	no	—
81	C₂×D₄×C₃₇	—	4	2	no	—
82	C₂×D₄×C₃₇	—	4	4	no	—
83	C₂²×C₄×C₃₇	—	2	2	no	—
84	C₂²×C₄×C₃₇	—	2	2	no	—
85	C₂²×C₄×C₃₇	—	2	2	no	—
86	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
91	SD₁₆×C₃₇	—	4	16	no	—
92	C₂×C₈×C₃₇	—	2	4	no	—
93	C₂×C₈×C₃₇	—	2	4	no	—
94	C₂×C₈×C₃₇	C₂×D₄	1	8	yes	—
95	C₄⋊C₄×C₃₇	—	2	8	no	—
96	C₄⋊C₄×C₃₇	—	2	2	no	—
97	C₄⋊C₄×C₃₇	—	2	4	no	—
98	C₄⋊C₄×C₃₇	—	2	2	no	—
99	C₄⋊C₄×C₃₇	—	4	4	no	—
100	C₄⋊C₄×C₃₇	—	4	4	no	—
101	C₄⋊C₄×C₃₇	—	4	2	no	—
102	C₂²⋊C₄×C₃₇	—	4	8	no	—
103	C₂²⋊C₄×C₃₇	—	4	8	no	—
104	C₄²×C₃₇	—	2	2	no	—
105	C₄²×C₃₇	—	2	1	no	—
106	C₄²×C₃₇	—	2	1	no	—
107	C₄²×C₃₇	—	4	2	no	—
108	C₄²×C₃₇	—	4	4	no	—
109	C₄²×C₃₇	SD₁₆	1	8	yes	complemented
110	C₄²×C₃₇	C₂×D₄	1	4	yes	—
111	C₂³×C₃₇	—	4	2	no	—
112	C₂³×C₃₇	—	4	2	no	—
113	C₂³×C₃₇	—	4	2	no	—
114	C₂³×C₃₇	—	4	4	no	—
115	C₂³×C₃₇	—	4	4	no	—
116	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
117	Q₈×C₃₇	—	2	4	no	—
118	Q₈×C₃₇	—	2	4	no	—
119	Q₈×C₃₇	—	4	2	no	—
120	Q₈×C₃₇	—	4	2	no	—
121	Q₈×C₃₇	—	8	4	no	—
122	D₄×C₃₇	—	4	4	no	—
123	D₄×C₃₇	—	4	4	no	—
124	D₄×C₃₇	—	8	4	no	—
125	D₄×C₃₇	—	8	4	no	—
126	C₂×C₄×C₃₇	—	2	2	no	—
127	C₂×C₄×C₃₇	—	2	2	no	—
128	C₂×C₄×C₃₇	—	2	2	no	—
129	C₂×C₄×C₃₇	—	2	2	no	—
130	C₂×C₄×C₃₇	—	2	2	no	—
131	C₂×C₄×C₃₇	—	2	4	no	—
132	C₂×C₄×C₃₇	—	4	2	no	—
133	C₂×C₄×C₃₇	—	4	2	no	—
134	C₂×C₄×C₃₇	—	4	2	no	—
135	C₂×C₄×C₃₇	—	4	2	no	—
136	C₂×C₄×C₃₇	—	4	4	no	—
137	C₂×C₄×C₃₇	—	4	4	no	—
138	C₂×C₄×C₃₇	#127	1	2	yes	—
139	C₂×C₄×C₃₇	#127	1	2	yes	—
140	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
141	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
142	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
143	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
144	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
145	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
147	C₈×C₃₇	—	2	8	no	—
148	#28374	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₃₇	—	8	4	no	—
150	C₂²×C₃₇	—	8	2	no	—
151	C₂²×C₃₇	—	8	2	no	—
152	C₂²×C₃₇	—	8	4	no	—
153	C₂²×C₃₇	—	8	2	no	—
154	C₂²×C₃₇	#525	1	1	yes	characteristic
155	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
156	C₂²×C₃₇	#537	1	1	yes	characteristic
157	C₂²×C₃₇	C₂²×SD₁₆	1	2	yes	—
158	C₂²×C₃₇	#575	1	2	yes	—
159	C₄×C₃₇	—	2	4	no	—
160	C₄×C₃₇	—	2	2	no	—
161	C₄×C₃₇	—	2	4	no	—
162	C₄×C₃₇	—	4	4	no	—
163	C₄×C₃₇	—	8	2	no	—
164	C₄×C₃₇	—	8	2	no	—
165	C₄×C₃₇	#492	1	4	yes	—
166	C₄×C₃₇	#546	1	2	yes	—
167	C₄×C₃₇	#548	1	2	yes	—
168	C₂×#475	C₂×C₃₇	1	2	yes	complemented
169	C₂×#460	C₂×C₃₇	1	2	yes	—
170	#2858	C₂×C₃₇	1	1	yes	characteristic
171	C₄×C₄⋊Q₈	C₂×C₃₇	1	1	yes	characteristic, complemented
172	#2094	C₂×C₃₇	1	2	yes	complemented
173	#2013	C₂×C₃₇	1	2	yes	complemented
174	#1968	C₂×C₃₇	1	4	yes	complemented
175	C₄×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic, complemented
176	C₂×C₃₇	—	16	2	no	—
177	C₂×C₃₇	#2540	1	1	yes	characteristic
178	C₂×C₃₇	#3103	1	2	yes	—
179	C₂×C₃₇	C₂×#492	1	1	yes	characteristic
180	C₂×C₃₇	#3153	1	1	yes	characteristic
181	C₂×C₃₇	#3246	1	2	yes	—
182	C₂²×SD₁₆	C₂²×C₃₇	1	4	yes	—
183	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
184	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
185	C₂×#127	C₂²×C₃₇	1	1	yes	characteristic
186	C₂×#124	—	2	2	no	—
187	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
188	#475	—	2	8	no	—
189	#460	—	2	8	no	—
190	C₂×#106	C₂²×C₃₇	1	4	yes	—
191	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
192	C₂×#103	C₂²×C₃₇	1	4	yes	—
193	C₂×#102	C₂²×C₃₇	1	4	yes	—
194	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
195	#391	C₂²×C₃₇	1	2	yes	—
196	#390	—	2	2	no	—
197	#390	C₂²×C₃₇	1	2	yes	—
198	#390	C₂²×C₃₇	1	2	yes	—
199	#389	—	2	2	no	—
200	#384	C₂²×C₃₇	1	1	yes	characteristic
201	C₄×C₄⋊C₄	—	2	2	no	—
202	C₄³	C₂²×C₃₇	1	1	yes	characteristic
203	#336	C₂²×C₃₇	1	2	yes	—
204	C₃₇	#28374	1	1	yes	characteristic, direct product factor, Sylow
205	C₂²×Q₈	—	2	1	no	—
206	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
207	C₂²×D₄	—	2	1	no	—
208	C₂²×D₄	—	2	1	no	—
209	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
210	C₂×SD₁₆	—	2	8	no	—
211	C₂×SD₁₆	—	2	8	no	—
212	C₂×SD₁₆	—	4	4	no	—
213	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
214	C₄⋊Q₈	—	2	4	no	—
215	C₄⋊Q₈	—	2	4	no	—
216	#127	—	2	4	no	—
217	#124	—	4	8	no	—
218	C₄×Q₈	—	2	4	no	—
219	C₄×Q₈	—	2	4	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
225	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
226	C₂×C₂²⋊C₄	—	2	4	no	—
227	C₂×C₂²⋊C₄	—	2	4	no	—
228	C₂×C₄²	—	2	2	no	—
229	C₂×C₄²	D₄×C₃₇	1	2	yes	—
230	C₂×C₄²	D₄×C₃₇	1	2	yes	—
231	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
232	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
234	#106	—	2	8	no	—
235	C₄⋊C₈	D₄×C₃₇	1	8	yes	—
236	#103	—	2	8	no	—
237	#102	—	2	8	no	—
238	C₄×C₈	—	4	2	no	—
239	#95	—	2	4	no	—
240	C₂⁴	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	4	4	no	—
248	C₂×Q₈	#41×C₃₇	1	4	yes	—
249	C₂×D₄	—	2	4	no	—
250	C₂×D₄	—	2	4	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	4	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	4	2	no	—
256	C₂×D₄	—	4	4	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
263	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
264	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
265	SD₁₆	—	4	16	no	—
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	C₂×D₄×C₃₇	1	8	yes	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	8	no	—
277	C₂²⋊C₄	—	4	8	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	4	2	no	—
282	C₄²	—	4	4	no	—
283	C₄²	SD₁₆×C₃₇	1	8	yes	complemented
284	C₄²	C₂×D₄×C₃₇	1	4	yes	—
285	C₂³	—	4	2	no	—
286	C₂³	—	4	2	no	—
287	C₂³	—	4	2	no	—
288	C₂³	—	4	4	no	—
289	C₂³	—	4	4	no	—
290	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
291	Q₈	—	2	4	no	—
292	Q₈	—	2	4	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	8	4	no	—
296	D₄	—	4	4	no	—
297	D₄	—	4	4	no	—
298	D₄	—	8	4	no	—
299	D₄	—	8	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	#127×C₃₇	1	2	yes	—
313	C₂×C₄	#127×C₃₇	1	2	yes	—
314	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
315	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
316	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
317	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
318	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
319	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
320	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
321	C₈	—	2	8	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	4	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	4	no	—
327	C₂²	C₃₇×#525	1	1	yes	characteristic
328	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
329	C₂²	C₃₇×#537	1	1	yes	characteristic
330	C₂²	C₂²×SD₁₆×C₃₇	1	2	yes	—
331	C₂²	C₃₇×#575	1	2	yes	—
332	C₄	—	2	2	no	—
333	C₄	—	2	4	no	—
334	C₄	—	2	4	no	—
335	C₄	—	4	4	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	C₃₇×#492	1	4	yes	—
339	C₄	C₃₇×#546	1	2	yes	—
340	C₄	C₃₇×#548	1	2	yes	—
341	C₂	—	16	2	no	—
342	C₂	C₃₇×#2540	1	1	yes	characteristic
343	C₂	C₃₇×#3103	1	2	yes	—
344	C₂	C₂×C₃₇×#492	1	1	yes	characteristic
345	C₂	C₃₇×#3153	1	1	yes	characteristic
346	C₂	C₃₇×#3246	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
1. C₂×C₃₇×#475	C₂	semidirect
4. C₄×C₄⋊Q₈×C₃₇	C₂	semidirect
5. C₃₇×#2094	C₂	semidirect
6. C₃₇×#2013	C₂	semidirect
7. C₃₇×#1968	C₂	semidirect
8. C₄×C₄⋊C₈×C₃₇	C₂	semidirect
109. C₄²×C₃₇	SD₁₆	semidirect
148. #28374	C₃₇	direct (N × H)
168. C₂×#475	C₂×C₃₇	semidirect
171. C₄×C₄⋊Q₈	C₂×C₃₇	semidirect
172. #2094	C₂×C₃₇	semidirect
173. #2013	C₂×C₃₇	semidirect
174. #1968	C₂×C₃₇	semidirect
175. C₄×C₄⋊C₈	C₂×C₃₇	semidirect
204. C₃₇	#28374	direct (N × H)
283. C₄²	SD₁₆×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#475	2
C₂×C₃₇×#460	2
C₃₇×#2858	2
C₄×C₄⋊Q₈×C₃₇	2
C₃₇×#2094	2
C₃₇×#2013	2
C₃₇×#1968	2
C₄×C₄⋊C₈×C₃₇	2
#28374	37

Subgroups — C37×#28374 #22649627

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28374 #22649627