Subgroups — C₃₇×#28267 #22649520

Order 9472.

862 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2580	C₂	1	2	yes	maximal, complemented
4	C₃₇×#2065	C₂	1	4	yes	maximal, complemented
5	C₃₇×#1970	C₂	1	2	yes	maximal
6	C₃₇×#1942	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#1804	C₂	1	2	yes	maximal, complemented
9	C₂²×D₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₃₇	C₂²	1	2	yes	—
12	C₈×Q₈×C₃₇	—	2	4	no	—
13	C₄×D₈×C₃₇	—	2	4	no	—
14	C₂×#107×C₃₇	C₂²	1	2	yes	—
15	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×#106×C₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
19	C₂×#102×C₃₇	C₂²	1	2	yes	—
20	C₂×#102×C₃₇	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₃₇×#399	—	2	2	no	—
24	C₃₇×#393	—	2	2	no	—
25	C₃₇×#390	C₂²	1	4	yes	—
26	C₃₇×#389	C₂²	1	4	yes	—
27	C₃₇×#385	—	2	4	no	—
28	C₃₇×#382	C₂²	1	2	yes	—
29	C₃₇×#340	C₂²	1	4	yes	—
30	C₂²×Q₈×C₃₇	D₄	1	1	yes	characteristic
31	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
32	C₂³×C₄×C₃₇	—	2	2	no	—
33	C₂×D₈×C₃₇	—	2	2	no	—
34	C₂×D₈×C₃₇	—	2	2	no	—
35	C₂×D₈×C₃₇	Q₈	1	4	yes	complemented
36	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
39	C₄×Q₈×C₃₇	—	2	2	no	—
40	C₄×Q₈×C₃₇	—	2	2	no	—
41	C₄×D₄×C₃₇	—	2	4	no	—
42	C₄×D₄×C₃₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
49	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
51	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
53	#107×C₃₇	—	2	2	no	—
54	#107×C₃₇	—	2	4	no	—
55	#106×C₃₇	—	4	2	no	—
56	C₄⋊C₈×C₃₇	—	2	4	no	—
57	C₄⋊C₈×C₃₇	—	2	2	no	—
58	#102×C₃₇	—	2	4	no	—
59	#102×C₃₇	—	4	4	no	—
60	C₄×C₈×C₃₇	—	2	4	no	—
61	C₄×C₈×C₃₇	D₄	1	4	yes	—
62	#95×C₃₇	—	2	2	no	—
63	#95×C₃₇	—	2	2	no	—
64	#95×C₃₇	—	2	4	no	—
65	C₂⁴×C₃₇	—	2	2	no	—
66	C₂×Q₈×C₃₇	—	2	2	no	—
67	C₂×Q₈×C₃₇	—	2	4	no	—
68	C₂×D₄×C₃₇	—	2	2	no	—
69	C₂×D₄×C₃₇	—	2	2	no	—
70	C₂×D₄×C₃₇	—	2	2	no	—
71	C₂×D₄×C₃₇	—	2	2	no	—
72	C₂×D₄×C₃₇	—	2	2	no	—
73	C₂×D₄×C₃₇	C₂×Q₈	1	4	yes	—
74	C₂²×C₄×C₃₇	—	2	4	no	—
75	C₂²×C₄×C₃₇	—	2	2	no	—
76	C₂²×C₄×C₃₇	—	4	4	no	—
77	C₂²×C₄×C₃₇	—	8	4	no	—
78	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
83	D₈×C₃₇	—	2	8	no	—
84	C₂×C₈×C₃₇	—	2	1	no	—
85	C₂×C₈×C₃₇	—	2	1	no	—
86	C₂×C₈×C₃₇	—	2	2	no	—
87	C₂×C₈×C₃₇	—	2	2	no	—
88	C₂×C₈×C₃₇	—	2	2	no	—
89	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₃₇	#41	1	2	yes	—
93	C₄⋊C₄×C₃₇	—	2	4	no	—
94	C₄⋊C₄×C₃₇	—	2	2	no	—
95	C₄⋊C₄×C₃₇	—	2	2	no	—
96	C₄⋊C₄×C₃₇	—	2	4	no	—
97	C₄⋊C₄×C₃₇	—	2	2	no	—
98	C₄⋊C₄×C₃₇	—	2	4	no	—
99	C₄⋊C₄×C₃₇	—	8	4	no	—
100	C₂²⋊C₄×C₃₇	—	4	4	no	—
101	C₂²⋊C₄×C₃₇	—	4	8	no	—
102	C₂²⋊C₄×C₃₇	—	8	4	no	—
103	C₄²×C₃₇	—	2	2	no	—
104	C₄²×C₃₇	—	2	2	no	—
105	C₄²×C₃₇	C₂×D₄	1	2	yes	—
106	C₄²×C₃₇	#41	1	2	yes	—
107	C₂³×C₃₇	—	2	4	no	—
108	C₂³×C₃₇	—	4	2	no	—
109	C₂³×C₃₇	—	4	2	no	—
110	C₂³×C₃₇	—	4	4	no	—
111	C₂³×C₃₇	—	4	4	no	—
112	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₃₇	—	2	8	no	—
114	D₄×C₃₇	—	2	4	no	—
115	D₄×C₃₇	—	2	4	no	—
116	D₄×C₃₇	—	4	2	no	—
117	D₄×C₃₇	—	4	2	no	—
118	C₂×C₄×C₃₇	—	2	2	no	—
119	C₂×C₄×C₃₇	—	2	2	no	—
120	C₂×C₄×C₃₇	—	2	2	no	—
121	C₂×C₄×C₃₇	—	2	2	no	—
122	C₂×C₄×C₃₇	—	2	2	no	—
123	C₂×C₄×C₃₇	—	2	2	no	—
124	C₂×C₄×C₃₇	—	2	2	no	—
125	C₂×C₄×C₃₇	—	4	2	no	—
126	C₂×C₄×C₃₇	—	4	4	no	—
127	C₂×C₄×C₃₇	—	4	4	no	—
128	C₂×C₄×C₃₇	—	4	4	no	—
129	C₂×C₄×C₃₇	—	4	4	no	—
130	C₂×C₄×C₃₇	—	8	8	no	—
131	C₂×C₄×C₃₇	#121	1	2	yes	—
132	C₂×C₄×C₃₇	#121	1	2	yes	—
133	C₂×C₄×C₃₇	#135	1	4	yes	—
134	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
140	C₈×C₃₇	—	2	2	no	—
141	C₈×C₃₇	—	4	2	no	—
142	C₈×C₃₇	C₂²⋊Q₈	1	4	yes	—
143	#28267	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₃₇	—	4	8	no	—
145	C₂²×C₃₇	—	4	4	no	—
146	C₂²×C₃₇	—	8	4	no	—
147	C₂²×C₃₇	—	8	2	no	—
148	C₂²×C₃₇	—	8	2	no	—
149	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
150	C₂²×C₃₇	D₄×Q₈	1	1	yes	characteristic
151	C₂²×C₃₇	#550	1	1	yes	characteristic
152	C₂²×C₃₇	C₂×#135	1	2	yes	—
153	C₂²×C₃₇	#576	1	2	yes	—
154	C₄×C₃₇	—	2	2	no	—
155	C₄×C₃₇	—	2	4	no	—
156	C₄×C₃₇	—	2	2	no	—
157	C₄×C₃₇	—	4	2	no	—
158	C₄×C₃₇	—	8	4	no	—
159	C₄×C₃₇	—	8	2	no	—
160	C₄×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₄×C₃₇	#537	1	1	yes	characteristic
162	C₄×C₃₇	#554	1	1	yes	characteristic
163	C₄×C₃₇	#556	1	1	yes	characteristic
164	C₂×C₈×Q₈	C₂×C₃₇	1	1	yes	characteristic, complemented
165	C₂×C₄×D₈	C₂×C₃₇	1	1	yes	characteristic
166	#2580	C₂×C₃₇	1	2	yes	complemented
167	#2065	C₂×C₃₇	1	4	yes	complemented
168	#1970	C₂×C₃₇	1	2	yes	—
169	#1942	C₂×C₃₇	1	2	yes	complemented
170	#1846	C₂×C₃₇	1	1	yes	characteristic, complemented
171	#1804	C₂×C₃₇	1	2	yes	complemented
172	C₂×C₃₇	—	8	4	no	—
173	C₂×C₃₇	#2538	1	1	yes	characteristic
174	C₂×C₃₇	#3050	1	1	yes	characteristic
175	C₂×C₃₇	#3058	1	1	yes	characteristic
176	C₂×C₃₇	#3380	1	2	yes	—
177	C₂×C₃₇	#3396	1	2	yes	—
178	C₂²×D₈	C₂²×C₃₇	1	1	yes	characteristic
179	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
180	C₂×C₄×D₄	C₂²×C₃₇	1	2	yes	—
181	C₈×Q₈	—	2	4	no	—
182	C₄×D₈	—	2	4	no	—
183	C₂×#107	C₂²×C₃₇	1	2	yes	—
184	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
185	C₂×#106	C₂²×C₃₇	1	2	yes	—
186	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
187	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
188	C₂×#102	C₂²×C₃₇	1	2	yes	—
189	C₂×#102	C₂²×C₃₇	1	4	yes	—
190	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
191	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
192	#399	—	2	2	no	—
193	#393	—	2	2	no	—
194	#390	C₂²×C₃₇	1	4	yes	—
195	#389	C₂²×C₃₇	1	4	yes	—
196	#385	—	2	4	no	—
197	#382	C₂²×C₃₇	1	2	yes	—
198	#340	C₂²×C₃₇	1	4	yes	—
199	C₃₇	#28267	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²×Q₈	D₄×C₃₇	1	1	yes	characteristic
201	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
202	C₂³×C₄	—	2	2	no	—
203	C₂×D₈	—	2	2	no	—
204	C₂×D₈	—	2	2	no	—
205	C₂×D₈	Q₈×C₃₇	1	4	yes	complemented
206	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
207	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×D₄	—	2	4	no	—
212	C₄×D₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
215	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
217	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
218	C₂×C₂²⋊C₄	—	2	2	no	—
219	C₂×C₂²⋊C₄	—	2	4	no	—
220	C₂×C₂²⋊C₄	—	2	4	no	—
221	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
222	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
223	#107	—	2	2	no	—
224	#107	—	2	4	no	—
225	#106	—	4	2	no	—
226	C₄⋊C₈	—	2	4	no	—
227	C₄⋊C₈	—	2	2	no	—
228	#102	—	2	4	no	—
229	#102	—	4	4	no	—
230	C₄×C₈	—	2	4	no	—
231	C₄×C₈	D₄×C₃₇	1	4	yes	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	—	2	4	no	—
235	C₂⁴	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	C₂×Q₈×C₃₇	1	4	yes	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	8	4	no	—
248	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
252	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
253	D₈	—	2	8	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
260	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
261	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
262	C₂×C₈	#41×C₃₇	1	2	yes	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	8	4	no	—
270	C₂²⋊C₄	—	4	4	no	—
271	C₂²⋊C₄	—	4	8	no	—
272	C₂²⋊C₄	—	8	4	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	C₂×D₄×C₃₇	1	2	yes	—
276	C₄²	#41×C₃₇	1	2	yes	—
277	C₂³	—	2	4	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	4	no	—
281	C₂³	—	4	4	no	—
282	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
283	Q₈	—	2	8	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	4	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	8	no	—
301	C₂×C₄	#121×C₃₇	1	2	yes	—
302	C₂×C₄	#121×C₃₇	1	2	yes	—
303	C₂×C₄	#135×C₃₇	1	4	yes	—
304	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
305	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
308	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
309	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	4	2	no	—
312	C₈	C₂²⋊Q₈×C₃₇	1	4	yes	—
313	C₂²	—	4	8	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	8	4	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
319	C₂²	D₄×Q₈×C₃₇	1	1	yes	characteristic
320	C₂²	C₃₇×#550	1	1	yes	characteristic
321	C₂²	C₂×#135×C₃₇	1	2	yes	—
322	C₂²	C₃₇×#576	1	2	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	2	4	no	—
325	C₄	—	2	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	8	4	no	—
328	C₄	—	8	2	no	—
329	C₄	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
330	C₄	C₃₇×#537	1	1	yes	characteristic
331	C₄	C₃₇×#554	1	1	yes	characteristic
332	C₄	C₃₇×#556	1	1	yes	characteristic
333	C₂	—	8	4	no	—
334	C₂	C₃₇×#2538	1	1	yes	characteristic
335	C₂	C₃₇×#3050	1	1	yes	characteristic
336	C₂	C₃₇×#3058	1	1	yes	characteristic
337	C₂	C₃₇×#3380	1	2	yes	—
338	C₂	C₃₇×#3396	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂²×D₈×C₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄×Q₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×Q₈×C₃₇	C₂	semidirect
3. C₃₇×#2580	C₂	semidirect
4. C₃₇×#2065	C₂	semidirect
6. C₃₇×#1942	C₂	semidirect
7. C₃₇×#1846	C₂	semidirect
8. C₃₇×#1804	C₂	semidirect
35. C₂×D₈×C₃₇	Q₈	semidirect
143. #28267	C₃₇	direct (N × H)
164. C₂×C₈×Q₈	C₂×C₃₇	semidirect
166. #2580	C₂×C₃₇	semidirect
167. #2065	C₂×C₃₇	semidirect
169. #1942	C₂×C₃₇	semidirect
170. #1846	C₂×C₃₇	semidirect
171. #1804	C₂×C₃₇	semidirect
199. C₃₇	#28267	direct (N × H)
205. C₂×D₈	Q₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₃₇	2
C₂×C₄×D₈×C₃₇	2
C₃₇×#2580	2
C₃₇×#2065	2
C₃₇×#1970	2
C₃₇×#1942	2
C₃₇×#1846	2
C₃₇×#1804	2
#28267	37

Subgroups — C37×#28267 #22649520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28267 #22649520