Subgroups — C₃₇×#28485 #22649738

Order 9472.

1398 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₇×#466	C₂	1	4	yes	maximal, complemented
3	C₂×C₈×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#2541	C₂	1	2	yes	maximal, complemented
5	C₃₇×#1999	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1982	C₂	1	2	yes	maximal, complemented
7	C₄×#107×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₃₇	C₂²	1	2	yes	—
9	C₂²×D₈×C₃₇	C₂²	1	2	yes	—
10	C₂³×C₈×C₃₇	C₂²	1	2	yes	—
11	C₂×#127×C₃₇	—	2	2	no	—
12	C₂×#127×C₃₇	C₂²	1	2	yes	—
13	C₂×#121×C₃₇	C₂²	1	8	yes	—
14	C₂×C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
15	C₃₇×#493	—	2	4	no	—
16	C₃₇×#466	—	2	16	no	—
17	C₈×D₄×C₃₇	—	2	4	no	—
18	C₂×#107×C₃₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×#102×C₃₇	C₂²	1	8	yes	—
21	C₂×C₂²⋊C₈×C₃₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₃₇×#392	—	2	4	no	—
24	C₃₇×#390	—	2	2	no	—
25	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
26	C₃₇×#340	C₂²	1	2	yes	—
27	C₂²×D₄×C₃₇	—	2	8	no	—
28	C₂²×D₄×C₃₇	—	2	2	no	—
29	C₂²×D₄×C₃₇	—	2	2	no	—
30	C₂²×D₄×C₃₇	D₄	1	1	yes	characteristic
31	C₂²×D₄×C₃₇	C₂³	1	4	yes	—
32	C₂³×C₄×C₃₇	C₂³	1	2	yes	—
33	C₂×D₈×C₃₇	—	2	8	no	—
34	C₂×D₈×C₃₇	—	2	4	no	—
35	C₂×D₈×C₃₇	—	2	4	no	—
36	C₂×D₈×C₃₇	—	4	4	no	—
37	C₂²×C₈×C₃₇	—	2	4	no	—
38	C₂²×C₈×C₃₇	—	2	8	no	—
39	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
42	#127×C₃₇	—	2	4	no	—
43	#127×C₃₇	—	2	4	no	—
44	#127×C₃₇	—	4	8	no	—
45	#121×C₃₇	—	2	16	no	—
46	#121×C₃₇	—	4	8	no	—
47	C₄×D₄×C₃₇	—	2	2	no	—
48	C₄×D₄×C₃₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
51	C₂×C₂²⋊C₄×C₃₇	—	2	8	no	—
52	C₂×C₂²⋊C₄×C₃₇	D₄	1	2	yes	—
53	C₂×C₄²×C₃₇	—	2	2	no	—
54	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
55	#107×C₃₇	D₄	1	8	yes	complemented
56	C₄⋊C₈×C₃₇	—	2	2	no	—
57	#102×C₃₇	—	2	16	no	—
58	C₂²⋊C₈×C₃₇	—	2	4	no	—
59	C₄×C₈×C₃₇	D₄	1	4	yes	complemented
60	#95×C₃₇	—	2	2	no	—
61	C₂⁴×C₃₇	—	2	4	no	—
62	C₂⁴×C₃₇	C₂×D₄	1	2	yes	—
63	C₂×D₄×C₃₇	—	2	2	no	—
64	C₂×D₄×C₃₇	—	2	2	no	—
65	C₂×D₄×C₃₇	—	2	8	no	—
66	C₂×D₄×C₃₇	—	2	4	no	—
67	C₂×D₄×C₃₇	—	2	4	no	—
68	C₂×D₄×C₃₇	—	2	8	no	—
69	C₂×D₄×C₃₇	—	4	16	no	—
70	C₂×D₄×C₃₇	—	4	16	no	—
71	C₂×D₄×C₃₇	—	4	4	no	—
72	C₂×D₄×C₃₇	—	4	4	no	—
73	C₂×D₄×C₃₇	—	4	4	no	—
74	C₂×D₄×C₃₇	—	4	4	no	—
75	C₂×D₄×C₃₇	—	8	4	no	—
76	C₂×D₄×C₃₇	—	8	8	no	—
77	C₂×D₄×C₃₇	D₈	1	4	yes	complemented
78	C₂²×C₄×C₃₇	—	2	4	no	—
79	C₂²×C₄×C₃₇	—	2	2	no	—
80	C₂²×C₄×C₃₇	—	2	4	no	—
81	C₂²×C₄×C₃₇	—	2	2	no	—
82	C₂²×C₄×C₃₇	—	2	4	no	—
83	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
87	D₈×C₃₇	—	4	8	no	—
88	D₈×C₃₇	—	4	8	no	—
89	C₂×C₈×C₃₇	—	2	2	no	—
90	C₂×C₈×C₃₇	—	2	2	no	—
91	C₂×C₈×C₃₇	—	2	4	no	—
92	C₂×C₈×C₃₇	—	4	8	no	—
93	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
96	C₄⋊C₄×C₃₇	—	2	2	no	—
97	C₄⋊C₄×C₃₇	—	2	4	no	—
98	C₄⋊C₄×C₃₇	C₂×D₄	1	8	yes	—
99	C₂²⋊C₄×C₃₇	—	2	4	no	—
100	C₂²⋊C₄×C₃₇	—	4	16	no	—
101	C₄²×C₃₇	—	4	4	no	—
102	C₄²×C₃₇	C₂×D₄	1	2	yes	—
103	C₄²×C₃₇	#41	1	2	yes	—
104	C₂³×C₃₇	—	2	2	no	—
105	C₂³×C₃₇	—	2	4	no	—
106	C₂³×C₃₇	—	2	2	no	—
107	C₂³×C₃₇	—	2	4	no	—
108	C₂³×C₃₇	—	4	4	no	—
109	C₂³×C₃₇	—	4	8	no	—
110	C₂³×C₃₇	—	4	4	no	—
111	C₂³×C₃₇	—	4	4	no	—
112	C₂³×C₃₇	—	4	8	no	—
113	C₂³×C₃₇	C₂×D₈	1	4	yes	—
114	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
115	D₄×C₃₇	—	2	8	no	—
116	D₄×C₃₇	—	4	4	no	—
117	D₄×C₃₇	—	4	4	no	—
118	D₄×C₃₇	—	4	8	no	—
119	D₄×C₃₇	—	8	8	no	—
120	D₄×C₃₇	—	8	32	no	—
121	D₄×C₃₇	—	8	8	no	—
122	C₂×C₄×C₃₇	—	2	2	no	—
123	C₂×C₄×C₃₇	—	2	2	no	—
124	C₂×C₄×C₃₇	—	2	4	no	—
125	C₂×C₄×C₃₇	—	2	8	no	—
126	C₂×C₄×C₃₇	—	4	4	no	—
127	C₂×C₄×C₃₇	—	4	4	no	—
128	C₂×C₄×C₃₇	—	4	2	no	—
129	C₂×C₄×C₃₇	—	4	2	no	—
130	C₂×C₄×C₃₇	—	4	4	no	—
131	C₂×C₄×C₃₇	#121	1	2	yes	—
132	C₂×C₄×C₃₇	#121	1	2	yes	—
133	C₂×C₄×C₃₇	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₃₇	#135	1	2	yes	—
135	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
137	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
139	C₈×C₃₇	—	2	2	no	—
140	C₈×C₃₇	—	4	2	no	—
141	C₈×C₃₇	#127	1	4	yes	complemented
142	#28485	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₃₇	—	2	8	no	—
144	C₂²×C₃₇	—	2	4	no	—
145	C₂²×C₃₇	—	4	4	no	—
146	C₂²×C₃₇	—	4	2	no	—
147	C₂²×C₃₇	—	4	2	no	—
148	C₂²×C₃₇	—	8	8	no	—
149	C₂²×C₃₇	—	8	8	no	—
150	C₂²×C₃₇	—	8	4	no	—
151	C₂²×C₃₇	—	8	4	no	—
152	C₂²×C₃₇	—	8	4	no	—
153	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
154	C₂²×C₃₇	D₄²	1	2	yes	—
155	C₂²×C₃₇	C₂²×D₈	1	1	yes	characteristic
156	C₂²×C₃₇	C₂×#135	1	1	yes	characteristic
157	C₂²×C₃₇	#576	1	2	yes	—
158	C₄×C₃₇	—	2	2	no	—
159	C₄×C₃₇	—	4	8	no	—
160	C₄×C₃₇	—	4	2	no	—
161	C₄×C₃₇	#466	1	4	yes	complemented
162	C₄×C₃₇	C₂×#127	1	1	yes	characteristic
163	C₄×C₃₇	#535	1	1	yes	characteristic
164	C₄×C₃₇	#550	1	2	yes	—
165	C₂×#493	C₂×C₃₇	1	1	yes	characteristic, complemented
166	C₂×#466	C₂×C₃₇	1	4	yes	complemented
167	C₂×C₈×D₄	C₂×C₃₇	1	1	yes	characteristic, complemented
168	#2541	C₂×C₃₇	1	2	yes	complemented
169	#1999	C₂×C₃₇	1	4	yes	complemented
170	#1982	C₂×C₃₇	1	2	yes	complemented
171	C₄×#107	C₂×C₃₇	1	1	yes	characteristic, complemented
172	C₂×C₃₇	—	4	4	no	—
173	C₂×C₃₇	—	16	4	no	—
174	C₂×C₃₇	#2541	1	1	yes	characteristic
175	C₂×C₃₇	C₂×#466	1	1	yes	characteristic
176	C₂×C₃₇	#3058	1	1	yes	characteristic
177	C₂×C₃₇	D₄×D₈	1	2	yes	—
178	C₂×C₃₇	#3287	1	2	yes	—
179	C₂²×D₈	C₂²×C₃₇	1	2	yes	—
180	C₂²×D₈	C₂²×C₃₇	1	2	yes	—
181	C₂³×C₈	C₂²×C₃₇	1	2	yes	—
182	C₂×#127	—	2	2	no	—
183	C₂×#127	C₂²×C₃₇	1	2	yes	—
184	C₂×#121	C₂²×C₃₇	1	8	yes	—
185	C₂×C₄×D₄	C₂²×C₃₇	1	1	yes	characteristic
186	#493	—	2	4	no	—
187	#466	—	2	16	no	—
188	C₈×D₄	—	2	4	no	—
189	C₂×#107	C₂²×C₃₇	1	2	yes	—
190	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
191	C₂×#102	C₂²×C₃₇	1	8	yes	—
192	C₂×C₂²⋊C₈	C₂²×C₃₇	1	2	yes	—
193	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
194	#392	—	2	4	no	—
195	#390	—	2	2	no	—
196	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
197	#340	C₂²×C₃₇	1	2	yes	—
198	C₃₇	#28485	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×D₄	—	2	8	no	—
200	C₂²×D₄	—	2	2	no	—
201	C₂²×D₄	—	2	2	no	—
202	C₂²×D₄	D₄×C₃₇	1	1	yes	characteristic
203	C₂²×D₄	C₂³×C₃₇	1	4	yes	—
204	C₂³×C₄	C₂³×C₃₇	1	2	yes	—
205	C₂×D₈	—	2	8	no	—
206	C₂×D₈	—	2	4	no	—
207	C₂×D₈	—	2	4	no	—
208	C₂×D₈	—	4	4	no	—
209	C₂²×C₈	—	2	4	no	—
210	C₂²×C₈	—	2	8	no	—
211	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
212	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
214	#127	—	2	4	no	—
215	#127	—	2	4	no	—
216	#127	—	4	8	no	—
217	#121	—	2	16	no	—
218	#121	—	4	8	no	—
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	—	2	2	no	—
221	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
222	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
223	C₂×C₂²⋊C₄	—	2	8	no	—
224	C₂×C₂²⋊C₄	D₄×C₃₇	1	2	yes	—
225	C₂×C₄²	—	2	2	no	—
226	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
227	#107	D₄×C₃₇	1	8	yes	complemented
228	C₄⋊C₈	—	2	2	no	—
229	#102	—	2	16	no	—
230	C₂²⋊C₈	—	2	4	no	—
231	C₄×C₈	D₄×C₃₇	1	4	yes	complemented
232	#95	—	2	2	no	—
233	C₂⁴	—	2	4	no	—
234	C₂⁴	C₂×D₄×C₃₇	1	2	yes	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	8	no	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	8	no	—
241	C₂×D₄	—	4	16	no	—
242	C₂×D₄	—	4	16	no	—
243	C₂×D₄	—	4	4	no	—
244	C₂×D₄	—	4	4	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂×D₄	—	8	4	no	—
248	C₂×D₄	—	8	8	no	—
249	C₂×D₄	D₈×C₃₇	1	4	yes	complemented
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
256	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
258	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
259	D₈	—	4	8	no	—
260	D₈	—	4	8	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	4	8	no	—
265	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
267	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	C₂×D₄×C₃₇	1	8	yes	—
271	C₂²⋊C₄	—	2	4	no	—
272	C₂²⋊C₄	—	4	16	no	—
273	C₄²	—	4	4	no	—
274	C₄²	C₂×D₄×C₃₇	1	2	yes	—
275	C₄²	#41×C₃₇	1	2	yes	—
276	C₂³	—	2	2	no	—
277	C₂³	—	2	4	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	4	no	—
280	C₂³	—	4	4	no	—
281	C₂³	—	4	8	no	—
282	C₂³	—	4	4	no	—
283	C₂³	—	4	4	no	—
284	C₂³	—	4	8	no	—
285	C₂³	C₂×D₈×C₃₇	1	4	yes	—
286	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
287	D₄	—	2	8	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	8	no	—
291	D₄	—	8	8	no	—
292	D₄	—	8	32	no	—
293	D₄	—	8	8	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	#121×C₃₇	1	2	yes	—
304	C₂×C₄	#121×C₃₇	1	2	yes	—
305	C₂×C₄	C₂×D₈×C₃₇	1	2	yes	—
306	C₂×C₄	#135×C₃₇	1	2	yes	—
307	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
309	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
310	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
311	C₈	—	2	2	no	—
312	C₈	—	4	2	no	—
313	C₈	#127×C₃₇	1	4	yes	complemented
314	C₂²	—	2	8	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	4	4	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	8	8	no	—
320	C₂²	—	8	8	no	—
321	C₂²	—	8	4	no	—
322	C₂²	—	8	4	no	—
323	C₂²	—	8	4	no	—
324	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
325	C₂²	D₄²×C₃₇	1	2	yes	—
326	C₂²	C₂²×D₈×C₃₇	1	1	yes	characteristic
327	C₂²	C₂×#135×C₃₇	1	1	yes	characteristic
328	C₂²	C₃₇×#576	1	2	yes	—
329	C₄	—	2	2	no	—
330	C₄	—	4	8	no	—
331	C₄	—	4	2	no	—
332	C₄	C₃₇×#466	1	4	yes	complemented
333	C₄	C₂×#127×C₃₇	1	1	yes	characteristic
334	C₄	C₃₇×#535	1	1	yes	characteristic
335	C₄	C₃₇×#550	1	2	yes	—
336	C₂	—	4	4	no	—
337	C₂	—	16	4	no	—
338	C₂	C₃₇×#2541	1	1	yes	characteristic
339	C₂	C₂×C₃₇×#466	1	1	yes	characteristic
340	C₂	C₃₇×#3058	1	1	yes	characteristic
341	C₂	D₄×D₈×C₃₇	1	2	yes	—
342	C₂	C₃₇×#3287	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×#493	C₂	semidirect
2. C₂×C₃₇×#466	C₂	semidirect
3. C₂×C₈×D₄×C₃₇	C₂	semidirect
4. C₃₇×#2541	C₂	semidirect
5. C₃₇×#1999	C₂	semidirect
6. C₃₇×#1982	C₂	semidirect
7. C₄×#107×C₃₇	C₂	semidirect
55. #107×C₃₇	D₄	semidirect
59. C₄×C₈×C₃₇	D₄	semidirect
77. C₂×D₄×C₃₇	D₈	semidirect
141. C₈×C₃₇	#127	semidirect
142. #28485	C₃₇	direct (N × H)
161. C₄×C₃₇	#466	semidirect
165. C₂×#493	C₂×C₃₇	semidirect
166. C₂×#466	C₂×C₃₇	semidirect
167. C₂×C₈×D₄	C₂×C₃₇	semidirect
168. #2541	C₂×C₃₇	semidirect
169. #1999	C₂×C₃₇	semidirect
170. #1982	C₂×C₃₇	semidirect
171. C₄×#107	C₂×C₃₇	semidirect
198. C₃₇	#28485	direct (N × H)
227. #107	D₄×C₃₇	semidirect
231. C₄×C₈	D₄×C₃₇	semidirect
249. C₂×D₄	D₈×C₃₇	semidirect
313. C₈	#127×C₃₇	semidirect
332. C₄	C₃₇×#466	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#493	2
C₂×C₃₇×#466	2
C₂×C₈×D₄×C₃₇	2
C₃₇×#2541	2
C₃₇×#1999	2
C₃₇×#1982	2
C₄×#107×C₃₇	2
#28485	37

Subgroups — C37×#28485 #22649738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28485 #22649738