Subgroups — C₃₇×#28488 #22649741

Order 9472.

1366 conjugacy classes of subgroups.

The subgroup lattice has 332 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×#469	C₂	1	4	yes	maximal, complemented
3	C₂×C₃₇×#436	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×#106×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	C₂²×M₁₆×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×#469	—	2	16	no	—
17	C₃₇×#436	—	2	4	no	—
18	C₂×#106×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	C₂×M₁₆×C₃₇	—	2	4	no	—
38	C₂×M₁₆×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	#106×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	4	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	C₂²×C₄×C₃₇	—	2	4	no	—
78	C₂²×C₄×C₃₇	—	2	2	no	—
79	C₂²×C₄×C₃₇	—	2	4	no	—
80	C₂²×C₄×C₃₇	—	2	2	no	—
81	C₂²×C₄×C₃₇	—	2	4	no	—
82	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
86	D₈×C₃₇	—	4	8	no	—
87	D₈×C₃₇	—	4	8	no	—
88	M₁₆×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	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
95	C₄⋊C₄×C₃₇	—	2	2	no	—
96	C₄⋊C₄×C₃₇	—	2	4	no	—
97	C₄⋊C₄×C₃₇	C₂×D₄	1	8	yes	—
98	C₂²⋊C₄×C₃₇	—	2	4	no	—
99	C₂²⋊C₄×C₃₇	—	4	16	no	—
100	C₄²×C₃₇	—	4	4	no	—
101	C₄²×C₃₇	C₂×D₄	1	2	yes	—
102	C₄²×C₃₇	#41	1	2	yes	—
103	C₂³×C₃₇	—	2	2	no	—
104	C₂³×C₃₇	—	2	4	no	—
105	C₂³×C₃₇	—	2	4	no	—
106	C₂³×C₃₇	—	2	4	no	—
107	C₂³×C₃₇	—	4	4	no	—
108	C₂³×C₃₇	—	4	8	no	—
109	C₂³×C₃₇	—	4	4	no	—
110	C₂³×C₃₇	—	4	4	no	—
111	C₂³×C₃₇	—	4	8	no	—
112	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
113	D₄×C₃₇	—	4	4	no	—
114	D₄×C₃₇	—	4	4	no	—
115	D₄×C₃₇	—	4	8	no	—
116	D₄×C₃₇	—	4	4	no	—
117	D₄×C₃₇	—	8	8	no	—
118	D₄×C₃₇	—	8	32	no	—
119	D₄×C₃₇	—	8	8	no	—
120	C₂×C₄×C₃₇	—	2	2	no	—
121	C₂×C₄×C₃₇	—	2	2	no	—
122	C₂×C₄×C₃₇	—	2	4	no	—
123	C₂×C₄×C₃₇	—	2	8	no	—
124	C₂×C₄×C₃₇	—	4	4	no	—
125	C₂×C₄×C₃₇	—	4	4	no	—
126	C₂×C₄×C₃₇	—	4	2	no	—
127	C₂×C₄×C₃₇	—	4	2	no	—
128	C₂×C₄×C₃₇	—	4	4	no	—
129	C₂×C₄×C₃₇	#121	1	2	yes	—
130	C₂×C₄×C₃₇	#121	1	2	yes	—
131	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
132	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
134	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
136	C₈×C₃₇	—	2	2	no	—
137	C₈×C₃₇	—	4	2	no	—
138	C₈×C₃₇	#127	1	4	yes	complemented
139	#28488	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂²×C₃₇	—	4	8	no	—
141	C₂²×C₃₇	—	4	2	no	—
142	C₂²×C₃₇	—	4	2	no	—
143	C₂²×C₃₇	—	4	2	no	—
144	C₂²×C₃₇	—	8	8	no	—
145	C₂²×C₃₇	—	8	8	no	—
146	C₂²×C₃₇	—	8	4	no	—
147	C₂²×C₃₇	—	8	4	no	—
148	C₂²×C₃₇	—	8	4	no	—
149	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
150	C₂²×C₃₇	D₄²	1	2	yes	—
151	C₂²×C₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
152	C₂²×C₃₇	#576	1	2	yes	—
153	C₄×C₃₇	—	2	2	no	—
154	C₄×C₃₇	—	4	8	no	—
155	C₄×C₃₇	—	4	2	no	—
156	C₄×C₃₇	#469	1	4	yes	complemented
157	C₄×C₃₇	C₂×#127	1	1	yes	characteristic
158	C₄×C₃₇	#535	1	1	yes	characteristic
159	C₄×C₃₇	#550	1	2	yes	—
160	C₂×#493	C₂×C₃₇	1	1	yes	characteristic, complemented
161	C₂×#469	C₂×C₃₇	1	4	yes	complemented
162	C₂×#436	C₂×C₃₇	1	1	yes	characteristic, complemented
163	#2541	C₂×C₃₇	1	2	yes	complemented
164	#1999	C₂×C₃₇	1	4	yes	complemented
165	#1982	C₂×C₃₇	1	2	yes	complemented
166	C₄×#106	C₂×C₃₇	1	1	yes	characteristic, complemented
167	C₂×C₃₇	—	8	2	no	—
168	C₂×C₃₇	—	16	4	no	—
169	C₂×C₃₇	#2541	1	1	yes	characteristic
170	C₂×C₃₇	C₂×#469	1	1	yes	characteristic
171	C₂×C₃₇	#3058	1	1	yes	characteristic
172	C₂×C₃₇	#3274	1	4	yes	—
173	C₂²×D₈	C₂²×C₃₇	1	2	yes	—
174	C₂²×D₈	C₂²×C₃₇	1	2	yes	—
175	C₂²×M₁₆	C₂²×C₃₇	1	2	yes	—
176	C₂×#127	—	2	2	no	—
177	C₂×#127	C₂²×C₃₇	1	2	yes	—
178	C₂×#121	C₂²×C₃₇	1	8	yes	—
179	C₂×C₄×D₄	C₂²×C₃₇	1	1	yes	characteristic
180	#493	—	2	4	no	—
181	#469	—	2	16	no	—
182	#436	—	2	4	no	—
183	C₂×#106	C₂²×C₃₇	1	2	yes	—
184	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
185	C₂×#102	C₂²×C₃₇	1	8	yes	—
186	C₂×C₂²⋊C₈	C₂²×C₃₇	1	2	yes	—
187	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
188	#392	—	2	4	no	—
189	#390	—	2	2	no	—
190	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
191	#340	C₂²×C₃₇	1	2	yes	—
192	C₃₇	#28488	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²×D₄	—	2	8	no	—
194	C₂²×D₄	—	2	2	no	—
195	C₂²×D₄	—	2	2	no	—
196	C₂²×D₄	D₄×C₃₇	1	1	yes	characteristic
197	C₂²×D₄	C₂³×C₃₇	1	4	yes	—
198	C₂³×C₄	C₂³×C₃₇	1	2	yes	—
199	C₂×D₈	—	2	8	no	—
200	C₂×D₈	—	2	4	no	—
201	C₂×D₈	—	2	4	no	—
202	C₂×D₈	—	4	4	no	—
203	C₂×M₁₆	—	2	4	no	—
204	C₂×M₁₆	—	2	8	no	—
205	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
206	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
208	#127	—	2	4	no	—
209	#127	—	2	4	no	—
210	#127	—	4	8	no	—
211	#121	—	2	16	no	—
212	#121	—	4	8	no	—
213	C₄×D₄	—	2	2	no	—
214	C₄×D₄	—	2	2	no	—
215	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
217	C₂×C₂²⋊C₄	—	2	8	no	—
218	C₂×C₂²⋊C₄	D₄×C₃₇	1	2	yes	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
221	#106	D₄×C₃₇	1	8	yes	complemented
222	C₄⋊C₈	—	2	2	no	—
223	#102	—	2	16	no	—
224	C₂²⋊C₈	—	2	4	no	—
225	C₄×C₈	D₄×C₃₇	1	4	yes	complemented
226	#95	—	2	2	no	—
227	C₂⁴	—	2	4	no	—
228	C₂⁴	C₂×D₄×C₃₇	1	2	yes	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	8	no	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	8	no	—
235	C₂×D₄	—	4	16	no	—
236	C₂×D₄	—	4	16	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	4	4	no	—
239	C₂×D₄	—	4	4	no	—
240	C₂×D₄	—	4	4	no	—
241	C₂×D₄	—	8	4	no	—
242	C₂×D₄	—	8	8	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	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	C₂⁴×C₃₇	1	1	yes	characteristic
252	D₈	—	4	8	no	—
253	D₈	—	4	8	no	—
254	M₁₆	—	4	8	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
259	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
260	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	C₂×D₄×C₃₇	1	8	yes	—
264	C₂²⋊C₄	—	2	4	no	—
265	C₂²⋊C₄	—	4	16	no	—
266	C₄²	—	4	4	no	—
267	C₄²	C₂×D₄×C₃₇	1	2	yes	—
268	C₄²	#41×C₃₇	1	2	yes	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	4	no	—
271	C₂³	—	2	4	no	—
272	C₂³	—	2	4	no	—
273	C₂³	—	4	4	no	—
274	C₂³	—	4	8	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	4	8	no	—
278	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
279	D₄	—	4	4	no	—
280	D₄	—	4	4	no	—
281	D₄	—	4	8	no	—
282	D₄	—	4	4	no	—
283	D₄	—	8	8	no	—
284	D₄	—	8	32	no	—
285	D₄	—	8	8	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	#121×C₃₇	1	2	yes	—
296	C₂×C₄	#121×C₃₇	1	2	yes	—
297	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	4	yes	—
298	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
300	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
301	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
302	C₈	—	2	2	no	—
303	C₈	—	4	2	no	—
304	C₈	#127×C₃₇	1	4	yes	complemented
305	C₂²	—	4	8	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	8	8	no	—
310	C₂²	—	8	8	no	—
311	C₂²	—	8	4	no	—
312	C₂²	—	8	4	no	—
313	C₂²	—	8	4	no	—
314	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
315	C₂²	D₄²×C₃₇	1	2	yes	—
316	C₂²	C₂×AGL(1,ℤ/8)×C₃₇	1	2	yes	—
317	C₂²	C₃₇×#576	1	2	yes	—
318	C₄	—	2	2	no	—
319	C₄	—	4	8	no	—
320	C₄	—	4	2	no	—
321	C₄	C₃₇×#469	1	4	yes	complemented
322	C₄	C₂×#127×C₃₇	1	1	yes	characteristic
323	C₄	C₃₇×#535	1	1	yes	characteristic
324	C₄	C₃₇×#550	1	2	yes	—
325	C₂	—	8	2	no	—
326	C₂	—	16	4	no	—
327	C₂	C₃₇×#2541	1	1	yes	characteristic
328	C₂	C₂×C₃₇×#469	1	1	yes	characteristic
329	C₂	C₃₇×#3058	1	1	yes	characteristic
330	C₂	C₃₇×#3274	1	4	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×#469	C₂	semidirect
3. C₂×C₃₇×#436	C₂	semidirect
4. C₃₇×#2541	C₂	semidirect
5. C₃₇×#1999	C₂	semidirect
6. C₃₇×#1982	C₂	semidirect
7. C₄×#106×C₃₇	C₂	semidirect
55. #106×C₃₇	D₄	semidirect
59. C₄×C₈×C₃₇	D₄	semidirect
138. C₈×C₃₇	#127	semidirect
139. #28488	C₃₇	direct (N × H)
156. C₄×C₃₇	#469	semidirect
160. C₂×#493	C₂×C₃₇	semidirect
161. C₂×#469	C₂×C₃₇	semidirect
162. C₂×#436	C₂×C₃₇	semidirect
163. #2541	C₂×C₃₇	semidirect
164. #1999	C₂×C₃₇	semidirect
165. #1982	C₂×C₃₇	semidirect
166. C₄×#106	C₂×C₃₇	semidirect
192. C₃₇	#28488	direct (N × H)
221. #106	D₄×C₃₇	semidirect
225. C₄×C₈	D₄×C₃₇	semidirect
304. C₈	#127×C₃₇	semidirect
321. C₄	C₃₇×#469	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×#469	2
C₂×C₃₇×#436	2
C₃₇×#2541	2
C₃₇×#1999	2
C₃₇×#1982	2
C₄×#106×C₃₇	2
#28488	37

Subgroups — C37×#28488 #22649741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28488 #22649741