Subgroups — C₃₇×#62637 #22683890

Order 9472.

2026 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3584	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₇×#578	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#3417	C₂	1	8	yes	maximal, complemented
4	C₃₇×#3416	C₂	1	8	yes	maximal, complemented
5	C₃₇×#3414	C₂	1	6	yes	maximal, complemented
6	C₂×C₃₇×#510	C₂	1	6	yes	maximal, complemented
7	C₃₇×#3407	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#585	—	2	1	no	—
9	C₂×2₋¹⁺⁴×C₃₇	—	2	1	no	—
10	C₃₇×#578	—	2	4	no	—
11	C₃₇×#578	C₂²	1	8	yes	complemented
12	C₃₇×#578	C₂²	1	4	yes	complemented
13	C₃₇×#577	—	2	4	no	—
14	C₃₇×#576	C₂²	1	4	yes	complemented
15	C₃₇×#575	—	2	3	no	—
16	C₂×#137×C₃₇	—	2	3	no	—
17	C₂×#135×C₃₇	C₂²	1	6	yes	—
18	C₂×#135×C₃₇	C₂²	1	3	yes	complemented
19	C₂²×Q₁₆×C₃₇	C₂²	1	3	yes	complemented
20	C₂×#131×C₃₇	C₂²	1	1	yes	characteristic
21	C₃₇×#510	C₂²	1	12	yes	complemented
22	C₃₇×#510	C₂²	1	24	yes	complemented
23	C₃₇×#510	C₂²	1	12	yes	complemented
24	C₃₇×#509	C₂²	1	4	yes	complemented
25	C₃₇×#509	C₂²	1	12	yes	complemented
26	C₃₇×#508	C₂²	1	24	yes	complemented
27	C₂×Q₃₂×C₃₇	C₂²	1	12	yes	complemented
28	C₂×SD₃₂×C₃₇	C₂²	1	12	yes	complemented
29	C₃₇×#504	C₂²	1	8	yes	complemented
30	C₂×M₃₂×C₃₇	C₂²	1	6	yes	complemented
31	2₋¹⁺⁴×C₃₇	—	2	4	no	—
32	2₋¹⁺⁴×C₃₇	—	2	8	no	—
33	2₋¹⁺⁴×C₃₇	—	4	4	no	—
34	2₋¹⁺⁴×C₃₇	—	4	1	no	—
35	H(5,𝔽₂)×C₃₇	—	2	4	no	—
36	H(5,𝔽₂)×C₃₇	—	4	3	no	—
37	C₂×#41×C₃₇	—	2	6	no	—
38	C₂×#41×C₃₇	—	2	3	no	—
39	C₂×#41×C₃₇	—	4	4	no	—
40	C₂×#41×C₃₇	—	4	3	no	—
41	C₂×#41×C₃₇	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₃₇	—	2	3	no	—
43	C₂²×Q₈×C₃₇	—	4	1	no	—
44	#137×C₃₇	—	2	24	no	—
45	#137×C₃₇	—	2	12	no	—
46	AGL(1,ℤ/8)×C₃₇	—	2	12	no	—
47	#135×C₃₇	—	2	6	no	—
48	#135×C₃₇	—	2	6	no	—
49	#135×C₃₇	—	2	6	no	—
50	#135×C₃₇	C₂³	1	4	yes	—
51	#135×C₃₇	C₂³	1	12	yes	—
52	#135×C₃₇	C₂³	1	12	yes	—
53	C₂×Q₁₆×C₃₇	—	2	3	no	—
54	C₂×Q₁₆×C₃₇	—	2	6	no	—
55	C₂×Q₁₆×C₃₇	C₂³	1	3	yes	—
56	C₂×Q₁₆×C₃₇	C₂³	1	12	yes	—
57	C₂×Q₁₆×C₃₇	C₂³	1	6	yes	—
58	C₂×SD₁₆×C₃₇	—	2	6	no	—
59	C₂×SD₁₆×C₃₇	—	2	3	no	—
60	C₂×D₈×C₃₇	C₂³	1	6	yes	—
61	C₂×D₈×C₃₇	C₂³	1	1	yes	characteristic
62	#131×C₃₇	D₄	1	4	yes	—
63	#131×C₃₇	C₂³	1	4	yes	—
64	C₂×M₁₆×C₃₇	D₄	1	3	yes	—
65	C₂²×C₈×C₃₇	C₂³	1	3	yes	—
66	Q₃₂×C₃₇	C₂³	1	24	yes	—
67	SD₃₂×C₃₇	C₂³	1	24	yes	—
68	SD₃₂×C₃₇	C₂³	1	8	yes	—
69	D₁₆×C₃₇	C₂³	1	8	yes	—
70	M₃₂×C₃₇	C₂³	1	4	yes	—
71	M₃₂×C₃₇	C₂³	1	12	yes	—
72	C₂×C₁₆×C₃₇	C₂³	1	12	yes	—
73	#41×C₃₇	—	2	4	no	—
74	#41×C₃₇	—	2	12	no	—
75	#41×C₃₇	—	2	12	no	—
76	#41×C₃₇	—	4	4	no	—
77	#41×C₃₇	—	4	12	no	—
78	#41×C₃₇	—	4	24	no	—
79	#41×C₃₇	—	4	6	no	—
80	#41×C₃₇	—	4	6	no	—
81	#41×C₃₇	—	4	6	no	—
82	#41×C₃₇	D₈	1	4	yes	—
83	#41×C₃₇	C₂×D₄	1	4	yes	—
84	C₂×Q₈×C₃₇	—	2	3	no	—
85	C₂×Q₈×C₃₇	—	2	6	no	—
86	C₂×Q₈×C₃₇	—	2	12	no	—
87	C₂×Q₈×C₃₇	—	4	8	no	—
88	C₂×Q₈×C₃₇	—	4	4	no	—
89	C₂×Q₈×C₃₇	—	4	3	no	—
90	C₂×Q₈×C₃₇	—	4	6	no	—
91	C₂×Q₈×C₃₇	D₈	1	1	yes	characteristic
92	C₂×D₄×C₃₇	—	2	1	no	—
93	C₂×D₄×C₃₇	—	2	6	no	—
94	C₂×D₄×C₃₇	—	4	3	no	—
95	C₂×D₄×C₃₇	—	4	6	no	—
96	C₂×D₄×C₃₇	—	4	12	no	—
97	C₂×D₄×C₃₇	D₈	1	3	yes	—
98	C₂²×C₄×C₃₇	—	4	3	no	—
99	C₂²×C₄×C₃₇	—	4	3	no	—
100	C₂²×C₄×C₃₇	—	4	6	no	—
101	C₂²×C₄×C₃₇	C₂×D₄	1	3	yes	—
102	Q₁₆×C₃₇	—	2	3	no	—
103	Q₁₆×C₃₇	—	2	12	no	—
104	Q₁₆×C₃₇	—	2	6	no	—
105	Q₁₆×C₃₇	C₂⁴	1	6	yes	—
106	Q₁₆×C₃₇	C₂⁴	1	4	yes	—
107	SD₁₆×C₃₇	—	2	4	no	—
108	SD₁₆×C₃₇	—	2	12	no	—
109	SD₁₆×C₃₇	—	2	12	no	—
110	D₈×C₃₇	—	2	1	no	—
111	D₈×C₃₇	—	2	6	no	—
112	D₈×C₃₇	C₂⁴	1	2	yes	—
113	D₈×C₃₇	C₂⁴	1	4	yes	—
114	M₁₆×C₃₇	C₂×D₄	1	12	yes	—
115	C₂×C₈×C₃₇	C₂×D₄	1	3	yes	—
116	C₂×C₈×C₃₇	C₂×D₄	1	6	yes	—
117	C₂×C₈×C₃₇	C₂⁴	1	6	yes	—
118	C₂×C₈×C₃₇	C₂⁴	1	1	yes	characteristic
119	C₁₆×C₃₇	C₂⁴	1	8	yes	—
120	C₂³×C₃₇	—	4	1	no	—
121	C₂³×C₃₇	—	4	6	no	—
122	C₂³×C₃₇	C₂×D₈	1	3	yes	—
123	Q₈×C₃₇	—	2	4	no	—
124	Q₈×C₃₇	—	2	6	no	—
125	Q₈×C₃₇	—	4	3	no	—
126	Q₈×C₃₇	—	4	12	no	—
127	Q₈×C₃₇	—	4	6	no	—
128	Q₈×C₃₇	C₂×D₈	1	4	yes	—
129	D₄×C₃₇	—	2	4	no	—
130	D₄×C₃₇	—	2	2	no	—
131	D₄×C₃₇	—	4	4	no	—
132	D₄×C₃₇	—	4	12	no	—
133	D₄×C₃₇	—	4	12	no	—
134	D₄×C₃₇	—	4	6	no	—
135	D₄×C₃₇	—	4	1	no	—
136	D₄×C₃₇	C₂×D₈	1	12	yes	—
137	C₂×C₄×C₃₇	—	4	3	no	—
138	C₂×C₄×C₃₇	—	4	6	no	—
139	C₂×C₄×C₃₇	—	4	12	no	—
140	C₂×C₄×C₃₇	—	4	4	no	—
141	C₂×C₄×C₃₇	—	4	12	no	—
142	C₂×C₄×C₃₇	—	4	12	no	—
143	C₂×C₄×C₃₇	C₂×D₈	1	3	yes	—
144	C₂×C₄×C₃₇	C₂×D₈	1	6	yes	—
145	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₃₇	C₂²×D₄	1	6	yes	—
147	C₈×C₃₇	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₃₇	C₂²×D₄	1	6	yes	—
149	C₈×C₃₇	C₂⁵	1	1	yes	characteristic
150	#62637	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂²×C₃₇	—	4	3	no	—
152	C₂²×C₃₇	—	4	2	no	—
153	C₂²×C₃₇	—	4	4	no	—
154	C₂²×C₃₇	—	8	2	no	—
155	C₂²×C₃₇	—	16	6	no	—
156	C₂²×C₃₇	C₂²×D₈	1	1	yes	characteristic
157	C₂²×C₃₇	C₂²×D₈	1	6	yes	—
158	C₄×C₃₇	—	4	4	no	—
159	C₄×C₃₇	—	4	6	no	—
160	C₄×C₃₇	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₃₇	C₂²×D₈	1	6	yes	—
162	C₄×C₃₇	C₂³×D₄	1	1	yes	characteristic
163	#3584	C₂×C₃₇	1	1	yes	characteristic, complemented
164	C₂×#578	C₂×C₃₇	1	1	yes	characteristic, complemented
165	#3417	C₂×C₃₇	1	8	yes	complemented
166	#3416	C₂×C₃₇	1	8	yes	complemented
167	#3414	C₂×C₃₇	1	6	yes	complemented
168	C₂×#510	C₂×C₃₇	1	6	yes	complemented
169	#3407	C₂×C₃₇	1	1	yes	characteristic, complemented
170	C₂×C₃₇	—	2	1	no	—
171	C₂×C₃₇	—	2	6	no	—
172	C₂×C₃₇	—	8	2	no	—
173	C₂×C₃₇	—	8	4	no	—
174	C₂×C₃₇	C₂³×D₈	1	1	yes	characteristic
175	#585	—	2	1	no	—
176	C₂×2₋¹⁺⁴	—	2	1	no	—
177	#578	—	2	4	no	—
178	#578	C₂²×C₃₇	1	4	yes	complemented
179	#578	C₂²×C₃₇	1	8	yes	complemented
180	#577	—	2	4	no	—
181	#576	C₂²×C₃₇	1	4	yes	complemented
182	#575	—	2	3	no	—
183	C₂×#137	—	2	3	no	—
184	C₂×#135	C₂²×C₃₇	1	6	yes	—
185	C₂×#135	C₂²×C₃₇	1	3	yes	complemented
186	C₂²×Q₁₆	C₂²×C₃₇	1	3	yes	complemented
187	C₂×#131	C₂²×C₃₇	1	1	yes	characteristic
188	#510	C₂²×C₃₇	1	12	yes	complemented
189	#510	C₂²×C₃₇	1	24	yes	complemented
190	#510	C₂²×C₃₇	1	12	yes	complemented
191	#509	C₂²×C₃₇	1	4	yes	complemented
192	#509	C₂²×C₃₇	1	12	yes	complemented
193	#508	C₂²×C₃₇	1	24	yes	complemented
194	C₂×Q₃₂	C₂²×C₃₇	1	12	yes	complemented
195	C₂×SD₃₂	C₂²×C₃₇	1	12	yes	complemented
196	#504	C₂²×C₃₇	1	8	yes	complemented
197	C₂×M₃₂	C₂²×C₃₇	1	6	yes	complemented
198	C₃₇	#62637	1	1	yes	characteristic, direct product factor, Sylow
199	2₋¹⁺⁴	—	2	4	no	—
200	2₋¹⁺⁴	—	2	8	no	—
201	2₋¹⁺⁴	—	4	4	no	—
202	2₋¹⁺⁴	—	4	1	no	—
203	H(5,𝔽₂)	—	2	4	no	—
204	H(5,𝔽₂)	—	4	3	no	—
205	C₂×#41	—	2	6	no	—
206	C₂×#41	—	2	3	no	—
207	C₂×#41	—	4	4	no	—
208	C₂×#41	—	4	3	no	—
209	C₂×#41	D₄×C₃₇	1	1	yes	characteristic
210	C₂²×Q₈	—	2	3	no	—
211	C₂²×Q₈	—	4	1	no	—
212	#137	—	2	24	no	—
213	#137	—	2	12	no	—
214	AGL(1,ℤ/8)	—	2	12	no	—
215	#135	—	2	6	no	—
216	#135	—	2	6	no	—
217	#135	—	2	6	no	—
218	#135	C₂³×C₃₇	1	4	yes	—
219	#135	C₂³×C₃₇	1	12	yes	—
220	#135	C₂³×C₃₇	1	12	yes	—
221	C₂×Q₁₆	—	2	3	no	—
222	C₂×Q₁₆	—	2	6	no	—
223	C₂×Q₁₆	C₂³×C₃₇	1	3	yes	—
224	C₂×Q₁₆	C₂³×C₃₇	1	12	yes	—
225	C₂×Q₁₆	C₂³×C₃₇	1	6	yes	—
226	C₂×SD₁₆	—	2	6	no	—
227	C₂×SD₁₆	—	2	3	no	—
228	C₂×D₈	C₂³×C₃₇	1	6	yes	—
229	C₂×D₈	C₂³×C₃₇	1	1	yes	characteristic
230	#131	D₄×C₃₇	1	4	yes	—
231	#131	C₂³×C₃₇	1	4	yes	—
232	C₂×M₁₆	D₄×C₃₇	1	3	yes	—
233	C₂²×C₈	C₂³×C₃₇	1	3	yes	—
234	Q₃₂	C₂³×C₃₇	1	24	yes	—
235	SD₃₂	C₂³×C₃₇	1	24	yes	—
236	SD₃₂	C₂³×C₃₇	1	8	yes	—
237	D₁₆	C₂³×C₃₇	1	8	yes	—
238	M₃₂	C₂³×C₃₇	1	4	yes	—
239	M₃₂	C₂³×C₃₇	1	12	yes	—
240	C₂×C₁₆	C₂³×C₃₇	1	12	yes	—
241	#41	—	2	4	no	—
242	#41	—	2	12	no	—
243	#41	—	2	12	no	—
244	#41	—	4	4	no	—
245	#41	—	4	12	no	—
246	#41	—	4	24	no	—
247	#41	—	4	6	no	—
248	#41	—	4	6	no	—
249	#41	—	4	6	no	—
250	#41	D₈×C₃₇	1	4	yes	—
251	#41	C₂×D₄×C₃₇	1	4	yes	—
252	C₂×Q₈	—	2	3	no	—
253	C₂×Q₈	—	2	6	no	—
254	C₂×Q₈	—	2	12	no	—
255	C₂×Q₈	—	4	8	no	—
256	C₂×Q₈	—	4	4	no	—
257	C₂×Q₈	—	4	3	no	—
258	C₂×Q₈	—	4	6	no	—
259	C₂×Q₈	D₈×C₃₇	1	1	yes	characteristic
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	6	no	—
262	C₂×D₄	—	4	3	no	—
263	C₂×D₄	—	4	6	no	—
264	C₂×D₄	—	4	12	no	—
265	C₂×D₄	D₈×C₃₇	1	3	yes	—
266	C₂²×C₄	—	4	3	no	—
267	C₂²×C₄	—	4	3	no	—
268	C₂²×C₄	—	4	6	no	—
269	C₂²×C₄	C₂×D₄×C₃₇	1	3	yes	—
270	Q₁₆	—	2	3	no	—
271	Q₁₆	—	2	12	no	—
272	Q₁₆	—	2	6	no	—
273	Q₁₆	C₂⁴×C₃₇	1	6	yes	—
274	Q₁₆	C₂⁴×C₃₇	1	4	yes	—
275	SD₁₆	—	2	4	no	—
276	SD₁₆	—	2	12	no	—
277	SD₁₆	—	2	12	no	—
278	D₈	—	2	1	no	—
279	D₈	—	2	6	no	—
280	D₈	C₂⁴×C₃₇	1	2	yes	—
281	D₈	C₂⁴×C₃₇	1	4	yes	—
282	M₁₆	C₂×D₄×C₃₇	1	12	yes	—
283	C₂×C₈	C₂×D₄×C₃₇	1	3	yes	—
284	C₂×C₈	C₂×D₄×C₃₇	1	6	yes	—
285	C₂×C₈	C₂⁴×C₃₇	1	6	yes	—
286	C₂×C₈	C₂⁴×C₃₇	1	1	yes	characteristic
287	C₁₆	C₂⁴×C₃₇	1	8	yes	—
288	C₂³	—	4	1	no	—
289	C₂³	—	4	6	no	—
290	C₂³	C₂×D₈×C₃₇	1	3	yes	—
291	Q₈	—	2	4	no	—
292	Q₈	—	2	6	no	—
293	Q₈	—	4	3	no	—
294	Q₈	—	4	12	no	—
295	Q₈	—	4	6	no	—
296	Q₈	C₂×D₈×C₃₇	1	4	yes	—
297	D₄	—	2	4	no	—
298	D₄	—	2	2	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	12	no	—
301	D₄	—	4	12	no	—
302	D₄	—	4	6	no	—
303	D₄	—	4	1	no	—
304	D₄	C₂×D₈×C₃₇	1	12	yes	—
305	C₂×C₄	—	4	3	no	—
306	C₂×C₄	—	4	6	no	—
307	C₂×C₄	—	4	12	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	12	no	—
310	C₂×C₄	—	4	12	no	—
311	C₂×C₄	C₂×D₈×C₃₇	1	3	yes	—
312	C₂×C₄	C₂×D₈×C₃₇	1	6	yes	—
313	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₃₇	1	6	yes	—
315	C₈	C₂²×D₄×C₃₇	1	1	yes	characteristic
316	C₈	C₂²×D₄×C₃₇	1	6	yes	—
317	C₈	C₂⁵×C₃₇	1	1	yes	characteristic
318	C₂²	—	4	3	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	4	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	16	6	no	—
323	C₂²	C₂²×D₈×C₃₇	1	1	yes	characteristic
324	C₂²	C₂²×D₈×C₃₇	1	6	yes	—
325	C₄	—	4	4	no	—
326	C₄	—	4	6	no	—
327	C₄	C₂²×D₈×C₃₇	1	1	yes	characteristic
328	C₄	C₂²×D₈×C₃₇	1	6	yes	—
329	C₄	C₂³×D₄×C₃₇	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	6	no	—
332	C₂	—	8	2	no	—
333	C₂	—	8	4	no	—
334	C₂	C₂³×D₈×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₇	C₂³×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₃₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₇ ◁ C₂×#41×C₃₇ ◁ C₂×#131×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#3584	C₂	semidirect
2. C₂×C₃₇×#578	C₂	semidirect
3. C₃₇×#3417	C₂	semidirect
4. C₃₇×#3416	C₂	semidirect
5. C₃₇×#3414	C₂	semidirect
6. C₂×C₃₇×#510	C₂	semidirect
7. C₃₇×#3407	C₂	semidirect
11. C₃₇×#578	C₂²	semidirect
12. C₃₇×#578	C₂²	semidirect
14. C₃₇×#576	C₂²	semidirect
18. C₂×#135×C₃₇	C₂²	semidirect
19. C₂²×Q₁₆×C₃₇	C₂²	semidirect
21. C₃₇×#510	C₂²	semidirect
22. C₃₇×#510	C₂²	semidirect
23. C₃₇×#510	C₂²	semidirect
24. C₃₇×#509	C₂²	semidirect
25. C₃₇×#509	C₂²	semidirect
26. C₃₇×#508	C₂²	semidirect
27. C₂×Q₃₂×C₃₇	C₂²	semidirect
28. C₂×SD₃₂×C₃₇	C₂²	semidirect
29. C₃₇×#504	C₂²	semidirect
30. C₂×M₃₂×C₃₇	C₂²	semidirect
150. #62637	C₃₇	direct (N × H)
163. #3584	C₂×C₃₇	semidirect
164. C₂×#578	C₂×C₃₇	semidirect
165. #3417	C₂×C₃₇	semidirect
166. #3416	C₂×C₃₇	semidirect
167. #3414	C₂×C₃₇	semidirect
168. C₂×#510	C₂×C₃₇	semidirect
169. #3407	C₂×C₃₇	semidirect
178. #578	C₂²×C₃₇	semidirect
179. #578	C₂²×C₃₇	semidirect
181. #576	C₂²×C₃₇	semidirect
185. C₂×#135	C₂²×C₃₇	semidirect
186. C₂²×Q₁₆	C₂²×C₃₇	semidirect
188. #510	C₂²×C₃₇	semidirect
189. #510	C₂²×C₃₇	semidirect
190. #510	C₂²×C₃₇	semidirect
191. #509	C₂²×C₃₇	semidirect
192. #509	C₂²×C₃₇	semidirect
193. #508	C₂²×C₃₇	semidirect
194. C₂×Q₃₂	C₂²×C₃₇	semidirect
195. C₂×SD₃₂	C₂²×C₃₇	semidirect
196. #504	C₂²×C₃₇	semidirect
197. C₂×M₃₂	C₂²×C₃₇	semidirect
198. C₃₇	#62637	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#3584	2
C₂×C₃₇×#578	2
C₃₇×#3417	2
C₃₇×#3416	2
C₃₇×#3414	2
C₂×C₃₇×#510	2
C₃₇×#3407	2
#62637	37

Subgroups — C37×#62637 #22683890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#62637 #22683890