Subgroups — C₃₇×#28260 #22649513

Order 9472.

990 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	C₂×C₃₇×#460	C₂	1	2	yes	maximal
2	C₂×C₃₇×#461	C₂	1	2	yes	maximal, complemented
3	C₃₇×#2328	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2327	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#2055	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1843	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#1805	C₂	1	2	yes	maximal, complemented
8	C₃₇×#1804	C₂	1	2	yes	maximal, complemented
9	C₂²×SD₁₆×C₃₇	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×#127×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₃₇	C₂²	1	2	yes	—
14	C₃₇×#461	—	2	8	no	—
15	C₃₇×#460	—	2	8	no	—
16	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	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	C₃₇×#391	C₂²	1	2	yes	—
22	C₃₇×#390	C₂²	1	2	yes	—
23	C₃₇×#387	—	2	4	no	—
24	C₃₇×#386	—	2	4	no	—
25	C₃₇×#382	C₂²	1	1	yes	characteristic
26	C₃₇×#382	C₂²	1	1	yes	characteristic
27	C₃₇×#376	C₂²	1	1	yes	characteristic
28	C₃₇×#340	C₂²	1	4	yes	—
29	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
30	C₂²×D₄×C₃₇	—	2	2	no	—
31	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
32	C₂³×C₄×C₃₇	—	2	2	no	—
33	C₂×SD₁₆×C₃₇	—	2	8	no	—
34	C₂×SD₁₆×C₃₇	—	2	8	no	—
35	C₂×SD₁₆×C₃₇	—	4	4	no	—
36	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
37	C₄⋊Q₈×C₃₇	—	2	2	no	—
38	C₄⋊Q₈×C₃₇	D₄	1	4	yes	complemented
39	#127×C₃₇	—	2	2	no	—
40	#127×C₃₇	D₄	1	4	yes	—
41	C₄×Q₈×C₃₇	—	2	4	no	—
42	C₄×Q₈×C₃₇	—	2	4	no	—
43	C₄×D₄×C₃₇	—	2	4	no	—
44	C₄×D₄×C₃₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
51	C₂×C₄²×C₃₇	—	2	2	no	—
52	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₃₇	—	2	4	no	—
56	C₄⋊C₈×C₃₇	—	2	4	no	—
57	C₄⋊C₈×C₃₇	—	4	2	no	—
58	#103×C₃₇	—	2	8	no	—
59	#102×C₃₇	—	2	8	no	—
60	#95×C₃₇	—	2	2	no	—
61	#95×C₃₇	—	2	2	no	—
62	#95×C₃₇	—	2	4	no	—
63	#95×C₃₇	D₄	1	4	yes	—
64	C₂⁴×C₃₇	—	2	2	no	—
65	C₂×Q₈×C₃₇	—	2	2	no	—
66	C₂×Q₈×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	C₂×D₄	1	4	yes	—
71	C₂×D₄×C₃₇	—	2	8	no	—
72	C₂×D₄×C₃₇	—	2	2	no	—
73	C₂×D₄×C₃₇	—	2	2	no	—
74	C₂×D₄×C₃₇	—	2	2	no	—
75	C₂×D₄×C₃₇	—	2	2	no	—
76	C₂×D₄×C₃₇	—	2	2	no	—
77	C₂×D₄×C₃₇	—	8	4	no	—
78	C₂×D₄×C₃₇	C₂×D₄	1	4	yes	—
79	C₂²×C₄×C₃₇	—	2	2	no	—
80	C₂²×C₄×C₃₇	—	2	2	no	—
81	C₂²×C₄×C₃₇	—	2	2	no	—
82	C₂²×C₄×C₃₇	—	4	4	no	—
83	C₂²×C₄×C₃₇	—	8	4	no	—
84	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
88	SD₁₆×C₃₇	—	4	16	no	—
89	C₂×C₈×C₃₇	—	2	4	no	—
90	C₂×C₈×C₃₇	—	2	4	no	—
91	C₂×C₈×C₃₇	—	2	4	no	—
92	C₄⋊C₄×C₃₇	—	2	8	no	—
93	C₄⋊C₄×C₃₇	—	2	2	no	—
94	C₄⋊C₄×C₃₇	—	2	2	no	—
95	C₄⋊C₄×C₃₇	—	2	2	no	—
96	C₄⋊C₄×C₃₇	—	2	2	no	—
97	C₄⋊C₄×C₃₇	—	4	4	no	—
98	C₂²⋊C₄×C₃₇	—	4	4	no	—
99	C₂²⋊C₄×C₃₇	—	8	4	no	—
100	C₄²×C₃₇	—	2	1	no	—
101	C₄²×C₃₇	—	2	1	no	—
102	C₄²×C₃₇	—	4	4	no	—
103	C₄²×C₃₇	C₂×D₄	1	2	yes	—
104	C₄²×C₃₇	C₂×D₄	1	2	yes	—
105	C₄²×C₃₇	#41	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	4	no	—
114	Q₈×C₃₇	—	2	4	no	—
115	Q₈×C₃₇	—	4	2	no	—
116	Q₈×C₃₇	—	4	2	no	—
117	D₄×C₃₇	—	2	4	no	—
118	D₄×C₃₇	—	2	4	no	—
119	D₄×C₃₇	—	4	2	no	—
120	D₄×C₃₇	—	4	2	no	—
121	D₄×C₃₇	—	8	8	no	—
122	C₂×C₄×C₃₇	—	2	4	no	—
123	C₂×C₄×C₃₇	—	2	4	no	—
124	C₂×C₄×C₃₇	—	2	4	no	—
125	C₂×C₄×C₃₇	—	2	2	no	—
126	C₂×C₄×C₃₇	—	2	4	no	—
127	C₂×C₄×C₃₇	—	4	2	no	—
128	C₂×C₄×C₃₇	—	4	2	no	—
129	C₂×C₄×C₃₇	—	4	2	no	—
130	C₂×C₄×C₃₇	—	4	2	no	—
131	C₂×C₄×C₃₇	—	4	4	no	—
132	C₂×C₄×C₃₇	—	4	4	no	—
133	C₂×C₄×C₃₇	—	8	8	no	—
134	C₂×C₄×C₃₇	#121	1	4	yes	—
135	C₂×C₄×C₃₇	#121	1	4	yes	—
136	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
142	C₈×C₃₇	—	4	4	no	—
143	#28260	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	C₂×#121	1	1	yes	characteristic
151	C₂²×C₃₇	#537	1	1	yes	characteristic
152	C₂²×C₃₇	#577	1	4	yes	—
153	C₄×C₃₇	—	2	4	no	—
154	C₄×C₃₇	—	2	4	no	—
155	C₄×C₃₇	—	4	2	no	—
156	C₄×C₃₇	—	4	4	no	—
157	C₄×C₃₇	—	8	2	no	—
158	C₄×C₃₇	—	8	2	no	—
159	C₄×C₃₇	D₄²	1	1	yes	characteristic
160	C₄×C₃₇	#547	1	1	yes	characteristic
161	C₄×C₃₇	#547	1	1	yes	characteristic
162	C₄×C₃₇	#555	1	1	yes	characteristic
163	C₂×#460	C₂×C₃₇	1	2	yes	—
164	C₂×#461	C₂×C₃₇	1	2	yes	complemented
165	#2328	C₂×C₃₇	1	1	yes	characteristic
166	#2327	C₂×C₃₇	1	1	yes	characteristic, complemented
167	#2055	C₂×C₃₇	1	4	yes	complemented
168	#1843	C₂×C₃₇	1	1	yes	characteristic, complemented
169	#1805	C₂×C₃₇	1	2	yes	complemented
170	#1804	C₂×C₃₇	1	2	yes	complemented
171	C₂×C₃₇	—	8	4	no	—
172	C₂×C₃₇	#2537	1	1	yes	characteristic
173	C₂×C₃₇	#3042	1	2	yes	—
174	C₂×C₃₇	#3043	1	2	yes	—
175	C₂×C₃₇	#3260	1	2	yes	—
176	C₂²×SD₁₆	C₂²×C₃₇	1	4	yes	—
177	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
178	C₂×#127	C₂²×C₃₇	1	1	yes	characteristic
179	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
180	C₂×C₄×D₄	C₂²×C₃₇	1	2	yes	—
181	#461	—	2	8	no	—
182	#460	—	2	8	no	—
183	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
184	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
185	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
186	C₂×#103	C₂²×C₃₇	1	4	yes	—
187	C₂×#102	C₂²×C₃₇	1	4	yes	—
188	#391	C₂²×C₃₇	1	2	yes	—
189	#390	C₂²×C₃₇	1	2	yes	—
190	#387	—	2	4	no	—
191	#386	—	2	4	no	—
192	#382	C₂²×C₃₇	1	1	yes	characteristic
193	#382	C₂²×C₃₇	1	1	yes	characteristic
194	#376	C₂²×C₃₇	1	1	yes	characteristic
195	#340	C₂²×C₃₇	1	4	yes	—
196	C₃₇	#28260	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
198	C₂²×D₄	—	2	2	no	—
199	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
200	C₂³×C₄	—	2	2	no	—
201	C₂×SD₁₆	—	2	8	no	—
202	C₂×SD₁₆	—	2	8	no	—
203	C₂×SD₁₆	—	4	4	no	—
204	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	D₄×C₃₇	1	4	yes	complemented
207	#127	—	2	2	no	—
208	#127	D₄×C₃₇	1	4	yes	—
209	C₄×Q₈	—	2	4	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₄×D₄	—	2	4	no	—
212	C₄×D₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
216	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₂²⋊C₄	—	2	4	no	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	2	4	no	—
225	C₄⋊C₈	—	4	2	no	—
226	#103	—	2	8	no	—
227	#102	—	2	8	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	4	no	—
231	#95	D₄×C₃₇	1	4	yes	—
232	C₂⁴	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	C₂×D₄×C₃₇	1	4	yes	—
239	C₂×D₄	—	2	8	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	8	4	no	—
246	C₂×D₄	C₂×D₄×C₃₇	1	4	yes	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	4	4	no	—
251	C₂²×C₄	—	8	4	no	—
252	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
254	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
255	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
256	SD₁₆	—	4	16	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₂²⋊C₄	—	4	4	no	—
267	C₂²⋊C₄	—	8	4	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	4	4	no	—
271	C₄²	C₂×D₄×C₃₇	1	2	yes	—
272	C₄²	C₂×D₄×C₃₇	1	2	yes	—
273	C₄²	#41×C₃₇	1	2	yes	—
274	C₄²	#41×C₃₇	1	2	yes	—
275	C₂³	—	2	4	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	4	no	—
279	C₂³	—	4	4	no	—
280	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
281	Q₈	—	2	4	no	—
282	Q₈	—	2	4	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	4	2	no	—
285	D₄	—	2	4	no	—
286	D₄	—	2	4	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	8	8	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	8	8	no	—
302	C₂×C₄	#121×C₃₇	1	4	yes	—
303	C₂×C₄	#121×C₃₇	1	4	yes	—
304	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×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	—	4	4	no	—
311	C₂²	—	4	8	no	—
312	C₂²	—	4	4	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
317	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
318	C₂²	C₃₇×#537	1	1	yes	characteristic
319	C₂²	C₃₇×#577	1	4	yes	—
320	C₄	—	2	4	no	—
321	C₄	—	2	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	4	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	D₄²×C₃₇	1	1	yes	characteristic
327	C₄	C₃₇×#547	1	1	yes	characteristic
328	C₄	C₃₇×#547	1	1	yes	characteristic
329	C₄	C₃₇×#555	1	1	yes	characteristic
330	C₂	—	8	4	no	—
331	C₂	C₃₇×#2537	1	1	yes	characteristic
332	C₂	C₃₇×#3042	1	2	yes	—
333	C₂	C₃₇×#3043	1	2	yes	—
334	C₂	C₃₇×#3260	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 ◁ C₃₇×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₇×#461	C₂	semidirect
4. C₃₇×#2327	C₂	semidirect
5. C₃₇×#2055	C₂	semidirect
6. C₃₇×#1843	C₂	semidirect
7. C₃₇×#1805	C₂	semidirect
8. C₃₇×#1804	C₂	semidirect
38. C₄⋊Q₈×C₃₇	D₄	semidirect
143. #28260	C₃₇	direct (N × H)
164. C₂×#461	C₂×C₃₇	semidirect
166. #2327	C₂×C₃₇	semidirect
167. #2055	C₂×C₃₇	semidirect
168. #1843	C₂×C₃₇	semidirect
169. #1805	C₂×C₃₇	semidirect
170. #1804	C₂×C₃₇	semidirect
196. C₃₇	#28260	direct (N × H)
206. C₄⋊Q₈	D₄×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×#460	2
C₂×C₃₇×#461	2
C₃₇×#2328	2
C₃₇×#2327	2
C₃₇×#2055	2
C₃₇×#1843	2
C₃₇×#1805	2
C₃₇×#1804	2
#28260	37

Subgroups — C37×#28260 #22649513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28260 #22649513