Subgroups — C₂×C₃₇×#1675 #22640357

Order 9472.

726 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₇×#476	C₂	1	2	yes	maximal
3	C₂×C₈×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#1675	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₇×#333	C₂	1	2	yes	maximal, complemented
7	C₂³×C₈×C₃₇	—	2	1	no	—
8	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
9	C₂×C₄×D₄×C₃₇	C₂²	1	1	yes	characteristic
10	C₃₇×#500	—	2	2	no	—
11	C₃₇×#500	C₂²	1	4	yes	complemented
12	C₃₇×#476	—	2	4	no	—
13	C₃₇×#476	C₂²	1	8	yes	—
14	C₈×D₄×C₃₇	—	2	2	no	—
15	C₈×D₄×C₃₇	C₂²	1	4	yes	—
16	C₂×#107×C₃₇	—	2	1	no	—
17	C₂×#107×C₃₇	—	2	1	no	—
18	C₂×#106×C₃₇	—	2	2	no	—
19	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
21	C₂×#102×C₃₇	—	2	2	no	—
22	C₂×C₂²⋊C₈×C₃₇	—	2	1	no	—
23	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
24	C₃₇×#335	C₂²	1	4	yes	complemented
25	C₃₇×#333	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₃₇	—	2	2	no	—
27	C₂²×D₄×C₃₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₇	—	2	1	no	—
29	C₂²×C₈×C₃₇	—	2	1	no	—
30	C₂²×C₈×C₃₇	—	2	2	no	—
31	C₂²×C₈×C₃₇	—	2	4	no	—
32	C₂²×C₈×C₃₇	—	4	4	no	—
33	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
35	C₄⋊Q₈×C₃₇	—	2	2	no	—
36	C₄⋊Q₈×C₃₇	—	2	2	no	—
37	C₄⋊Q₈×C₃₇	—	2	2	no	—
38	C₄⋊Q₈×C₃₇	C₂³	1	4	yes	—
39	C₄×D₄×C₃₇	—	2	1	no	—
40	C₄×D₄×C₃₇	—	2	1	no	—
41	C₄×D₄×C₃₇	D₄	1	2	yes	—
42	C₄×D₄×C₃₇	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₃₇	—	2	1	no	—
47	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
48	#107×C₃₇	—	2	4	no	—
49	#107×C₃₇	—	2	4	no	—
50	#106×C₃₇	—	2	8	no	—
51	C₄⋊C₈×C₃₇	—	2	1	no	—
52	C₄⋊C₈×C₃₇	—	2	2	no	—
53	C₄⋊C₈×C₃₇	C₂³	1	2	yes	—
54	C₄⋊C₈×C₃₇	C₂³	1	4	yes	—
55	#102×C₃₇	—	2	8	no	—
56	C₂²⋊C₈×C₃₇	—	2	4	no	—
57	C₄×C₈×C₃₇	D₄	1	2	yes	—
58	C₄×C₈×C₃₇	C₂³	1	2	yes	—
59	C₂⁴×C₃₇	—	2	1	no	—
60	C₂×Q₈×C₃₇	—	2	4	no	—
61	C₂×Q₈×C₃₇	—	4	2	no	—
62	C₂×Q₈×C₃₇	—	8	4	no	—
63	C₂×D₄×C₃₇	—	2	1	no	—
64	C₂×D₄×C₃₇	—	2	2	no	—
65	C₂×D₄×C₃₇	Q₁₆	1	4	yes	—
66	C₂×D₄×C₃₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₇	—	2	1	no	—
68	C₂²×C₄×C₃₇	—	2	2	no	—
69	C₂²×C₄×C₃₇	—	2	2	no	—
70	C₂²×C₄×C₃₇	—	4	2	no	—
71	C₂²×C₄×C₃₇	—	4	2	no	—
72	C₂²×C₄×C₃₇	—	4	2	no	—
73	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₇	#41	1	1	yes	characteristic
76	C₂×C₈×C₃₇	—	2	2	no	—
77	C₂×C₈×C₃₇	—	2	2	no	—
78	C₂×C₈×C₃₇	—	2	4	no	—
79	C₂×C₈×C₃₇	—	4	2	no	—
80	C₂×C₈×C₃₇	—	4	4	no	—
81	C₂×C₈×C₃₇	—	4	8	no	—
82	C₂×C₈×C₃₇	SD₁₆	1	4	yes	—
83	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
85	C₄⋊C₄×C₃₇	—	2	1	no	—
86	C₄⋊C₄×C₃₇	—	2	4	no	—
87	C₄⋊C₄×C₃₇	—	2	4	no	—
88	C₄⋊C₄×C₃₇	—	4	2	no	—
89	C₄⋊C₄×C₃₇	—	4	2	no	—
90	C₄⋊C₄×C₃₇	C₂×D₄	1	2	yes	—
91	C₂²⋊C₄×C₃₇	—	2	2	no	—
92	C₂²⋊C₄×C₃₇	—	2	2	no	—
93	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₃₇	#41	1	1	yes	characteristic
96	C₄²×C₃₇	C₂⁴	1	1	yes	characteristic
97	C₂³×C₃₇	—	2	2	no	—
98	C₂³×C₃₇	—	2	4	no	—
99	C₂³×C₃₇	—	4	2	no	—
100	C₂³×C₃₇	—	4	2	no	—
101	C₂³×C₃₇	#121	1	1	yes	characteristic
102	Q₈×C₃₇	—	8	4	no	—
103	Q₈×C₃₇	—	8	4	no	—
104	D₄×C₃₇	—	2	2	no	—
105	D₄×C₃₇	—	2	4	no	—
106	D₄×C₃₇	C₂×Q₁₆	1	4	yes	—
107	C₂×C₄×C₃₇	—	2	2	no	—
108	C₂×C₄×C₃₇	—	2	2	no	—
109	C₂×C₄×C₃₇	—	4	2	no	—
110	C₂×C₄×C₃₇	—	4	2	no	—
111	C₂×C₄×C₃₇	—	4	2	no	—
112	C₂×C₄×C₃₇	—	4	4	no	—
113	C₂×C₄×C₃₇	—	4	4	no	—
114	C₂×C₄×C₃₇	—	8	4	no	—
115	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
116	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
118	C₂×C₄×C₃₇	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
120	C₂×C₄×C₃₇	#135	1	2	yes	—
121	C₂×C₄×C₃₇	#137	1	2	yes	—
122	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
125	C₈×C₃₇	—	2	2	no	—
126	C₈×C₃₇	—	4	2	no	—
127	C₈×C₃₇	—	4	4	no	—
128	C₈×C₃₇	C₂×SD₁₆	1	4	yes	—
129	C₂×#1675	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₃₇	—	2	4	no	—
131	C₂²×C₃₇	—	4	8	no	—
132	C₂²×C₃₇	—	4	2	no	—
133	C₂²×C₃₇	—	4	2	no	—
134	C₂²×C₃₇	#461	1	2	yes	—
135	C₂²×C₃₇	#467	1	2	yes	—
136	C₂²×C₃₇	#472	1	2	yes	—
137	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
138	C₄×C₃₇	—	2	1	no	—
139	C₄×C₃₇	—	2	1	no	—
140	C₄×C₃₇	—	4	2	no	—
141	C₄×C₃₇	—	8	4	no	—
142	C₄×C₃₇	#461	1	2	yes	—
143	C₄×C₃₇	#467	1	2	yes	—
144	C₄×C₃₇	C₂²×SD₁₆	1	1	yes	characteristic
145	C₄×C₃₇	C₂²×Q₁₆	1	1	yes	characteristic
146	C₄×C₃₇	C₂×#135	1	1	yes	characteristic
147	C₄×C₃₇	C₂×#137	1	1	yes	characteristic
148	C₂×#500	C₂×C₃₇	1	1	yes	characteristic, complemented
149	C₂×#476	C₂×C₃₇	1	2	yes	—
150	C₂×C₈×D₄	C₂×C₃₇	1	1	yes	characteristic
151	#1675	C₂×C₃₇	1	8	yes	direct product factor
152	C₂×#335	C₂×C₃₇	1	1	yes	characteristic, complemented
153	C₂×#333	C₂×C₃₇	1	2	yes	complemented
154	C₂×C₃₇	—	4	4	no	—
155	C₂×C₃₇	#1675	1	4	yes	direct product factor
156	C₂×C₃₇	C₂×#461	1	1	yes	characteristic
157	C₂×C₃₇	C₂×#467	1	1	yes	characteristic
158	C₂×C₃₇	C₂×#472	1	1	yes	characteristic
159	C₂³×C₈	—	2	1	no	—
160	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
161	C₂×C₄×D₄	C₂²×C₃₇	1	1	yes	characteristic
162	#500	—	2	2	no	—
163	#500	C₂²×C₃₇	1	4	yes	complemented
164	#476	—	2	4	no	—
165	#476	C₂²×C₃₇	1	8	yes	—
166	C₈×D₄	—	2	2	no	—
167	C₈×D₄	C₂²×C₃₇	1	4	yes	—
168	C₂×#107	—	2	1	no	—
169	C₂×#107	—	2	1	no	—
170	C₂×#106	—	2	2	no	—
171	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
172	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
173	C₂×#102	—	2	2	no	—
174	C₂×C₂²⋊C₈	—	2	1	no	—
175	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
176	#335	C₂²×C₃₇	1	4	yes	complemented
177	#333	C₂²×C₃₇	1	8	yes	complemented
178	C₃₇	C₂×#1675	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²×Q₈	—	2	2	no	—
180	C₂²×D₄	D₄×C₃₇	1	1	yes	characteristic
181	C₂³×C₄	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	4	no	—
185	C₂²×C₈	—	4	4	no	—
186	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
187	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	C₂³×C₃₇	1	4	yes	—
192	C₄×D₄	—	2	1	no	—
193	C₄×D₄	—	2	1	no	—
194	C₄×D₄	D₄×C₃₇	1	2	yes	—
195	C₄×D₄	C₂³×C₃₇	1	2	yes	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
199	C₂×C₂²⋊C₄	—	2	1	no	—
200	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
201	#107	—	2	4	no	—
202	#107	—	2	4	no	—
203	#106	—	2	8	no	—
204	C₄⋊C₈	—	2	1	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	C₂³×C₃₇	1	2	yes	—
207	C₄⋊C₈	C₂³×C₃₇	1	4	yes	—
208	#102	—	2	8	no	—
209	C₂²⋊C₈	—	2	4	no	—
210	C₄×C₈	D₄×C₃₇	1	2	yes	—
211	C₄×C₈	C₂³×C₃₇	1	2	yes	—
212	C₂⁴	—	2	1	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	4	2	no	—
215	C₂×Q₈	—	8	4	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	Q₁₆×C₃₇	1	4	yes	—
219	C₂×D₄	C₂×D₄×C₃₇	1	2	yes	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
228	C₂²×C₄	#41×C₃₇	1	1	yes	characteristic
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	4	no	—
234	C₂×C₈	—	4	8	no	—
235	C₂×C₈	SD₁₆×C₃₇	1	4	yes	—
236	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
237	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
238	C₄⋊C₄	—	2	1	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	C₂×D₄×C₃₇	1	2	yes	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	2	2	no	—
246	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
247	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
248	C₄²	#41×C₃₇	1	1	yes	characteristic
249	C₄²	C₂⁴×C₃₇	1	1	yes	characteristic
250	C₂³	—	2	2	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	#121×C₃₇	1	1	yes	characteristic
255	Q₈	—	8	4	no	—
256	Q₈	—	8	4	no	—
257	D₄	—	2	2	no	—
258	D₄	—	2	4	no	—
259	D₄	C₂×Q₁₆×C₃₇	1	4	yes	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
269	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
270	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
271	C₂×C₄	C₂×SD₁₆×C₃₇	1	2	yes	—
272	C₂×C₄	C₂×Q₁₆×C₃₇	1	2	yes	—
273	C₂×C₄	#135×C₃₇	1	2	yes	—
274	C₂×C₄	#137×C₃₇	1	2	yes	—
275	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
277	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₈	—	4	4	no	—
281	C₈	C₂×SD₁₆×C₃₇	1	4	yes	—
282	C₂²	—	2	4	no	—
283	C₂²	—	4	8	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	C₃₇×#461	1	2	yes	—
287	C₂²	C₃₇×#467	1	2	yes	—
288	C₂²	C₃₇×#472	1	2	yes	—
289	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	2	no	—
293	C₄	—	8	4	no	—
294	C₄	C₃₇×#461	1	2	yes	—
295	C₄	C₃₇×#467	1	2	yes	—
296	C₄	C₂²×SD₁₆×C₃₇	1	1	yes	characteristic
297	C₄	C₂²×Q₁₆×C₃₇	1	1	yes	characteristic
298	C₄	C₂×#135×C₃₇	1	1	yes	characteristic
299	C₄	C₂×#137×C₃₇	1	1	yes	characteristic
300	C₂	—	4	4	no	—
301	C₂	C₃₇×#1675	1	4	yes	direct product factor
302	C₂	C₂×C₃₇×#461	1	1	yes	characteristic
303	C₂	C₂×C₃₇×#467	1	1	yes	characteristic
304	C₂	C₂×C₃₇×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₇
prime radical	C₂²×D₄×C₃₇	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×#500	C₂	semidirect
4. C₃₇×#1675	C₂	direct (N × H)
5. C₂×C₃₇×#335	C₂	semidirect
6. C₂×C₃₇×#333	C₂	semidirect
11. C₃₇×#500	C₂²	semidirect
24. C₃₇×#335	C₂²	semidirect
25. C₃₇×#333	C₂²	semidirect
129. C₂×#1675	C₃₇	direct (N × H)
148. C₂×#500	C₂×C₃₇	semidirect
151. #1675	C₂×C₃₇	direct (N × H)
152. C₂×#335	C₂×C₃₇	semidirect
153. C₂×#333	C₂×C₃₇	semidirect
155. C₂×C₃₇	#1675	direct (N × H)
163. #500	C₂²×C₃₇	semidirect
176. #335	C₂²×C₃₇	semidirect
177. #333	C₂²×C₃₇	semidirect
178. C₃₇	C₂×#1675	direct (N × H)
301. C₂	C₃₇×#1675	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#500	2
C₂×C₃₇×#476	2
C₂×C₈×D₄×C₃₇	2
C₃₇×#1675	2
C₂×C₃₇×#335	2
C₂×C₃₇×#333	2
C₂×#1675	37

Subgroups — C2×C37×#1675 #22640357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×#1675 #22640357