Subgroups — C₃₇×#28813 #22650066

Order 9472.

822 conjugacy classes of subgroups.

The subgroup lattice has 308 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	2	yes	maximal
2	C₃₇×#2551	C₂	1	2	yes	maximal
3	C₃₇×#2057	C₂	1	4	yes	maximal
4	C₃₇×#1947	C₂	1	2	yes	maximal
5	C₃₇×#1942	C₂	1	2	yes	maximal
6	C₃₇×#1845	C₂	1	1	yes	maximal, characteristic
7	C₄×#107×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₃₇	—	2	2	no	—
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	4	yes	—
11	C₃₇×#500	—	2	8	no	—
12	C₂×#107×C₃₇	C₂²	1	2	yes	—
13	C₂×#107×C₃₇	C₂²	1	4	yes	—
14	C₂×#107×C₃₇	C₂²	1	2	yes	—
15	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×#107×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₃₇×#395	—	2	4	no	—
21	C₃₇×#389	—	2	2	no	—
22	C₃₇×#389	C₂²	1	4	yes	—
23	C₃₇×#384	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
26	C₃₇×#340	C₂²	1	4	yes	—
27	C₃₇×#340	C₂²	1	2	yes	—
28	C₃₇×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₃₇	—	2	4	no	—
30	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
32	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₃₇	—	2	8	no	—
34	C₄⋊Q₈×C₃₇	—	4	4	no	—
35	C₄⋊Q₈×C₃₇	—	4	8	no	—
36	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₃₇	—	2	2	no	—
46	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
48	#107×C₃₇	—	2	4	no	—
49	#107×C₃₇	—	2	8	no	—
50	#107×C₃₇	—	4	2	no	—
51	#107×C₃₇	D₄	1	4	yes	—
52	#107×C₃₇	Q₈	1	4	yes	complemented
53	C₄⋊C₈×C₃₇	Q₈	1	4	yes	complemented
54	C₄×C₈×C₃₇	—	2	4	no	—
55	C₄×C₈×C₃₇	Q₈	1	4	yes	complemented
56	#95×C₃₇	—	2	2	no	—
57	#95×C₃₇	—	2	4	no	—
58	#95×C₃₇	D₄	1	2	yes	—
59	C₂×Q₈×C₃₇	—	4	8	no	—
60	C₂×Q₈×C₃₇	—	4	8	no	—
61	C₂×Q₈×C₃₇	—	8	4	no	—
62	C₂²×C₄×C₃₇	—	2	2	no	—
63	C₂²×C₄×C₃₇	—	2	2	no	—
64	C₂²×C₄×C₃₇	—	2	4	no	—
65	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₃₇	—	2	2	no	—
71	C₂×C₈×C₃₇	—	2	1	no	—
72	C₂×C₈×C₃₇	—	2	1	no	—
73	C₂×C₈×C₃₇	—	2	2	no	—
74	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
77	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
79	C₄⋊C₄×C₃₇	—	2	2	no	—
80	C₄⋊C₄×C₃₇	—	2	2	no	—
81	C₄⋊C₄×C₃₇	—	2	4	no	—
82	C₄⋊C₄×C₃₇	—	2	4	no	—
83	C₄⋊C₄×C₃₇	—	4	8	no	—
84	C₄⋊C₄×C₃₇	—	4	8	no	—
85	C₄⋊C₄×C₃₇	—	4	4	no	—
86	C₄⋊C₄×C₃₇	—	8	2	no	—
87	C₄⋊C₄×C₃₇	D₈	1	4	yes	—
88	C₄⋊C₄×C₃₇	Q₁₆	1	4	yes	—
89	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
90	C₄⋊C₄×C₃₇	C₂×Q₈	1	4	yes	—
91	C₄²×C₃₇	—	2	2	no	—
92	C₄²×C₃₇	—	2	2	no	—
93	C₄²×C₃₇	—	4	4	no	—
94	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
95	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
96	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₃₇	—	8	16	no	—
98	C₂×C₄×C₃₇	—	2	2	no	—
99	C₂×C₄×C₃₇	—	2	2	no	—
100	C₂×C₄×C₃₇	—	2	1	no	—
101	C₂×C₄×C₃₇	—	2	2	no	—
102	C₂×C₄×C₃₇	—	2	1	no	—
103	C₂×C₄×C₃₇	—	2	2	no	—
104	C₂×C₄×C₃₇	—	2	4	no	—
105	C₂×C₄×C₃₇	—	2	4	no	—
106	C₂×C₄×C₃₇	—	4	2	no	—
107	C₂×C₄×C₃₇	—	4	2	no	—
108	C₂×C₄×C₃₇	—	4	2	no	—
109	C₂×C₄×C₃₇	—	4	2	no	—
110	C₂×C₄×C₃₇	—	4	4	no	—
111	C₂×C₄×C₃₇	—	4	4	no	—
112	C₂×C₄×C₃₇	—	4	4	no	—
113	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
114	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₃₇	C₂×D₈	1	4	yes	—
116	C₂×C₄×C₃₇	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₃₇	C₂×Q₁₆	1	4	yes	—
118	C₂×C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
119	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
125	C₈×C₃₇	—	2	2	no	—
126	C₈×C₃₇	—	2	4	no	—
127	C₈×C₃₇	C₄⋊Q₈	1	4	yes	complemented
128	#28813	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
130	C₂²×C₃₇	D₄²	1	1	yes	characteristic
131	C₂²×C₃₇	D₄×Q₈	1	1	yes	characteristic
132	C₂²×C₃₇	C₂²×D₈	1	1	yes	characteristic
133	C₂²×C₃₇	C₂²×Q₁₆	1	1	yes	characteristic
134	C₂²×C₃₇	#576	1	1	yes	characteristic
135	C₂²×C₃₇	#578	1	1	yes	characteristic
136	C₄×C₃₇	—	2	2	no	—
137	C₄×C₃₇	—	2	4	no	—
138	C₄×C₃₇	—	2	4	no	—
139	C₄×C₃₇	—	4	4	no	—
140	C₄×C₃₇	—	4	4	no	—
141	C₄×C₃₇	—	8	4	no	—
142	C₄×C₃₇	#500	1	4	yes	complemented
143	C₄×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
144	C₄×C₃₇	#544	1	1	yes	characteristic
145	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
146	C₄×C₃₇	Q₈²	1	1	yes	characteristic
147	C₂×#500	C₂×C₃₇	1	2	yes	—
148	#2551	C₂×C₃₇	1	2	yes	—
149	#2057	C₂×C₃₇	1	4	yes	—
150	#1947	C₂×C₃₇	1	2	yes	—
151	#1942	C₂×C₃₇	1	2	yes	—
152	#1845	C₂×C₃₇	1	1	yes	characteristic
153	C₄×#107	C₂×C₃₇	1	1	yes	characteristic
154	C₈×C₄⋊C₄	C₂×C₃₇	1	1	yes	characteristic
155	C₂×C₃₇	#2551	1	1	yes	characteristic
156	C₂×C₃₇	C₂×#500	1	1	yes	characteristic
157	C₂×C₃₇	#3164	1	1	yes	characteristic
158	C₂×C₃₇	D₄×D₈	1	1	yes	characteristic
159	C₂×C₃₇	D₄×Q₁₆	1	1	yes	characteristic
160	C₂×C₃₇	Q₈×D₈	1	1	yes	characteristic
161	C₂×C₃₇	Q₈×Q₁₆	1	1	yes	characteristic
162	C₂×C₄⋊Q₈	—	2	2	no	—
163	C₂×C₄⋊Q₈	C₂²×C₃₇	1	4	yes	—
164	#500	—	2	8	no	—
165	C₂×#107	C₂²×C₃₇	1	4	yes	—
166	C₂×#107	C₂²×C₃₇	1	2	yes	—
167	C₂×#107	C₂²×C₃₇	1	2	yes	—
168	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
169	C₂×#107	C₂²×C₃₇	1	1	yes	characteristic
170	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
173	#395	—	2	4	no	—
174	#389	—	2	2	no	—
175	#389	C₂²×C₃₇	1	4	yes	—
176	#384	C₂²×C₃₇	1	2	yes	—
177	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
178	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
179	#340	C₂²×C₃₇	1	2	yes	—
180	#340	C₂²×C₃₇	1	4	yes	—
181	#336	C₂²×C₃₇	1	2	yes	—
182	C₃₇	#28813	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×Q₈	—	2	4	no	—
184	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
186	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
187	C₄⋊Q₈	—	2	8	no	—
188	C₄⋊Q₈	—	4	4	no	—
189	C₄⋊Q₈	—	4	8	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
195	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
196	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
197	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
198	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
201	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
202	#107	—	2	4	no	—
203	#107	—	2	8	no	—
204	#107	—	4	2	no	—
205	#107	D₄×C₃₇	1	4	yes	—
206	#107	Q₈×C₃₇	1	4	yes	complemented
207	C₄⋊C₈	Q₈×C₃₇	1	4	yes	complemented
208	C₄×C₈	—	2	4	no	—
209	C₄×C₈	Q₈×C₃₇	1	4	yes	complemented
210	#95	—	2	2	no	—
211	#95	—	2	4	no	—
212	#95	D₄×C₃₇	1	2	yes	—
213	C₂×Q₈	—	4	8	no	—
214	C₂×Q₈	—	4	8	no	—
215	C₂×Q₈	—	8	4	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
223	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
229	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
230	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
231	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
232	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	8	no	—
238	C₄⋊C₄	—	4	8	no	—
239	C₄⋊C₄	—	4	4	no	—
240	C₄⋊C₄	—	8	2	no	—
241	C₄⋊C₄	D₈×C₃₇	1	4	yes	—
242	C₄⋊C₄	Q₁₆×C₃₇	1	4	yes	—
243	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
244	C₄⋊C₄	C₂×Q₈×C₃₇	1	4	yes	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	4	no	—
248	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
249	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
250	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
251	Q₈	—	8	16	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
268	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
269	C₂×C₄	C₂×D₈×C₃₇	1	4	yes	—
270	C₂×C₄	C₂×D₈×C₃₇	1	2	yes	—
271	C₂×C₄	C₂×Q₁₆×C₃₇	1	4	yes	—
272	C₂×C₄	C₂×Q₁₆×C₃₇	1	2	yes	—
273	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
274	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
275	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
276	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
277	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
278	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	2	4	no	—
281	C₈	C₄⋊Q₈×C₃₇	1	4	yes	complemented
282	C₂²	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
283	C₂²	D₄²×C₃₇	1	1	yes	characteristic
284	C₂²	D₄×Q₈×C₃₇	1	1	yes	characteristic
285	C₂²	C₂²×D₈×C₃₇	1	1	yes	characteristic
286	C₂²	C₂²×Q₁₆×C₃₇	1	1	yes	characteristic
287	C₂²	C₃₇×#576	1	1	yes	characteristic
288	C₂²	C₃₇×#578	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	4	4	no	—
293	C₄	—	4	4	no	—
294	C₄	—	8	4	no	—
295	C₄	C₃₇×#500	1	4	yes	complemented
296	C₄	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
297	C₄	C₃₇×#544	1	1	yes	characteristic
298	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
299	C₄	Q₈²×C₃₇	1	1	yes	characteristic
300	C₂	C₃₇×#2551	1	1	yes	characteristic
301	C₂	C₂×C₃₇×#500	1	1	yes	characteristic
302	C₂	C₃₇×#3164	1	1	yes	characteristic
303	C₂	D₄×D₈×C₃₇	1	1	yes	characteristic
304	C₂	D₄×Q₁₆×C₃₇	1	1	yes	characteristic
305	C₂	Q₈×D₈×C₃₇	1	1	yes	characteristic
306	C₂	Q₈×Q₁₆×C₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
52. #107×C₃₇	Q₈	semidirect
53. C₄⋊C₈×C₃₇	Q₈	semidirect
55. C₄×C₈×C₃₇	Q₈	semidirect
127. C₈×C₃₇	C₄⋊Q₈	semidirect
128. #28813	C₃₇	direct (N × H)
142. C₄×C₃₇	#500	semidirect
182. C₃₇	#28813	direct (N × H)
206. #107	Q₈×C₃₇	semidirect
207. C₄⋊C₈	Q₈×C₃₇	semidirect
209. C₄×C₈	Q₈×C₃₇	semidirect
281. C₈	C₄⋊Q₈×C₃₇	semidirect
295. C₄	C₃₇×#500	semidirect

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
C₃₇×#2551	2
C₃₇×#2057	2
C₃₇×#1947	2
C₃₇×#1942	2
C₃₇×#1845	2
C₄×#107×C₃₇	2
C₈×C₄⋊C₄×C₃₇	2
#28813	37

Subgroups — C37×#28813 #22650066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28813 #22650066