Subgroups — C₃₇×#28815 #22650068

Order 9472.

722 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#501	C₂	1	2	yes	maximal
2	C₃₇×#2550	C₂	1	2	yes	maximal
3	C₃₇×#2059	C₂	1	2	yes	maximal
4	C₃₇×#2058	C₂	1	2	yes	maximal
5	C₃₇×#1949	C₂	1	2	yes	maximal
6	C₃₇×#1944	C₂	1	2	yes	maximal
7	C₃₇×#1846	C₂	1	1	yes	maximal, characteristic
8	C₄×#106×C₃₇	C₂	1	1	yes	maximal, characteristic
9	C₃₇×#1770	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
11	C₂×#125×C₃₇	C₂²	1	2	yes	—
12	C₃₇×#501	—	2	8	no	—
13	C₂×#107×C₃₇	C₂²	1	2	yes	—
14	C₂×#107×C₃₇	C₂²	1	2	yes	—
15	C₂×#106×C₃₇	C₂²	1	2	yes	—
16	C₂×#106×C₃₇	C₂²	1	1	yes	characteristic
17	C₂×#106×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×#106×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×#106×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×#97×C₃₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₃₇×#398	—	2	2	no	—
24	C₃₇×#395	—	2	2	no	—
25	C₃₇×#391	—	2	2	no	—
26	C₃₇×#389	—	2	2	no	—
27	C₃₇×#389	C₂²	1	2	yes	—
28	C₃₇×#389	C₂²	1	2	yes	—
29	C₃₇×#382	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
32	C₃₇×#340	C₂²	1	2	yes	—
33	C₃₇×#340	C₂²	1	2	yes	—
34	C₃₇×#340	C₂²	1	2	yes	—
35	C₃₇×#336	C₂²	1	2	yes	—
36	C₂²×Q₈×C₃₇	—	2	2	no	—
37	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₃₇	—	2	4	no	—
41	C₄⋊Q₈×C₃₇	—	4	2	no	—
42	#125×C₃₇	—	2	4	no	—
43	#125×C₃₇	—	4	2	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₃₇	—	2	2	no	—
55	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
57	#107×C₃₇	—	2	4	no	—
58	#107×C₃₇	—	2	4	no	—
59	#106×C₃₇	—	2	4	no	—
60	#106×C₃₇	—	4	1	no	—
61	#106×C₃₇	—	4	1	no	—
62	#106×C₃₇	D₄	1	4	yes	—
63	#106×C₃₇	Q₈	1	4	yes	complemented
64	C₄⋊C₈×C₃₇	Q₈	1	4	yes	complemented
65	#97×C₃₇	—	2	4	no	—
66	C₄×C₈×C₃₇	Q₈	1	4	yes	complemented
67	#95×C₃₇	—	2	2	no	—
68	#95×C₃₇	—	2	2	no	—
69	#95×C₃₇	—	2	2	no	—
70	#95×C₃₇	—	2	2	no	—
71	#95×C₃₇	—	2	2	no	—
72	#95×C₃₇	D₄	1	1	yes	characteristic
73	#95×C₃₇	D₄	1	1	yes	characteristic
74	C₂×Q₈×C₃₇	—	4	4	no	—
75	C₂×Q₈×C₃₇	—	4	4	no	—
76	C₂×Q₈×C₃₇	—	8	2	no	—
77	C₂²×C₄×C₃₇	—	2	2	no	—
78	C₂²×C₄×C₃₇	—	2	2	no	—
79	C₂²×C₄×C₃₇	—	2	2	no	—
80	C₂²×C₄×C₃₇	—	2	2	no	—
81	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
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	C₂×C₈×C₃₇	—	2	2	no	—
87	C₂×C₈×C₃₇	—	2	1	no	—
88	C₂×C₈×C₃₇	—	2	1	no	—
89	C₂×C₈×C₃₇	—	2	2	no	—
90	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₃₇	C₂×Q₈	1	4	yes	—
93	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
95	C₄⋊C₄×C₃₇	—	2	2	no	—
96	C₄⋊C₄×C₃₇	—	2	2	no	—
97	C₄⋊C₄×C₃₇	—	2	2	no	—
98	C₄⋊C₄×C₃₇	—	2	2	no	—
99	C₄⋊C₄×C₃₇	—	2	2	no	—
100	C₄⋊C₄×C₃₇	—	2	2	no	—
101	C₄⋊C₄×C₃₇	—	4	2	no	—
102	C₄⋊C₄×C₃₇	—	4	4	no	—
103	C₄⋊C₄×C₃₇	—	4	4	no	—
104	C₄⋊C₄×C₃₇	—	4	4	no	—
105	C₄⋊C₄×C₃₇	—	4	4	no	—
106	C₄⋊C₄×C₃₇	—	8	2	no	—
107	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
108	C₄⋊C₄×C₃₇	C₂×Q₈	1	4	yes	—
109	C₄²×C₃₇	—	2	2	no	—
110	C₄²×C₃₇	—	2	2	no	—
111	C₄²×C₃₇	—	8	2	no	—
112	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
113	C₄²×C₃₇	C₂×Q₈	1	2	yes	—
114	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₃₇	—	8	8	no	—
116	C₂×C₄×C₃₇	—	2	2	no	—
117	C₂×C₄×C₃₇	—	2	2	no	—
118	C₂×C₄×C₃₇	—	2	2	no	—
119	C₂×C₄×C₃₇	—	2	2	no	—
120	C₂×C₄×C₃₇	—	2	4	no	—
121	C₂×C₄×C₃₇	—	2	4	no	—
122	C₂×C₄×C₃₇	—	2	2	no	—
123	C₂×C₄×C₃₇	—	2	2	no	—
124	C₂×C₄×C₃₇	—	2	2	no	—
125	C₂×C₄×C₃₇	—	2	2	no	—
126	C₂×C₄×C₃₇	—	4	2	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	2	no	—
132	C₂×C₄×C₃₇	—	4	2	no	—
133	C₂×C₄×C₃₇	—	4	2	no	—
134	C₂×C₄×C₃₇	—	4	2	no	—
135	C₂×C₄×C₃₇	—	4	2	no	—
136	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
137	C₂×C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
138	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
144	C₈×C₃₇	—	2	2	no	—
145	C₈×C₃₇	—	2	4	no	—
146	C₈×C₃₇	C₄⋊Q₈	1	4	yes	—
147	#28815	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
149	C₂²×C₃₇	D₄²	1	1	yes	characteristic
150	C₂²×C₃₇	D₄×Q₈	1	1	yes	characteristic
151	C₂²×C₃₇	#575	1	2	yes	—
152	C₂²×C₃₇	#577	1	2	yes	—
153	C₄×C₃₇	—	2	2	no	—
154	C₄×C₃₇	—	2	2	no	—
155	C₄×C₃₇	—	4	4	no	—
156	C₄×C₃₇	—	4	2	no	—
157	C₄×C₃₇	—	4	2	no	—
158	C₄×C₃₇	—	4	4	no	—
159	C₄×C₃₇	—	8	2	no	—
160	C₄×C₃₇	—	8	2	no	—
161	C₄×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
162	C₄×C₃₇	#544	1	1	yes	characteristic
163	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
164	C₄×C₃₇	Q₈²	1	1	yes	characteristic
165	C₂×#501	C₂×C₃₇	1	2	yes	—
166	#2550	C₂×C₃₇	1	2	yes	—
167	#2059	C₂×C₃₇	1	2	yes	—
168	#2058	C₂×C₃₇	1	2	yes	—
169	#1949	C₂×C₃₇	1	2	yes	—
170	#1944	C₂×C₃₇	1	2	yes	—
171	#1846	C₂×C₃₇	1	1	yes	characteristic
172	C₄×#106	C₂×C₃₇	1	1	yes	characteristic
173	#1770	C₂×C₃₇	1	1	yes	characteristic
174	C₂×C₃₇	#2551	1	1	yes	characteristic
175	C₂×C₃₇	#3162	1	1	yes	characteristic
176	C₂×C₃₇	#3163	1	1	yes	characteristic
177	C₂×C₃₇	#3275	1	2	yes	—
178	C₂×C₃₇	#3388	1	2	yes	—
179	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
180	C₂×#125	C₂²×C₃₇	1	2	yes	—
181	#501	—	2	8	no	—
182	C₂×#107	C₂²×C₃₇	1	2	yes	—
183	C₂×#107	C₂²×C₃₇	1	2	yes	—
184	C₂×#106	C₂²×C₃₇	1	2	yes	—
185	C₂×#106	C₂²×C₃₇	1	1	yes	characteristic
186	C₂×#106	C₂²×C₃₇	1	1	yes	characteristic
187	C₂×#106	C₂²×C₃₇	1	1	yes	characteristic
188	C₂×#106	C₂²×C₃₇	1	1	yes	characteristic
189	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
190	C₂×#97	C₂²×C₃₇	1	2	yes	—
191	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
192	#398	—	2	2	no	—
193	#395	—	2	2	no	—
194	#391	—	2	2	no	—
195	#389	—	2	2	no	—
196	#389	C₂²×C₃₇	1	2	yes	—
197	#389	C₂²×C₃₇	1	2	yes	—
198	#382	C₂²×C₃₇	1	2	yes	—
199	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
200	C₄×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
201	#340	C₂²×C₃₇	1	2	yes	—
202	#340	C₂²×C₃₇	1	2	yes	—
203	#340	C₂²×C₃₇	1	2	yes	—
204	#336	C₂²×C₃₇	1	2	yes	—
205	C₃₇	#28815	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²×Q₈	—	2	2	no	—
207	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
209	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
210	C₄⋊Q₈	—	2	4	no	—
211	C₄⋊Q₈	—	4	2	no	—
212	#125	—	2	4	no	—
213	#125	—	4	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
220	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
221	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
222	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
226	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
227	#107	—	2	4	no	—
228	#107	—	2	4	no	—
229	#106	—	2	4	no	—
230	#106	—	4	1	no	—
231	#106	—	4	1	no	—
232	#106	D₄×C₃₇	1	4	yes	—
233	#106	Q₈×C₃₇	1	4	yes	complemented
234	C₄⋊C₈	Q₈×C₃₇	1	4	yes	complemented
235	#97	—	2	4	no	—
236	C₄×C₈	Q₈×C₃₇	1	4	yes	complemented
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	D₄×C₃₇	1	1	yes	characteristic
243	#95	D₄×C₃₇	1	1	yes	characteristic
244	C₂×Q₈	—	4	4	no	—
245	C₂×Q₈	—	4	4	no	—
246	C₂×Q₈	—	8	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
255	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
261	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
262	C₂×C₈	C₂×Q₈×C₃₇	1	4	yes	—
263	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
264	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	4	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	8	2	no	—
277	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
278	C₄⋊C₄	C₂×Q₈×C₃₇	1	4	yes	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	8	2	no	—
282	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
283	C₄²	C₂×Q₈×C₃₇	1	2	yes	—
284	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
285	Q₈	—	8	8	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
307	C₂×C₄	C₄⋊Q₈×C₃₇	1	2	yes	—
308	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
309	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
310	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
311	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
312	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
313	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	2	4	no	—
316	C₈	C₄⋊Q₈×C₃₇	1	4	yes	—
317	C₂²	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
318	C₂²	D₄²×C₃₇	1	1	yes	characteristic
319	C₂²	D₄×Q₈×C₃₇	1	1	yes	characteristic
320	C₂²	C₃₇×#575	1	2	yes	—
321	C₂²	C₃₇×#577	1	2	yes	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	4	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	4	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	2	no	—
330	C₄	C₂×C₄⋊Q₈×C₃₇	1	1	yes	characteristic
331	C₄	C₃₇×#544	1	1	yes	characteristic
332	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
333	C₄	Q₈²×C₃₇	1	1	yes	characteristic
334	C₂	C₃₇×#2551	1	1	yes	characteristic
335	C₂	C₃₇×#3162	1	1	yes	characteristic
336	C₂	C₃₇×#3163	1	1	yes	characteristic
337	C₂	C₃₇×#3275	1	2	yes	—
338	C₂	C₃₇×#3388	1	2	yes	—

	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
63. #106×C₃₇	Q₈	semidirect
64. C₄⋊C₈×C₃₇	Q₈	semidirect
66. C₄×C₈×C₃₇	Q₈	semidirect
147. #28815	C₃₇	direct (N × H)
205. C₃₇	#28815	direct (N × H)
233. #106	Q₈×C₃₇	semidirect
234. C₄⋊C₈	Q₈×C₃₇	semidirect
236. C₄×C₈	Q₈×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×#501	2
C₃₇×#2550	2
C₃₇×#2059	2
C₃₇×#2058	2
C₃₇×#1949	2
C₃₇×#1944	2
C₃₇×#1846	2
C₄×#106×C₃₇	2
C₃₇×#1770	2
#28815	37

Subgroups — C37×#28815 #22650068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28815 #22650068