Subgroups — C₃₇×#28447 #22649700

Order 9472.

1130 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#497	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₇×#464	C₂	1	4	yes	maximal, complemented
3	C₃₇×#2858	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#2016	C₂	1	2	yes	maximal, complemented
6	C₃₇×#2014	C₂	1	2	yes	maximal
7	C₃₇×#1959	C₂	1	2	yes	maximal, complemented
8	C₃₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₃₇	C₂²	1	4	yes	—
10	C₂²×SD₁₆×C₃₇	C₂²	1	4	yes	—
11	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×#127×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×#124×C₃₇	C₂²	1	2	yes	—
15	C₂×#124×C₃₇	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
17	C₃₇×#497	—	2	8	no	—
18	C₃₇×#464	—	2	16	no	—
19	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
20	C₂×#103×C₃₇	C₂²	1	4	yes	—
21	C₂×#102×C₃₇	C₂²	1	4	yes	—
22	C₂×#97×C₃₇	C₂²	1	2	yes	—
23	C₃₇×#391	C₂²	1	2	yes	—
24	C₃₇×#390	—	2	2	no	—
25	C₃₇×#390	—	2	2	no	—
26	C₃₇×#389	—	2	2	no	—
27	C₃₇×#384	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₃₇	—	2	2	no	—
29	C₄³×C₃₇	C₂²	1	1	yes	characteristic
30	C₃₇×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
32	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
33	C₂²×D₄×C₃₇	—	2	2	no	—
34	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
35	C₂×Q₁₆×C₃₇	—	2	8	no	—
36	C₂×Q₁₆×C₃₇	—	2	8	no	—
37	C₂×Q₁₆×C₃₇	—	4	4	no	—
38	C₂×SD₁₆×C₃₇	—	2	8	no	—
39	C₂×SD₁₆×C₃₇	—	2	8	no	—
40	C₂×SD₁₆×C₃₇	—	4	4	no	—
41	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
42	C₄⋊Q₈×C₃₇	—	2	2	no	—
43	C₄⋊Q₈×C₃₇	—	4	2	no	—
44	C₄⋊Q₈×C₃₇	D₄	1	4	yes	complemented
45	#127×C₃₇	—	4	2	no	—
46	#124×C₃₇	—	2	8	no	—
47	#124×C₃₇	—	2	4	no	—
48	#124×C₃₇	—	2	4	no	—
49	C₄×Q₈×C₃₇	—	2	8	no	—
50	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
55	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
56	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
58	C₂×C₄²×C₃₇	—	2	2	no	—
59	C₂×C₄²×C₃₇	D₄	1	2	yes	—
60	C₂×C₄²×C₃₇	D₄	1	2	yes	—
61	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₃₇	D₄	1	8	yes	complemented
65	#103×C₃₇	—	2	8	no	—
66	#102×C₃₇	—	2	8	no	—
67	#97×C₃₇	—	2	4	no	—
68	#95×C₃₇	—	2	4	no	—
69	C₂⁴×C₃₇	—	2	2	no	—
70	C₂×Q₈×C₃₇	—	2	2	no	—
71	C₂×Q₈×C₃₇	—	2	4	no	—
72	C₂×Q₈×C₃₇	—	2	4	no	—
73	C₂×Q₈×C₃₇	—	2	2	no	—
74	C₂×Q₈×C₃₇	—	2	2	no	—
75	C₂×Q₈×C₃₇	—	2	4	no	—
76	C₂×Q₈×C₃₇	—	2	4	no	—
77	C₂×Q₈×C₃₇	C₂×D₄	1	4	yes	—
78	C₂×D₄×C₃₇	—	2	4	no	—
79	C₂×D₄×C₃₇	—	2	2	no	—
80	C₂×D₄×C₃₇	—	2	2	no	—
81	C₂×D₄×C₃₇	—	2	4	no	—
82	C₂×D₄×C₃₇	—	4	4	no	—
83	C₂×D₄×C₃₇	—	4	4	no	—
84	C₂×D₄×C₃₇	—	4	4	no	—
85	C₂²×C₄×C₃₇	—	2	2	no	—
86	C₂²×C₄×C₃₇	—	2	2	no	—
87	C₂²×C₄×C₃₇	—	2	2	no	—
88	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
93	Q₁₆×C₃₇	—	4	16	no	—
94	SD₁₆×C₃₇	—	4	16	no	—
95	C₂×C₈×C₃₇	—	2	4	no	—
96	C₂×C₈×C₃₇	—	2	4	no	—
97	C₂×C₈×C₃₇	C₂×D₄	1	8	yes	—
98	C₄⋊C₄×C₃₇	—	2	8	no	—
99	C₄⋊C₄×C₃₇	—	2	4	no	—
100	C₄⋊C₄×C₃₇	—	4	4	no	—
101	C₄⋊C₄×C₃₇	—	4	4	no	—
102	C₄⋊C₄×C₃₇	—	4	4	no	—
103	C₂²⋊C₄×C₃₇	—	4	8	no	—
104	C₂²⋊C₄×C₃₇	—	4	4	no	—
105	C₂²⋊C₄×C₃₇	—	4	4	no	—
106	C₄²×C₃₇	—	2	4	no	—
107	C₄²×C₃₇	—	2	2	no	—
108	C₄²×C₃₇	—	4	2	no	—
109	C₄²×C₃₇	—	4	4	no	—
110	C₄²×C₃₇	C₂×D₄	1	2	yes	—
111	C₄²×C₃₇	C₂×D₄	1	4	yes	—
112	C₄²×C₃₇	#41	1	2	yes	—
113	C₂³×C₃₇	—	4	2	no	—
114	C₂³×C₃₇	—	4	2	no	—
115	C₂³×C₃₇	—	4	2	no	—
116	C₂³×C₃₇	—	4	4	no	—
117	C₂³×C₃₇	—	4	4	no	—
118	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
119	Q₈×C₃₇	—	2	8	no	—
120	Q₈×C₃₇	—	4	4	no	—
121	Q₈×C₃₇	—	4	4	no	—
122	Q₈×C₃₇	—	4	4	no	—
123	D₄×C₃₇	—	4	4	no	—
124	D₄×C₃₇	—	4	4	no	—
125	D₄×C₃₇	—	8	8	no	—
126	C₂×C₄×C₃₇	—	2	2	no	—
127	C₂×C₄×C₃₇	—	2	2	no	—
128	C₂×C₄×C₃₇	—	2	2	no	—
129	C₂×C₄×C₃₇	—	2	2	no	—
130	C₂×C₄×C₃₇	—	2	2	no	—
131	C₂×C₄×C₃₇	—	2	2	no	—
132	C₂×C₄×C₃₇	—	2	4	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	—	4	2	no	—
137	C₂×C₄×C₃₇	—	4	2	no	—
138	C₂×C₄×C₃₇	—	4	2	no	—
139	C₂×C₄×C₃₇	—	4	4	no	—
140	C₂×C₄×C₃₇	#121	1	4	yes	—
141	C₂×C₄×C₃₇	#127	1	2	yes	—
142	C₂×C₄×C₃₇	#127	1	2	yes	—
143	C₂×C₄×C₃₇	#137	1	4	yes	—
144	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
146	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
148	C₈×C₃₇	—	2	8	no	—
149	#28447	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂²×C₃₇	—	8	2	no	—
151	C₂²×C₃₇	—	8	2	no	—
152	C₂²×C₃₇	—	8	4	no	—
153	C₂²×C₃₇	—	8	4	no	—
154	C₂²×C₃₇	—	8	2	no	—
155	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
156	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
157	C₂²×C₃₇	#535	1	1	yes	characteristic
158	C₂²×C₃₇	C₂×#137	1	2	yes	—
159	C₂²×C₃₇	#575	1	2	yes	—
160	C₄×C₃₇	—	2	2	no	—
161	C₄×C₃₇	—	2	2	no	—
162	C₄×C₃₇	—	2	4	no	—
163	C₄×C₃₇	—	2	4	no	—
164	C₄×C₃₇	—	4	4	no	—
165	C₄×C₃₇	—	8	2	no	—
166	C₄×C₃₇	—	8	2	no	—
167	C₄×C₃₇	D₄²	1	2	yes	—
168	C₄×C₃₇	#550	1	2	yes	—
169	C₂×#497	C₂×C₃₇	1	2	yes	complemented
170	C₂×#464	C₂×C₃₇	1	4	yes	complemented
171	#2858	C₂×C₃₇	1	1	yes	characteristic
172	C₄×C₄⋊Q₈	C₂×C₃₇	1	1	yes	characteristic, complemented
173	#2016	C₂×C₃₇	1	2	yes	complemented
174	#2014	C₂×C₃₇	1	2	yes	—
175	#1959	C₂×C₃₇	1	2	yes	complemented
176	#1767	C₂×C₃₇	1	1	yes	characteristic, complemented
177	C₂×C₃₇	—	16	2	no	—
178	C₂×C₃₇	#2541	1	1	yes	characteristic
179	C₂×C₃₇	#3037	1	2	yes	—
180	C₂×C₃₇	#3150	1	1	yes	characteristic
181	C₂×C₃₇	#3152	1	1	yes	characteristic
182	C₂×C₃₇	#3215	1	2	yes	—
183	C₂²×Q₁₆	C₂²×C₃₇	1	4	yes	—
184	C₂²×SD₁₆	C₂²×C₃₇	1	4	yes	—
185	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
186	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
187	C₂×#127	C₂²×C₃₇	1	1	yes	characteristic
188	C₂×#124	C₂²×C₃₇	1	2	yes	—
189	C₂×#124	C₂²×C₃₇	1	2	yes	—
190	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
191	#497	—	2	8	no	—
192	#464	—	2	16	no	—
193	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
194	C₂×#103	C₂²×C₃₇	1	4	yes	—
195	C₂×#102	C₂²×C₃₇	1	4	yes	—
196	C₂×#97	C₂²×C₃₇	1	2	yes	—
197	#391	C₂²×C₃₇	1	2	yes	—
198	#390	—	2	2	no	—
199	#390	—	2	2	no	—
200	#389	—	2	2	no	—
201	#384	C₂²×C₃₇	1	1	yes	characteristic
202	C₄×C₄⋊C₄	—	2	2	no	—
203	C₄³	C₂²×C₃₇	1	1	yes	characteristic
204	#336	C₂²×C₃₇	1	2	yes	—
205	C₃₇	#28447	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
207	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
208	C₂²×D₄	—	2	2	no	—
209	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
210	C₂×Q₁₆	—	2	8	no	—
211	C₂×Q₁₆	—	2	8	no	—
212	C₂×Q₁₆	—	4	4	no	—
213	C₂×SD₁₆	—	2	8	no	—
214	C₂×SD₁₆	—	2	8	no	—
215	C₂×SD₁₆	—	4	4	no	—
216	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
217	C₄⋊Q₈	—	2	2	no	—
218	C₄⋊Q₈	—	4	2	no	—
219	C₄⋊Q₈	D₄×C₃₇	1	4	yes	complemented
220	#127	—	4	2	no	—
221	#124	—	2	8	no	—
222	#124	—	2	4	no	—
223	#124	—	2	4	no	—
224	C₄×Q₈	—	2	8	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
230	C₂×C₂²⋊C₄	—	2	4	no	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₂²⋊C₄	—	2	2	no	—
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	D₄×C₃₇	1	2	yes	—
235	C₂×C₄²	D₄×C₃₇	1	2	yes	—
236	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
237	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
238	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
239	C₄⋊C₈	D₄×C₃₇	1	8	yes	complemented
240	#103	—	2	8	no	—
241	#102	—	2	8	no	—
242	#97	—	2	4	no	—
243	#95	—	2	4	no	—
244	C₂⁴	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	4	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	C₂×D₄×C₃₇	1	4	yes	—
253	C₂×D₄	—	2	4	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	4	4	no	—
258	C₂×D₄	—	4	4	no	—
259	C₂×D₄	—	4	4	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
264	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
267	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
268	Q₁₆	—	4	16	no	—
269	SD₁₆	—	4	16	no	—
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	C₂×D₄×C₃₇	1	8	yes	—
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₂²⋊C₄	—	4	8	no	—
279	C₂²⋊C₄	—	4	4	no	—
280	C₂²⋊C₄	—	4	4	no	—
281	C₄²	—	2	4	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	—	4	4	no	—
285	C₄²	C₂×D₄×C₃₇	1	2	yes	—
286	C₄²	C₂×D₄×C₃₇	1	4	yes	—
287	C₄²	#41×C₃₇	1	2	yes	—
288	C₂³	—	4	2	no	—
289	C₂³	—	4	2	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	4	4	no	—
292	C₂³	—	4	4	no	—
293	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
294	Q₈	—	2	8	no	—
295	Q₈	—	4	4	no	—
296	Q₈	—	4	4	no	—
297	Q₈	—	4	4	no	—
298	D₄	—	4	4	no	—
299	D₄	—	4	4	no	—
300	D₄	—	8	8	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	#121×C₃₇	1	4	yes	—
316	C₂×C₄	#127×C₃₇	1	2	yes	—
317	C₂×C₄	#127×C₃₇	1	2	yes	—
318	C₂×C₄	#137×C₃₇	1	4	yes	—
319	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
320	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
321	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
322	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
323	C₈	—	2	8	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	4	no	—
327	C₂²	—	8	4	no	—
328	C₂²	—	8	2	no	—
329	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
330	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
331	C₂²	C₃₇×#535	1	1	yes	characteristic
332	C₂²	C₂×#137×C₃₇	1	2	yes	—
333	C₂²	C₃₇×#575	1	2	yes	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	4	no	—
337	C₄	—	2	4	no	—
338	C₄	—	4	4	no	—
339	C₄	—	8	2	no	—
340	C₄	—	8	2	no	—
341	C₄	D₄²×C₃₇	1	2	yes	—
342	C₄	C₃₇×#550	1	2	yes	—
343	C₂	—	16	2	no	—
344	C₂	C₃₇×#2541	1	1	yes	characteristic
345	C₂	C₃₇×#3037	1	2	yes	—
346	C₂	C₃₇×#3150	1	1	yes	characteristic
347	C₂	C₃₇×#3152	1	1	yes	characteristic
348	C₂	C₃₇×#3215	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 ◁ C₄³×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₇×#497	C₂	semidirect
2. C₂×C₃₇×#464	C₂	semidirect
4. C₄×C₄⋊Q₈×C₃₇	C₂	semidirect
5. C₃₇×#2016	C₂	semidirect
7. C₃₇×#1959	C₂	semidirect
8. C₃₇×#1767	C₂	semidirect
44. C₄⋊Q₈×C₃₇	D₄	semidirect
64. C₄⋊C₈×C₃₇	D₄	semidirect
149. #28447	C₃₇	direct (N × H)
169. C₂×#497	C₂×C₃₇	semidirect
170. C₂×#464	C₂×C₃₇	semidirect
172. C₄×C₄⋊Q₈	C₂×C₃₇	semidirect
173. #2016	C₂×C₃₇	semidirect
175. #1959	C₂×C₃₇	semidirect
176. #1767	C₂×C₃₇	semidirect
205. C₃₇	#28447	direct (N × H)
219. C₄⋊Q₈	D₄×C₃₇	semidirect
239. C₄⋊C₈	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×#497	2
C₂×C₃₇×#464	2
C₃₇×#2858	2
C₄×C₄⋊Q₈×C₃₇	2
C₃₇×#2016	2
C₃₇×#2014	2
C₃₇×#1959	2
C₃₇×#1767	2
#28447	37

Subgroups — C37×#28447 #22649700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28447 #22649700