Subgroups — C₃₇×#28496 #22649749

Order 9472.

1022 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3153	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₇×#495	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₇×#473	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₇×#472	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₇×#2019	C₂	1	4	yes	maximal, complemented
7	C₃₇×#1967	C₂	1	2	yes	maximal, complemented
8	C₄×#108×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#137×C₃₇	C₂²	1	4	yes	—
10	C₂×AGL(1,ℤ/8)×C₃₇	C₂²	1	4	yes	—
11	C₂²×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
12	C₂²×SD₁₆×C₃₇	C₂²	1	2	yes	—
13	C₂²×D₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×#131×C₃₇	C₂²	1	2	yes	—
15	C₃₇×#532	C₂²	1	1	yes	characteristic
16	C₂×#124×C₃₇	C₂²	1	2	yes	—
17	C₃₇×#497	—	2	2	no	—
18	C₃₇×#496	—	2	2	no	—
19	C₃₇×#495	—	2	4	no	—
20	C₃₇×#495	—	2	2	no	—
21	C₃₇×#495	—	2	2	no	—
22	C₃₇×#473	—	2	8	no	—
23	C₃₇×#472	—	2	8	no	—
24	C₃₇×#436	—	2	2	no	—
25	C₈×D₄×C₃₇	—	2	2	no	—
26	C₂×#108×C₃₇	C₂²	1	2	yes	—
27	C₃₇×#423	C₂²	1	1	yes	characteristic
28	C₂×#101×C₃₇	C₂²	1	4	yes	—
29	C₂×#100×C₃₇	C₂²	1	4	yes	—
30	C₃₇×#408	C₂²	1	2	yes	—
31	C₄×M₁₆×C₃₇	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
33	C₃₇×#341	C₂²	1	2	yes	—
34	C₂×#41×C₃₇	C₂³	1	2	yes	—
35	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
36	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
37	#137×C₃₇	—	2	8	no	—
38	#137×C₃₇	—	4	4	no	—
39	AGL(1,ℤ/8)×C₃₇	—	2	8	no	—
40	AGL(1,ℤ/8)×C₃₇	—	4	4	no	—
41	C₂×Q₁₆×C₃₇	—	2	4	no	—
42	C₂×Q₁₆×C₃₇	—	2	4	no	—
43	C₂×Q₁₆×C₃₇	—	2	1	no	—
44	C₂×Q₁₆×C₃₇	—	2	1	no	—
45	C₂×SD₁₆×C₃₇	—	2	4	no	—
46	C₂×SD₁₆×C₃₇	—	2	4	no	—
47	C₂×SD₁₆×C₃₇	—	2	4	no	—
48	C₂×SD₁₆×C₃₇	—	4	4	no	—
49	C₂×D₈×C₃₇	—	2	4	no	—
50	C₂×D₈×C₃₇	—	2	4	no	—
51	C₂×D₈×C₃₇	—	2	1	no	—
52	C₂×D₈×C₃₇	—	2	1	no	—
53	#131×C₃₇	—	2	4	no	—
54	#131×C₃₇	—	2	4	no	—
55	C₂×M₁₆×C₃₇	—	2	2	no	—
56	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
57	C₂×M₁₆×C₃₇	C₂³	1	4	yes	—
58	C₂²×C₈×C₃₇	—	2	2	no	—
59	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₃₇	D₄	1	1	yes	characteristic
62	#127×C₃₇	D₄	1	1	yes	characteristic
63	#124×C₃₇	—	2	2	no	—
64	#124×C₃₇	—	2	4	no	—
65	#124×C₃₇	—	2	2	no	—
66	#124×C₃₇	D₄	1	2	yes	—
67	#121×C₃₇	—	2	2	no	—
68	C₄×D₄×C₃₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
70	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
71	#108×C₃₇	D₄	1	8	yes	complemented
72	C₄⋊C₈×C₃₇	—	2	2	no	—
73	#101×C₃₇	—	2	8	no	—
74	#100×C₃₇	—	2	8	no	—
75	C₂²⋊C₈×C₃₇	—	2	4	no	—
76	#97×C₃₇	—	2	2	no	—
77	C₄×C₈×C₃₇	—	2	2	no	—
78	C₄×C₈×C₃₇	D₄	1	2	yes	—
79	C₄×C₈×C₃₇	D₄	1	2	yes	—
80	C₂⁴×C₃₇	—	2	2	no	—
81	#41×C₃₇	—	2	2	no	—
82	#41×C₃₇	—	2	2	no	—
83	#41×C₃₇	—	2	2	no	—
84	#41×C₃₇	—	2	2	no	—
85	C₂×Q₈×C₃₇	—	2	4	no	—
86	C₂×Q₈×C₃₇	—	2	2	no	—
87	C₂×Q₈×C₃₇	—	2	2	no	—
88	C₂×Q₈×C₃₇	—	2	4	no	—
89	C₂×Q₈×C₃₇	C₂×D₄	1	2	yes	—
90	C₂×D₄×C₃₇	—	2	2	no	—
91	C₂×D₄×C₃₇	—	2	2	no	—
92	C₂×D₄×C₃₇	—	2	4	no	—
93	C₂×D₄×C₃₇	—	2	2	no	—
94	C₂×D₄×C₃₇	—	2	2	no	—
95	C₂×D₄×C₃₇	—	2	4	no	—
96	C₂×D₄×C₃₇	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₃₇	—	2	2	no	—
98	C₂²×C₄×C₃₇	—	2	2	no	—
99	C₂²×C₄×C₃₇	#41	1	2	yes	—
100	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
101	Q₁₆×C₃₇	—	4	8	no	—
102	Q₁₆×C₃₇	—	4	2	no	—
103	Q₁₆×C₃₇	—	4	2	no	—
104	SD₁₆×C₃₇	—	4	8	no	—
105	SD₁₆×C₃₇	—	4	8	no	—
106	SD₁₆×C₃₇	—	4	8	no	—
107	D₈×C₃₇	—	4	8	no	—
108	D₈×C₃₇	—	4	2	no	—
109	D₈×C₃₇	—	4	2	no	—
110	M₁₆×C₃₇	—	2	4	no	—
111	M₁₆×C₃₇	—	2	2	no	—
112	M₁₆×C₃₇	—	2	2	no	—
113	M₁₆×C₃₇	—	4	4	no	—
114	M₁₆×C₃₇	C₂×D₄	1	8	yes	—
115	C₂×C₈×C₃₇	—	2	4	no	—
116	C₂×C₈×C₃₇	—	2	2	no	—
117	C₂×C₈×C₃₇	—	2	2	no	—
118	C₂×C₈×C₃₇	—	4	4	no	—
119	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₃₇	C₂×D₄	1	4	yes	—
123	C₄⋊C₄×C₃₇	—	2	2	no	—
124	C₂²⋊C₄×C₃₇	—	2	4	no	—
125	C₂²⋊C₄×C₃₇	—	4	4	no	—
126	C₂²⋊C₄×C₃₇	—	4	4	no	—
127	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₃₇	C₂×D₄	1	2	yes	—
129	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
130	C₂³×C₃₇	—	2	2	no	—
131	C₂³×C₃₇	—	4	2	no	—
132	C₂³×C₃₇	—	4	4	no	—
133	C₂³×C₃₇	—	8	4	no	—
134	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
135	Q₈×C₃₇	—	2	2	no	—
136	Q₈×C₃₇	—	2	2	no	—
137	Q₈×C₃₇	—	4	2	no	—
138	Q₈×C₃₇	—	4	2	no	—
139	Q₈×C₃₇	—	4	4	no	—
140	D₄×C₃₇	—	2	2	no	—
141	D₄×C₃₇	—	2	2	no	—
142	D₄×C₃₇	—	4	4	no	—
143	D₄×C₃₇	—	4	2	no	—
144	D₄×C₃₇	—	4	2	no	—
145	D₄×C₃₇	—	4	2	no	—
146	D₄×C₃₇	—	4	2	no	—
147	D₄×C₃₇	—	4	4	no	—
148	C₂×C₄×C₃₇	—	2	2	no	—
149	C₂×C₄×C₃₇	—	4	2	no	—
150	C₂×C₄×C₃₇	—	4	2	no	—
151	C₂×C₄×C₃₇	—	4	2	no	—
152	C₂×C₄×C₃₇	—	4	2	no	—
153	C₂×C₄×C₃₇	—	4	4	no	—
154	C₂×C₄×C₃₇	#121	1	2	yes	—
155	C₂×C₄×C₃₇	#121	1	2	yes	—
156	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
158	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
160	C₈×C₃₇	—	2	2	no	—
161	C₈×C₃₇	—	2	8	no	—
162	C₈×C₃₇	—	4	2	no	—
163	C₈×C₃₇	#127	1	2	yes	—
164	C₈×C₃₇	#127	1	2	yes	—
165	#28496	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂²×C₃₇	—	2	2	no	—
167	C₂²×C₃₇	—	4	2	no	—
168	C₂²×C₃₇	—	4	4	no	—
169	C₂²×C₃₇	—	8	2	no	—
170	C₂²×C₃₇	—	8	4	no	—
171	C₂²×C₃₇	—	16	4	no	—
172	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
173	C₂²×C₃₇	#550	1	2	yes	—
174	C₄×C₃₇	—	2	2	no	—
175	C₄×C₃₇	—	2	2	no	—
176	C₄×C₃₇	—	4	2	no	—
177	C₄×C₃₇	—	8	2	no	—
178	C₄×C₃₇	C₂×#127	1	1	yes	characteristic
179	C₄×C₃₇	#535	1	1	yes	characteristic
180	C₄×C₃₇	D₄²	1	2	yes	—
181	#3153	C₂×C₃₇	1	2	yes	complemented
182	C₂×#495	C₂×C₃₇	1	1	yes	characteristic, complemented
183	C₂×#473	C₂×C₃₇	1	2	yes	complemented
184	C₂×#472	C₂×C₃₇	1	2	yes	complemented
185	#2931	C₂×C₃₇	1	1	yes	characteristic, complemented
186	#2019	C₂×C₃₇	1	4	yes	complemented
187	#1967	C₂×C₃₇	1	2	yes	complemented
188	C₄×#108	C₂×C₃₇	1	1	yes	characteristic, complemented
189	C₂×C₃₇	—	2	2	no	—
190	C₂×C₃₇	—	8	2	no	—
191	C₂×C₃₇	—	16	2	no	—
192	C₂×C₃₇	#2541	1	1	yes	characteristic
193	C₂×C₃₇	#3067	1	2	yes	—
194	C₂×#137	C₂²×C₃₇	1	4	yes	—
195	C₂×AGL(1,ℤ/8)	C₂²×C₃₇	1	4	yes	—
196	C₂²×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic
197	C₂²×SD₁₆	C₂²×C₃₇	1	2	yes	—
198	C₂²×D₈	C₂²×C₃₇	1	1	yes	characteristic
199	C₂×#131	C₂²×C₃₇	1	2	yes	—
200	#532	C₂²×C₃₇	1	1	yes	characteristic
201	C₂×#124	C₂²×C₃₇	1	2	yes	—
202	#497	—	2	2	no	—
203	#496	—	2	2	no	—
204	#495	—	2	4	no	—
205	#495	—	2	2	no	—
206	#495	—	2	2	no	—
207	#473	—	2	8	no	—
208	#472	—	2	8	no	—
209	#436	—	2	2	no	—
210	C₈×D₄	—	2	2	no	—
211	C₂×#108	C₂²×C₃₇	1	2	yes	—
212	#423	C₂²×C₃₇	1	1	yes	characteristic
213	C₂×#101	C₂²×C₃₇	1	4	yes	—
214	C₂×#100	C₂²×C₃₇	1	4	yes	—
215	#408	C₂²×C₃₇	1	2	yes	—
216	C₄×M₁₆	C₂²×C₃₇	1	2	yes	—
217	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
218	#341	C₂²×C₃₇	1	2	yes	—
219	C₃₇	#28496	1	1	yes	characteristic, direct product factor, Sylow
220	C₂×#41	C₂³×C₃₇	1	2	yes	—
221	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
222	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
223	#137	—	2	8	no	—
224	#137	—	4	4	no	—
225	AGL(1,ℤ/8)	—	2	8	no	—
226	AGL(1,ℤ/8)	—	4	4	no	—
227	C₂×Q₁₆	—	2	4	no	—
228	C₂×Q₁₆	—	2	4	no	—
229	C₂×Q₁₆	—	2	1	no	—
230	C₂×Q₁₆	—	2	1	no	—
231	C₂×SD₁₆	—	2	4	no	—
232	C₂×SD₁₆	—	2	4	no	—
233	C₂×SD₁₆	—	2	4	no	—
234	C₂×SD₁₆	—	4	4	no	—
235	C₂×D₈	—	2	4	no	—
236	C₂×D₈	—	2	4	no	—
237	C₂×D₈	—	2	1	no	—
238	C₂×D₈	—	2	1	no	—
239	#131	—	2	4	no	—
240	#131	—	2	4	no	—
241	C₂×M₁₆	—	2	2	no	—
242	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
243	C₂×M₁₆	C₂³×C₃₇	1	4	yes	—
244	C₂²×C₈	—	2	2	no	—
245	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
246	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
247	C₄⋊Q₈	D₄×C₃₇	1	1	yes	characteristic
248	#127	D₄×C₃₇	1	1	yes	characteristic
249	#124	—	2	2	no	—
250	#124	—	2	4	no	—
251	#124	—	2	2	no	—
252	#124	D₄×C₃₇	1	2	yes	—
253	#121	—	2	2	no	—
254	C₄×D₄	—	2	2	no	—
255	C₂×C₂²⋊C₄	—	2	4	no	—
256	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
257	#108	D₄×C₃₇	1	8	yes	complemented
258	C₄⋊C₈	—	2	2	no	—
259	#101	—	2	8	no	—
260	#100	—	2	8	no	—
261	C₂²⋊C₈	—	2	4	no	—
262	#97	—	2	2	no	—
263	C₄×C₈	—	2	2	no	—
264	C₄×C₈	D₄×C₃₇	1	2	yes	—
265	C₄×C₈	D₄×C₃₇	1	2	yes	—
266	C₂⁴	—	2	2	no	—
267	#41	—	2	2	no	—
268	#41	—	2	2	no	—
269	#41	—	2	2	no	—
270	#41	—	2	2	no	—
271	C₂×Q₈	—	2	4	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	2	2	no	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	C₂×D₄×C₃₇	1	2	yes	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	2	2	no	—
278	C₂×D₄	—	2	4	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	2	4	no	—
282	C₂×D₄	C₂×D₄×C₃₇	1	2	yes	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	#41×C₃₇	1	2	yes	—
286	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
287	Q₁₆	—	4	8	no	—
288	Q₁₆	—	4	2	no	—
289	Q₁₆	—	4	2	no	—
290	SD₁₆	—	4	8	no	—
291	SD₁₆	—	4	8	no	—
292	SD₁₆	—	4	8	no	—
293	D₈	—	4	8	no	—
294	D₈	—	4	2	no	—
295	D₈	—	4	2	no	—
296	M₁₆	—	2	4	no	—
297	M₁₆	—	2	2	no	—
298	M₁₆	—	2	2	no	—
299	M₁₆	—	4	4	no	—
300	M₁₆	C₂×D₄×C₃₇	1	8	yes	—
301	C₂×C₈	—	2	4	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	4	4	no	—
305	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
306	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
307	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
308	C₂×C₈	C₂×D₄×C₃₇	1	4	yes	—
309	C₄⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	2	4	no	—
311	C₂²⋊C₄	—	4	4	no	—
312	C₂²⋊C₄	—	4	4	no	—
313	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
314	C₄²	C₂×D₄×C₃₇	1	2	yes	—
315	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
316	C₂³	—	2	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	4	4	no	—
319	C₂³	—	8	4	no	—
320	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
321	Q₈	—	2	2	no	—
322	Q₈	—	2	2	no	—
323	Q₈	—	4	2	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	4	4	no	—
326	D₄	—	2	2	no	—
327	D₄	—	2	2	no	—
328	D₄	—	4	4	no	—
329	D₄	—	4	2	no	—
330	D₄	—	4	2	no	—
331	D₄	—	4	2	no	—
332	D₄	—	4	2	no	—
333	D₄	—	4	4	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	4	no	—
340	C₂×C₄	#121×C₃₇	1	2	yes	—
341	C₂×C₄	#121×C₃₇	1	2	yes	—
342	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
343	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
344	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
345	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
346	C₈	—	2	2	no	—
347	C₈	—	2	8	no	—
348	C₈	—	4	2	no	—
349	C₈	#127×C₃₇	1	2	yes	—
350	C₈	#127×C₃₇	1	2	yes	—
351	C₂²	—	2	2	no	—
352	C₂²	—	4	2	no	—
353	C₂²	—	4	4	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	4	no	—
356	C₂²	—	16	4	no	—
357	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
358	C₂²	C₃₇×#550	1	2	yes	—
359	C₄	—	2	2	no	—
360	C₄	—	2	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	8	2	no	—
363	C₄	C₂×#127×C₃₇	1	1	yes	characteristic
364	C₄	C₃₇×#535	1	1	yes	characteristic
365	C₄	D₄²×C₃₇	1	2	yes	—
366	C₂	—	2	2	no	—
367	C₂	—	8	2	no	—
368	C₂	—	16	2	no	—
369	C₂	C₃₇×#2541	1	1	yes	characteristic
370	C₂	C₃₇×#3067	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#3153	C₂	semidirect
2. C₂×C₃₇×#495	C₂	semidirect
3. C₂×C₃₇×#473	C₂	semidirect
4. C₂×C₃₇×#472	C₂	semidirect
5. C₃₇×#2931	C₂	semidirect
6. C₃₇×#2019	C₂	semidirect
7. C₃₇×#1967	C₂	semidirect
8. C₄×#108×C₃₇	C₂	semidirect
71. #108×C₃₇	D₄	semidirect
165. #28496	C₃₇	direct (N × H)
181. #3153	C₂×C₃₇	semidirect
182. C₂×#495	C₂×C₃₇	semidirect
183. C₂×#473	C₂×C₃₇	semidirect
184. C₂×#472	C₂×C₃₇	semidirect
185. #2931	C₂×C₃₇	semidirect
186. #2019	C₂×C₃₇	semidirect
187. #1967	C₂×C₃₇	semidirect
188. C₄×#108	C₂×C₃₇	semidirect
219. C₃₇	#28496	direct (N × H)
257. #108	D₄×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#3153	2
C₂×C₃₇×#495	2
C₂×C₃₇×#473	2
C₂×C₃₇×#472	2
C₃₇×#2931	2
C₃₇×#2019	2
C₃₇×#1967	2
C₄×#108×C₃₇	2
#28496	37

Subgroups — C37×#28496 #22649749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28496 #22649749