Subgroups — C₃₇×#28764 #22650017

Order 9472.

778 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#492	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#3139	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#3138	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#2046	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1977	C₂	1	4	yes	maximal, complemented
7	C₃₇×#1966	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#1965	C₂	1	1	yes	maximal, characteristic
9	C₄×#108×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×SD₁₆×C₃₇	C₂²	1	2	yes	—
11	C₂²×SD₁₆×C₃₇	C₂²	1	2	yes	—
12	C₃₇×#533	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₃₇	C₂²	1	1	yes	characteristic
15	C₃₇×#492	—	2	2	no	—
16	C₃₇×#492	—	2	2	no	—
17	C₃₇×#491	—	2	2	no	—
18	C₃₇×#490	—	2	2	no	—
19	C₃₇×#487	—	2	2	no	—
20	C₃₇×#486	—	2	2	no	—
21	C₃₇×#446	—	2	2	no	—
22	C₈×Q₈×C₃₇	—	2	2	no	—
23	C₂×#108×C₃₇	C₂²	1	2	yes	—
24	C₃₇×#424	C₂²	1	2	yes	—
25	C₃₇×#423	C₂²	1	1	yes	characteristic
26	C₃₇×#419	C₂²	1	4	yes	—
27	C₃₇×#418	C₂²	1	4	yes	—
28	C₂×#101×C₃₇	C₂²	1	4	yes	—
29	C₂×#100×C₃₇	C₂²	1	4	yes	—
30	C₃₇×#405	C₂²	1	2	yes	—
31	C₄×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
32	C₄×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
34	C₃₇×#341	C₂²	1	2	yes	—
35	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
36	C₂²×D₄×C₃₇	—	2	2	no	—
37	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
38	C₂×SD₁₆×C₃₇	—	2	4	no	—
39	C₂×SD₁₆×C₃₇	—	2	2	no	—
40	C₂×SD₁₆×C₃₇	—	2	2	no	—
41	C₂×SD₁₆×C₃₇	—	2	8	no	—
42	C₂×SD₁₆×C₃₇	—	4	4	no	—
43	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
44	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
45	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₃₇	—	2	1	no	—
49	C₄⋊Q₈×C₃₇	—	2	1	no	—
50	C₄⋊Q₈×C₃₇	—	2	2	no	—
51	C₄⋊Q₈×C₃₇	D₄	1	2	yes	—
52	#127×C₃₇	—	2	1	no	—
53	#127×C₃₇	—	2	1	no	—
54	#127×C₃₇	—	2	2	no	—
55	#125×C₃₇	D₄	1	2	yes	—
56	C₂²⋊Q₈×C₃₇	—	2	2	no	—
57	C₄×Q₈×C₃₇	—	2	2	no	—
58	#117×C₃₇	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
60	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
61	#108×C₃₇	—	2	4	no	—
62	C₄⋊C₈×C₃₇	—	2	4	no	—
63	C₄⋊C₈×C₃₇	—	2	2	no	—
64	#103×C₃₇	—	2	8	no	—
65	#102×C₃₇	—	2	8	no	—
66	#101×C₃₇	—	4	4	no	—
67	#100×C₃₇	—	4	4	no	—
68	#97×C₃₇	—	2	2	no	—
69	#97×C₃₇	—	2	1	no	—
70	#97×C₃₇	—	2	1	no	—
71	C₄×C₈×C₃₇	—	2	2	no	—
72	C₄×C₈×C₃₇	—	2	1	no	—
73	C₄×C₈×C₃₇	—	2	1	no	—
74	C₄×C₈×C₃₇	D₄	1	2	yes	—
75	C₄×C₈×C₃₇	D₄	1	2	yes	—
76	C₂⁴×C₃₇	—	2	2	no	—
77	C₂×Q₈×C₃₇	—	2	2	no	—
78	C₂×Q₈×C₃₇	—	2	2	no	—
79	C₂×Q₈×C₃₇	—	2	2	no	—
80	C₂×Q₈×C₃₇	—	2	4	no	—
81	C₂×Q₈×C₃₇	—	2	4	no	—
82	C₂×D₄×C₃₇	—	2	2	no	—
83	C₂×D₄×C₃₇	—	2	2	no	—
84	C₂×D₄×C₃₇	—	2	4	no	—
85	C₂×D₄×C₃₇	—	2	4	no	—
86	C₂×D₄×C₃₇	—	4	2	no	—
87	C₂×D₄×C₃₇	—	4	2	no	—
88	C₂×D₄×C₃₇	—	8	4	no	—
89	C₂²×C₄×C₃₇	—	2	2	no	—
90	C₂²×C₄×C₃₇	#41	1	2	yes	—
91	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
92	SD₁₆×C₃₇	—	4	8	no	—
93	SD₁₆×C₃₇	—	4	4	no	—
94	SD₁₆×C₃₇	—	4	4	no	—
95	M₁₆×C₃₇	—	2	2	no	—
96	M₁₆×C₃₇	—	2	2	no	—
97	M₁₆×C₃₇	—	2	2	no	—
98	M₁₆×C₃₇	—	2	4	no	—
99	M₁₆×C₃₇	—	2	2	no	—
100	C₂×C₈×C₃₇	—	2	2	no	—
101	C₂×C₈×C₃₇	—	2	2	no	—
102	C₂×C₈×C₃₇	—	2	2	no	—
103	C₂×C₈×C₃₇	—	2	2	no	—
104	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₃₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₃₇	#41	1	4	yes	—
108	C₄⋊C₄×C₃₇	—	2	4	no	—
109	C₄⋊C₄×C₃₇	—	2	2	no	—
110	C₄⋊C₄×C₃₇	—	4	2	no	—
111	C₄⋊C₄×C₃₇	—	4	2	no	—
112	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
113	C₂²⋊C₄×C₃₇	—	2	2	no	—
114	C₄²×C₃₇	—	2	2	no	—
115	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₃₇	C₂×D₄	1	2	yes	—
117	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
118	C₂³×C₃₇	—	4	2	no	—
119	C₂³×C₃₇	—	4	4	no	—
120	C₂³×C₃₇	—	8	4	no	—
121	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
122	Q₈×C₃₇	—	4	4	no	—
123	Q₈×C₃₇	—	4	2	no	—
124	Q₈×C₃₇	—	4	2	no	—
125	Q₈×C₃₇	—	4	4	no	—
126	D₄×C₃₇	—	4	2	no	—
127	D₄×C₃₇	—	4	2	no	—
128	D₄×C₃₇	—	4	4	no	—
129	D₄×C₃₇	—	8	4	no	—
130	D₄×C₃₇	—	8	4	no	—
131	C₂×C₄×C₃₇	—	2	4	no	—
132	C₂×C₄×C₃₇	—	2	4	no	—
133	C₂×C₄×C₃₇	—	4	2	no	—
134	C₂×C₄×C₃₇	—	4	4	no	—
135	C₂×C₄×C₃₇	#121	1	2	yes	—
136	C₂×C₄×C₃₇	#121	1	2	yes	—
137	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
140	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
141	C₈×C₃₇	—	2	2	no	—
142	C₈×C₃₇	—	4	2	no	—
143	C₈×C₃₇	—	4	2	no	—
144	C₈×C₃₇	—	4	2	no	—
145	C₈×C₃₇	#124	1	2	yes	—
146	C₈×C₃₇	#124	1	2	yes	—
147	#28764	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₃₇	—	2	2	no	—
149	C₂²×C₃₇	—	8	2	no	—
150	C₂²×C₃₇	—	8	4	no	—
151	C₂²×C₃₇	—	16	4	no	—
152	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
153	C₂²×C₃₇	#559	1	2	yes	—
154	C₄×C₃₇	—	2	2	no	—
155	C₄×C₃₇	—	4	4	no	—
156	C₄×C₃₇	—	8	2	no	—
157	C₄×C₃₇	#471	1	4	yes	—
158	C₄×C₃₇	C₂×#124	1	1	yes	characteristic
159	C₄×C₃₇	#535	1	1	yes	characteristic
160	C₄×C₃₇	#550	1	2	yes	—
161	C₂×#492	C₂×C₃₇	1	1	yes	characteristic
162	#3139	C₂×C₃₇	1	1	yes	characteristic, complemented
163	#3138	C₂×C₃₇	1	1	yes	characteristic
164	#2960	C₂×C₃₇	1	1	yes	characteristic, complemented
165	#2046	C₂×C₃₇	1	4	yes	complemented
166	#1977	C₂×C₃₇	1	4	yes	complemented
167	#1966	C₂×C₃₇	1	1	yes	characteristic, complemented
168	#1965	C₂×C₃₇	1	1	yes	characteristic
169	C₄×#108	C₂×C₃₇	1	1	yes	characteristic, complemented
170	C₂×C₃₇	—	2	2	no	—
171	C₂×C₃₇	—	16	2	no	—
172	C₂×C₃₇	#2548	1	1	yes	characteristic
173	C₂×C₃₇	C₂×#471	1	2	yes	—
174	C₂²×SD₁₆	C₂²×C₃₇	1	2	yes	—
175	C₂²×SD₁₆	C₂²×C₃₇	1	2	yes	—
176	#533	C₂²×C₃₇	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
178	C₂×#127	C₂²×C₃₇	1	1	yes	characteristic
179	#492	—	2	2	no	—
180	#492	—	2	2	no	—
181	#491	—	2	2	no	—
182	#490	—	2	2	no	—
183	#487	—	2	2	no	—
184	#486	—	2	2	no	—
185	#446	—	2	2	no	—
186	C₈×Q₈	—	2	2	no	—
187	C₂×#108	C₂²×C₃₇	1	2	yes	—
188	#424	C₂²×C₃₇	1	2	yes	—
189	#423	C₂²×C₃₇	1	1	yes	characteristic
190	#419	C₂²×C₃₇	1	4	yes	—
191	#418	C₂²×C₃₇	1	4	yes	—
192	C₂×#101	C₂²×C₃₇	1	4	yes	—
193	C₂×#100	C₂²×C₃₇	1	4	yes	—
194	#405	C₂²×C₃₇	1	2	yes	—
195	C₄×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
196	C₄×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
198	#341	C₂²×C₃₇	1	2	yes	—
199	C₃₇	#28764	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
201	C₂²×D₄	—	2	2	no	—
202	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
203	C₂×SD₁₆	—	2	4	no	—
204	C₂×SD₁₆	—	2	2	no	—
205	C₂×SD₁₆	—	2	2	no	—
206	C₂×SD₁₆	—	2	8	no	—
207	C₂×SD₁₆	—	4	4	no	—
208	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
209	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
210	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
211	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	D₄×C₃₇	1	2	yes	—
217	#127	—	2	1	no	—
218	#127	—	2	1	no	—
219	#127	—	2	2	no	—
220	#125	D₄×C₃₇	1	2	yes	—
221	C₂²⋊Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	#117	C₂³×C₃₇	1	2	yes	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
226	#108	—	2	4	no	—
227	C₄⋊C₈	—	2	4	no	—
228	C₄⋊C₈	—	2	2	no	—
229	#103	—	2	8	no	—
230	#102	—	2	8	no	—
231	#101	—	4	4	no	—
232	#100	—	4	4	no	—
233	#97	—	2	2	no	—
234	#97	—	2	1	no	—
235	#97	—	2	1	no	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	D₄×C₃₇	1	2	yes	—
240	C₄×C₈	D₄×C₃₇	1	2	yes	—
241	C₂⁴	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	2	4	no	—
250	C₂×D₄	—	2	4	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	8	4	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	#41×C₃₇	1	2	yes	—
256	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
257	SD₁₆	—	4	8	no	—
258	SD₁₆	—	4	4	no	—
259	SD₁₆	—	4	4	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	2	2	no	—
262	M₁₆	—	2	2	no	—
263	M₁₆	—	2	4	no	—
264	M₁₆	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
270	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₃₇	1	1	yes	characteristic
272	C₂×C₈	#41×C₃₇	1	4	yes	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₃₇	1	2	yes	—
282	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
283	C₂³	—	4	2	no	—
284	C₂³	—	4	4	no	—
285	C₂³	—	8	4	no	—
286	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
287	Q₈	—	4	4	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	4	no	—
294	D₄	—	8	4	no	—
295	D₄	—	8	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	#121×C₃₇	1	2	yes	—
301	C₂×C₄	#121×C₃₇	1	2	yes	—
302	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
304	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
305	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
306	C₈	—	2	2	no	—
307	C₈	—	4	2	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	2	no	—
310	C₈	#124×C₃₇	1	2	yes	—
311	C₈	#124×C₃₇	1	2	yes	—
312	C₂²	—	2	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	4	no	—
315	C₂²	—	16	4	no	—
316	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
317	C₂²	C₃₇×#559	1	2	yes	—
318	C₄	—	2	2	no	—
319	C₄	—	4	4	no	—
320	C₄	—	8	2	no	—
321	C₄	C₃₇×#471	1	4	yes	—
322	C₄	C₂×#124×C₃₇	1	1	yes	characteristic
323	C₄	C₃₇×#535	1	1	yes	characteristic
324	C₄	C₃₇×#550	1	2	yes	—
325	C₂	—	2	2	no	—
326	C₂	—	16	2	no	—
327	C₂	C₃₇×#2548	1	1	yes	characteristic
328	C₂	C₂×C₃₇×#471	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂×#127×C₃₇	C₂²
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
2. C₃₇×#3139	C₂	semidirect
4. C₃₇×#2960	C₂	semidirect
5. C₃₇×#2046	C₂	semidirect
6. C₃₇×#1977	C₂	semidirect
7. C₃₇×#1966	C₂	semidirect
9. C₄×#108×C₃₇	C₂	semidirect
147. #28764	C₃₇	direct (N × H)
162. #3139	C₂×C₃₇	semidirect
164. #2960	C₂×C₃₇	semidirect
165. #2046	C₂×C₃₇	semidirect
166. #1977	C₂×C₃₇	semidirect
167. #1966	C₂×C₃₇	semidirect
169. C₄×#108	C₂×C₃₇	semidirect
199. C₃₇	#28764	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×#492	2
C₃₇×#3139	2
C₃₇×#3138	2
C₃₇×#2960	2
C₃₇×#2046	2
C₃₇×#1977	2
C₃₇×#1966	2
C₃₇×#1965	2
C₄×#108×C₃₇	2
#28764	37

Subgroups — C37×#28764 #22650017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28764 #22650017