Subgroups — C₃₇×#28634 #22649887

Order 9472.

754 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	C₃₇×#3152	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#3142	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₇×#491	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#2932	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#2052	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1998	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#1966	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₇×#1908	C₂	1	4	yes	maximal, complemented
9	C₃₇×#1778	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137×C₃₇	C₂²	1	4	yes	—
11	C₂×#131×C₃₇	C₂²	1	2	yes	—
12	C₃₇×#532	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×#125×C₃₇	C₂²	1	1	yes	characteristic
15	C₃₇×#497	—	2	2	no	—
16	C₃₇×#494	—	2	1	no	—
17	C₃₇×#492	—	2	1	no	—
18	C₃₇×#491	—	2	1	no	—
19	C₃₇×#491	—	2	4	no	—
20	C₃₇×#490	—	2	1	no	—
21	C₃₇×#488	—	2	2	no	—
22	C₃₇×#435	—	2	4	no	—
23	C₂×#108×C₃₇	C₂²	1	2	yes	—
24	C₃₇×#423	C₂²	1	1	yes	characteristic
25	C₃₇×#417	C₂²	1	4	yes	—
26	C₂×#103×C₃₇	C₂²	1	2	yes	—
27	C₂×#103×C₃₇	C₂²	1	2	yes	—
28	C₂×#101×C₃₇	C₂²	1	4	yes	—
29	C₃₇×#408	C₂²	1	2	yes	—
30	C₄×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
31	C₄×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
32	C₂×#97×C₃₇	C₂²	1	1	yes	characteristic
33	C₃₇×#343	C₂²	1	4	yes	—
34	C₃₇×#341	C₂²	1	2	yes	—
35	C₂×#41×C₃₇	C₂³	1	2	yes	—
36	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
37	#137×C₃₇	—	2	8	no	—
38	#137×C₃₇	—	4	4	no	—
39	C₂×Q₁₆×C₃₇	—	2	4	no	—
40	C₂×SD₁₆×C₃₇	—	2	4	no	—
41	#131×C₃₇	—	4	4	no	—
42	C₂×M₁₆×C₃₇	—	2	2	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	—	2	2	no	—
47	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₃₇	—	2	2	no	—
50	C₄⋊Q₈×C₃₇	—	2	1	no	—
51	C₄⋊Q₈×C₃₇	—	2	1	no	—
52	C₄⋊Q₈×C₃₇	D₄	1	1	yes	characteristic
53	#127×C₃₇	D₄	1	1	yes	characteristic
54	#125×C₃₇	—	2	1	no	—
55	#125×C₃₇	—	2	1	no	—
56	#125×C₃₇	—	2	2	no	—
57	#124×C₃₇	D₄	1	2	yes	—
58	#121×C₃₇	—	2	2	no	—
59	C₄×D₄×C₃₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
64	#108×C₃₇	—	2	4	no	—
65	C₄⋊C₈×C₃₇	—	2	2	no	—
66	#103×C₃₇	—	2	4	no	—
67	#103×C₃₇	—	2	4	no	—
68	#103×C₃₇	—	2	4	no	—
69	#102×C₃₇	—	2	4	no	—
70	#101×C₃₇	—	4	4	no	—
71	C₂²⋊C₈×C₃₇	—	2	4	no	—
72	#97×C₃₇	—	2	1	no	—
73	#97×C₃₇	—	2	1	no	—
74	#97×C₃₇	D₄	1	4	yes	—
75	C₄×C₈×C₃₇	—	2	1	no	—
76	C₄×C₈×C₃₇	—	2	1	no	—
77	#41×C₃₇	—	2	4	no	—
78	#41×C₃₇	—	2	4	no	—
79	C₂×Q₈×C₃₇	—	2	2	no	—
80	C₂×Q₈×C₃₇	—	2	2	no	—
81	C₂×Q₈×C₃₇	—	2	4	no	—
82	C₂×Q₈×C₃₇	—	2	4	no	—
83	C₂×Q₈×C₃₇	C₂×D₄	1	2	yes	—
84	C₂×D₄×C₃₇	—	2	2	no	—
85	C₂×D₄×C₃₇	—	2	2	no	—
86	C₂×D₄×C₃₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₃₇	—	2	2	no	—
88	C₂²×C₄×C₃₇	—	2	2	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	Q₁₆×C₃₇	—	4	8	no	—
93	SD₁₆×C₃₇	—	4	8	no	—
94	M₁₆×C₃₇	—	2	2	no	—
95	M₁₆×C₃₇	—	2	2	no	—
96	M₁₆×C₃₇	—	2	2	no	—
97	M₁₆×C₃₇	—	2	4	no	—
98	M₁₆×C₃₇	—	4	4	no	—
99	M₁₆×C₃₇	#41	1	4	yes	—
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	—	2	2	no	—
105	C₂×C₈×C₃₇	—	4	4	no	—
106	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
108	C₄⋊C₄×C₃₇	—	2	2	no	—
109	C₄⋊C₄×C₃₇	—	2	2	no	—
110	C₄⋊C₄×C₃₇	—	2	2	no	—
111	C₄⋊C₄×C₃₇	—	4	2	no	—
112	C₄⋊C₄×C₃₇	—	4	2	no	—
113	C₄⋊C₄×C₃₇	—	4	2	no	—
114	C₄⋊C₄×C₃₇	—	4	2	no	—
115	C₂²⋊C₄×C₃₇	—	2	4	no	—
116	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₃₇	C₂×D₄	1	2	yes	—
119	C₂³×C₃₇	—	2	2	no	—
120	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
121	Q₈×C₃₇	—	2	4	no	—
122	Q₈×C₃₇	—	4	2	no	—
123	Q₈×C₃₇	—	4	2	no	—
124	Q₈×C₃₇	—	4	4	no	—
125	D₄×C₃₇	—	2	4	no	—
126	D₄×C₃₇	—	4	4	no	—
127	D₄×C₃₇	—	8	2	no	—
128	C₂×C₄×C₃₇	—	2	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	4	no	—
135	C₂×C₄×C₃₇	—	8	2	no	—
136	C₂×C₄×C₃₇	#121	1	2	yes	—
137	C₂×C₄×C₃₇	#121	1	2	yes	—
138	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
141	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
142	C₈×C₃₇	—	2	2	no	—
143	C₈×C₃₇	—	2	2	no	—
144	C₈×C₃₇	—	2	4	no	—
145	C₈×C₃₇	—	4	2	no	—
146	C₈×C₃₇	—	4	2	no	—
147	#28634	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₃₇	—	2	2	no	—
149	C₂²×C₃₇	—	4	2	no	—
150	C₂²×C₃₇	—	4	4	no	—
151	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
152	C₂²×C₃₇	#559	1	2	yes	—
153	C₄×C₃₇	—	2	2	no	—
154	C₄×C₃₇	—	2	2	no	—
155	C₄×C₃₇	—	4	2	no	—
156	C₄×C₃₇	—	8	2	no	—
157	C₄×C₃₇	—	8	2	no	—
158	C₄×C₃₇	#532	1	1	yes	characteristic
159	C₄×C₃₇	#535	1	1	yes	characteristic
160	C₄×C₃₇	#546	1	2	yes	—
161	#3152	C₂×C₃₇	1	1	yes	characteristic
162	#3142	C₂×C₃₇	1	1	yes	characteristic
163	C₂×#491	C₂×C₃₇	1	1	yes	characteristic, complemented
164	#2932	C₂×C₃₇	1	1	yes	characteristic
165	#2052	C₂×C₃₇	1	4	yes	complemented
166	#1998	C₂×C₃₇	1	1	yes	characteristic, complemented
167	#1966	C₂×C₃₇	1	1	yes	characteristic, complemented
168	#1908	C₂×C₃₇	1	4	yes	complemented
169	#1778	C₂×C₃₇	1	1	yes	characteristic, complemented
170	C₂×C₃₇	—	2	2	no	—
171	C₂×C₃₇	—	8	2	no	—
172	C₂×C₃₇	#2544	1	1	yes	characteristic
173	C₂×C₃₇	#3069	1	2	yes	—
174	C₂×#137	C₂²×C₃₇	1	4	yes	—
175	C₂×#131	C₂²×C₃₇	1	2	yes	—
176	#532	C₂²×C₃₇	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
178	C₂×#125	C₂²×C₃₇	1	1	yes	characteristic
179	#497	—	2	2	no	—
180	#494	—	2	1	no	—
181	#492	—	2	1	no	—
182	#491	—	2	1	no	—
183	#491	—	2	4	no	—
184	#490	—	2	1	no	—
185	#488	—	2	2	no	—
186	#435	—	2	4	no	—
187	C₂×#108	C₂²×C₃₇	1	2	yes	—
188	#423	C₂²×C₃₇	1	1	yes	characteristic
189	#417	C₂²×C₃₇	1	4	yes	—
190	C₂×#103	C₂²×C₃₇	1	2	yes	—
191	C₂×#103	C₂²×C₃₇	1	2	yes	—
192	C₂×#101	C₂²×C₃₇	1	4	yes	—
193	#408	C₂²×C₃₇	1	2	yes	—
194	C₄×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
195	C₄×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
196	C₂×#97	C₂²×C₃₇	1	1	yes	characteristic
197	#343	C₂²×C₃₇	1	4	yes	—
198	#341	C₂²×C₃₇	1	2	yes	—
199	C₃₇	#28634	1	1	yes	characteristic, direct product factor, Sylow
200	C₂×#41	C₂³×C₃₇	1	2	yes	—
201	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
202	#137	—	2	8	no	—
203	#137	—	4	4	no	—
204	C₂×Q₁₆	—	2	4	no	—
205	C₂×SD₁₆	—	2	4	no	—
206	#131	—	4	4	no	—
207	C₂×M₁₆	—	2	2	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	—	2	2	no	—
212	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
213	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
214	C₄⋊Q₈	—	2	2	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	D₄×C₃₇	1	1	yes	characteristic
218	#127	D₄×C₃₇	1	1	yes	characteristic
219	#125	—	2	1	no	—
220	#125	—	2	1	no	—
221	#125	—	2	2	no	—
222	#124	D₄×C₃₇	1	2	yes	—
223	#121	—	2	2	no	—
224	C₄×D₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
228	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
229	#108	—	2	4	no	—
230	C₄⋊C₈	—	2	2	no	—
231	#103	—	2	4	no	—
232	#103	—	2	4	no	—
233	#103	—	2	4	no	—
234	#102	—	2	4	no	—
235	#101	—	4	4	no	—
236	C₂²⋊C₈	—	2	4	no	—
237	#97	—	2	1	no	—
238	#97	—	2	1	no	—
239	#97	D₄×C₃₇	1	4	yes	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	2	1	no	—
242	#41	—	2	4	no	—
243	#41	—	2	4	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	C₂×D₄×C₃₇	1	2	yes	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	C₂×D₄×C₃₇	1	2	yes	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	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	Q₁₆	—	4	8	no	—
258	SD₁₆	—	4	8	no	—
259	M₁₆	—	2	2	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	2	2	no	—
262	M₁₆	—	2	4	no	—
263	M₁₆	—	4	4	no	—
264	M₁₆	#41×C₃₇	1	4	yes	—
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	—	2	2	no	—
270	C₂×C₈	—	4	4	no	—
271	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
272	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₃₇	1	2	yes	—
284	C₂³	—	2	2	no	—
285	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
286	Q₈	—	2	4	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	4	no	—
290	D₄	—	2	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	8	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	#121×C₃₇	1	2	yes	—
302	C₂×C₄	#121×C₃₇	1	2	yes	—
303	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
305	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
306	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
307	C₈	—	2	2	no	—
308	C₈	—	2	2	no	—
309	C₈	—	2	4	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	4	no	—
315	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
316	C₂²	C₃₇×#559	1	2	yes	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	4	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	C₃₇×#532	1	1	yes	characteristic
323	C₄	C₃₇×#535	1	1	yes	characteristic
324	C₄	C₃₇×#546	1	2	yes	—
325	C₂	—	2	2	no	—
326	C₂	—	8	2	no	—
327	C₂	C₃₇×#2544	1	1	yes	characteristic
328	C₂	C₃₇×#3069	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₃₇×#532	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
3. C₂×C₃₇×#491	C₂	semidirect
5. C₃₇×#2052	C₂	semidirect
6. C₃₇×#1998	C₂	semidirect
7. C₃₇×#1966	C₂	semidirect
8. C₃₇×#1908	C₂	semidirect
9. C₃₇×#1778	C₂	semidirect
147. #28634	C₃₇	direct (N × H)
163. C₂×#491	C₂×C₃₇	semidirect
165. #2052	C₂×C₃₇	semidirect
166. #1998	C₂×C₃₇	semidirect
167. #1966	C₂×C₃₇	semidirect
168. #1908	C₂×C₃₇	semidirect
169. #1778	C₂×C₃₇	semidirect
199. C₃₇	#28634	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#3152	2
C₃₇×#3142	2
C₂×C₃₇×#491	2
C₃₇×#2932	2
C₃₇×#2052	2
C₃₇×#1998	2
C₃₇×#1966	2
C₃₇×#1908	2
C₃₇×#1778	2
#28634	37

Subgroups — C37×#28634 #22649887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28634 #22649887