Subgroups — C₃₇×#28741 #22649994

Order 9472.

846 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#2592	C₂	1	2	yes	maximal, complemented
4	C₃₇×#2056	C₂	1	2	yes	maximal, complemented
5	C₃₇×#2054	C₂	1	2	yes	maximal
6	C₃₇×#1983	C₂	1	2	yes	maximal, complemented
7	C₃₇×#1973	C₂	1	4	yes	maximal, complemented
8	C₄×#107×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆×C₃₇	C₂²	1	2	yes	—
11	C₂²×D₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×#124×C₃₇	C₂²	1	2	yes	—
13	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₃₇×#495	—	2	4	no	—
15	C₃₇×#446	—	2	4	no	—
16	C₂×#107×C₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
19	C₂×#103×C₃₇	C₂²	1	4	yes	—
20	C₂×#102×C₃₇	C₂²	1	4	yes	—
21	C₂×#97×C₃₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₃₇×#397	—	2	2	no	—
24	C₃₇×#393	—	2	2	no	—
25	C₃₇×#391	C₂²	1	4	yes	—
26	C₃₇×#390	C₂²	1	4	yes	—
27	C₃₇×#388	—	2	4	no	—
28	C₄×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
29	C₃₇×#340	C₂²	1	2	yes	—
30	C₂²×Q₈×C₃₇	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
32	C₂²×D₄×C₃₇	C₂³	1	2	yes	—
33	C₂×Q₁₆×C₃₇	—	2	2	no	—
34	C₂×Q₁₆×C₃₇	—	4	2	no	—
35	C₂×SD₁₆×C₃₇	—	2	4	no	—
36	C₂×SD₁₆×C₃₇	—	2	8	no	—
37	C₂×D₈×C₃₇	—	2	2	no	—
38	C₂×D₈×C₃₇	—	4	2	no	—
39	C₂²×C₈×C₃₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
42	#124×C₃₇	—	2	4	no	—
43	#124×C₃₇	—	2	4	no	—
44	C₄×Q₈×C₃₇	—	2	2	no	—
45	C₄×Q₈×C₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₇	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
49	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₃₇	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
52	C₂×C₂²⋊C₄×C₃₇	—	2	4	no	—
53	C₂×C₄²×C₃₇	—	2	2	no	—
54	C₂×C₄²×C₃₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
56	#107×C₃₇	—	2	4	no	—
57	C₄⋊C₈×C₃₇	—	2	4	no	—
58	C₄⋊C₈×C₃₇	—	2	2	no	—
59	#103×C₃₇	—	4	4	no	—
60	#102×C₃₇	—	4	4	no	—
61	#97×C₃₇	—	2	4	no	—
62	C₄×C₈×C₃₇	D₄	1	4	yes	—
63	#95×C₃₇	—	2	2	no	—
64	#95×C₃₇	—	2	4	no	—
65	#95×C₃₇	—	2	4	no	—
66	C₂⁴×C₃₇	—	2	2	no	—
67	C₂×Q₈×C₃₇	—	2	2	no	—
68	C₂×Q₈×C₃₇	—	2	2	no	—
69	C₂×Q₈×C₃₇	—	2	2	no	—
70	C₂×Q₈×C₃₇	—	2	4	no	—
71	C₂×Q₈×C₃₇	—	2	4	no	—
72	C₂×Q₈×C₃₇	—	4	2	no	—
73	C₂×D₄×C₃₇	—	2	2	no	—
74	C₂×D₄×C₃₇	—	2	2	no	—
75	C₂×D₄×C₃₇	—	2	4	no	—
76	C₂×D₄×C₃₇	—	2	4	no	—
77	C₂²×C₄×C₃₇	—	2	2	no	—
78	C₂²×C₄×C₃₇	—	2	4	no	—
79	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
84	Q₁₆×C₃₇	—	4	4	no	—
85	SD₁₆×C₃₇	—	4	8	no	—
86	D₈×C₃₇	—	4	4	no	—
87	C₂×C₈×C₃₇	—	2	2	no	—
88	C₂×C₈×C₃₇	—	2	2	no	—
89	C₂×C₈×C₃₇	—	2	4	no	—
90	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₃₇	#41	1	4	yes	—
93	C₄⋊C₄×C₃₇	—	2	2	no	—
94	C₄⋊C₄×C₃₇	—	2	4	no	—
95	C₄⋊C₄×C₃₇	—	2	4	no	—
96	C₄⋊C₄×C₃₇	—	2	4	no	—
97	C₂²⋊C₄×C₃₇	—	4	8	no	—
98	C₂²⋊C₄×C₃₇	—	4	4	no	—
99	C₂²⋊C₄×C₃₇	—	4	4	no	—
100	C₂²⋊C₄×C₃₇	—	8	4	no	—
101	C₄²×C₃₇	—	2	2	no	—
102	C₄²×C₃₇	—	2	2	no	—
103	C₄²×C₃₇	—	8	2	no	—
104	C₄²×C₃₇	C₂×D₄	1	2	yes	—
105	C₄²×C₃₇	#41	1	2	yes	—
106	C₂³×C₃₇	—	4	2	no	—
107	C₂³×C₃₇	—	4	4	no	—
108	C₂³×C₃₇	—	4	4	no	—
109	C₂³×C₃₇	—	4	2	no	—
110	C₂³×C₃₇	—	4	2	no	—
111	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₃₇	—	4	4	no	—
113	Q₈×C₃₇	—	4	2	no	—
114	Q₈×C₃₇	—	4	2	no	—
115	Q₈×C₃₇	—	4	4	no	—
116	D₄×C₃₇	—	4	2	no	—
117	D₄×C₃₇	—	4	2	no	—
118	D₄×C₃₇	—	4	4	no	—
119	C₂×C₄×C₃₇	—	2	2	no	—
120	C₂×C₄×C₃₇	—	2	1	no	—
121	C₂×C₄×C₃₇	—	2	1	no	—
122	C₂×C₄×C₃₇	—	2	2	no	—
123	C₂×C₄×C₃₇	—	2	2	no	—
124	C₂×C₄×C₃₇	—	2	4	no	—
125	C₂×C₄×C₃₇	—	2	2	no	—
126	C₂×C₄×C₃₇	—	2	2	no	—
127	C₂×C₄×C₃₇	—	4	2	no	—
128	C₂×C₄×C₃₇	—	4	4	no	—
129	C₂×C₄×C₃₇	—	4	4	no	—
130	C₂×C₄×C₃₇	—	4	4	no	—
131	C₂×C₄×C₃₇	—	4	4	no	—
132	C₂×C₄×C₃₇	#121	1	2	yes	—
133	C₂×C₄×C₃₇	#121	1	2	yes	—
134	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
136	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
138	C₈×C₃₇	—	2	2	no	—
139	C₈×C₃₇	—	4	2	no	—
140	C₈×C₃₇	#124	1	4	yes	—
141	#28741	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂²×C₃₇	—	8	4	no	—
143	C₂²×C₃₇	—	8	4	no	—
144	C₂²×C₃₇	—	8	2	no	—
145	C₂²×C₃₇	—	8	2	no	—
146	C₂²×C₃₇	—	8	2	no	—
147	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
148	C₂²×C₃₇	#550	1	2	yes	—
149	C₂²×C₃₇	#575	1	2	yes	—
150	C₂²×C₃₇	#576	1	1	yes	characteristic
151	C₂²×C₃₇	#578	1	1	yes	characteristic
152	C₄×C₃₇	—	2	2	no	—
153	C₄×C₃₇	—	2	2	no	—
154	C₄×C₃₇	—	4	2	no	—
155	C₄×C₃₇	—	4	2	no	—
156	C₄×C₃₇	—	8	4	no	—
157	C₄×C₃₇	—	8	2	no	—
158	C₄×C₃₇	C₂×#124	1	1	yes	characteristic
159	C₄×C₃₇	#535	1	1	yes	characteristic
160	C₄×C₃₇	#559	1	2	yes	—
161	C₂×#495	C₂×C₃₇	1	1	yes	characteristic
162	C₂×#446	C₂×C₃₇	1	1	yes	characteristic, complemented
163	#2592	C₂×C₃₇	1	2	yes	complemented
164	#2056	C₂×C₃₇	1	2	yes	complemented
165	#2054	C₂×C₃₇	1	2	yes	—
166	#1983	C₂×C₃₇	1	2	yes	complemented
167	#1973	C₂×C₃₇	1	4	yes	complemented
168	C₄×#107	C₂×C₃₇	1	1	yes	characteristic, complemented
169	C₂×C₃₇	—	16	2	no	—
170	C₂×C₃₇	#2548	1	1	yes	characteristic
171	C₂×C₃₇	#3054	1	1	yes	characteristic
172	C₂×C₃₇	#3061	1	1	yes	characteristic
173	C₂×C₃₇	#3402	1	2	yes	—
174	C₂×C₃₇	#3403	1	2	yes	—
175	C₂²×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic
176	C₂²×SD₁₆	C₂²×C₃₇	1	2	yes	—
177	C₂²×D₈	C₂²×C₃₇	1	1	yes	characteristic
178	C₂×#124	C₂²×C₃₇	1	2	yes	—
179	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
180	#495	—	2	4	no	—
181	#446	—	2	4	no	—
182	C₂×#107	C₂²×C₃₇	1	2	yes	—
183	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
184	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
185	C₂×#103	C₂²×C₃₇	1	4	yes	—
186	C₂×#102	C₂²×C₃₇	1	4	yes	—
187	C₂×#97	C₂²×C₃₇	1	2	yes	—
188	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
189	#397	—	2	2	no	—
190	#393	—	2	2	no	—
191	#391	C₂²×C₃₇	1	4	yes	—
192	#390	C₂²×C₃₇	1	4	yes	—
193	#388	—	2	4	no	—
194	C₄×C₄⋊C₄	C₂²×C₃₇	1	2	yes	—
195	#340	C₂²×C₃₇	1	2	yes	—
196	C₃₇	#28741	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²×Q₈	D₄×C₃₇	1	1	yes	characteristic
198	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
199	C₂²×D₄	C₂³×C₃₇	1	2	yes	—
200	C₂×Q₁₆	—	2	2	no	—
201	C₂×Q₁₆	—	4	2	no	—
202	C₂×SD₁₆	—	2	4	no	—
203	C₂×SD₁₆	—	2	8	no	—
204	C₂×D₈	—	2	2	no	—
205	C₂×D₈	—	4	2	no	—
206	C₂²×C₈	C₂³×C₃₇	1	2	yes	—
207	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
209	#124	—	2	4	no	—
210	#124	—	2	4	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₂×C₄⋊C₄	D₄×C₃₇	1	2	yes	—
214	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂³×C₃₇	1	4	yes	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₂²⋊C₄	—	2	4	no	—
219	C₂×C₂²⋊C₄	—	2	4	no	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	C₂³×C₃₇	1	2	yes	—
222	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
223	#107	—	2	4	no	—
224	C₄⋊C₈	—	2	4	no	—
225	C₄⋊C₈	—	2	2	no	—
226	#103	—	4	4	no	—
227	#102	—	4	4	no	—
228	#97	—	2	4	no	—
229	C₄×C₈	D₄×C₃₇	1	4	yes	—
230	#95	—	2	2	no	—
231	#95	—	2	4	no	—
232	#95	—	2	4	no	—
233	C₂⁴	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	4	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	4	no	—
243	C₂×D₄	—	2	4	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
250	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
251	Q₁₆	—	4	4	no	—
252	SD₁₆	—	4	8	no	—
253	D₈	—	4	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
259	C₂×C₈	#41×C₃₇	1	4	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	—	4	8	no	—
265	C₂²⋊C₄	—	4	4	no	—
266	C₂²⋊C₄	—	4	4	no	—
267	C₂²⋊C₄	—	8	4	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	8	2	no	—
271	C₄²	C₂×D₄×C₃₇	1	2	yes	—
272	C₄²	#41×C₃₇	1	2	yes	—
273	C₂³	—	4	2	no	—
274	C₂³	—	4	4	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
279	Q₈	—	4	4	no	—
280	Q₈	—	4	2	no	—
281	Q₈	—	4	2	no	—
282	Q₈	—	4	4	no	—
283	D₄	—	4	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	4	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	#121×C₃₇	1	2	yes	—
300	C₂×C₄	#121×C₃₇	1	2	yes	—
301	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
302	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
303	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
304	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
305	C₈	—	2	2	no	—
306	C₈	—	4	2	no	—
307	C₈	#124×C₃₇	1	4	yes	—
308	C₂²	—	8	4	no	—
309	C₂²	—	8	4	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
314	C₂²	C₃₇×#550	1	2	yes	—
315	C₂²	C₃₇×#575	1	2	yes	—
316	C₂²	C₃₇×#576	1	1	yes	characteristic
317	C₂²	C₃₇×#578	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	8	4	no	—
323	C₄	—	8	2	no	—
324	C₄	C₂×#124×C₃₇	1	1	yes	characteristic
325	C₄	C₃₇×#535	1	1	yes	characteristic
326	C₄	C₃₇×#559	1	2	yes	—
327	C₂	—	16	2	no	—
328	C₂	C₃₇×#2548	1	1	yes	characteristic
329	C₂	C₃₇×#3054	1	1	yes	characteristic
330	C₂	C₃₇×#3061	1	1	yes	characteristic
331	C₂	C₃₇×#3402	1	2	yes	—
332	C₂	C₃₇×#3403	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂²×D₈×C₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄×Q₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₇×#446	C₂	semidirect
3. C₃₇×#2592	C₂	semidirect
4. C₃₇×#2056	C₂	semidirect
6. C₃₇×#1983	C₂	semidirect
7. C₃₇×#1973	C₂	semidirect
8. C₄×#107×C₃₇	C₂	semidirect
141. #28741	C₃₇	direct (N × H)
162. C₂×#446	C₂×C₃₇	semidirect
163. #2592	C₂×C₃₇	semidirect
164. #2056	C₂×C₃₇	semidirect
166. #1983	C₂×C₃₇	semidirect
167. #1973	C₂×C₃₇	semidirect
168. C₄×#107	C₂×C₃₇	semidirect
196. C₃₇	#28741	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×#495	2
C₂×C₃₇×#446	2
C₃₇×#2592	2
C₃₇×#2056	2
C₃₇×#2054	2
C₃₇×#1983	2
C₃₇×#1973	2
C₄×#107×C₃₇	2
#28741	37

Subgroups — C37×#28741 #22649994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28741 #22649994