Subgroups — C₃₇×#28441 #22649694

Order 9472.

1402 conjugacy classes of subgroups.

The subgroup lattice has 320 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	2	yes	maximal, complemented
2	C₂×C₃₇×#460	C₂	1	4	yes	maximal, complemented
3	C₃₇×#2867	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#2014	C₂	1	4	yes	maximal, complemented
6	C₃₇×#1957	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₃₇	C₂²	1	8	yes	—
9	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×#127×C₃₇	—	2	2	no	—
11	C₂×#127×C₃₇	—	2	2	no	—
12	C₂×#127×C₃₇	C₂²	1	2	yes	—
13	C₂×#127×C₃₇	C₂²	1	2	yes	—
14	C₂×#127×C₃₇	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
16	C₃₇×#492	—	2	8	no	—
17	C₃₇×#460	—	2	16	no	—
18	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
19	C₂×#102×C₃₇	C₂²	1	8	yes	—
20	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
21	C₃₇×#391	C₂²	1	2	yes	—
22	C₃₇×#389	—	2	2	no	—
23	C₃₇×#384	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₃₇	—	2	2	no	—
25	C₄³×C₃₇	C₂²	1	1	yes	characteristic
26	C₃₇×#336	C₂²	1	2	yes	—
27	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
28	C₂²×D₄×C₃₇	—	2	2	no	—
29	C₂²×D₄×C₃₇	—	2	4	no	—
30	C₂²×D₄×C₃₇	—	2	4	no	—
31	C₂²×D₄×C₃₇	—	2	2	no	—
32	C₂²×D₄×C₃₇	C₂³	1	4	yes	—
33	C₂×SD₁₆×C₃₇	—	2	16	no	—
34	C₂×SD₁₆×C₃₇	—	2	16	no	—
35	C₂×SD₁₆×C₃₇	—	4	8	no	—
36	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
37	C₄⋊Q₈×C₃₇	—	2	2	no	—
38	C₄⋊Q₈×C₃₇	D₄	1	4	yes	complemented
39	#127×C₃₇	—	2	8	no	—
40	#127×C₃₇	—	2	4	no	—
41	#127×C₃₇	—	2	4	no	—
42	#127×C₃₇	—	4	4	no	—
43	#127×C₃₇	—	4	8	no	—
44	#127×C₃₇	—	4	8	no	—
45	C₄×Q₈×C₃₇	—	2	8	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₃₇	—	2	2	no	—
51	C₂×C₄²×C₃₇	D₄	1	2	yes	—
52	C₂×C₄²×C₃₇	D₄	1	2	yes	—
53	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₃₇	D₄	1	8	yes	complemented
57	#102×C₃₇	—	2	16	no	—
58	C₄×C₈×C₃₇	—	2	4	no	—
59	#95×C₃₇	—	2	4	no	—
60	C₂⁴×C₃₇	—	2	4	no	—
61	C₂×Q₈×C₃₇	—	2	2	no	—
62	C₂×Q₈×C₃₇	—	2	4	no	—
63	C₂×Q₈×C₃₇	—	2	4	no	—
64	C₂×Q₈×C₃₇	C₂×D₄	1	4	yes	—
65	C₂×D₄×C₃₇	—	2	8	no	—
66	C₂×D₄×C₃₇	—	2	4	no	—
67	C₂×D₄×C₃₇	—	2	4	no	—
68	C₂×D₄×C₃₇	—	2	8	no	—
69	C₂×D₄×C₃₇	—	4	8	no	—
70	C₂×D₄×C₃₇	—	4	8	no	—
71	C₂×D₄×C₃₇	—	4	8	no	—
72	C₂×D₄×C₃₇	—	4	8	no	—
73	C₂×D₄×C₃₇	—	4	4	no	—
74	C₂×D₄×C₃₇	—	4	4	no	—
75	C₂×D₄×C₃₇	—	4	8	no	—
76	C₂×D₄×C₃₇	—	4	8	no	—
77	C₂×D₄×C₃₇	—	4	8	no	—
78	C₂×D₄×C₃₇	—	4	8	no	—
79	C₂²×C₄×C₃₇	—	2	2	no	—
80	C₂²×C₄×C₃₇	—	2	2	no	—
81	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
86	SD₁₆×C₃₇	—	4	32	no	—
87	C₂×C₈×C₃₇	—	2	4	no	—
88	C₂×C₈×C₃₇	—	2	4	no	—
89	C₂×C₈×C₃₇	C₂×D₄	1	8	yes	—
90	C₄⋊C₄×C₃₇	—	2	8	no	—
91	C₄⋊C₄×C₃₇	—	2	4	no	—
92	C₄⋊C₄×C₃₇	—	4	4	no	—
93	C₄⋊C₄×C₃₇	—	4	4	no	—
94	C₄²×C₃₇	—	2	4	no	—
95	C₄²×C₃₇	—	2	2	no	—
96	C₄²×C₃₇	—	4	4	no	—
97	C₄²×C₃₇	SD₁₆	1	8	yes	complemented
98	C₄²×C₃₇	C₂×D₄	1	2	yes	—
99	C₄²×C₃₇	C₂×D₄	1	4	yes	—
100	C₄²×C₃₇	#41	1	2	yes	—
101	C₂³×C₃₇	—	4	4	no	—
102	C₂³×C₃₇	—	4	4	no	—
103	C₂³×C₃₇	—	4	4	no	—
104	C₂³×C₃₇	—	4	8	no	—
105	C₂³×C₃₇	—	4	8	no	—
106	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₃₇	—	2	8	no	—
108	Q₈×C₃₇	—	4	4	no	—
109	D₄×C₃₇	—	4	8	no	—
110	D₄×C₃₇	—	4	8	no	—
111	D₄×C₃₇	—	8	8	no	—
112	D₄×C₃₇	—	8	16	no	—
113	D₄×C₃₇	—	8	16	no	—
114	D₄×C₃₇	—	8	8	no	—
115	C₂×C₄×C₃₇	—	2	2	no	—
116	C₂×C₄×C₃₇	—	2	2	no	—
117	C₂×C₄×C₃₇	—	2	2	no	—
118	C₂×C₄×C₃₇	—	2	4	no	—
119	C₂×C₄×C₃₇	—	4	2	no	—
120	C₂×C₄×C₃₇	—	4	2	no	—
121	C₂×C₄×C₃₇	—	4	2	no	—
122	C₂×C₄×C₃₇	—	4	4	no	—
123	C₂×C₄×C₃₇	#121	1	4	yes	—
124	C₂×C₄×C₃₇	#127	1	2	yes	—
125	C₂×C₄×C₃₇	#127	1	2	yes	—
126	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
127	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
128	C₂×C₄×C₃₇	C₂×SD₁₆	1	4	yes	—
129	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
130	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
134	C₈×C₃₇	—	2	8	no	—
135	#28441	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₃₇	—	8	4	no	—
137	C₂²×C₃₇	—	8	4	no	—
138	C₂²×C₃₇	—	8	8	no	—
139	C₂²×C₃₇	—	8	8	no	—
140	C₂²×C₃₇	—	8	4	no	—
141	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
142	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
143	C₂²×C₃₇	#535	1	1	yes	characteristic
144	C₂²×C₃₇	C₂²×SD₁₆	1	2	yes	—
145	C₂²×C₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
146	C₄×C₃₇	—	2	2	no	—
147	C₄×C₃₇	—	2	4	no	—
148	C₄×C₃₇	—	4	4	no	—
149	C₄×C₃₇	—	8	2	no	—
150	C₄×C₃₇	#460	1	8	yes	complemented
151	C₄×C₃₇	#492	1	4	yes	complemented
152	C₄×C₃₇	D₄²	1	2	yes	—
153	C₄×C₃₇	#550	1	2	yes	—
154	C₂×#492	C₂×C₃₇	1	2	yes	complemented
155	C₂×#460	C₂×C₃₇	1	4	yes	complemented
156	#2867	C₂×C₃₇	1	1	yes	characteristic
157	C₄×C₄⋊Q₈	C₂×C₃₇	1	1	yes	characteristic, complemented
158	#2014	C₂×C₃₇	1	4	yes	complemented
159	#1957	C₂×C₃₇	1	2	yes	complemented
160	C₄×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic, complemented
161	C₂×C₃₇	—	16	4	no	—
162	C₂×C₃₇	#2541	1	1	yes	characteristic
163	C₂×C₃₇	C₂×#460	1	2	yes	—
164	C₂×C₃₇	C₂×#492	1	1	yes	characteristic
165	C₂×C₃₇	#3151	1	1	yes	characteristic
166	C₂×C₃₇	#3208	1	2	yes	—
167	C₂²×SD₁₆	C₂²×C₃₇	1	8	yes	—
168	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
169	C₂×#127	—	2	2	no	—
170	C₂×#127	—	2	2	no	—
171	C₂×#127	C₂²×C₃₇	1	2	yes	—
172	C₂×#127	C₂²×C₃₇	1	2	yes	—
173	C₂×#127	C₂²×C₃₇	1	2	yes	—
174	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
175	#492	—	2	8	no	—
176	#460	—	2	16	no	—
177	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
178	C₂×#102	C₂²×C₃₇	1	8	yes	—
179	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
180	#391	C₂²×C₃₇	1	2	yes	—
181	#389	—	2	2	no	—
182	#384	C₂²×C₃₇	1	1	yes	characteristic
183	C₄×C₄⋊C₄	—	2	2	no	—
184	C₄³	C₂²×C₃₇	1	1	yes	characteristic
185	#336	C₂²×C₃₇	1	2	yes	—
186	C₃₇	#28441	1	1	yes	characteristic, direct product factor, Sylow
187	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
188	C₂²×D₄	—	2	2	no	—
189	C₂²×D₄	—	2	4	no	—
190	C₂²×D₄	—	2	4	no	—
191	C₂²×D₄	—	2	2	no	—
192	C₂²×D₄	C₂³×C₃₇	1	4	yes	—
193	C₂×SD₁₆	—	2	16	no	—
194	C₂×SD₁₆	—	2	16	no	—
195	C₂×SD₁₆	—	4	8	no	—
196	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	D₄×C₃₇	1	4	yes	complemented
199	#127	—	2	8	no	—
200	#127	—	2	4	no	—
201	#127	—	2	4	no	—
202	#127	—	4	4	no	—
203	#127	—	4	8	no	—
204	#127	—	4	8	no	—
205	C₄×Q₈	—	2	8	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	D₄×C₃₇	1	2	yes	—
212	C₂×C₄²	D₄×C₃₇	1	2	yes	—
213	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
216	C₄⋊C₈	D₄×C₃₇	1	8	yes	complemented
217	#102	—	2	16	no	—
218	C₄×C₈	—	2	4	no	—
219	#95	—	2	4	no	—
220	C₂⁴	—	2	4	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	C₂×D₄×C₃₇	1	4	yes	—
225	C₂×D₄	—	2	8	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	2	8	no	—
229	C₂×D₄	—	4	8	no	—
230	C₂×D₄	—	4	8	no	—
231	C₂×D₄	—	4	8	no	—
232	C₂×D₄	—	4	8	no	—
233	C₂×D₄	—	4	4	no	—
234	C₂×D₄	—	4	4	no	—
235	C₂×D₄	—	4	8	no	—
236	C₂×D₄	—	4	8	no	—
237	C₂×D₄	—	4	8	no	—
238	C₂×D₄	—	4	8	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
245	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
246	SD₁₆	—	4	32	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	C₂×D₄×C₃₇	1	8	yes	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	4	4	no	—
257	C₄²	SD₁₆×C₃₇	1	8	yes	complemented
258	C₄²	C₂×D₄×C₃₇	1	2	yes	—
259	C₄²	C₂×D₄×C₃₇	1	4	yes	—
260	C₄²	#41×C₃₇	1	2	yes	—
261	C₂³	—	4	4	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	4	no	—
264	C₂³	—	4	8	no	—
265	C₂³	—	4	8	no	—
266	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
267	Q₈	—	2	8	no	—
268	Q₈	—	4	4	no	—
269	D₄	—	4	8	no	—
270	D₄	—	4	8	no	—
271	D₄	—	8	8	no	—
272	D₄	—	8	16	no	—
273	D₄	—	8	16	no	—
274	D₄	—	8	8	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	#121×C₃₇	1	4	yes	—
284	C₂×C₄	#127×C₃₇	1	2	yes	—
285	C₂×C₄	#127×C₃₇	1	2	yes	—
286	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
287	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
288	C₂×C₄	C₂×SD₁₆×C₃₇	1	4	yes	—
289	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	4	yes	—
290	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
291	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
292	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
293	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
294	C₈	—	2	8	no	—
295	C₂²	—	8	4	no	—
296	C₂²	—	8	4	no	—
297	C₂²	—	8	8	no	—
298	C₂²	—	8	8	no	—
299	C₂²	—	8	4	no	—
300	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
301	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
302	C₂²	C₃₇×#535	1	1	yes	characteristic
303	C₂²	C₂²×SD₁₆×C₃₇	1	2	yes	—
304	C₂²	C₂×AGL(1,ℤ/8)×C₃₇	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	4	no	—
308	C₄	—	8	2	no	—
309	C₄	C₃₇×#460	1	8	yes	complemented
310	C₄	C₃₇×#492	1	4	yes	complemented
311	C₄	D₄²×C₃₇	1	2	yes	—
312	C₄	C₃₇×#550	1	2	yes	—
313	C₂	—	16	4	no	—
314	C₂	C₃₇×#2541	1	1	yes	characteristic
315	C₂	C₂×C₃₇×#460	1	2	yes	—
316	C₂	C₂×C₃₇×#492	1	1	yes	characteristic
317	C₂	C₃₇×#3151	1	1	yes	characteristic
318	C₂	C₃₇×#3208	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₃₇×#2867	C₂
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×#492	C₂	semidirect
2. C₂×C₃₇×#460	C₂	semidirect
4. C₄×C₄⋊Q₈×C₃₇	C₂	semidirect
5. C₃₇×#2014	C₂	semidirect
6. C₃₇×#1957	C₂	semidirect
7. C₄×C₄⋊C₈×C₃₇	C₂	semidirect
38. C₄⋊Q₈×C₃₇	D₄	semidirect
56. C₄⋊C₈×C₃₇	D₄	semidirect
97. C₄²×C₃₇	SD₁₆	semidirect
135. #28441	C₃₇	direct (N × H)
150. C₄×C₃₇	#460	semidirect
151. C₄×C₃₇	#492	semidirect
154. C₂×#492	C₂×C₃₇	semidirect
155. C₂×#460	C₂×C₃₇	semidirect
157. C₄×C₄⋊Q₈	C₂×C₃₇	semidirect
158. #2014	C₂×C₃₇	semidirect
159. #1957	C₂×C₃₇	semidirect
160. C₄×C₄⋊C₈	C₂×C₃₇	semidirect
186. C₃₇	#28441	direct (N × H)
198. C₄⋊Q₈	D₄×C₃₇	semidirect
216. C₄⋊C₈	D₄×C₃₇	semidirect
257. C₄²	SD₁₆×C₃₇	semidirect
309. C₄	C₃₇×#460	semidirect
310. C₄	C₃₇×#492	semidirect

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
C₂×C₃₇×#460	2
C₃₇×#2867	2
C₄×C₄⋊Q₈×C₃₇	2
C₃₇×#2014	2
C₃₇×#1957	2
C₄×C₄⋊C₈×C₃₇	2
#28441	37

Subgroups — C37×#28441 #22649694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28441 #22649694