Subgroups — C₃₇×#28440 #22649693

Order 9472.

1090 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#494	C₂	1	2	yes	maximal
2	C₂×C₃₇×#462	C₂	1	4	yes	maximal
3	C₃₇×#2868	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#2016	C₂	1	4	yes	maximal
6	C₃₇×#1956	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₃₇	C₂²	1	8	yes	—
9	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
13	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
14	C₃₇×#494	—	2	8	no	—
15	C₃₇×#462	—	2	16	no	—
16	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
17	C₂×#103×C₃₇	C₂²	1	8	yes	—
18	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
19	C₃₇×#391	C₂²	1	2	yes	—
20	C₃₇×#389	—	2	2	no	—
21	C₃₇×#384	—	2	2	no	—
22	C₃₇×#384	—	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	4	yes	—
28	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
29	C₂×Q₁₆×C₃₇	—	2	16	no	—
30	C₂×Q₁₆×C₃₇	—	2	16	no	—
31	C₂×Q₁₆×C₃₇	—	4	8	no	—
32	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
33	C₄⋊Q₈×C₃₇	—	2	8	no	—
34	C₄⋊Q₈×C₃₇	—	2	4	no	—
35	C₄⋊Q₈×C₃₇	—	2	2	no	—
36	C₄⋊Q₈×C₃₇	—	2	4	no	—
37	C₄⋊Q₈×C₃₇	—	4	4	no	—
38	C₄⋊Q₈×C₃₇	D₄	1	4	yes	—
39	C₄×Q₈×C₃₇	—	2	8	no	—
40	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₃₇	—	2	2	no	—
49	C₂×C₄²×C₃₇	D₄	1	2	yes	—
50	C₂×C₄²×C₃₇	D₄	1	2	yes	—
51	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₃₇	D₄	1	8	yes	—
55	#103×C₃₇	—	2	16	no	—
56	C₄×C₈×C₃₇	—	2	4	no	—
57	#95×C₃₇	—	2	4	no	—
58	C₂×Q₈×C₃₇	—	2	2	no	—
59	C₂×Q₈×C₃₇	—	2	4	no	—
60	C₂×Q₈×C₃₇	—	2	8	no	—
61	C₂×Q₈×C₃₇	—	2	4	no	—
62	C₂×Q₈×C₃₇	—	2	4	no	—
63	C₂×Q₈×C₃₇	—	2	4	no	—
64	C₂×Q₈×C₃₇	—	2	8	no	—
65	C₂×Q₈×C₃₇	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₃₇	—	2	2	no	—
67	C₂²×C₄×C₃₇	—	2	2	no	—
68	C₂²×C₄×C₃₇	—	2	4	no	—
69	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
74	Q₁₆×C₃₇	—	4	32	no	—
75	C₂×C₈×C₃₇	—	2	4	no	—
76	C₂×C₈×C₃₇	—	2	4	no	—
77	C₂×C₈×C₃₇	C₂×D₄	1	8	yes	—
78	C₄⋊C₄×C₃₇	—	2	8	no	—
79	C₄⋊C₄×C₃₇	—	2	4	no	—
80	C₄⋊C₄×C₃₇	—	4	8	no	—
81	C₄⋊C₄×C₃₇	—	4	8	no	—
82	C₄⋊C₄×C₃₇	—	4	4	no	—
83	C₄⋊C₄×C₃₇	—	4	4	no	—
84	C₄⋊C₄×C₃₇	—	4	4	no	—
85	C₄⋊C₄×C₃₇	—	4	4	no	—
86	C₄²×C₃₇	—	2	4	no	—
87	C₄²×C₃₇	—	2	2	no	—
88	C₄²×C₃₇	—	4	4	no	—
89	C₄²×C₃₇	Q₁₆	1	8	yes	complemented
90	C₄²×C₃₇	C₂×D₄	1	2	yes	—
91	C₄²×C₃₇	C₂×D₄	1	4	yes	—
92	C₄²×C₃₇	#41	1	2	yes	—
93	C₂³×C₃₇	C₂²×D₄	1	1	yes	characteristic
94	Q₈×C₃₇	—	2	8	no	—
95	Q₈×C₃₇	—	4	4	no	—
96	Q₈×C₃₇	—	4	8	no	—
97	Q₈×C₃₇	—	4	8	no	—
98	C₂×C₄×C₃₇	—	2	2	no	—
99	C₂×C₄×C₃₇	—	2	2	no	—
100	C₂×C₄×C₃₇	—	2	2	no	—
101	C₂×C₄×C₃₇	—	2	4	no	—
102	C₂×C₄×C₃₇	—	4	2	no	—
103	C₂×C₄×C₃₇	—	4	2	no	—
104	C₂×C₄×C₃₇	—	4	4	no	—
105	C₂×C₄×C₃₇	—	4	2	no	—
106	C₂×C₄×C₃₇	—	4	4	no	—
107	C₂×C₄×C₃₇	—	4	4	no	—
108	C₂×C₄×C₃₇	—	4	4	no	—
109	C₂×C₄×C₃₇	#121	1	4	yes	—
110	C₂×C₄×C₃₇	#127	1	2	yes	—
111	C₂×C₄×C₃₇	#127	1	2	yes	—
112	C₂×C₄×C₃₇	C₂×Q₁₆	1	4	yes	—
113	C₂×C₄×C₃₇	C₂×Q₁₆	1	4	yes	—
114	C₂×C₄×C₃₇	C₂×Q₁₆	1	4	yes	—
115	C₂×C₄×C₃₇	#137	1	4	yes	—
116	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₃₇	C₂²×D₄	1	2	yes	—
118	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₃₇	H(5,𝔽₂)	1	2	yes	—
120	C₈×C₃₇	—	2	8	no	—
121	#28440	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
123	C₂²×C₃₇	C₂×#127	1	1	yes	characteristic
124	C₂²×C₃₇	#535	1	1	yes	characteristic
125	C₂²×C₃₇	C₂²×Q₁₆	1	2	yes	—
126	C₂²×C₃₇	C₂×#137	1	2	yes	—
127	C₄×C₃₇	—	2	2	no	—
128	C₄×C₃₇	—	2	4	no	—
129	C₄×C₃₇	—	4	4	no	—
130	C₄×C₃₇	—	8	2	no	—
131	C₄×C₃₇	—	8	4	no	—
132	C₄×C₃₇	#462	1	8	yes	complemented
133	C₄×C₃₇	#494	1	4	yes	complemented
134	C₄×C₃₇	D₄²	1	2	yes	—
135	C₄×C₃₇	#550	1	2	yes	—
136	C₂×#494	C₂×C₃₇	1	2	yes	—
137	C₂×#462	C₂×C₃₇	1	4	yes	—
138	#2868	C₂×C₃₇	1	1	yes	characteristic
139	C₄×C₄⋊Q₈	C₂×C₃₇	1	1	yes	characteristic
140	#2016	C₂×C₃₇	1	4	yes	—
141	#1956	C₂×C₃₇	1	2	yes	—
142	C₄×C₄⋊C₈	C₂×C₃₇	1	1	yes	characteristic
143	C₂×C₃₇	#2541	1	1	yes	characteristic
144	C₂×C₃₇	C₂×#462	1	2	yes	—
145	C₂×C₃₇	C₂×#494	1	1	yes	characteristic
146	C₂×C₃₇	#3152	1	1	yes	characteristic
147	C₂×C₃₇	#3209	1	2	yes	—
148	C₂²×Q₁₆	C₂²×C₃₇	1	8	yes	—
149	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
150	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
151	C₂×C₄⋊Q₈	C₂²×C₃₇	1	1	yes	characteristic
152	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
153	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
154	#494	—	2	8	no	—
155	#462	—	2	16	no	—
156	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
157	C₂×#103	C₂²×C₃₇	1	8	yes	—
158	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
159	#391	C₂²×C₃₇	1	2	yes	—
160	#389	—	2	2	no	—
161	#384	—	2	2	no	—
162	#384	—	2	2	no	—
163	#384	C₂²×C₃₇	1	1	yes	characteristic
164	C₄×C₄⋊C₄	—	2	2	no	—
165	C₄³	C₂²×C₃₇	1	1	yes	characteristic
166	#336	C₂²×C₃₇	1	2	yes	—
167	C₃₇	#28440	1	1	yes	characteristic, direct product factor, Sylow
168	C₂²×Q₈	C₂³×C₃₇	1	4	yes	—
169	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
170	C₂×Q₁₆	—	2	16	no	—
171	C₂×Q₁₆	—	2	16	no	—
172	C₂×Q₁₆	—	4	8	no	—
173	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
174	C₄⋊Q₈	—	2	8	no	—
175	C₄⋊Q₈	—	2	4	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	—	2	4	no	—
178	C₄⋊Q₈	—	4	4	no	—
179	C₄⋊Q₈	D₄×C₃₇	1	4	yes	—
180	C₄×Q₈	—	2	8	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	D₄×C₃₇	1	2	yes	—
191	C₂×C₄²	D₄×C₃₇	1	2	yes	—
192	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
195	C₄⋊C₈	D₄×C₃₇	1	8	yes	—
196	#103	—	2	16	no	—
197	C₄×C₈	—	2	4	no	—
198	#95	—	2	4	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	2	8	no	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	2	4	no	—
205	C₂×Q₈	—	2	8	no	—
206	C₂×Q₈	C₂×D₄×C₃₇	1	4	yes	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
212	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
214	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
215	Q₁₆	—	4	32	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	C₂×D₄×C₃₇	1	8	yes	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	8	no	—
222	C₄⋊C₄	—	4	8	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	4	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄²	—	2	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	4	4	no	—
230	C₄²	Q₁₆×C₃₇	1	8	yes	complemented
231	C₄²	C₂×D₄×C₃₇	1	2	yes	—
232	C₄²	C₂×D₄×C₃₇	1	4	yes	—
233	C₄²	#41×C₃₇	1	2	yes	—
234	C₂³	C₂²×D₄×C₃₇	1	1	yes	characteristic
235	Q₈	—	2	8	no	—
236	Q₈	—	4	4	no	—
237	Q₈	—	4	8	no	—
238	Q₈	—	4	8	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	#121×C₃₇	1	4	yes	—
251	C₂×C₄	#127×C₃₇	1	2	yes	—
252	C₂×C₄	#127×C₃₇	1	2	yes	—
253	C₂×C₄	C₂×Q₁₆×C₃₇	1	4	yes	—
254	C₂×C₄	C₂×Q₁₆×C₃₇	1	4	yes	—
255	C₂×C₄	C₂×Q₁₆×C₃₇	1	4	yes	—
256	C₂×C₄	#137×C₃₇	1	4	yes	—
257	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₄×C₃₇	1	2	yes	—
259	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
260	C₂×C₄	H(5,𝔽₂)×C₃₇	1	2	yes	—
261	C₈	—	2	8	no	—
262	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
263	C₂²	C₂×#127×C₃₇	1	1	yes	characteristic
264	C₂²	C₃₇×#535	1	1	yes	characteristic
265	C₂²	C₂²×Q₁₆×C₃₇	1	2	yes	—
266	C₂²	C₂×#137×C₃₇	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	2	4	no	—
269	C₄	—	4	4	no	—
270	C₄	—	8	2	no	—
271	C₄	—	8	4	no	—
272	C₄	C₃₇×#462	1	8	yes	complemented
273	C₄	C₃₇×#494	1	4	yes	complemented
274	C₄	D₄²×C₃₇	1	2	yes	—
275	C₄	C₃₇×#550	1	2	yes	—
276	C₂	C₃₇×#2541	1	1	yes	characteristic
277	C₂	C₂×C₃₇×#462	1	2	yes	—
278	C₂	C₂×C₃₇×#494	1	1	yes	characteristic
279	C₂	C₃₇×#3152	1	1	yes	characteristic
280	C₂	C₃₇×#3209	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
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
89. C₄²×C₃₇	Q₁₆	semidirect
121. #28440	C₃₇	direct (N × H)
132. C₄×C₃₇	#462	semidirect
133. C₄×C₃₇	#494	semidirect
167. C₃₇	#28440	direct (N × H)
230. C₄²	Q₁₆×C₃₇	semidirect
272. C₄	C₃₇×#462	semidirect
273. C₄	C₃₇×#494	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₇×#494	2
C₂×C₃₇×#462	2
C₃₇×#2868	2
C₄×C₄⋊Q₈×C₃₇	2
C₃₇×#2016	2
C₃₇×#1956	2
C₄×C₄⋊C₈×C₃₇	2
#28440	37

Subgroups — C37×#28440 #22649693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28440 #22649693