Subgroups — C₃₇×#28300 #22649553

Order 9472.

666 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#3078	C₂	1	2	yes	maximal
2	C₃₇×#2961	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₇×#441	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#2064	C₂	1	4	yes	maximal
5	C₃₇×#1979	C₂	1	2	yes	maximal
6	C₃₇×#1951	C₂	1	2	yes	maximal
7	C₃₇×#1849	C₂	1	1	yes	maximal, characteristic
8	C₃₇×#1801	C₂	1	2	yes	maximal
9	C₂²×Q₁₆×C₃₇	C₂²	1	1	yes	characteristic
10	C₃₇×#533	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₃₇	C₂²	1	2	yes	—
12	C₃₇×#479	—	2	4	no	—
13	C₃₇×#477	—	2	2	no	—
14	C₃₇×#475	—	2	2	no	—
15	C₃₇×#446	—	2	4	no	—
16	C₃₇×#441	—	2	4	no	—
17	C₂×#108×C₃₇	C₂²	1	2	yes	—
18	C₂×#108×C₃₇	C₂²	1	2	yes	—
19	C₃₇×#428	C₂²	1	4	yes	—
20	C₂×#106×C₃₇	C₂²	1	1	yes	characteristic
21	C₃₇×#424	C₂²	1	2	yes	—
22	C₃₇×#423	C₂²	1	2	yes	—
23	C₃₇×#423	C₂²	1	1	yes	characteristic
24	C₃₇×#419	C₂²	1	4	yes	—
25	C₂×#103×C₃₇	C₂²	1	2	yes	—
26	C₂×#101×C₃₇	C₂²	1	4	yes	—
27	C₃₇×#405	C₂²	1	2	yes	—
28	C₂×#97×C₃₇	C₂²	1	1	yes	characteristic
29	C₃₇×#343	C₂²	1	4	yes	—
30	C₂²×Q₈×C₃₇	C₂³	1	2	yes	—
31	C₂×Q₁₆×C₃₇	—	2	2	no	—
32	C₂×Q₁₆×C₃₇	—	2	2	no	—
33	C₂×Q₁₆×C₃₇	Q₈	1	4	yes	complemented
34	C₂×M₁₆×C₃₇	C₂³	1	2	yes	—
35	C₂×M₁₆×C₃₇	C₂³	1	4	yes	—
36	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₃₇	D₄	1	2	yes	—
39	#125×C₃₇	D₄	1	2	yes	—
40	C₂²⋊Q₈×C₃₇	—	2	2	no	—
41	C₄×Q₈×C₃₇	—	2	2	no	—
42	C₄×Q₈×C₃₇	—	2	2	no	—
43	C₄×Q₈×C₃₇	—	2	2	no	—
44	C₄×Q₈×C₃₇	—	2	2	no	—
45	C₄×Q₈×C₃₇	—	2	2	no	—
46	#117×C₃₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₃₇	—	2	2	no	—
50	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
51	#108×C₃₇	—	2	4	no	—
52	#108×C₃₇	—	4	2	no	—
53	#107×C₃₇	—	2	4	no	—
54	#106×C₃₇	—	2	2	no	—
55	#106×C₃₇	—	2	4	no	—
56	C₄⋊C₈×C₃₇	—	2	4	no	—
57	C₄⋊C₈×C₃₇	—	2	2	no	—
58	C₄⋊C₈×C₃₇	—	2	2	no	—
59	C₄⋊C₈×C₃₇	—	2	2	no	—
60	#103×C₃₇	—	2	4	no	—
61	#103×C₃₇	—	2	4	no	—
62	#103×C₃₇	—	2	4	no	—
63	#101×C₃₇	—	4	4	no	—
64	#97×C₃₇	—	2	2	no	—
65	#97×C₃₇	D₄	1	4	yes	—
66	C₄×C₈×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	2	no	—
71	C₂×Q₈×C₃₇	—	2	2	no	—
72	C₂×Q₈×C₃₇	—	2	2	no	—
73	C₂×Q₈×C₃₇	C₂×Q₈	1	4	yes	—
74	C₂²×C₄×C₃₇	—	2	2	no	—
75	C₂²×C₄×C₃₇	—	2	2	no	—
76	C₂²×C₄×C₃₇	#41	1	2	yes	—
77	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
78	Q₁₆×C₃₇	—	2	8	no	—
79	M₁₆×C₃₇	—	2	4	no	—
80	M₁₆×C₃₇	—	2	2	no	—
81	M₁₆×C₃₇	—	2	4	no	—
82	M₁₆×C₃₇	—	2	2	no	—
83	C₂×C₈×C₃₇	—	2	1	no	—
84	C₂×C₈×C₃₇	—	2	1	no	—
85	C₂×C₈×C₃₇	—	2	2	no	—
86	C₂×C₈×C₃₇	—	2	4	no	—
87	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₃₇	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₃₇	#41	1	2	yes	—
91	C₄⋊C₄×C₃₇	—	2	4	no	—
92	C₄⋊C₄×C₃₇	—	2	2	no	—
93	C₄⋊C₄×C₃₇	—	2	2	no	—
94	C₄⋊C₄×C₃₇	—	2	2	no	—
95	C₄⋊C₄×C₃₇	—	4	2	no	—
96	C₄⋊C₄×C₃₇	—	4	2	no	—
97	C₄⋊C₄×C₃₇	C₂×D₄	1	4	yes	—
98	C₂²⋊C₄×C₃₇	—	2	2	no	—
99	C₄²×C₃₇	—	2	2	no	—
100	C₄²×C₃₇	—	4	2	no	—
101	C₄²×C₃₇	—	4	2	no	—
102	C₄²×C₃₇	C₂×D₄	1	2	yes	—
103	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₃₇	C₂×#41	1	1	yes	characteristic
106	Q₈×C₃₇	—	2	4	no	—
107	Q₈×C₃₇	—	2	4	no	—
108	Q₈×C₃₇	—	4	2	no	—
109	Q₈×C₃₇	—	4	2	no	—
110	Q₈×C₃₇	—	8	2	no	—
111	C₂×C₄×C₃₇	—	2	4	no	—
112	C₂×C₄×C₃₇	—	2	4	no	—
113	C₂×C₄×C₃₇	—	4	2	no	—
114	C₂×C₄×C₃₇	—	4	2	no	—
115	C₂×C₄×C₃₇	—	4	2	no	—
116	C₂×C₄×C₃₇	—	4	2	no	—
117	C₂×C₄×C₃₇	—	8	2	no	—
118	C₂×C₄×C₃₇	#121	1	2	yes	—
119	C₂×C₄×C₃₇	#121	1	2	yes	—
120	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₃₇	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
126	C₈×C₃₇	—	2	2	no	—
127	C₈×C₃₇	—	4	2	no	—
128	C₈×C₃₇	—	4	4	no	—
129	C₈×C₃₇	C₂²⋊Q₈	1	4	yes	—
130	#28300	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₃₇	—	2	2	no	—
132	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
133	C₂²×C₃₇	#554	1	1	yes	characteristic
134	C₂²×C₃₇	#556	1	1	yes	characteristic
135	C₄×C₃₇	—	2	2	no	—
136	C₄×C₃₇	—	2	2	no	—
137	C₄×C₃₇	—	4	4	no	—
138	C₄×C₃₇	—	4	4	no	—
139	C₄×C₃₇	—	8	2	no	—
140	C₄×C₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
141	C₄×C₃₇	#537	1	1	yes	characteristic
142	C₄×C₃₇	D₄×Q₈	1	1	yes	characteristic
143	C₄×C₃₇	#550	1	1	yes	characteristic
144	#3078	C₂×C₃₇	1	2	yes	—
145	#2961	C₂×C₃₇	1	1	yes	characteristic
146	C₂×#441	C₂×C₃₇	1	1	yes	characteristic
147	#2064	C₂×C₃₇	1	4	yes	—
148	#1979	C₂×C₃₇	1	2	yes	—
149	#1951	C₂×C₃₇	1	2	yes	—
150	#1849	C₂×C₃₇	1	1	yes	characteristic
151	#1801	C₂×C₃₇	1	2	yes	—
152	C₂×C₃₇	—	2	1	no	—
153	C₂×C₃₇	—	2	1	no	—
154	C₂×C₃₇	#2538	1	1	yes	characteristic
155	C₂×C₃₇	#3069	1	2	yes	—
156	C₂²×Q₁₆	C₂²×C₃₇	1	1	yes	characteristic
157	#533	C₂²×C₃₇	1	1	yes	characteristic
158	C₂×C₄×Q₈	C₂²×C₃₇	1	2	yes	—
159	#479	—	2	4	no	—
160	#477	—	2	2	no	—
161	#475	—	2	2	no	—
162	#446	—	2	4	no	—
163	#441	—	2	4	no	—
164	C₂×#108	C₂²×C₃₇	1	2	yes	—
165	C₂×#108	C₂²×C₃₇	1	2	yes	—
166	#428	C₂²×C₃₇	1	4	yes	—
167	C₂×#106	C₂²×C₃₇	1	1	yes	characteristic
168	#424	C₂²×C₃₇	1	2	yes	—
169	#423	C₂²×C₃₇	1	2	yes	—
170	#423	C₂²×C₃₇	1	1	yes	characteristic
171	#419	C₂²×C₃₇	1	4	yes	—
172	C₂×#103	C₂²×C₃₇	1	2	yes	—
173	C₂×#101	C₂²×C₃₇	1	4	yes	—
174	#405	C₂²×C₃₇	1	2	yes	—
175	C₂×#97	C₂²×C₃₇	1	1	yes	characteristic
176	#343	C₂²×C₃₇	1	4	yes	—
177	C₃₇	#28300	1	1	yes	characteristic, direct product factor, Sylow
178	C₂²×Q₈	C₂³×C₃₇	1	2	yes	—
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	—	2	2	no	—
181	C₂×Q₁₆	Q₈×C₃₇	1	4	yes	complemented
182	C₂×M₁₆	C₂³×C₃₇	1	2	yes	—
183	C₂×M₁₆	C₂³×C₃₇	1	4	yes	—
184	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
186	C₄⋊Q₈	D₄×C₃₇	1	2	yes	—
187	#125	D₄×C₃₇	1	2	yes	—
188	C₂²⋊Q₈	—	2	2	no	—
189	C₄×Q₈	—	2	2	no	—
190	C₄×Q₈	—	2	2	no	—
191	C₄×Q₈	—	2	2	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	—	2	2	no	—
194	#117	C₂³×C₃₇	1	2	yes	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	C₂³×C₃₇	1	2	yes	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
199	#108	—	2	4	no	—
200	#108	—	4	2	no	—
201	#107	—	2	4	no	—
202	#106	—	2	2	no	—
203	#106	—	2	4	no	—
204	C₄⋊C₈	—	2	4	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	—	2	2	no	—
208	#103	—	2	4	no	—
209	#103	—	2	4	no	—
210	#103	—	2	4	no	—
211	#101	—	4	4	no	—
212	#97	—	2	2	no	—
213	#97	D₄×C₃₇	1	4	yes	—
214	C₄×C₈	—	2	2	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	C₂×Q₈×C₃₇	1	4	yes	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	#41×C₃₇	1	2	yes	—
225	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
226	Q₁₆	—	2	8	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	2	2	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
236	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
237	C₂×C₈	C₂×Q₈×C₃₇	1	2	yes	—
238	C₂×C₈	#41×C₃₇	1	2	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	C₂×D₄×C₃₇	1	4	yes	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	4	2	no	—
249	C₄²	—	4	2	no	—
250	C₄²	C₂×D₄×C₃₇	1	2	yes	—
251	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
252	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
253	C₂³	C₂×#41×C₃₇	1	1	yes	characteristic
254	Q₈	—	2	4	no	—
255	Q₈	—	2	4	no	—
256	Q₈	—	4	2	no	—
257	Q₈	—	4	2	no	—
258	Q₈	—	8	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	#121×C₃₇	1	2	yes	—
267	C₂×C₄	#121×C₃₇	1	2	yes	—
268	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
269	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
270	C₂×C₄	C₂²×Q₈×C₃₇	1	1	yes	characteristic
271	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
272	C₂×C₄	H(5,𝔽₂)×C₃₇	1	1	yes	characteristic
273	C₂×C₄	2₋¹⁺⁴×C₃₇	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	4	no	—
277	C₈	C₂²⋊Q₈×C₃₇	1	4	yes	—
278	C₂²	—	2	2	no	—
279	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
280	C₂²	C₃₇×#554	1	1	yes	characteristic
281	C₂²	C₃₇×#556	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	4	no	—
285	C₄	—	4	4	no	—
286	C₄	—	8	2	no	—
287	C₄	C₂×C₂²⋊Q₈×C₃₇	1	1	yes	characteristic
288	C₄	C₃₇×#537	1	1	yes	characteristic
289	C₄	D₄×Q₈×C₃₇	1	1	yes	characteristic
290	C₄	C₃₇×#550	1	1	yes	characteristic
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	C₃₇×#2538	1	1	yes	characteristic
294	C₂	C₃₇×#3069	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
prime radical	C₂³×C₃₇	C₂×#41
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
33. C₂×Q₁₆×C₃₇	Q₈	semidirect
130. #28300	C₃₇	direct (N × H)
177. C₃₇	#28300	direct (N × H)
181. C₂×Q₁₆	Q₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃₇×#3078	2
C₃₇×#2961	2
C₂×C₃₇×#441	2
C₃₇×#2064	2
C₃₇×#1979	2
C₃₇×#1951	2
C₃₇×#1849	2
C₃₇×#1801	2
#28300	37

Subgroups — C37×#28300 #22649553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#28300 #22649553