Subgroups — C₂×C₃₇×#1694 #22640401

Order 9472.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#462	C₂	1	2	yes	maximal
3	C₂×C₃₇×#446	C₂	1	1	yes	maximal, characteristic
4	C₃₇×#1694	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₇×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₇×#332	C₂	1	2	yes	maximal
7	C₂²×Q₁₆×C₃₇	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₃₇	C₂²	1	2	yes	—
9	C₂×C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
10	C₃₇×#500	—	2	2	no	—
11	C₃₇×#500	C₂²	1	4	yes	—
12	C₃₇×#462	—	2	4	no	—
13	C₃₇×#462	C₂²	1	8	yes	—
14	C₃₇×#446	—	2	2	no	—
15	C₃₇×#446	C₂²	1	4	yes	—
16	C₂×#107×C₃₇	—	2	1	no	—
17	C₂×#107×C₃₇	—	2	1	no	—
18	C₂×C₄⋊C₈×C₃₇	—	2	1	no	—
19	C₂×C₄⋊C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
21	C₂×#103×C₃₇	—	2	2	no	—
22	C₂×#97×C₃₇	—	2	1	no	—
23	C₂×C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
24	C₃₇×#334	C₂²	1	4	yes	—
25	C₃₇×#332	C₂²	1	8	yes	—
26	C₂²×Q₈×C₃₇	—	2	2	no	—
27	C₂²×Q₈×C₃₇	D₄	1	1	yes	characteristic
28	C₂×Q₁₆×C₃₇	—	2	8	no	—
29	C₂×Q₁₆×C₃₇	—	4	8	no	—
30	C₂²×C₈×C₃₇	—	2	2	no	—
31	C₂²×C₈×C₃₇	—	2	1	no	—
32	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₇	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₃₇	—	2	2	no	—
35	C₄⋊Q₈×C₃₇	—	2	2	no	—
36	C₄⋊Q₈×C₃₇	—	2	2	no	—
37	C₄⋊Q₈×C₃₇	C₂³	1	4	yes	—
38	C₄×Q₈×C₃₇	—	2	1	no	—
39	C₄×Q₈×C₃₇	—	2	1	no	—
40	C₄×Q₈×C₃₇	D₄	1	2	yes	—
41	C₄×Q₈×C₃₇	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
43	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₃₇	—	2	1	no	—
47	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
48	#107×C₃₇	—	2	4	no	—
49	#107×C₃₇	—	2	4	no	—
50	C₄⋊C₈×C₃₇	—	2	4	no	—
51	C₄⋊C₈×C₃₇	—	2	1	no	—
52	C₄⋊C₈×C₃₇	—	2	2	no	—
53	C₄⋊C₈×C₃₇	C₂³	1	2	yes	—
54	C₄⋊C₈×C₃₇	C₂³	1	4	yes	—
55	#103×C₃₇	—	2	8	no	—
56	#97×C₃₇	—	2	4	no	—
57	C₄×C₈×C₃₇	D₄	1	2	yes	—
58	C₄×C₈×C₃₇	C₂³	1	2	yes	—
59	C₂×Q₈×C₃₇	—	2	1	no	—
60	C₂×Q₈×C₃₇	—	2	4	no	—
61	C₂×Q₈×C₃₇	—	2	4	no	—
62	C₂×Q₈×C₃₇	—	4	2	no	—
63	C₂×Q₈×C₃₇	—	4	4	no	—
64	C₂×Q₈×C₃₇	—	4	4	no	—
65	C₂×Q₈×C₃₇	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₃₇	—	2	1	no	—
67	C₂²×C₄×C₃₇	—	2	1	no	—
68	C₂²×C₄×C₃₇	—	4	2	no	—
69	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₇	#41	1	1	yes	characteristic
72	Q₁₆×C₃₇	—	4	16	no	—
73	C₂×C₈×C₃₇	—	2	2	no	—
74	C₂×C₈×C₃₇	—	2	2	no	—
75	C₂×C₈×C₃₇	—	2	4	no	—
76	C₂×C₈×C₃₇	—	4	4	no	—
77	C₂×C₈×C₃₇	—	4	2	no	—
78	C₂×C₈×C₃₇	Q₁₆	1	4	yes	complemented
79	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
81	C₄⋊C₄×C₃₇	—	2	2	no	—
82	C₄⋊C₄×C₃₇	—	2	1	no	—
83	C₄⋊C₄×C₃₇	—	2	2	no	—
84	C₄⋊C₄×C₃₇	—	2	4	no	—
85	C₄⋊C₄×C₃₇	—	2	4	no	—
86	C₄⋊C₄×C₃₇	—	4	2	no	—
87	C₄⋊C₄×C₃₇	—	4	2	no	—
88	C₄⋊C₄×C₃₇	C₂×D₄	1	2	yes	—
89	C₄²×C₃₇	—	2	2	no	—
90	C₄²×C₃₇	—	2	2	no	—
91	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₃₇	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₃₇	#41	1	1	yes	characteristic
94	C₄²×C₃₇	C₂⁴	1	1	yes	characteristic
95	C₂³×C₃₇	#121	1	1	yes	characteristic
96	Q₈×C₃₇	—	2	4	no	—
97	Q₈×C₃₇	—	4	2	no	—
98	Q₈×C₃₇	—	4	4	no	—
99	Q₈×C₃₇	—	4	4	no	—
100	Q₈×C₃₇	—	8	4	no	—
101	C₂×C₄×C₃₇	—	2	2	no	—
102	C₂×C₄×C₃₇	—	2	2	no	—
103	C₂×C₄×C₃₇	—	2	2	no	—
104	C₂×C₄×C₃₇	—	4	2	no	—
105	C₂×C₄×C₃₇	—	4	2	no	—
106	C₂×C₄×C₃₇	—	4	4	no	—
107	C₂×C₄×C₃₇	—	8	4	no	—
108	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
109	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
110	C₂×C₄×C₃₇	#121	1	1	yes	characteristic
111	C₂×C₄×C₃₇	C₂×Q₁₆	1	2	yes	—
112	C₂×C₄×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
113	C₂×C₄×C₃₇	#137	1	2	yes	—
114	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₇	C₂×#41	1	1	yes	characteristic
117	C₈×C₃₇	—	2	2	no	—
118	C₈×C₃₇	—	4	4	no	—
119	C₈×C₃₇	—	4	2	no	—
120	C₈×C₃₇	C₂×Q₁₆	1	4	yes	complemented
121	C₂×#1694	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂²×C₃₇	#462	1	2	yes	—
123	C₂²×C₃₇	#469	1	2	yes	—
124	C₂²×C₃₇	#473	1	2	yes	—
125	C₂²×C₃₇	C₂×#121	1	1	yes	characteristic
126	C₄×C₃₇	—	2	1	no	—
127	C₄×C₃₇	—	2	1	no	—
128	C₄×C₃₇	—	4	2	no	—
129	C₄×C₃₇	—	4	2	no	—
130	C₄×C₃₇	—	8	4	no	—
131	C₄×C₃₇	#462	1	2	yes	—
132	C₄×C₃₇	#469	1	2	yes	—
133	C₄×C₃₇	C₂²×Q₁₆	1	1	yes	characteristic
134	C₄×C₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
135	C₄×C₃₇	C₂×#137	1	1	yes	characteristic
136	C₂×#500	C₂×C₃₇	1	1	yes	characteristic
137	C₂×#462	C₂×C₃₇	1	2	yes	—
138	C₂×#446	C₂×C₃₇	1	1	yes	characteristic
139	#1694	C₂×C₃₇	1	8	yes	direct product factor
140	C₂×#334	C₂×C₃₇	1	1	yes	characteristic
141	C₂×#332	C₂×C₃₇	1	2	yes	—
142	C₂×C₃₇	#1694	1	4	yes	direct product factor
143	C₂×C₃₇	C₂×#462	1	1	yes	characteristic
144	C₂×C₃₇	C₂×#469	1	1	yes	characteristic
145	C₂×C₃₇	C₂×#473	1	1	yes	characteristic
146	C₂²×Q₁₆	—	2	2	no	—
147	C₂×C₄⋊Q₈	C₂²×C₃₇	1	2	yes	—
148	C₂×C₄×Q₈	C₂²×C₃₇	1	1	yes	characteristic
149	#500	—	2	2	no	—
150	#500	C₂²×C₃₇	1	4	yes	—
151	#462	—	2	4	no	—
152	#462	C₂²×C₃₇	1	8	yes	—
153	#446	—	2	2	no	—
154	#446	C₂²×C₃₇	1	4	yes	—
155	C₂×#107	—	2	1	no	—
156	C₂×#107	—	2	1	no	—
157	C₂×C₄⋊C₈	—	2	1	no	—
158	C₂×C₄⋊C₈	C₂²×C₃₇	1	1	yes	characteristic
159	C₂×C₄⋊C₈	C₂²×C₃₇	1	2	yes	—
160	C₂×#103	—	2	2	no	—
161	C₂×#97	—	2	1	no	—
162	C₂×C₄×C₈	C₂²×C₃₇	1	1	yes	characteristic
163	#334	C₂²×C₃₇	1	4	yes	—
164	#332	C₂²×C₃₇	1	8	yes	—
165	C₃₇	C₂×#1694	1	1	yes	characteristic, direct product factor, Sylow
166	C₂²×Q₈	—	2	2	no	—
167	C₂²×Q₈	D₄×C₃₇	1	1	yes	characteristic
168	C₂×Q₁₆	—	2	8	no	—
169	C₂×Q₁₆	—	4	8	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
173	C₂²×C₈	D₄×C₃₇	1	1	yes	characteristic
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	—	2	2	no	—
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	C₂³×C₃₇	1	4	yes	—
178	C₄×Q₈	—	2	1	no	—
179	C₄×Q₈	—	2	1	no	—
180	C₄×Q₈	D₄×C₃₇	1	2	yes	—
181	C₄×Q₈	C₂³×C₃₇	1	2	yes	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	D₄×C₃₇	1	1	yes	characteristic
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
188	#107	—	2	4	no	—
189	#107	—	2	4	no	—
190	C₄⋊C₈	—	2	4	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	C₂³×C₃₇	1	2	yes	—
194	C₄⋊C₈	C₂³×C₃₇	1	4	yes	—
195	#103	—	2	8	no	—
196	#97	—	2	4	no	—
197	C₄×C₈	D₄×C₃₇	1	2	yes	—
198	C₄×C₈	C₂³×C₃₇	1	2	yes	—
199	C₂×Q₈	—	2	1	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×Q₈	—	4	4	no	—
205	C₂×Q₈	C₂×D₄×C₃₇	1	2	yes	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₃₇	1	1	yes	characteristic
211	C₂²×C₄	#41×C₃₇	1	1	yes	characteristic
212	Q₁₆	—	4	16	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	4	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	Q₁₆×C₃₇	1	4	yes	complemented
219	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
220	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	C₂×D₄×C₃₇	1	2	yes	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
232	C₄²	C₂×D₄×C₃₇	1	1	yes	characteristic
233	C₄²	#41×C₃₇	1	1	yes	characteristic
234	C₄²	C₂⁴×C₃₇	1	1	yes	characteristic
235	C₂³	#121×C₃₇	1	1	yes	characteristic
236	Q₈	—	2	4	no	—
237	Q₈	—	4	2	no	—
238	Q₈	—	4	4	no	—
239	Q₈	—	4	4	no	—
240	Q₈	—	8	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
249	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
250	C₂×C₄	#121×C₃₇	1	1	yes	characteristic
251	C₂×C₄	C₂×Q₁₆×C₃₇	1	2	yes	—
252	C₂×C₄	AGL(1,ℤ/8)×C₃₇	1	4	yes	—
253	C₂×C₄	#137×C₃₇	1	2	yes	—
254	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₄×C₃₇	1	1	yes	characteristic
256	C₂×C₄	C₂×#41×C₃₇	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	4	4	no	—
259	C₈	—	4	2	no	—
260	C₈	C₂×Q₁₆×C₃₇	1	4	yes	complemented
261	C₂²	C₃₇×#462	1	2	yes	—
262	C₂²	C₃₇×#469	1	2	yes	—
263	C₂²	C₃₇×#473	1	2	yes	—
264	C₂²	C₂×#121×C₃₇	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	8	4	no	—
270	C₄	C₃₇×#462	1	2	yes	—
271	C₄	C₃₇×#469	1	2	yes	—
272	C₄	C₂²×Q₁₆×C₃₇	1	1	yes	characteristic
273	C₄	C₂×AGL(1,ℤ/8)×C₃₇	1	2	yes	—
274	C₄	C₂×#137×C₃₇	1	1	yes	characteristic
275	C₂	C₃₇×#1694	1	4	yes	direct product factor
276	C₂	C₂×C₃₇×#462	1	1	yes	characteristic
277	C₂	C₂×C₃₇×#469	1	1	yes	characteristic
278	C₂	C₂×C₃₇×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₇×#1694	C₂	direct (N × H)
78. C₂×C₈×C₃₇	Q₁₆	semidirect
120. C₈×C₃₇	C₂×Q₁₆	semidirect
121. C₂×#1694	C₃₇	direct (N × H)
139. #1694	C₂×C₃₇	direct (N × H)
142. C₂×C₃₇	#1694	direct (N × H)
165. C₃₇	C₂×#1694	direct (N × H)
218. C₂×C₈	Q₁₆×C₃₇	semidirect
260. C₈	C₂×Q₁₆×C₃₇	semidirect
275. C₂	C₃₇×#1694	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×#500	2
C₂×C₃₇×#462	2
C₂×C₃₇×#446	2
C₃₇×#1694	2
C₂×C₃₇×#334	2
C₂×C₃₇×#332	2
C₂×#1694	37

Subgroups — C2×C37×#1694 #22640401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×#1694 #22640401