Subgroups — C₄×C₃₇×#433 #22641717

Order 9472.

970 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#433	C₂	1	2	yes	maximal
2	C₄×#117×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#1768	C₂	1	2	yes	maximal, complemented
4	C₄×C₄⋊C₈×C₃₇	C₂	1	2	yes	maximal, complemented
5	C₃₇×#1750	C₂	1	2	yes	maximal
6	C₄×C₂²⋊C₈×C₃₇	C₂	1	2	yes	maximal
7	C₃₇×#1746	C₂	1	2	yes	maximal, complemented
8	C₄×#97×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄²×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#117×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₃₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₃₇	C₂²	1	1	yes	characteristic
13	C₃₇×#433	C₄	1	16	yes	direct product factor
14	C₂×C₄⋊C₈×C₃₇	C₂²	1	4	yes	—
15	C₂×C₂²⋊C₈×C₃₇	C₂²	1	4	yes	—
16	C₂×#97×C₃₇	C₂²	1	2	yes	—
17	C₂×#97×C₃₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₇	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₃₇	C₄	1	4	yes	—
23	C₄×C₂²⋊C₄×C₃₇	C₄	1	4	yes	—
24	C₃₇×#376	C₂²	1	1	yes	characteristic
25	C₃₇×#376	C₂²	1	1	yes	characteristic
26	C₄³×C₃₇	C₂²	1	1	yes	characteristic
27	C₄³×C₃₇	C₂²	1	1	yes	characteristic
28	C₃₇×#336	C₂²	1	4	yes	—
29	C₃₇×#336	C₂²	1	4	yes	—
30	C₂³×C₄×C₃₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
33	C₂²×C₈×C₃₇	C₂³	1	4	yes	—
34	#117×C₃₇	—	2	4	no	—
35	#117×C₃₇	C₂×C₄	1	8	yes	—
36	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
38	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₃₇	—	2	4	no	—
41	C₂×C₄²×C₃₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₃₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₃₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₃₇	C₂×C₄	1	16	yes	complemented
51	C₂²⋊C₈×C₃₇	C₂×C₄	1	16	yes	—
52	#97×C₃₇	—	2	4	no	—
53	#97×C₃₇	C₂×C₄	1	8	yes	complemented
54	C₄×C₈×C₃₇	—	2	4	no	—
55	C₄×C₈×C₃₇	—	2	4	no	—
56	C₄×C₈×C₃₇	—	2	4	no	—
57	C₄×C₈×C₃₇	C₂×C₄	1	8	yes	complemented
58	#95×C₃₇	C₂×C₄	1	4	yes	—
59	#95×C₃₇	C₂×C₄	1	4	yes	—
60	C₂⁴×C₃₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₇	—	2	2	no	—
62	C₂²×C₄×C₃₇	—	2	2	no	—
63	C₂²×C₄×C₃₇	—	2	4	no	—
64	C₂²×C₄×C₃₇	—	2	8	no	—
65	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₃₇	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₃₇	—	2	4	no	—
75	C₂×C₈×C₃₇	—	2	4	no	—
76	C₂×C₈×C₃₇	—	2	4	no	—
77	C₂×C₈×C₃₇	—	2	4	no	—
78	C₂×C₈×C₃₇	C₂²×C₄	1	8	yes	—
79	C₂×C₈×C₃₇	C₂²×C₄	1	8	yes	—
80	C₄⋊C₄×C₃₇	—	2	4	no	—
81	C₄⋊C₄×C₃₇	C₄²	1	8	yes	—
82	C₂²⋊C₄×C₃₇	—	2	4	no	—
83	C₂²⋊C₄×C₃₇	C₄²	1	8	yes	—
84	C₄²×C₃₇	—	2	4	no	—
85	C₄²×C₃₇	—	2	8	no	—
86	C₄²×C₃₇	—	2	8	no	—
87	C₄²×C₃₇	—	2	2	no	—
88	C₄²×C₃₇	—	2	2	no	—
89	C₄²×C₃₇	C₂²×C₄	1	4	yes	—
90	C₄²×C₃₇	C₂²×C₄	1	4	yes	—
91	C₄²×C₃₇	#41	1	4	yes	—
92	C₄²×C₃₇	#41	1	4	yes	—
93	C₂³×C₃₇	—	2	1	no	—
94	C₂³×C₃₇	—	2	1	no	—
95	C₂³×C₃₇	—	2	4	no	—
96	C₂³×C₃₇	C₂×C₄²	1	2	yes	—
97	C₂³×C₃₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₃₇	—	2	1	no	—
99	C₂×C₄×C₃₇	—	2	1	no	—
100	C₂×C₄×C₃₇	—	2	4	no	—
101	C₂×C₄×C₃₇	—	2	4	no	—
102	C₂×C₄×C₃₇	—	2	4	no	—
103	C₂×C₄×C₃₇	—	2	8	no	—
104	C₂×C₄×C₃₇	—	2	2	no	—
105	C₂×C₄×C₃₇	—	2	2	no	—
106	C₂×C₄×C₃₇	—	2	4	no	—
107	C₂×C₄×C₃₇	—	2	4	no	—
108	C₂×C₄×C₃₇	—	4	8	no	—
109	C₂×C₄×C₃₇	C₂×C₄²	1	4	yes	—
110	C₂×C₄×C₃₇	C₂×C₄²	1	4	yes	—
111	C₂×C₄×C₃₇	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₃₇	#117	1	4	yes	—
113	C₂×C₄×C₃₇	#117	1	4	yes	—
114	C₂×C₄×C₃₇	#131	1	8	yes	—
115	C₂×C₄×C₃₇	C₂³×C₄	1	2	yes	—
116	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
117	C₂×C₄×C₃₇	C₂×#41	1	2	yes	—
118	C₈×C₃₇	—	2	8	no	—
119	C₈×C₃₇	—	2	8	no	—
120	C₄×#433	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₃₇	—	2	4	no	—
122	C₂²×C₃₇	—	2	2	no	—
123	C₂²×C₃₇	—	4	2	no	—
124	C₂²×C₃₇	—	4	2	no	—
125	C₂²×C₃₇	C₂²×C₄²	1	1	yes	characteristic
126	C₂²×C₃₇	C₂×#117	1	1	yes	characteristic
127	C₂²×C₃₇	C₂×#117	1	1	yes	characteristic
128	C₂²×C₃₇	C₂×#131	1	4	yes	—
129	C₄×C₃₇	—	2	4	no	—
130	C₄×C₃₇	—	2	4	no	—
131	C₄×C₃₇	—	2	4	no	—
132	C₄×C₃₇	—	2	2	no	—
133	C₄×C₃₇	—	4	2	no	—
134	C₄×C₃₇	—	4	2	no	—
135	C₄×C₃₇	#433	1	8	yes	direct product factor
136	C₄×C₃₇	C₄×#41	1	4	yes	—
137	C₂×#433	C₂×C₃₇	1	2	yes	—
138	C₄×#117	C₂×C₃₇	1	1	yes	characteristic
139	#1768	C₂×C₃₇	1	2	yes	complemented
140	C₄×C₄⋊C₈	C₂×C₃₇	1	2	yes	complemented
141	#1750	C₂×C₃₇	1	2	yes	—
142	C₄×C₂²⋊C₈	C₂×C₃₇	1	2	yes	—
143	#1746	C₂×C₃₇	1	2	yes	complemented
144	C₄×#97	C₂×C₃₇	1	1	yes	characteristic, complemented
145	C₄²×C₈	C₂×C₃₇	1	1	yes	characteristic, complemented
146	C₂×C₃₇	—	4	2	no	—
147	C₂×C₃₇	C₄×#117	1	1	yes	characteristic
148	C₂×C₃₇	C₄×#131	1	2	yes	—
149	C₂×C₃₇	C₂×#433	1	2	yes	—
150	C₂×C₃₇	#2922	1	2	yes	—
151	C₂×#117	C₂²×C₃₇	1	1	yes	characteristic
152	C₂×#117	C₂²×C₃₇	1	1	yes	characteristic
153	C₂²×C₄²	C₂²×C₃₇	1	1	yes	characteristic
154	#433	C₄×C₃₇	1	16	yes	direct product factor
155	C₂×C₄⋊C₈	C₂²×C₃₇	1	4	yes	—
156	C₂×C₂²⋊C₈	C₂²×C₃₇	1	4	yes	—
157	C₂×#97	C₂²×C₃₇	1	2	yes	—
158	C₂×#97	C₂²×C₃₇	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
160	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
162	C₂×C₄×C₈	C₂²×C₃₇	1	2	yes	—
163	C₄×C₄⋊C₄	C₄×C₃₇	1	4	yes	—
164	C₄×C₂²⋊C₄	C₄×C₃₇	1	4	yes	—
165	#376	C₂²×C₃₇	1	1	yes	characteristic
166	#376	C₂²×C₃₇	1	1	yes	characteristic
167	C₄³	C₂²×C₃₇	1	1	yes	characteristic
168	C₄³	C₂²×C₃₇	1	1	yes	characteristic
169	#336	C₂²×C₃₇	1	4	yes	—
170	#336	C₂²×C₃₇	1	4	yes	—
171	C₃₇	C₄×#433	1	1	yes	characteristic, direct product factor, Sylow
172	C₂³×C₄	C₂×C₄×C₃₇	1	2	yes	—
173	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
175	C₂²×C₈	C₂³×C₃₇	1	4	yes	—
176	#117	—	2	4	no	—
177	#117	C₂×C₄×C₃₇	1	8	yes	—
178	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×C₄×C₃₇	1	2	yes	—
180	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	2	yes	—
181	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	2	yes	—
182	C₂×C₄²	—	2	4	no	—
183	C₂×C₄²	C₂×C₄×C₃₇	1	4	yes	—
184	C₂×C₄²	C₂×C₄×C₃₇	1	4	yes	—
185	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
186	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
187	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₃₇	1	1	yes	characteristic
192	C₄⋊C₈	C₂×C₄×C₃₇	1	16	yes	complemented
193	C₂²⋊C₈	C₂×C₄×C₃₇	1	16	yes	—
194	#97	—	2	4	no	—
195	#97	C₂×C₄×C₃₇	1	8	yes	complemented
196	C₄×C₈	—	2	4	no	—
197	C₄×C₈	—	2	4	no	—
198	C₄×C₈	—	2	4	no	—
199	C₄×C₈	C₂×C₄×C₃₇	1	8	yes	complemented
200	#95	C₂×C₄×C₃₇	1	4	yes	—
201	#95	C₂×C₄×C₃₇	1	4	yes	—
202	C₂⁴	C₂²×C₄×C₃₇	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
208	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
209	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
210	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
212	C₂²×C₄	C₂²×C₄×C₃₇	1	4	yes	—
213	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
214	C₂²×C₄	C₂²×C₄×C₃₇	1	2	yes	—
215	C₂²×C₄	C₂⁴×C₃₇	1	1	yes	characteristic
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	C₂²×C₄×C₃₇	1	8	yes	—
221	C₂×C₈	C₂²×C₄×C₃₇	1	8	yes	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	C₄²×C₃₇	1	8	yes	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	C₄²×C₃₇	1	8	yes	—
226	C₄²	—	2	4	no	—
227	C₄²	—	2	8	no	—
228	C₄²	—	2	8	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	C₂²×C₄×C₃₇	1	4	yes	—
232	C₄²	C₂²×C₄×C₃₇	1	4	yes	—
233	C₄²	#41×C₃₇	1	4	yes	—
234	C₄²	#41×C₃₇	1	4	yes	—
235	C₂³	—	2	4	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	C₂×C₄²×C₃₇	1	2	yes	—
239	C₂³	C₂³×C₄×C₃₇	1	1	yes	characteristic
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	C₂×C₄²×C₃₇	1	4	yes	—
252	C₂×C₄	C₂×C₄²×C₃₇	1	4	yes	—
253	C₂×C₄	C₂×C₄²×C₃₇	1	2	yes	—
254	C₂×C₄	#117×C₃₇	1	4	yes	—
255	C₂×C₄	#117×C₃₇	1	4	yes	—
256	C₂×C₄	#131×C₃₇	1	8	yes	—
257	C₂×C₄	C₂³×C₄×C₃₇	1	2	yes	—
258	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
259	C₂×C₄	C₂×#41×C₃₇	1	2	yes	—
260	C₈	—	2	8	no	—
261	C₈	—	2	8	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₂²×C₄²×C₃₇	1	1	yes	characteristic
267	C₂²	C₂×#117×C₃₇	1	1	yes	characteristic
268	C₂²	C₂×#117×C₃₇	1	1	yes	characteristic
269	C₂²	C₂×#131×C₃₇	1	4	yes	—
270	C₄	—	2	4	no	—
271	C₄	—	2	4	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₄	C₃₇×#433	1	8	yes	direct product factor
277	C₄	C₄×#41×C₃₇	1	4	yes	—
278	C₂	—	4	2	no	—
279	C₂	C₄×#117×C₃₇	1	1	yes	characteristic
280	C₂	C₄×#131×C₃₇	1	2	yes	—
281	C₂	C₂×C₃₇×#433	1	2	yes	—
282	C₂	C₃₇×#2922	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²×C₃₇	C₂³
commutator	C₂²	C₂²×C₄²×C₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₇
socle	C₂³×C₃₇	C₂³×C₄
prime radical	C₂⁴×C₃₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₇×#1768	C₂	semidirect
4. C₄×C₄⋊C₈×C₃₇	C₂	semidirect
7. C₃₇×#1746	C₂	semidirect
8. C₄×#97×C₃₇	C₂	semidirect
9. C₄²×C₈×C₃₇	C₂	semidirect
13. C₃₇×#433	C₄	direct (N × H)
50. C₄⋊C₈×C₃₇	C₂×C₄	semidirect
53. #97×C₃₇	C₂×C₄	semidirect
57. C₄×C₈×C₃₇	C₂×C₄	semidirect
120. C₄×#433	C₃₇	direct (N × H)
135. C₄×C₃₇	#433	direct (N × H)
139. #1768	C₂×C₃₇	semidirect
140. C₄×C₄⋊C₈	C₂×C₃₇	semidirect
143. #1746	C₂×C₃₇	semidirect
144. C₄×#97	C₂×C₃₇	semidirect
145. C₄²×C₈	C₂×C₃₇	semidirect
154. #433	C₄×C₃₇	direct (N × H)
171. C₃₇	C₄×#433	direct (N × H)
192. C₄⋊C₈	C₂×C₄×C₃₇	semidirect
195. #97	C₂×C₄×C₃₇	semidirect
199. C₄×C₈	C₂×C₄×C₃₇	semidirect
276. C₄	C₃₇×#433	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×#433	2
C₄×#117×C₃₇	2
C₃₇×#1768	2
C₄×C₄⋊C₈×C₃₇	2
C₃₇×#1750	2
C₄×C₂²⋊C₈×C₃₇	2
C₃₇×#1746	2
C₄×#97×C₃₇	2
C₄²×C₈×C₃₇	2
C₄×#433	37

Subgroups — C4×C37×#433 #22641717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₃₇×#433 #22641717