Subgroups — C₃₇×#33427 #22654680

Order 9472.

524 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#2873	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#2166	C₂	1	4	yes	maximal, complemented
3	C₃₇×#2165	C₂	1	4	yes	maximal, complemented
4	C₃₇×#2153	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₇×#349	C₂	1	2	yes	maximal
6	C₃₇×#2110	C₂	1	2	yes	maximal, complemented
7	C₂²×M₁₆×C₃₇	C₄	1	1	yes	characteristic
8	C₂²×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
9	C₂×#117×C₃₇	C₄	1	1	yes	characteristic
10	C₂×M₃₂×C₃₇	C₂²	1	4	yes	—
11	C₃₇×#405	C₄	1	1	yes	characteristic
12	C₃₇×#405	C₄	1	2	yes	—
13	C₃₇×#405	C₄	1	1	yes	characteristic
14	C₃₇×#405	C₂²	1	2	yes	—
15	C₃₇×#405	C₂²	1	1	yes	characteristic
16	C₃₇×#405	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₃₇	C₄	1	2	yes	—
18	C₄×M₁₆×C₃₇	C₂²	1	2	yes	—
19	C₃₇×#364	C₂²	1	8	yes	complemented
20	C₃₇×#349	C₂²	1	4	yes	—
21	C₃₇×#349	C₂²	1	4	yes	—
22	C₃₇×#347	C₂²	1	8	yes	complemented
23	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×M₁₆×C₃₇	—	2	4	no	—
25	C₂×M₁₆×C₃₇	—	2	4	no	—
26	C₂×M₁₆×C₃₇	C₂×C₄	1	2	yes	—
27	C₂×M₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₃₇	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₃₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₃₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
36	#117×C₃₇	—	2	1	no	—
37	#117×C₃₇	—	2	1	no	—
38	#117×C₃₇	C₂×C₄	1	2	yes	—
39	#117×C₃₇	C₂×C₄	1	1	yes	characteristic
40	#117×C₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
42	C₂×C₂²⋊C₄×C₃₇	—	2	1	no	—
43	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
44	M₃₂×C₃₇	—	2	4	no	—
45	M₃₂×C₃₇	C₂³	1	8	yes	—
46	C₂×C₁₆×C₃₇	—	2	4	no	—
47	#97×C₃₇	C₂×C₄	1	2	yes	—
48	#97×C₃₇	C₂×C₄	1	2	yes	—
49	#97×C₃₇	C₂×C₄	1	2	yes	—
50	#97×C₃₇	C₂³	1	2	yes	—
51	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₃₇	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₃₇	C₂³	1	2	yes	—
55	C₂⁴×C₃₇	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₃₇	—	2	2	no	—
57	C₂²×C₄×C₃₇	—	2	2	no	—
58	C₂²×C₄×C₃₇	—	2	2	no	—
59	C₂²×C₄×C₃₇	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₃₇	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₃₇	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
65	M₁₆×C₃₇	—	2	2	no	—
66	M₁₆×C₃₇	—	2	2	no	—
67	M₁₆×C₃₇	—	2	2	no	—
68	M₁₆×C₃₇	—	2	4	no	—
69	M₁₆×C₃₇	—	2	2	no	—
70	M₁₆×C₃₇	—	2	2	no	—
71	M₁₆×C₃₇	M₁₆	1	4	yes	—
72	C₂×C₈×C₃₇	—	2	2	no	—
73	C₂×C₈×C₃₇	—	2	4	no	—
74	C₂×C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₃₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₃₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₃₇	#41	1	4	yes	—
80	C₂×C₈×C₃₇	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₃₇	—	2	2	no	—
82	C₄⋊C₄×C₃₇	—	2	1	no	—
83	C₄⋊C₄×C₃₇	—	2	1	no	—
84	C₂²⋊C₄×C₃₇	—	2	2	no	—
85	C₂²⋊C₄×C₃₇	—	2	1	no	—
86	C₂²⋊C₄×C₃₇	—	2	1	no	—
87	C₄²×C₃₇	—	2	2	no	—
88	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₃₇	C₂²×C₄	1	2	yes	—
90	C₁₆×C₃₇	—	2	8	no	—
91	C₂³×C₃₇	—	2	2	no	—
92	C₂³×C₃₇	—	8	1	no	—
93	C₂³×C₃₇	C₂×M₁₆	1	1	yes	characteristic
94	C₂³×C₃₇	C₂×M₁₆	1	1	yes	characteristic
95	C₂³×C₃₇	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₃₇	—	2	2	no	—
97	C₂×C₄×C₃₇	—	2	1	no	—
98	C₂×C₄×C₃₇	—	2	1	no	—
99	C₂×C₄×C₃₇	—	2	1	no	—
100	C₂×C₄×C₃₇	—	2	1	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	—	2	2	no	—
105	C₂×C₄×C₃₇	—	8	2	no	—
106	C₂×C₄×C₃₇	#117	1	2	yes	—
107	C₂×C₄×C₃₇	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₃₇	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₃₇	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₃₇	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
112	C₈×C₃₇	—	2	1	no	—
113	C₈×C₃₇	—	2	1	no	—
114	C₈×C₃₇	—	2	2	no	—
115	C₈×C₃₇	#117	1	2	yes	—
116	C₈×C₃₇	C₂×M₁₆	1	4	yes	—
117	C₈×C₃₇	C₂×#41	1	2	yes	—
118	#33427	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₃₇	—	2	1	no	—
120	C₂²×C₃₇	—	2	1	no	—
121	C₂²×C₃₇	—	4	1	no	—
122	C₂²×C₃₇	—	4	1	no	—
123	C₂²×C₃₇	—	4	1	no	—
124	C₂²×C₃₇	—	8	2	no	—
125	C₂²×C₃₇	#432	1	2	yes	—
126	C₂²×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
127	C₄×C₃₇	—	2	1	no	—
128	C₄×C₃₇	—	2	1	no	—
129	C₄×C₃₇	—	4	2	no	—
130	C₄×C₃₇	—	4	2	no	—
131	C₄×C₃₇	#432	1	2	yes	—
132	C₄×C₃₇	C₂×#117	1	1	yes	characteristic
133	C₄×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
134	#2873	C₂×C₃₇	1	1	yes	characteristic
135	#2166	C₂×C₃₇	1	4	yes	complemented
136	#2165	C₂×C₃₇	1	4	yes	complemented
137	#2153	C₂×C₃₇	1	2	yes	complemented
138	C₂×#349	C₂×C₃₇	1	2	yes	—
139	#2110	C₂×C₃₇	1	2	yes	complemented
140	C₂×C₃₇	—	2	1	no	—
141	C₂×C₃₇	—	2	2	no	—
142	C₂×C₃₇	—	4	1	no	—
143	C₂×C₃₇	—	4	1	no	—
144	C₂×C₃₇	C₂×#432	1	1	yes	characteristic
145	C₂²×M₁₆	C₄×C₃₇	1	1	yes	characteristic
146	C₂²×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
147	C₂×#117	C₄×C₃₇	1	1	yes	characteristic
148	C₂×M₃₂	C₂²×C₃₇	1	4	yes	—
149	#405	C₄×C₃₇	1	1	yes	characteristic
150	#405	C₄×C₃₇	1	2	yes	—
151	#405	C₄×C₃₇	1	1	yes	characteristic
152	#405	C₂²×C₃₇	1	1	yes	characteristic
153	#405	C₂²×C₃₇	1	2	yes	—
154	#405	C₂²×C₃₇	1	1	yes	characteristic
155	C₄×M₁₆	C₄×C₃₇	1	2	yes	—
156	C₄×M₁₆	C₂²×C₃₇	1	2	yes	—
157	#364	C₂²×C₃₇	1	8	yes	complemented
158	#349	C₂²×C₃₇	1	4	yes	—
159	#349	C₂²×C₃₇	1	4	yes	—
160	#347	C₂²×C₃₇	1	8	yes	complemented
161	C₃₇	#33427	1	1	yes	characteristic, direct product factor, Sylow
162	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	C₂×C₄×C₃₇	1	2	yes	—
166	C₂×M₁₆	C₂×C₄×C₃₇	1	1	yes	characteristic
167	C₂×M₁₆	C₂×C₄×C₃₇	1	1	yes	characteristic
168	C₂×M₁₆	C₂×C₄×C₃₇	1	1	yes	characteristic
169	C₂×M₁₆	C₂×C₄×C₃₇	1	1	yes	characteristic
170	C₂×M₁₆	C₂³×C₃₇	1	1	yes	characteristic
171	C₂×M₁₆	C₂³×C₃₇	1	1	yes	characteristic
172	C₂²×C₈	C₂×C₄×C₃₇	1	2	yes	—
173	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
175	#117	—	2	1	no	—
176	#117	—	2	1	no	—
177	#117	C₂×C₄×C₃₇	1	1	yes	characteristic
178	#117	C₂×C₄×C₃₇	1	2	yes	—
179	#117	C₂×C₄×C₃₇	1	1	yes	characteristic
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₂²⋊C₄	—	2	1	no	—
182	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
183	M₃₂	—	2	4	no	—
184	M₃₂	C₂³×C₃₇	1	8	yes	—
185	C₂×C₁₆	—	2	4	no	—
186	#97	C₂×C₄×C₃₇	1	2	yes	—
187	#97	C₂×C₄×C₃₇	1	2	yes	—
188	#97	C₂×C₄×C₃₇	1	2	yes	—
189	#97	C₂³×C₃₇	1	2	yes	—
190	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
191	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
192	C₄×C₈	C₂×C₄×C₃₇	1	2	yes	—
193	C₄×C₈	C₂³×C₃₇	1	2	yes	—
194	C₂⁴	M₁₆×C₃₇	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	M₁₆×C₃₇	1	1	yes	characteristic
199	C₂²×C₄	M₁₆×C₃₇	1	1	yes	characteristic
200	C₂²×C₄	M₁₆×C₃₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	M₁₆×C₃₇	1	4	yes	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	C₂²×C₄×C₃₇	1	1	yes	characteristic
214	C₂×C₈	C₂²×C₄×C₃₇	1	1	yes	characteristic
215	C₂×C₈	C₂²×C₄×C₃₇	1	2	yes	—
216	C₂×C₈	C₂²×C₄×C₃₇	1	1	yes	characteristic
217	C₂×C₈	C₂²×C₄×C₃₇	1	2	yes	—
218	C₂×C₈	#41×C₃₇	1	4	yes	—
219	C₂×C₈	C₂⁴×C₃₇	1	1	yes	characteristic
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₂²⋊C₄	—	2	1	no	—
226	C₄²	—	2	2	no	—
227	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
228	C₄²	C₂²×C₄×C₃₇	1	2	yes	—
229	C₁₆	—	2	8	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	8	1	no	—
232	C₂³	C₂×M₁₆×C₃₇	1	1	yes	characteristic
233	C₂³	C₂×M₁₆×C₃₇	1	1	yes	characteristic
234	C₂³	C₂×M₁₆×C₃₇	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	#117×C₃₇	1	2	yes	—
246	C₂×C₄	C₂×M₁₆×C₃₇	1	1	yes	characteristic
247	C₂×C₄	C₂×M₁₆×C₃₇	1	1	yes	characteristic
248	C₂×C₄	C₂×M₁₆×C₃₇	1	2	yes	—
249	C₂×C₄	C₂×M₁₆×C₃₇	1	1	yes	characteristic
250	C₂×C₄	C₂³×C₄×C₃₇	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	#117×C₃₇	1	2	yes	—
255	C₈	C₂×M₁₆×C₃₇	1	4	yes	—
256	C₈	C₂×#41×C₃₇	1	2	yes	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	2	no	—
263	C₂²	C₃₇×#432	1	2	yes	—
264	C₂²	C₂²×M₁₆×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	C₃₇×#432	1	2	yes	—
270	C₄	C₂×#117×C₃₇	1	1	yes	characteristic
271	C₄	C₂²×M₁₆×C₃₇	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	2	no	—
274	C₂	—	4	1	no	—
275	C₂	—	4	1	no	—
276	C₂	C₂×C₃₇×#432	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₇	C₂×#117
commutator	C₂×C₄	C₂³×C₄×C₃₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₇
socle	C₂×C₃₇	C₂×#432
prime radical	C₂⁴×C₃₇	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₇ ◁ C₃₇×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#2166	C₂	semidirect
3. C₃₇×#2165	C₂	semidirect
4. C₃₇×#2153	C₂	semidirect
6. C₃₇×#2110	C₂	semidirect
19. C₃₇×#364	C₂²	semidirect
22. C₃₇×#347	C₂²	semidirect
118. #33427	C₃₇	direct (N × H)
135. #2166	C₂×C₃₇	semidirect
136. #2165	C₂×C₃₇	semidirect
137. #2153	C₂×C₃₇	semidirect
139. #2110	C₂×C₃₇	semidirect
157. #364	C₂²×C₃₇	semidirect
160. #347	C₂²×C₃₇	semidirect
161. C₃₇	#33427	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#2873	2
C₃₇×#2166	2
C₃₇×#2165	2
C₃₇×#2153	2
C₂×C₃₇×#349	2
C₃₇×#2110	2
#33427	37

Subgroups — C37×#33427 #22654680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#33427 #22654680