Subgroups — C₂×C₃₇×#1327 #22632708

Order 9472.

410 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#349	C₂	1	2	yes	maximal
3	C₃₇×#1327	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₃₇	C₄	1	1	yes	characteristic
5	C₂²×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₃₇	C₄	1	1	yes	characteristic
7	C₂×M₃₂×C₃₇	—	2	2	no	—
8	C₃₇×#423	—	2	2	no	—
9	C₃₇×#423	C₄	1	2	yes	—
10	C₃₇×#423	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₃₇	—	2	2	no	—
12	C₃₇×#349	—	2	2	no	—
13	C₃₇×#349	C₂²	1	4	yes	—
14	C₂³×C₄×C₃₇	—	2	1	no	—
15	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂×M₁₆×C₃₇	—	2	1	no	—
17	C₂×M₁₆×C₃₇	—	4	1	no	—
18	C₂×M₁₆×C₃₇	—	4	1	no	—
19	C₂×M₁₆×C₃₇	—	4	1	no	—
20	C₂×M₁₆×C₃₇	—	4	1	no	—
21	C₂×M₁₆×C₃₇	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₃₇	D₄	1	2	yes	—
24	C₂×M₁₆×C₃₇	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₃₇	—	2	1	no	—
26	C₂²×C₈×C₃₇	D₄	1	2	yes	—
27	C₂×C₄²×C₃₇	—	2	1	no	—
28	C₂×C₄²×C₃₇	—	2	1	no	—
29	C₂×C₄²×C₃₇	—	2	1	no	—
30	C₂×C₄²×C₃₇	C₈	1	1	yes	characteristic
31	C₂×C₄²×C₃₇	C₈	1	2	yes	—
32	C₂×C₄²×C₃₇	C₈	1	1	yes	characteristic
33	C₂×C₄²×C₃₇	C₂×C₄	1	2	yes	—
34	M₃₂×C₃₇	—	2	4	no	—
35	M₃₂×C₃₇	—	4	2	no	—
36	C₂×C₁₆×C₃₇	—	4	2	no	—
37	C₄⋊C₈×C₃₇	—	2	4	no	—
38	C₄⋊C₈×C₃₇	—	2	4	no	—
39	C₂⁴×C₃₇	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₃₇	—	2	1	no	—
41	C₂²×C₄×C₃₇	—	2	1	no	—
42	C₂²×C₄×C₃₇	—	2	1	no	—
43	C₂²×C₄×C₃₇	—	2	1	no	—
44	C₂²×C₄×C₃₇	—	2	1	no	—
45	C₂²×C₄×C₃₇	—	2	2	no	—
46	C₂²×C₄×C₃₇	—	2	1	no	—
47	C₂²×C₄×C₃₇	—	2	2	no	—
48	C₂²×C₄×C₃₇	—	4	4	no	—
49	C₂²×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₃₇	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
55	M₁₆×C₃₇	—	4	1	no	—
56	M₁₆×C₃₇	—	4	2	no	—
57	M₁₆×C₃₇	—	4	1	no	—
58	M₁₆×C₃₇	—	4	2	no	—
59	M₁₆×C₃₇	—	4	2	no	—
60	C₂×C₈×C₃₇	—	2	2	no	—
61	C₂×C₈×C₃₇	—	2	2	no	—
62	C₂×C₈×C₃₇	—	2	2	no	—
63	C₂×C₈×C₃₇	—	4	1	no	—
64	C₂×C₈×C₃₇	—	4	1	no	—
65	C₂×C₈×C₃₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₃₇	C₂×D₄	1	2	yes	—
67	C₄²×C₃₇	—	2	1	no	—
68	C₄²×C₃₇	—	2	2	no	—
69	C₄²×C₃₇	—	2	1	no	—
70	C₄²×C₃₇	—	2	2	no	—
71	C₄²×C₃₇	C₂×C₈	1	2	yes	—
72	C₄²×C₃₇	C₂×C₈	1	2	yes	—
73	C₁₆×C₃₇	—	4	4	no	—
74	C₂³×C₃₇	—	2	1	no	—
75	C₂³×C₃₇	—	2	1	no	—
76	C₂³×C₃₇	—	4	2	no	—
77	C₂³×C₃₇	C₂²⋊C₈	1	1	yes	characteristic
78	C₂³×C₃₇	C₂²⋊C₈	1	1	yes	characteristic
79	C₂³×C₃₇	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₃₇	—	2	4	no	—
81	C₂×C₄×C₃₇	—	2	1	no	—
82	C₂×C₄×C₃₇	—	2	1	no	—
83	C₂×C₄×C₃₇	—	2	2	no	—
84	C₂×C₄×C₃₇	—	2	1	no	—
85	C₂×C₄×C₃₇	—	2	1	no	—
86	C₂×C₄×C₃₇	—	2	1	no	—
87	C₂×C₄×C₃₇	—	2	1	no	—
88	C₂×C₄×C₃₇	—	2	1	no	—
89	C₂×C₄×C₃₇	—	2	1	no	—
90	C₂×C₄×C₃₇	—	2	1	no	—
91	C₂×C₄×C₃₇	—	2	1	no	—
92	C₂×C₄×C₃₇	—	2	2	no	—
93	C₂×C₄×C₃₇	—	4	4	no	—
94	C₂×C₄×C₃₇	—	4	4	no	—
95	C₂×C₄×C₃₇	—	4	2	no	—
96	C₂×C₄×C₃₇	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₃₇	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₃₇	#99	1	1	yes	characteristic
99	C₂×C₄×C₃₇	#99	1	1	yes	characteristic
100	C₂×C₄×C₃₇	#101	1	1	yes	characteristic
101	C₂×C₄×C₃₇	#101	1	1	yes	characteristic
102	C₂×C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
104	C₈×C₃₇	—	2	2	no	—
105	C₈×C₃₇	—	2	2	no	—
106	C₈×C₃₇	—	4	2	no	—
107	C₂×#1327	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂²×C₃₇	—	2	2	no	—
109	C₂²×C₃₇	—	2	1	no	—
110	C₂²×C₃₇	—	2	1	no	—
111	C₂²×C₃₇	—	4	1	no	—
112	C₂²×C₃₇	—	4	1	no	—
113	C₂²×C₃₇	—	4	2	no	—
114	C₂²×C₃₇	—	4	2	no	—
115	C₂²×C₃₇	#324	1	1	yes	characteristic
116	C₂²×C₃₇	#324	1	1	yes	characteristic
117	C₂²×C₃₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₄×C₃₇	—	2	4	no	—
119	C₄×C₃₇	—	2	1	no	—
120	C₄×C₃₇	—	2	1	no	—
121	C₄×C₃₇	—	4	2	no	—
122	C₄×C₃₇	#324	1	1	yes	characteristic
123	C₄×C₃₇	#324	1	1	yes	characteristic
124	C₄×C₃₇	C₂×#99	1	1	yes	characteristic
125	C₄×C₃₇	C₂×#101	1	1	yes	characteristic
126	C₂×#423	C₂×C₃₇	1	1	yes	characteristic
127	C₂×#349	C₂×C₃₇	1	2	yes	—
128	#1327	C₂×C₃₇	1	4	yes	direct product factor
129	C₂×C₃₇	—	2	1	no	—
130	C₂×C₃₇	—	2	1	no	—
131	C₂×C₃₇	—	4	1	no	—
132	C₂×C₃₇	—	4	1	no	—
133	C₂×C₃₇	#1327	1	2	yes	direct product factor
134	C₂×C₃₇	C₂×#324	1	1	yes	characteristic
135	C₂²×M₁₆	C₄×C₃₇	1	1	yes	characteristic
136	C₂²×M₁₆	C₂²×C₃₇	1	1	yes	characteristic
137	C₂²×C₄²	C₄×C₃₇	1	1	yes	characteristic
138	C₂×M₃₂	—	2	2	no	—
139	#423	—	2	2	no	—
140	#423	C₄×C₃₇	1	2	yes	—
141	#423	C₂²×C₃₇	1	2	yes	—
142	C₂×C₄⋊C₈	—	2	2	no	—
143	#349	—	2	2	no	—
144	#349	C₂²×C₃₇	1	4	yes	—
145	C₃₇	C₂×#1327	1	1	yes	characteristic, direct product factor, Sylow
146	C₂³×C₄	—	2	1	no	—
147	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
148	C₂×M₁₆	—	2	1	no	—
149	C₂×M₁₆	—	4	1	no	—
150	C₂×M₁₆	—	4	1	no	—
151	C₂×M₁₆	—	4	1	no	—
152	C₂×M₁₆	—	4	1	no	—
153	C₂×M₁₆	C₂×C₄×C₃₇	1	2	yes	—
154	C₂×M₁₆	C₂×C₄×C₃₇	1	1	yes	characteristic
155	C₂×M₁₆	D₄×C₃₇	1	2	yes	—
156	C₂×M₁₆	C₂³×C₃₇	1	1	yes	characteristic
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	D₄×C₃₇	1	2	yes	—
159	C₂×C₄²	—	2	1	no	—
160	C₂×C₄²	—	2	1	no	—
161	C₂×C₄²	—	2	1	no	—
162	C₂×C₄²	C₈×C₃₇	1	1	yes	characteristic
163	C₂×C₄²	C₈×C₃₇	1	2	yes	—
164	C₂×C₄²	C₈×C₃₇	1	1	yes	characteristic
165	C₂×C₄²	C₂×C₄×C₃₇	1	2	yes	—
166	M₃₂	—	2	4	no	—
167	M₃₂	—	4	2	no	—
168	C₂×C₁₆	—	4	2	no	—
169	C₄⋊C₈	—	2	4	no	—
170	C₄⋊C₈	—	2	4	no	—
171	C₂⁴	M₁₆×C₃₇	1	1	yes	characteristic
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	4	4	no	—
181	C₂²×C₄	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
182	C₂²×C₄	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₈×C₃₇	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₈×C₃₇	1	1	yes	characteristic
185	C₂²×C₄	M₁₆×C₃₇	1	1	yes	characteristic
186	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
187	M₁₆	—	4	1	no	—
188	M₁₆	—	4	2	no	—
189	M₁₆	—	4	1	no	—
190	M₁₆	—	4	2	no	—
191	M₁₆	—	4	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	C₂²⋊C₄×C₃₇	1	2	yes	—
198	C₂×C₈	C₂×D₄×C₃₇	1	2	yes	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	2	no	—
203	C₄²	C₂×C₈×C₃₇	1	2	yes	—
204	C₄²	C₂×C₈×C₃₇	1	2	yes	—
205	C₁₆	—	4	4	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	4	2	no	—
209	C₂³	C₂²⋊C₈×C₃₇	1	1	yes	characteristic
210	C₂³	C₂²⋊C₈×C₃₇	1	1	yes	characteristic
211	C₂³	C₂×M₁₆×C₃₇	1	1	yes	characteristic
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂²⋊C₈×C₃₇	1	1	yes	characteristic
229	C₂×C₄	C₂²⋊C₈×C₃₇	1	1	yes	characteristic
230	C₂×C₄	#99×C₃₇	1	1	yes	characteristic
231	C₂×C₄	#99×C₃₇	1	1	yes	characteristic
232	C₂×C₄	#101×C₃₇	1	1	yes	characteristic
233	C₂×C₄	#101×C₃₇	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₈×C₃₇	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	2	no	—
238	C₈	—	4	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	C₃₇×#324	1	1	yes	characteristic
247	C₂²	C₃₇×#324	1	1	yes	characteristic
248	C₂²	C₂×C₂²⋊C₈×C₃₇	1	1	yes	characteristic
249	C₄	—	2	4	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	2	no	—
253	C₄	C₃₇×#324	1	1	yes	characteristic
254	C₄	C₃₇×#324	1	1	yes	characteristic
255	C₄	C₂×#99×C₃₇	1	1	yes	characteristic
256	C₄	C₂×#101×C₃₇	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	1	no	—
259	C₂	—	4	1	no	—
260	C₂	—	4	1	no	—
261	C₂	C₃₇×#1327	1	2	yes	direct product factor
262	C₂	C₂×C₃₇×#324	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₇	#99
commutator	C₂×C₄	C₂²×C₈×C₃₇
Frattini, cosocle	C₂×M₁₆	C₂³×C₃₇
socle	C₂²×C₃₇	#324
prime radical	C₂⁴×C₃₇	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ C₂²×M₁₆×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₇×#1327	C₂	direct (N × H)
107. C₂×#1327	C₃₇	direct (N × H)
128. #1327	C₂×C₃₇	direct (N × H)
133. C₂×C₃₇	#1327	direct (N × H)
145. C₃₇	C₂×#1327	direct (N × H)
261. C₂	C₃₇×#1327	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×#423	2
C₂×C₃₇×#349	2
C₃₇×#1327	2
C₂×#1327	37

Subgroups — C2×C37×#1327 #22632708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₇×#1327 #22632708