Subgroups — C₁₃₇×#17319 #46866705

Order 35072.

834 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1452×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	#1451×C₁₃₇	C₂	1	8	yes	maximal, complemented
5	C₂²×M₁₆×C₁₃₇	C₄	1	2	yes	—
6	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
7	#431×C₁₃₇	C₄	1	4	yes	—
8	#431×C₁₃₇	C₄	1	4	yes	—
9	#431×C₁₃₇	C₂²	1	4	yes	—
10	#431×C₁₃₇	C₂²	1	4	yes	—
11	C₂×C₄⋊C₈×C₁₃₇	C₄	1	4	yes	—
12	C₂×C₂²⋊C₈×C₁₃₇	C₄	1	4	yes	—
13	C₄×M₁₆×C₁₃₇	C₄	1	4	yes	—
14	C₄×M₁₆×C₁₃₇	C₂²	1	4	yes	—
15	C₂×#97×C₁₃₇	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃₇	C₂²	1	4	yes	—
18	#322×C₁₃₇	C₂²	1	16	yes	complemented
19	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₁₃₇	—	2	8	no	—
22	C₂×M₁₆×C₁₃₇	C₂×C₄	1	8	yes	—
23	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
31	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
33	C₄⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
34	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
35	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	—
36	#97×C₁₃₇	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₁₃₇	C₂³	1	4	yes	—
40	C₄×C₈×C₁₃₇	C₂³	1	8	yes	—
41	C₂⁴×C₁₃₇	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₇	—	4	4	no	—
43	C₂²×C₄×C₁₃₇	C₄²	1	2	yes	—
44	C₂²×C₄×C₁₃₇	C₄²	1	2	yes	—
45	C₂²×C₄×C₁₃₇	C₄²	1	2	yes	—
46	C₂²×C₄×C₁₃₇	C₄²	1	2	yes	—
47	C₂²×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₇	M₁₆	1	8	yes	—
49	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
54	M₁₆×C₁₃₇	—	2	16	no	—
55	C₂×C₈×C₁₃₇	—	2	8	no	—
56	C₂×C₈×C₁₃₇	—	4	8	no	—
57	C₂×C₈×C₁₃₇	C₂²×C₄	1	8	yes	—
58	C₂×C₈×C₁₃₇	C₂²×C₄	1	16	yes	—
59	C₄²×C₁₃₇	C₄²	1	1	yes	characteristic
60	C₄²×C₁₃₇	C₄²	1	1	yes	characteristic
61	C₄²×C₁₃₇	C₄²	1	2	yes	—
62	C₄²×C₁₃₇	C₄²	1	2	yes	—
63	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
65	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
66	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
69	C₂³×C₁₃₇	—	4	2	no	—
70	C₂³×C₁₃₇	#97	1	4	yes	—
71	C₂³×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
72	C₂³×C₁₃₇	C₂×C₄²	1	2	yes	—
73	C₂×C₄×C₁₃₇	—	2	4	no	—
74	C₂×C₄×C₁₃₇	—	2	4	no	—
75	C₂×C₄×C₁₃₇	—	2	4	no	—
76	C₂×C₄×C₁₃₇	—	4	2	no	—
77	C₂×C₄×C₁₃₇	—	4	2	no	—
78	C₂×C₄×C₁₃₇	—	4	2	no	—
79	C₂×C₄×C₁₃₇	#97	1	8	yes	—
80	C₂×C₄×C₁₃₇	#97	1	4	yes	—
81	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
82	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
83	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
85	C₂×C₄×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₁₃₇	C₂×M₁₆	1	4	yes	—
87	C₂×C₄×C₁₃₇	C₂×M₁₆	1	8	yes	—
88	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃₇	C₂³×C₄	1	2	yes	—
90	C₈×C₁₃₇	—	2	8	no	—
91	C₈×C₁₃₇	—	2	16	no	—
92	C₂²×C₁₃₇	—	2	2	no	—
93	C₂²×C₁₃₇	—	2	2	no	—
94	C₂²×C₁₃₇	—	2	2	no	—
95	C₂²×C₁₃₇	—	4	4	no	—
96	C₂²×C₁₃₇	C₂×#97	1	2	yes	—
97	C₂²×C₁₃₇	C₂×#97	1	4	yes	—
98	C₂²×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
99	C₄×C₁₃₇	—	2	2	no	—
100	C₄×C₁₃₇	—	2	2	no	—
101	C₄×C₁₃₇	—	2	2	no	—
102	C₄×C₁₃₇	C₂×#97	1	2	yes	—
103	C₄×C₁₃₇	C₂×#97	1	4	yes	—
104	C₄×C₁₃₇	C₂²×M₁₆	1	2	yes	—
105	C₄×C₁₃₇	#568	1	4	yes	—
106	C₂×C₁₃₇	—	2	4	no	—
107	C₂×C₁₃₇	—	2	2	no	—
108	C₂×C₁₃₇	C₂²×#97	1	1	yes	characteristic
109	C₂×C₁₃₇	#2873	1	2	yes	—
110	#17319	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₁₃₇	#17319	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×#431	C₂×C₁₃₇	1	2	yes	—
113	C₂×C₄×M₁₆	C₂×C₁₃₇	1	1	yes	characteristic
114	#1452	C₂×C₁₃₇	1	4	yes	complemented
115	#1451	C₂×C₁₃₇	1	8	yes	complemented
116	C₂²×M₁₆	C₄×C₁₃₇	1	2	yes	—
117	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
118	#431	C₄×C₁₃₇	1	4	yes	—
119	#431	C₄×C₁₃₇	1	4	yes	—
120	#431	C₂²×C₁₃₇	1	4	yes	—
121	#431	C₂²×C₁₃₇	1	4	yes	—
122	C₂×C₄⋊C₈	C₄×C₁₃₇	1	4	yes	—
123	C₂×C₂²⋊C₈	C₄×C₁₃₇	1	4	yes	—
124	C₄×M₁₆	C₄×C₁₃₇	1	4	yes	—
125	C₄×M₁₆	C₂²×C₁₃₇	1	4	yes	—
126	C₂×#97	C₄×C₁₃₇	1	2	yes	—
127	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₁₃₇	1	4	yes	—
129	#322	C₂²×C₁₃₇	1	16	yes	complemented
130	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
131	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
132	C₂×M₁₆	—	2	8	no	—
133	C₂×M₁₆	C₂×C₄×C₁₃₇	1	8	yes	—
134	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
135	C₂²×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
136	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
137	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
138	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
139	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
140	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
141	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
142	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
143	C₄⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
144	C₄⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
145	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
146	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	—
147	#97	C₂×C₄×C₁₃₇	1	8	yes	—
148	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
149	C₄×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
150	C₄×C₈	C₂³×C₁₃₇	1	4	yes	—
151	C₄×C₈	C₂³×C₁₃₇	1	8	yes	—
152	C₂⁴	C₄²×C₁₃₇	1	1	yes	characteristic
153	C₂²×C₄	—	4	4	no	—
154	C₂²×C₄	C₄²×C₁₃₇	1	2	yes	—
155	C₂²×C₄	C₄²×C₁₃₇	1	2	yes	—
156	C₂²×C₄	C₄²×C₁₃₇	1	2	yes	—
157	C₂²×C₄	C₄²×C₁₃₇	1	2	yes	—
158	C₂²×C₄	C₄²×C₁₃₇	1	1	yes	characteristic
159	C₂²×C₄	M₁₆×C₁₃₇	1	8	yes	—
160	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
161	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
162	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
163	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
164	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
165	M₁₆	—	2	16	no	—
166	C₂×C₈	—	2	8	no	—
167	C₂×C₈	—	4	8	no	—
168	C₂×C₈	C₂²×C₄×C₁₃₇	1	8	yes	—
169	C₂×C₈	C₂²×C₄×C₁₃₇	1	16	yes	—
170	C₄²	C₄²×C₁₃₇	1	1	yes	characteristic
171	C₄²	C₄²×C₁₃₇	1	1	yes	characteristic
172	C₄²	C₄²×C₁₃₇	1	2	yes	—
173	C₄²	C₄²×C₁₃₇	1	2	yes	—
174	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
175	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
176	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
177	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
178	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
179	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
180	C₂³	—	4	2	no	—
181	C₂³	#97×C₁₃₇	1	4	yes	—
182	C₂³	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
183	C₂³	C₂×C₄²×C₁₃₇	1	2	yes	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	#97×C₁₃₇	1	8	yes	—
191	C₂×C₄	#97×C₁₃₇	1	4	yes	—
192	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
193	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
194	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
195	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
196	C₂×C₄	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
197	C₂×C₄	C₂×M₁₆×C₁₃₇	1	4	yes	—
198	C₂×C₄	C₂×M₁₆×C₁₃₇	1	8	yes	—
199	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
200	C₂×C₄	C₂³×C₄×C₁₃₇	1	2	yes	—
201	C₈	—	2	8	no	—
202	C₈	—	2	16	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	2	2	no	—
206	C₂²	—	4	4	no	—
207	C₂²	C₂×#97×C₁₃₇	1	2	yes	—
208	C₂²	C₂×#97×C₁₃₇	1	4	yes	—
209	C₂²	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	2	no	—
213	C₄	C₂×#97×C₁₃₇	1	2	yes	—
214	C₄	C₂×#97×C₁₃₇	1	4	yes	—
215	C₄	C₂²×M₁₆×C₁₃₇	1	2	yes	—
216	C₄	#568×C₁₃₇	1	4	yes	—
217	C₂	—	2	4	no	—
218	C₂	—	2	2	no	—
219	C₂	C₂²×#97×C₁₃₇	1	1	yes	characteristic
220	C₂	#2873×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₃₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₃₇
Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
socle	C₂²×C₁₃₇	C₂²×C₄²
prime radical	C₂⁴×C₁₃₇	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1452×C₁₃₇	C₂	semidirect
4. #1451×C₁₃₇	C₂	semidirect
18. #322×C₁₃₇	C₂²	semidirect
110. #17319	C₁₃₇	direct (N × H)
111. C₁₃₇	#17319	direct (N × H)
114. #1452	C₂×C₁₃₇	semidirect
115. #1451	C₂×C₁₃₇	semidirect
129. #322	C₂²×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₁₃₇	2
C₂×C₄×M₁₆×C₁₃₇	2
#1452×C₁₃₇	2
#1451×C₁₃₇	2
#17319	137

Subgroups — C137×#17319 #46866705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#17319 #46866705