Subgroups — C₁₃₇×#13182 #46862568

Order 35072.

280 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2161×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×#364×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1848×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1772×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1389×C₁₃₇	C₂	1	2	yes	maximal
6	C₂×M₃₂×C₁₃₇	C₂²	1	2	yes	—
7	C₂×#108×C₁₃₇	—	2	1	no	—
8	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
9	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
10	#427×C₁₃₇	C₂²	1	1	yes	characteristic
11	#423×C₁₃₇	—	2	1	no	—
12	#405×C₁₃₇	—	2	1	no	—
13	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	#368×C₁₃₇	—	2	4	no	—
15	#364×C₁₃₇	C₄	1	2	yes	complemented
16	#364×C₁₃₇	C₄	1	2	yes	complemented
17	#339×C₁₃₇	—	2	1	no	—
18	C₂×M₁₆×C₁₃₇	—	2	1	no	—
19	C₂×M₁₆×C₁₃₇	—	2	2	no	—
20	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
22	#117×C₁₃₇	—	2	1	no	—
23	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
24	M₃₂×C₁₃₇	—	2	2	no	—
25	M₃₂×C₁₃₇	C₂×C₄	1	4	yes	—
26	C₂×C₁₆×C₁₃₇	—	2	2	no	—
27	#108×C₁₃₇	—	2	2	no	—
28	#108×C₁₃₇	—	2	1	no	—
29	#108×C₁₃₇	—	2	2	no	—
30	#108×C₁₃₇	—	4	1	no	—
31	#108×C₁₃₇	—	4	1	no	—
32	#108×C₁₃₇	D₄	1	2	yes	—
33	#107×C₁₃₇	—	2	1	no	—
34	#106×C₁₃₇	Q₈	1	2	yes	—
35	C₄⋊C₈×C₁₃₇	—	4	2	no	—
36	#97×C₁₃₇	—	4	1	no	—
37	C₄×C₈×C₁₃₇	—	4	1	no	—
38	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₃₇	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₁₃₇	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₇	—	2	1	no	—
43	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₁₃₇	—	2	2	no	—
45	M₁₆×C₁₃₇	—	4	1	no	—
46	M₁₆×C₁₃₇	—	4	2	no	—
47	M₁₆×C₁₃₇	—	4	2	no	—
48	C₂×C₈×C₁₃₇	—	4	1	no	—
49	C₂×C₈×C₁₃₇	—	4	2	no	—
50	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃₇	D₈	1	1	yes	characteristic
53	C₂×C₈×C₁₃₇	D₈	1	1	yes	characteristic
54	C₂×C₈×C₁₃₇	Q₁₆	1	1	yes	characteristic
55	C₂×C₈×C₁₃₇	Q₁₆	1	1	yes	characteristic
56	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
61	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
62	C₄⋊C₄×C₁₃₇	—	2	2	no	—
63	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
64	C₄²×C₁₃₇	—	2	1	no	—
65	C₄²×C₁₃₇	—	4	1	no	—
66	C₄²×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
68	C₁₆×C₁₃₇	—	2	4	no	—
69	C₂³×C₁₃₇	#137	1	1	yes	characteristic
70	C₂×C₄×C₁₃₇	—	2	1	no	—
71	C₂×C₄×C₁₃₇	—	2	1	no	—
72	C₂×C₄×C₁₃₇	—	4	2	no	—
73	C₂×C₄×C₁₃₇	—	4	2	no	—
74	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₈×C₁₃₇	—	4	2	no	—
81	C₈×C₁₃₇	—	8	2	no	—
82	C₈×C₁₃₇	#107	1	1	yes	characteristic
83	C₈×C₁₃₇	#107	1	1	yes	characteristic
84	C₈×C₁₃₇	#107	1	1	yes	characteristic
85	C₈×C₁₃₇	#107	1	1	yes	characteristic
86	C₈×C₁₃₇	C₄×D₄	1	1	yes	characteristic
87	C₈×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
88	C₈×C₁₃₇	#125	1	1	yes	characteristic
89	C₈×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
90	C₂²×C₁₃₇	—	4	1	no	—
91	C₂²×C₁₃₇	#428	1	1	yes	characteristic
92	C₂²×C₁₃₇	#462	1	1	yes	characteristic
93	C₂²×C₁₃₇	#474	1	1	yes	characteristic
94	C₄×C₁₃₇	—	2	2	no	—
95	C₄×C₁₃₇	—	4	1	no	—
96	C₄×C₁₃₇	—	8	2	no	—
97	C₄×C₁₃₇	#389	1	1	yes	characteristic
98	C₄×C₁₃₇	C₂×#107	1	1	yes	characteristic
99	C₄×C₁₃₇	#459	1	1	yes	characteristic
100	C₄×C₁₃₇	#477	1	1	yes	characteristic
101	C₂×C₁₃₇	—	2	1	no	—
102	C₂×C₁₃₇	—	2	1	no	—
103	C₂×C₁₃₇	—	2	1	no	—
104	C₂×C₁₃₇	#1947	1	1	yes	characteristic
105	#13182	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₁₃₇	#13182	1	1	yes	characteristic, direct product factor, Sylow
107	#2161	C₂×C₁₃₇	1	2	yes	—
108	C₂×#364	C₂×C₁₃₇	1	1	yes	characteristic
109	#1848	C₂×C₁₃₇	1	1	yes	characteristic
110	#1772	C₂×C₁₃₇	1	1	yes	characteristic
111	#1389	C₂×C₁₃₇	1	2	yes	—
112	C₂×M₃₂	C₂²×C₁₃₇	1	2	yes	—
113	C₂×#108	—	2	1	no	—
114	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
115	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
116	#427	C₂²×C₁₃₇	1	1	yes	characteristic
117	#423	—	2	1	no	—
118	#405	—	2	1	no	—
119	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
120	#368	—	2	4	no	—
121	#364	C₄×C₁₃₇	1	2	yes	complemented
122	#364	C₄×C₁₃₇	1	2	yes	complemented
123	#339	—	2	1	no	—
124	C₂×M₁₆	—	2	1	no	—
125	C₂×M₁₆	—	2	2	no	—
126	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
127	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
128	#117	—	2	1	no	—
129	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
130	M₃₂	—	2	2	no	—
131	M₃₂	C₂×C₄×C₁₃₇	1	4	yes	—
132	C₂×C₁₆	—	2	2	no	—
133	#108	—	2	2	no	—
134	#108	—	2	1	no	—
135	#108	—	2	2	no	—
136	#108	—	4	1	no	—
137	#108	—	4	1	no	—
138	#108	D₄×C₁₃₇	1	2	yes	—
139	#107	—	2	1	no	—
140	#106	Q₈×C₁₃₇	1	2	yes	—
141	C₄⋊C₈	—	4	2	no	—
142	#97	—	4	1	no	—
143	C₄×C₈	—	4	1	no	—
144	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
145	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
146	C₄×C₈	Q₈×C₁₃₇	1	1	yes	characteristic
147	C₄×C₈	Q₈×C₁₃₇	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	4	1	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	4	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
157	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
158	C₂×C₈	D₈×C₁₃₇	1	1	yes	characteristic
159	C₂×C₈	D₈×C₁₃₇	1	1	yes	characteristic
160	C₂×C₈	Q₁₆×C₁₃₇	1	1	yes	characteristic
161	C₂×C₈	Q₁₆×C₁₃₇	1	1	yes	characteristic
162	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
163	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
164	C₂×C₈	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
165	C₂×C₈	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
166	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
167	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
168	C₄⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
173	C₄²	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
174	C₁₆	—	2	4	no	—
175	C₂³	#137×C₁₃₇	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
181	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
182	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
183	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
184	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	1	yes	characteristic
185	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
186	C₈	—	4	2	no	—
187	C₈	—	8	2	no	—
188	C₈	#107×C₁₃₇	1	1	yes	characteristic
189	C₈	#107×C₁₃₇	1	1	yes	characteristic
190	C₈	#107×C₁₃₇	1	1	yes	characteristic
191	C₈	#107×C₁₃₇	1	1	yes	characteristic
192	C₈	C₄×D₄×C₁₃₇	1	1	yes	characteristic
193	C₈	C₄×Q₈×C₁₃₇	1	1	yes	characteristic
194	C₈	#125×C₁₃₇	1	1	yes	characteristic
195	C₈	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
196	C₂²	—	4	1	no	—
197	C₂²	#428×C₁₃₇	1	1	yes	characteristic
198	C₂²	#462×C₁₃₇	1	1	yes	characteristic
199	C₂²	#474×C₁₃₇	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	8	2	no	—
203	C₄	#389×C₁₃₇	1	1	yes	characteristic
204	C₄	C₂×#107×C₁₃₇	1	1	yes	characteristic
205	C₄	#459×C₁₃₇	1	1	yes	characteristic
206	C₄	#477×C₁₃₇	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#1947×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₇	#459
commutator	C₂×C₈	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
socle	C₂×C₁₃₇	#1947
prime radical	C₂³×C₁₃₇	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. #364×C₁₃₇	C₄	semidirect
16. #364×C₁₃₇	C₄	semidirect
105. #13182	C₁₃₇	direct (N × H)
106. C₁₃₇	#13182	direct (N × H)
121. #364	C₄×C₁₃₇	semidirect
122. #364	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#2161×C₁₃₇	2
C₂×#364×C₁₃₇	2
#1848×C₁₃₇	2
#1772×C₁₃₇	2
#1389×C₁₃₇	2
#13182	137

Subgroups — C137×#13182 #46862568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#13182 #46862568