Subgroups — #3113×C₁₃₇ #31778124

Order 17536.

380 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#533×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#483×C₁₃₇	C₂	1	2	yes	maximal
4	#475×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#462×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#451×C₁₃₇	C₂	1	2	yes	maximal
7	#441×C₁₃₇	C₂	1	4	yes	maximal, complemented
8	#432×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₁₃₇	C₂²	1	4	yes	—
11	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	#125×C₁₃₇	—	2	1	no	—
14	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
15	C₂²⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
16	C₄×Q₈×C₁₃₇	—	2	1	no	—
17	C₄×Q₈×C₁₃₇	—	2	1	no	—
18	C₄×Q₈×C₁₃₇	—	2	1	no	—
19	C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
20	C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
21	C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
22	#117×C₁₃₇	—	2	1	no	—
23	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
24	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₁₃₇	—	2	1	no	—
26	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
27	#106×C₁₃₇	C₂²	1	4	yes	—
28	C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
29	#103×C₁₃₇	C₂²	1	4	yes	—
30	#103×C₁₃₇	C₂²	1	4	yes	—
31	C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
32	#97×C₁₃₇	C₂²	1	2	yes	—
33	C₂×Q₈×C₁₃₇	—	2	4	no	—
34	C₂×Q₈×C₁₃₇	—	2	1	no	—
35	C₂×Q₈×C₁₃₇	—	2	1	no	—
36	C₂×Q₈×C₁₃₇	C₂³	1	2	yes	—
37	C₂×Q₈×C₁₃₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₃₇	—	2	1	no	—
39	C₂²×C₄×C₁₃₇	—	2	1	no	—
40	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₇	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
43	Q₁₆×C₁₃₇	—	2	8	no	—
44	C₂×C₈×C₁₃₇	C₂³	1	4	yes	—
45	C₄⋊C₄×C₁₃₇	—	2	2	no	—
46	C₄⋊C₄×C₁₃₇	—	2	2	no	—
47	C₄⋊C₄×C₁₃₇	—	2	1	no	—
48	C₄⋊C₄×C₁₃₇	—	2	1	no	—
49	C₄⋊C₄×C₁₃₇	—	2	1	no	—
50	C₄⋊C₄×C₁₃₇	—	2	1	no	—
51	C₄⋊C₄×C₁₃₇	—	2	2	no	—
52	C₄⋊C₄×C₁₃₇	—	2	2	no	—
53	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
54	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
56	C₄²×C₁₃₇	—	2	2	no	—
57	C₄²×C₁₃₇	—	2	2	no	—
58	C₄²×C₁₃₇	—	2	2	no	—
59	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
60	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
61	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
62	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
63	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
64	Q₈×C₁₃₇	—	2	4	no	—
65	Q₈×C₁₃₇	—	2	2	no	—
66	Q₈×C₁₃₇	—	2	4	no	—
67	Q₈×C₁₃₇	—	4	2	no	—
68	Q₈×C₁₃₇	#41	1	4	yes	—
69	C₂×C₄×C₁₃₇	—	2	4	no	—
70	C₂×C₄×C₁₃₇	—	2	2	no	—
71	C₂×C₄×C₁₃₇	—	2	2	no	—
72	C₂×C₄×C₁₃₇	—	2	2	no	—
73	C₂×C₄×C₁₃₇	—	2	2	no	—
74	C₂×C₄×C₁₃₇	—	2	2	no	—
75	C₂×C₄×C₁₃₇	—	2	2	no	—
76	C₂×C₄×C₁₃₇	—	2	2	no	—
77	C₂×C₄×C₁₃₇	—	4	2	no	—
78	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
84	C₈×C₁₃₇	—	2	4	no	—
85	C₂²×C₁₃₇	—	2	1	no	—
86	C₂²×C₁₃₇	—	2	1	no	—
87	C₂²×C₁₃₇	#137	1	2	yes	—
88	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₁₃₇	—	2	4	no	—
90	C₄×C₁₃₇	—	2	1	no	—
91	C₄×C₁₃₇	—	2	1	no	—
92	C₄×C₁₃₇	—	2	1	no	—
93	C₄×C₁₃₇	—	2	1	no	—
94	C₄×C₁₃₇	—	4	2	no	—
95	C₄×C₁₃₇	—	4	2	no	—
96	C₄×C₁₃₇	—	4	2	no	—
97	C₄×C₁₃₇	#137	1	2	yes	—
98	C₄×C₁₃₇	C₂×#41	1	2	yes	—
99	C₂×C₁₃₇	—	2	2	no	—
100	C₂×C₁₃₇	#525	1	1	yes	characteristic
101	C₂×C₁₃₇	C₂×#137	1	1	yes	characteristic
102	C₂×C₁₃₇	C₂×#137	1	1	yes	characteristic
103	C₁₃₇	#3113	1	1	yes	characteristic, direct product factor, Sylow
104	#3113	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	#533	C₂×C₁₃₇	1	1	yes	characteristic
106	C₂×C₄×Q₈	C₂×C₁₃₇	1	1	yes	characteristic
107	#483	C₂×C₁₃₇	1	2	yes	—
108	#475	C₂×C₁₃₇	1	2	yes	complemented
109	#462	C₂×C₁₃₇	1	2	yes	complemented
110	#451	C₂×C₁₃₇	1	2	yes	—
111	#441	C₂×C₁₃₇	1	4	yes	complemented
112	#432	C₂×C₁₃₇	1	1	yes	characteristic
113	C₂²×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
114	C₂×Q₁₆	C₂²×C₁₃₇	1	4	yes	—
115	C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
116	C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
117	#125	—	2	1	no	—
118	C₂²⋊Q₈	—	2	1	no	—
119	C₂²⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
120	C₄×Q₈	—	2	1	no	—
121	C₄×Q₈	—	2	1	no	—
122	C₄×Q₈	—	2	1	no	—
123	C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
124	C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
125	C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
126	#117	—	2	1	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
129	C₂×C₄²	—	2	1	no	—
130	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
131	#106	C₂²×C₁₃₇	1	4	yes	—
132	C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
133	#103	C₂²×C₁₃₇	1	4	yes	—
134	#103	C₂²×C₁₃₇	1	4	yes	—
135	C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
136	#97	C₂²×C₁₃₇	1	2	yes	—
137	C₂×Q₈	—	2	4	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	C₂³×C₁₃₇	1	2	yes	—
141	C₂×Q₈	C₂³×C₁₃₇	1	2	yes	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	D₄×C₁₃₇	1	1	yes	characteristic
145	C₂²×C₄	D₄×C₁₃₇	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
147	Q₁₆	—	2	8	no	—
148	C₂×C₈	C₂³×C₁₃₇	1	4	yes	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	2	1	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
158	C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	D₄×C₁₃₇	1	1	yes	characteristic
164	C₄²	D₄×C₁₃₇	1	1	yes	characteristic
165	C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
166	C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
167	C₂³	C₂×D₄×C₁₃₇	1	1	yes	characteristic
168	Q₈	—	2	4	no	—
169	Q₈	—	2	2	no	—
170	Q₈	—	2	4	no	—
171	Q₈	—	4	2	no	—
172	Q₈	#41×C₁₃₇	1	4	yes	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
183	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
184	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
185	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
186	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
187	C₂×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
188	C₈	—	2	4	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	#137×C₁₃₇	1	2	yes	—
192	C₂²	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
193	C₄	—	2	4	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	2	no	—
201	C₄	#137×C₁₃₇	1	2	yes	—
202	C₄	C₂×#41×C₁₃₇	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	#525×C₁₃₇	1	1	yes	characteristic
205	C₂	C₂×#137×C₁₃₇	1	1	yes	characteristic
206	C₂	C₂×#137×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃₇
prime radical	C₂³×C₁₃₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. #475×C₁₃₇	C₂	semidirect
5. #462×C₁₃₇	C₂	semidirect
7. #441×C₁₃₇	C₂	semidirect
103. C₁₃₇	#3113	direct (N × H)
104. #3113	C₁₃₇	direct (N × H)
108. #475	C₂×C₁₃₇	semidirect
109. #462	C₂×C₁₃₇	semidirect
111. #441	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#533×C₁₃₇	2
C₂×C₄×Q₈×C₁₃₇	2
#483×C₁₃₇	2
#475×C₁₃₇	2
#462×C₁₃₇	2
#451×C₁₃₇	2
#441×C₁₃₇	2
#432×C₁₃₇	2
#3113	137

Subgroups — #3113×C137 #31778124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3113×C₁₃₇ #31778124