Subgroups — C₁₃₇×#21735 #46871121

Order 35072.

662 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3162×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2960×C₁₃₇	C₂	1	2	yes	maximal
3	#1995×C₁₃₇	C₂	1	2	yes	maximal
4	#1946×C₁₃₇	C₂	1	4	yes	maximal
5	#1849×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#1772×C₁₃₇	C₂	1	4	yes	maximal
7	#1726×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#533×C₁₃₇	C₂²	1	2	yes	—
9	#501×C₁₃₇	C₄	1	8	yes	complemented
10	#446×C₁₃₇	—	2	4	no	—
11	C₈×Q₈×C₁₃₇	—	2	4	no	—
12	C₂×#108×C₁₃₇	C₂²	1	4	yes	—
13	#427×C₁₃₇	C₂²	1	4	yes	—
14	#424×C₁₃₇	C₂²	1	4	yes	—
15	#423×C₁₃₇	C₂²	1	2	yes	—
16	#405×C₁₃₇	C₂²	1	4	yes	—
17	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
18	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	#376×C₁₃₇	C₂²	1	1	yes	characteristic
22	#339×C₁₃₇	C₂²	1	8	yes	—
23	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
24	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
26	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	4	yes	—
28	#125×C₁₃₇	C₂×C₄	1	4	yes	—
29	C₂²⋊Q₈×C₁₃₇	—	2	4	no	—
30	C₄×Q₈×C₁₃₇	—	2	4	no	—
31	#117×C₁₃₇	C₂³	1	4	yes	—
32	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
33	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
34	#108×C₁₃₇	—	2	8	no	—
35	#107×C₁₃₇	C₂×C₄	1	8	yes	—
36	#106×C₁₃₇	C₂×C₄	1	8	yes	—
37	C₄⋊C₈×C₁₃₇	—	2	8	no	—
38	C₄⋊C₈×C₁₃₇	—	2	4	no	—
39	#97×C₁₃₇	—	2	4	no	—
40	#97×C₁₃₇	—	2	4	no	—
41	#97×C₁₃₇	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₁₃₇	—	2	4	no	—
43	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
44	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
45	C₄×C₈×C₁₃₇	Q₈	1	8	yes	complemented
46	#95×C₁₃₇	—	2	2	no	—
47	C₂×Q₈×C₁₃₇	—	2	4	no	—
48	C₂²×C₄×C₁₃₇	—	2	2	no	—
49	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
50	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
51	M₁₆×C₁₃₇	—	2	8	no	—
52	C₂×C₈×C₁₃₇	—	2	2	no	—
53	C₂×C₈×C₁₃₇	—	2	4	no	—
54	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
59	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
60	C₂×C₈×C₁₃₇	#41	1	4	yes	—
61	C₄⋊C₄×C₁₃₇	—	2	8	no	—
62	C₄⋊C₄×C₁₃₇	—	2	4	no	—
63	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
64	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
65	C₄²×C₁₃₇	—	2	4	no	—
66	C₄²×C₁₃₇	—	2	2	no	—
67	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
68	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
69	C₄²×C₁₃₇	C₂×Q₈	1	4	yes	—
70	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
71	Q₈×C₁₃₇	—	4	8	no	—
72	C₂×C₄×C₁₃₇	—	2	8	no	—
73	C₂×C₄×C₁₃₇	—	2	4	no	—
74	C₂×C₄×C₁₃₇	—	4	2	no	—
75	C₂×C₄×C₁₃₇	—	4	4	no	—
76	C₂×C₄×C₁₃₇	C₄×D₄	1	4	yes	—
77	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
80	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
81	C₈×C₁₃₇	—	2	2	no	—
82	C₈×C₁₃₇	—	4	4	no	—
83	C₈×C₁₃₇	C₄×Q₈	1	8	yes	—
84	C₈×C₁₃₇	C₄⋊Q₈	1	2	yes	—
85	C₈×C₁₃₇	C₄⋊Q₈	1	2	yes	—
86	C₂²×C₁₃₇	—	2	2	no	—
87	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
88	C₂²×C₁₃₇	#533	1	2	yes	—
89	C₄×C₁₃₇	—	2	4	no	—
90	C₄×C₁₃₇	—	4	4	no	—
91	C₄×C₁₃₇	—	4	8	no	—
92	C₄×C₁₃₇	C₂×C₄×Q₈	1	2	yes	—
93	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
94	C₄×C₁₃₇	#532	1	1	yes	characteristic
95	C₂×C₁₃₇	—	2	2	no	—
96	C₂×C₁₃₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₁₃₇	#2955	1	2	yes	—
98	#21735	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₁₃₇	#21735	1	1	yes	characteristic, direct product factor, Sylow
100	#3162	C₂×C₁₃₇	1	1	yes	characteristic
101	#2960	C₂×C₁₃₇	1	2	yes	—
102	#1995	C₂×C₁₃₇	1	2	yes	—
103	#1946	C₂×C₁₃₇	1	4	yes	—
104	#1849	C₂×C₁₃₇	1	1	yes	characteristic
105	#1772	C₂×C₁₃₇	1	4	yes	—
106	#1726	C₂×C₁₃₇	1	1	yes	characteristic
107	#533	C₂²×C₁₃₇	1	2	yes	—
108	#501	C₄×C₁₃₇	1	8	yes	complemented
109	#446	—	2	4	no	—
110	C₈×Q₈	—	2	4	no	—
111	C₂×#108	C₂²×C₁₃₇	1	4	yes	—
112	#427	C₂²×C₁₃₇	1	4	yes	—
113	#424	C₂²×C₁₃₇	1	4	yes	—
114	#423	C₂²×C₁₃₇	1	2	yes	—
115	#405	C₂²×C₁₃₇	1	4	yes	—
116	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
117	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
120	#376	C₂²×C₁₃₇	1	1	yes	characteristic
121	#339	C₂²×C₁₃₇	1	8	yes	—
122	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
123	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
124	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
125	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
126	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	4	yes	—
127	#125	C₂×C₄×C₁₃₇	1	4	yes	—
128	C₂²⋊Q₈	—	2	4	no	—
129	C₄×Q₈	—	2	4	no	—
130	#117	C₂³×C₁₃₇	1	4	yes	—
131	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
132	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
133	#108	—	2	8	no	—
134	#107	C₂×C₄×C₁₃₇	1	8	yes	—
135	#106	C₂×C₄×C₁₃₇	1	8	yes	—
136	C₄⋊C₈	—	2	8	no	—
137	C₄⋊C₈	—	2	4	no	—
138	#97	—	2	4	no	—
139	#97	—	2	4	no	—
140	#97	C₂×C₄×C₁₃₇	1	4	yes	—
141	C₄×C₈	—	2	4	no	—
142	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
143	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
144	C₄×C₈	Q₈×C₁₃₇	1	8	yes	complemented
145	#95	—	2	2	no	—
146	C₂×Q₈	—	2	4	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
149	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
150	M₁₆	—	2	8	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
154	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
155	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
156	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
157	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
158	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
159	C₂×C₈	#41×C₁₃₇	1	4	yes	—
160	C₄⋊C₄	—	2	8	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	2	2	no	—
166	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
167	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
168	C₄²	C₂×Q₈×C₁₃₇	1	4	yes	—
169	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
170	Q₈	—	4	8	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	C₄×D₄×C₁₃₇	1	4	yes	—
176	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
178	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
179	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	4	4	no	—
182	C₈	C₄×Q₈×C₁₃₇	1	8	yes	—
183	C₈	C₄⋊Q₈×C₁₃₇	1	2	yes	—
184	C₈	C₄⋊Q₈×C₁₃₇	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
187	C₂²	#533×C₁₃₇	1	2	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	4	4	no	—
190	C₄	—	4	8	no	—
191	C₄	C₂×C₄×Q₈×C₁₃₇	1	2	yes	—
192	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
193	C₄	#532×C₁₃₇	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	C₄×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
196	C₂	#2955×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₃₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂²×C₁₃₇	C₂×C₄×D₄
prime radical	C₂³×C₁₃₇	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
9. #501×C₁₃₇	C₄	semidirect
45. C₄×C₈×C₁₃₇	Q₈	semidirect
98. #21735	C₁₃₇	direct (N × H)
99. C₁₃₇	#21735	direct (N × H)
108. #501	C₄×C₁₃₇	semidirect
144. C₄×C₈	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3162×C₁₃₇	2
#2960×C₁₃₇	2
#1995×C₁₃₇	2
#1946×C₁₃₇	2
#1849×C₁₃₇	2
#1772×C₁₃₇	2
#1726×C₁₃₇	2
#21735	137

Subgroups — C137×#21735 #46871121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21735 #46871121