Subgroups — C₁₃₇×#9771 #46859157

Order 35072.

414 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	C₂×#500×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1848×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1549×C₁₃₇	C₂	1	2	yes	maximal
4	#1548×C₁₃₇	C₂	1	2	yes	maximal
5	#1450×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
7	#500×C₁₃₇	—	2	1	no	—
8	#500×C₁₃₇	—	2	1	no	—
9	#500×C₁₃₇	C₄	1	4	yes	—
10	#431×C₁₃₇	C₂²	1	1	yes	characteristic
11	#431×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×#108×C₁₃₇	—	2	2	no	—
13	C₂×#107×C₁₃₇	—	2	2	no	—
14	#423×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	#333×C₁₃₇	—	2	4	no	—
17	#332×C₁₃₇	—	2	4	no	—
18	#322×C₁₃₇	—	2	1	no	—
19	C₈²×C₁₃₇	—	2	1	no	—
20	C₂²×Q₈×C₁₃₇	—	2	2	no	—
21	C₂×M₁₆×C₁₃₇	—	2	2	no	—
22	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
23	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
24	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
25	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
26	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
29	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
30	#108×C₁₃₇	—	4	4	no	—
31	#107×C₁₃₇	—	2	4	no	—
32	#107×C₁₃₇	—	2	4	no	—
33	C₄⋊C₈×C₁₃₇	—	2	2	no	—
34	C₄⋊C₈×C₁₃₇	—	2	2	no	—
35	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
36	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
37	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
38	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
39	C₄×C₈×C₁₃₇	—	2	1	no	—
40	C₄×C₈×C₁₃₇	—	2	1	no	—
41	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₁₃₇	—	2	4	no	—
45	C₂×Q₈×C₁₃₇	—	4	2	no	—
46	C₂×Q₈×C₁₃₇	—	8	4	no	—
47	C₂²×C₄×C₁₃₇	—	4	2	no	—
48	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
51	M₁₆×C₁₃₇	—	4	4	no	—
52	C₂×C₈×C₁₃₇	—	2	2	no	—
53	C₂×C₈×C₁₃₇	—	2	2	no	—
54	C₂×C₈×C₁₃₇	—	4	2	no	—
55	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₃₇	SD₁₆	1	4	yes	—
58	C₂×C₈×C₁₃₇	Q₁₆	1	4	yes	—
59	C₄⋊C₄×C₁₃₇	—	2	4	no	—
60	C₄⋊C₄×C₁₃₇	—	2	4	no	—
61	C₄⋊C₄×C₁₃₇	—	4	2	no	—
62	C₄⋊C₄×C₁₃₇	—	4	2	no	—
63	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
67	C₂³×C₁₃₇	#124	1	1	yes	characteristic
68	Q₈×C₁₃₇	—	8	4	no	—
69	Q₈×C₁₃₇	—	8	4	no	—
70	C₂×C₄×C₁₃₇	—	2	2	no	—
71	C₂×C₄×C₁₃₇	—	2	2	no	—
72	C₂×C₄×C₁₃₇	—	4	4	no	—
73	C₂×C₄×C₁₃₇	—	8	4	no	—
74	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
78	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
80	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
82	C₈×C₁₃₇	—	4	2	no	—
83	C₈×C₁₃₇	—	4	2	no	—
84	C₈×C₁₃₇	—	8	2	no	—
85	C₈×C₁₃₇	#103	1	4	yes	—
86	C₈×C₁₃₇	#103	1	4	yes	—
87	C₂²×C₁₃₇	—	2	1	no	—
88	C₂²×C₁₃₇	—	2	1	no	—
89	C₂²×C₁₃₇	#390	1	1	yes	characteristic
90	C₂²×C₁₃₇	#487	1	2	yes	—
91	C₄×C₁₃₇	—	2	1	no	—
92	C₄×C₁₃₇	—	2	1	no	—
93	C₄×C₁₃₇	—	8	4	no	—
94	C₄×C₁₃₇	C₂×#103	1	2	yes	—
95	C₄×C₁₃₇	#472	1	2	yes	—
96	C₄×C₁₃₇	#473	1	2	yes	—
97	C₄×C₁₃₇	#492	1	1	yes	characteristic
98	C₄×C₁₃₇	#494	1	1	yes	characteristic
99	C₂×C₁₃₇	—	2	2	no	—
100	C₂×C₁₃₇	#1956	1	1	yes	characteristic
101	C₂×C₁₃₇	#1978	1	1	yes	characteristic
102	C₂×C₁₃₇	#1979	1	1	yes	characteristic
103	#9771	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₁₃₇	#9771	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×#500	C₂×C₁₃₇	1	1	yes	characteristic
106	#1848	C₂×C₁₃₇	1	1	yes	characteristic
107	#1549	C₂×C₁₃₇	1	2	yes	—
108	#1548	C₂×C₁₃₇	1	2	yes	—
109	#1450	C₂×C₁₃₇	1	1	yes	characteristic
110	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
111	#500	—	2	1	no	—
112	#500	—	2	1	no	—
113	#500	C₄×C₁₃₇	1	4	yes	—
114	#431	C₂²×C₁₃₇	1	1	yes	characteristic
115	#431	C₂²×C₁₃₇	1	1	yes	characteristic
116	C₂×#108	—	2	2	no	—
117	C₂×#107	—	2	2	no	—
118	#423	C₂²×C₁₃₇	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
120	#333	—	2	4	no	—
121	#332	—	2	4	no	—
122	#322	—	2	1	no	—
123	C₈²	—	2	1	no	—
124	C₂²×Q₈	—	2	2	no	—
125	C₂×M₁₆	—	2	2	no	—
126	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	4	yes	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
134	#108	—	4	4	no	—
135	#107	—	2	4	no	—
136	#107	—	2	4	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
140	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
141	C₂²⋊C₈	—	2	1	no	—
142	C₂²⋊C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
146	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
147	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
148	C₂×Q₈	—	2	4	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	8	4	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
153	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
154	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
155	M₁₆	—	4	4	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
160	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
161	C₂×C₈	SD₁₆×C₁₃₇	1	4	yes	—
162	C₂×C₈	Q₁₆×C₁₃₇	1	4	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
168	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
169	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
170	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
171	C₂³	#124×C₁₃₇	1	1	yes	characteristic
172	Q₈	—	8	4	no	—
173	Q₈	—	8	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	8	4	no	—
178	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
179	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
181	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
182	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
183	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
184	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
185	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
186	C₈	—	4	2	no	—
187	C₈	—	4	2	no	—
188	C₈	—	8	2	no	—
189	C₈	#103×C₁₃₇	1	4	yes	—
190	C₈	#103×C₁₃₇	1	4	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#390×C₁₃₇	1	1	yes	characteristic
194	C₂²	#487×C₁₃₇	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	8	4	no	—
198	C₄	C₂×#103×C₁₃₇	1	2	yes	—
199	C₄	#472×C₁₃₇	1	2	yes	—
200	C₄	#473×C₁₃₇	1	2	yes	—
201	C₄	#492×C₁₃₇	1	1	yes	characteristic
202	C₄	#494×C₁₃₇	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	#1956×C₁₃₇	1	1	yes	characteristic
205	C₂	#1978×C₁₃₇	1	1	yes	characteristic
206	C₂	#1979×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#390
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
prime radical	C₂³×C₁₃₇	#124
LCS	G ▷ C₄² ▷ C₂² ▷ 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
103. #9771	C₁₃₇	direct (N × H)
104. C₁₃₇	#9771	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×#500×C₁₃₇	2
#1848×C₁₃₇	2
#1549×C₁₃₇	2
#1548×C₁₃₇	2
#1450×C₁₃₇	2
#9771	137

Subgroups — C137×#9771 #46859157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#9771 #46859157