Subgroups — C₁₃₇×#13179 #46862565

Order 35072.

286 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#368×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1848×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄×#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1381×C₁₃₇	C₂	1	2	yes	maximal
6	C₂²×C₁₆×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	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
11	#423×C₁₃₇	—	2	1	no	—
12	C₄×M₁₆×C₁₃₇	—	2	1	no	—
13	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	#368×C₁₃₇	—	2	4	no	—
15	C₄⋊C₁₆×C₁₃₇	C₄	1	2	yes	—
16	C₄⋊C₁₆×C₁₃₇	C₄	1	2	yes	—
17	#341×C₁₃₇	—	2	1	no	—
18	C₂×M₁₆×C₁₃₇	—	2	1	no	—
19	C₂×M₁₆×C₁₃₇	—	2	1	no	—
20	C₂×M₁₆×C₁₃₇	—	2	2	no	—
21	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₁₃₇	—	2	2	no	—
25	C₂×C₁₆×C₁₃₇	—	2	2	no	—
26	C₂×C₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
27	#108×C₁₃₇	—	2	2	no	—
28	#108×C₁₃₇	—	2	1	no	—
29	#108×C₁₃₇	—	2	1	no	—
30	#108×C₁₃₇	—	2	2	no	—
31	#108×C₁₃₇	—	4	2	no	—
32	#108×C₁₃₇	D₄	1	2	yes	—
33	#108×C₁₃₇	Q₈	1	2	yes	—
34	C₄⋊C₈×C₁₃₇	—	4	1	no	—
35	C₄⋊C₈×C₁₃₇	—	4	1	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	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₁₃₇	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
45	M₁₆×C₁₃₇	—	2	2	no	—
46	M₁₆×C₁₃₇	—	2	2	no	—
47	M₁₆×C₁₃₇	—	4	1	no	—
48	M₁₆×C₁₃₇	—	4	1	no	—
49	M₁₆×C₁₃₇	—	4	2	no	—
50	M₁₆×C₁₃₇	—	4	2	no	—
51	C₂×C₈×C₁₃₇	—	2	1	no	—
52	C₂×C₈×C₁₃₇	—	2	1	no	—
53	C₂×C₈×C₁₃₇	—	4	1	no	—
54	C₂×C₈×C₁₃₇	—	4	1	no	—
55	C₂×C₈×C₁₃₇	—	4	2	no	—
56	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₈×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
63	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
64	C₄²×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
65	C₄²×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
66	C₄²×C₁₃₇	D₈	1	1	yes	characteristic
67	C₄²×C₁₃₇	D₈	1	1	yes	characteristic
68	C₁₆×C₁₃₇	—	2	4	no	—
69	C₂³×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₁₃₇	—	2	2	no	—
71	C₂×C₄×C₁₃₇	—	4	1	no	—
72	C₂×C₄×C₁₃₇	#107	1	1	yes	characteristic
73	C₂×C₄×C₁₃₇	#107	1	1	yes	characteristic
74	C₂×C₄×C₁₃₇	#107	1	1	yes	characteristic
75	C₂×C₄×C₁₃₇	#107	1	1	yes	characteristic
76	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
78	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₁₃₇	#137	1	1	yes	characteristic
82	C₈×C₁₃₇	—	2	1	no	—
83	C₈×C₁₃₇	—	2	1	no	—
84	C₈×C₁₃₇	—	4	2	no	—
85	C₈×C₁₃₇	—	4	2	no	—
86	C₈×C₁₃₇	—	8	2	no	—
87	C₈×C₁₃₇	C₄×D₄	1	1	yes	characteristic
88	C₈×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
89	C₈×C₁₃₇	#125	1	1	yes	characteristic
90	C₈×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
91	C₂²×C₁₃₇	—	2	1	no	—
92	C₂²×C₁₃₇	—	2	1	no	—
93	C₂²×C₁₃₇	C₂×#107	1	1	yes	characteristic
94	C₂²×C₁₃₇	#462	1	1	yes	characteristic
95	C₂²×C₁₃₇	#477	1	1	yes	characteristic
96	C₄×C₁₃₇	—	2	1	no	—
97	C₄×C₁₃₇	—	2	1	no	—
98	C₄×C₁₃₇	—	2	1	no	—
99	C₄×C₁₃₇	#368	1	2	yes	complemented
100	C₄×C₁₃₇	#389	1	1	yes	characteristic
101	C₄×C₁₃₇	#428	1	1	yes	characteristic
102	C₄×C₁₃₇	#459	1	1	yes	characteristic
103	C₄×C₁₃₇	#474	1	1	yes	characteristic
104	C₂×C₁₃₇	—	2	1	no	—
105	C₂×C₁₃₇	—	2	1	no	—
106	C₂×C₁₃₇	#1947	1	1	yes	characteristic
107	C₂×C₁₃₇	C₂×#368	1	1	yes	characteristic
108	C₂×C₁₃₇	#2161	1	1	yes	characteristic
109	#13179	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₁₃₇	#13179	1	1	yes	characteristic, direct product factor, Sylow
111	C₂×#368	C₂×C₁₃₇	1	2	yes	—
112	C₂×C₄⋊C₁₆	C₂×C₁₃₇	1	1	yes	characteristic
113	#1848	C₂×C₁₃₇	1	1	yes	characteristic
114	C₄×#108	C₂×C₁₃₇	1	1	yes	characteristic
115	#1381	C₂×C₁₃₇	1	2	yes	—
116	C₂²×C₁₆	C₂²×C₁₃₇	1	2	yes	—
117	C₂×#108	—	2	1	no	—
118	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
119	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
120	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
121	#423	—	2	1	no	—
122	C₄×M₁₆	—	2	1	no	—
123	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
124	#368	—	2	4	no	—
125	C₄⋊C₁₆	C₄×C₁₃₇	1	2	yes	—
126	C₄⋊C₁₆	C₄×C₁₃₇	1	2	yes	—
127	#341	—	2	1	no	—
128	C₂×M₁₆	—	2	1	no	—
129	C₂×M₁₆	—	2	1	no	—
130	C₂×M₁₆	—	2	2	no	—
131	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
132	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
133	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
137	#108	—	2	2	no	—
138	#108	—	2	1	no	—
139	#108	—	2	1	no	—
140	#108	—	2	2	no	—
141	#108	—	4	2	no	—
142	#108	D₄×C₁₃₇	1	2	yes	—
143	#108	Q₈×C₁₃₇	1	2	yes	—
144	C₄⋊C₈	—	4	1	no	—
145	C₄⋊C₈	—	4	1	no	—
146	#97	—	4	1	no	—
147	C₄×C₈	—	4	1	no	—
148	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
149	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
150	C₄×C₈	Q₈×C₁₃₇	1	1	yes	characteristic
151	C₄×C₈	Q₈×C₁₃₇	1	1	yes	characteristic
152	C₂²×C₄	Q₁₆×C₁₃₇	1	1	yes	characteristic
153	C₂²×C₄	Q₁₆×C₁₃₇	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
155	M₁₆	—	2	2	no	—
156	M₁₆	—	2	2	no	—
157	M₁₆	—	4	1	no	—
158	M₁₆	—	4	1	no	—
159	M₁₆	—	4	2	no	—
160	M₁₆	—	4	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
167	C₂×C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
168	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
169	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
170	C₂×C₈	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
171	C₂×C₈	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
172	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
173	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
174	C₄²	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
175	C₄²	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
176	C₄²	D₈×C₁₃₇	1	1	yes	characteristic
177	C₄²	D₈×C₁₃₇	1	1	yes	characteristic
178	C₁₆	—	2	4	no	—
179	C₂³	C₂×Q₁₆×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	#107×C₁₃₇	1	1	yes	characteristic
183	C₂×C₄	#107×C₁₃₇	1	1	yes	characteristic
184	C₂×C₄	#107×C₁₃₇	1	1	yes	characteristic
185	C₂×C₄	#107×C₁₃₇	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
187	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
188	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
189	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
190	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
191	C₂×C₄	#137×C₁₃₇	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	4	2	no	—
195	C₈	—	4	2	no	—
196	C₈	—	8	2	no	—
197	C₈	C₄×D₄×C₁₃₇	1	1	yes	characteristic
198	C₈	C₄×Q₈×C₁₃₇	1	1	yes	characteristic
199	C₈	#125×C₁₃₇	1	1	yes	characteristic
200	C₈	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×#107×C₁₃₇	1	1	yes	characteristic
204	C₂²	#462×C₁₃₇	1	1	yes	characteristic
205	C₂²	#477×C₁₃₇	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	#368×C₁₃₇	1	2	yes	complemented
210	C₄	#389×C₁₃₇	1	1	yes	characteristic
211	C₄	#428×C₁₃₇	1	1	yes	characteristic
212	C₄	#459×C₁₃₇	1	1	yes	characteristic
213	C₄	#474×C₁₃₇	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	#1947×C₁₃₇	1	1	yes	characteristic
217	C₂	C₂×#368×C₁₃₇	1	1	yes	characteristic
218	C₂	#2161×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	C₂×#107
prime radical	C₂³×C₁₃₇	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
99. C₄×C₁₃₇	#368	semidirect
109. #13179	C₁₃₇	direct (N × H)
110. C₁₃₇	#13179	direct (N × H)
209. C₄	#368×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#368×C₁₃₇	2
C₂×C₄⋊C₁₆×C₁₃₇	2
#1848×C₁₃₇	2
C₄×#108×C₁₃₇	2
#1381×C₁₃₇	2
#13179	137

Subgroups — C137×#13179 #46862565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#13179 #46862565