Subgroups — C₁₃₇×#13244 #46862630

Order 35072.

402 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#368×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#362×C₁₃₇	C₂	1	2	yes	maximal
4	#2108×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1979×C₁₃₇	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
7	C₂×Q₃₂×C₁₃₇	—	2	2	no	—
8	C₂×Q₃₂×C₁₃₇	—	2	1	no	—
9	C₂×Q₃₂×C₁₃₇	—	2	1	no	—
10	C₂×Q₃₂×C₁₃₇	C₄	1	4	yes	complemented
11	C₂×M₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
14	#419×C₁₃₇	—	2	2	no	—
15	C₂×#101×C₁₃₇	—	2	2	no	—
16	#405×C₁₃₇	C₂²	1	1	yes	characteristic
17	#368×C₁₃₇	—	2	2	no	—
18	#362×C₁₃₇	—	2	4	no	—
19	#346×C₁₃₇	—	2	1	no	—
20	C₄×C₁₆×C₁₃₇	—	2	1	no	—
21	C₂²×Q₈×C₁₃₇	—	2	2	no	—
22	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
23	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
24	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
25	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
26	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
27	C₂×Q₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₁₃₇	—	2	2	no	—
29	C₂×M₁₆×C₁₃₇	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
31	#117×C₁₃₇	D₄	1	1	yes	characteristic
32	Q₃₂×C₁₃₇	—	2	4	no	—
33	Q₃₂×C₁₃₇	—	2	4	no	—
34	M₃₂×C₁₃₇	—	2	1	no	—
35	M₃₂×C₁₃₇	D₄	1	2	yes	—
36	C₂×C₁₆×C₁₃₇	—	2	1	no	—
37	C₂×C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₁₃₇	D₄	1	2	yes	—
39	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
41	#108×C₁₃₇	—	2	2	no	—
42	#108×C₁₃₇	—	2	2	no	—
43	#103×C₁₃₇	—	4	4	no	—
44	#101×C₁₃₇	—	4	4	no	—
45	#97×C₁₃₇	—	2	1	no	—
46	C₄×C₈×C₁₃₇	—	2	1	no	—
47	C₂×Q₈×C₁₃₇	—	2	4	no	—
48	C₂×Q₈×C₁₃₇	—	4	2	no	—
49	C₂×Q₈×C₁₃₇	—	4	2	no	—
50	C₂×Q₈×C₁₃₇	—	4	2	no	—
51	C₂×Q₈×C₁₃₇	—	4	2	no	—
52	C₂×Q₈×C₁₃₇	—	4	2	no	—
53	C₂²×C₄×C₁₃₇	—	4	2	no	—
54	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
55	Q₁₆×C₁₃₇	—	2	4	no	—
56	Q₁₆×C₁₃₇	—	2	4	no	—
57	Q₁₆×C₁₃₇	—	4	2	no	—
58	Q₁₆×C₁₃₇	—	4	2	no	—
59	M₁₆×C₁₃₇	—	2	1	no	—
60	M₁₆×C₁₃₇	—	4	2	no	—
61	M₁₆×C₁₃₇	—	4	2	no	—
62	M₁₆×C₁₃₇	D₈	1	2	yes	—
63	C₂×C₈×C₁₃₇	—	2	1	no	—
64	C₂×C₈×C₁₃₇	—	4	2	no	—
65	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
70	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
71	C₄⋊C₄×C₁₃₇	D₈	1	2	yes	—
72	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
73	C₄²×C₁₃₇	—	2	1	no	—
74	C₁₆×C₁₃₇	—	2	2	no	—
75	C₁₆×C₁₃₇	C₂²⋊C₄	1	2	yes	—
76	C₁₆×C₁₃₇	C₂²⋊C₄	1	2	yes	—
77	C₂³×C₁₃₇	#135	1	1	yes	characteristic
78	Q₈×C₁₃₇	—	4	4	no	—
79	Q₈×C₁₃₇	—	4	4	no	—
80	Q₈×C₁₃₇	—	8	2	no	—
81	Q₈×C₁₃₇	—	8	2	no	—
82	C₂×C₄×C₁₃₇	—	2	2	no	—
83	C₂×C₄×C₁₃₇	—	4	4	no	—
84	C₂×C₄×C₁₃₇	—	8	2	no	—
85	C₂×C₄×C₁₃₇	—	8	2	no	—
86	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
88	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
89	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃₇	#135	1	1	yes	characteristic
92	C₈×C₁₃₇	—	2	2	no	—
93	C₈×C₁₃₇	—	8	2	no	—
94	C₈×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₈×C₁₃₇	C₄×D₄	1	1	yes	characteristic
96	C₈×C₁₃₇	#121	1	1	yes	characteristic
97	C₈×C₁₃₇	#124	1	1	yes	characteristic
98	C₂²×C₁₃₇	—	2	2	no	—
99	C₂²×C₁₃₇	C₄×D₈	1	1	yes	characteristic
100	C₂²×C₁₃₇	#466	1	1	yes	characteristic
101	C₂²×C₁₃₇	#495	1	1	yes	characteristic
102	C₄×C₁₃₇	—	4	2	no	—
103	C₄×C₁₃₇	—	8	4	no	—
104	C₄×C₁₃₇	#390	1	1	yes	characteristic
105	C₄×C₁₃₇	C₄×D₈	1	1	yes	characteristic
106	C₄×C₁₃₇	#466	1	1	yes	characteristic
107	C₄×C₁₃₇	#493	1	1	yes	characteristic
108	C₂×C₁₃₇	—	2	1	no	—
109	C₂×C₁₃₇	—	2	1	no	—
110	C₂×C₁₃₇	#1970	1	1	yes	characteristic
111	C₂×C₁₃₇	#2222	1	2	yes	—
112	#13244	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₁₃₇	#13244	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×Q₃₂	C₂×C₁₃₇	1	1	yes	characteristic
115	C₂×#368	C₂×C₁₃₇	1	1	yes	characteristic
116	C₂×#362	C₂×C₁₃₇	1	2	yes	—
117	#2108	C₂×C₁₃₇	1	1	yes	characteristic
118	#1979	C₂×C₁₃₇	1	2	yes	—
119	C₂²×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
120	C₂×Q₃₂	—	2	1	no	—
121	C₂×Q₃₂	—	2	1	no	—
122	C₂×Q₃₂	—	2	2	no	—
123	C₂×Q₃₂	C₄×C₁₃₇	1	4	yes	complemented
124	C₂×M₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
125	C₂²×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
126	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
127	#419	—	2	2	no	—
128	C₂×#101	—	2	2	no	—
129	#405	C₂²×C₁₃₇	1	1	yes	characteristic
130	#368	—	2	2	no	—
131	#362	—	2	4	no	—
132	#346	—	2	1	no	—
133	C₄×C₁₆	—	2	1	no	—
134	C₂²×Q₈	—	2	2	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×Q₁₆	—	2	2	no	—
140	C₂×Q₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	D₄×C₁₃₇	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
144	#117	D₄×C₁₃₇	1	1	yes	characteristic
145	Q₃₂	—	2	4	no	—
146	Q₃₂	—	2	4	no	—
147	M₃₂	—	2	1	no	—
148	M₃₂	D₄×C₁₃₇	1	2	yes	—
149	C₂×C₁₆	—	2	1	no	—
150	C₂×C₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
151	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
152	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
153	C₂×C₁₆	D₄×C₁₃₇	1	2	yes	—
154	#108	—	2	2	no	—
155	#108	—	2	2	no	—
156	#103	—	4	4	no	—
157	#101	—	4	4	no	—
158	#97	—	2	1	no	—
159	C₄×C₈	—	2	1	no	—
160	C₂×Q₈	—	2	4	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
168	Q₁₆	—	2	4	no	—
169	Q₁₆	—	2	4	no	—
170	Q₁₆	—	4	2	no	—
171	Q₁₆	—	4	2	no	—
172	M₁₆	—	2	1	no	—
173	M₁₆	—	4	2	no	—
174	M₁₆	—	4	2	no	—
175	M₁₆	D₈×C₁₃₇	1	2	yes	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
179	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
180	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
182	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
183	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
184	C₄⋊C₄	D₈×C₁₃₇	1	2	yes	—
185	C₂²⋊C₄	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₁₆	—	2	2	no	—
188	C₁₆	C₂²⋊C₄×C₁₃₇	1	2	yes	—
189	C₁₆	C₂²⋊C₄×C₁₃₇	1	2	yes	—
190	C₂³	#135×C₁₃₇	1	1	yes	characteristic
191	Q₈	—	4	4	no	—
192	Q₈	—	4	4	no	—
193	Q₈	—	8	2	no	—
194	Q₈	—	8	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
200	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
201	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
202	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
203	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
204	C₂×C₄	#135×C₁₃₇	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	8	2	no	—
207	C₈	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
208	C₈	C₄×D₄×C₁₃₇	1	1	yes	characteristic
209	C₈	#121×C₁₃₇	1	1	yes	characteristic
210	C₈	#124×C₁₃₇	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	C₄×D₈×C₁₃₇	1	1	yes	characteristic
213	C₂²	#466×C₁₃₇	1	1	yes	characteristic
214	C₂²	#495×C₁₃₇	1	1	yes	characteristic
215	C₄	—	4	2	no	—
216	C₄	—	8	4	no	—
217	C₄	#390×C₁₃₇	1	1	yes	characteristic
218	C₄	C₄×D₈×C₁₃₇	1	1	yes	characteristic
219	C₄	#466×C₁₃₇	1	1	yes	characteristic
220	C₄	#493×C₁₃₇	1	1	yes	characteristic
221	C₂	—	2	1	no	—
222	C₂	—	2	1	no	—
223	C₂	#1970×C₁₃₇	1	1	yes	characteristic
224	C₂	#2222×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#466
commutator	C₂×C₈	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
prime radical	C₂³×C₁₃₇	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ #405×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₂×Q₃₂×C₁₃₇	C₄	semidirect
112. #13244	C₁₃₇	direct (N × H)
113. C₁₃₇	#13244	direct (N × H)
123. C₂×Q₃₂	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₃₂×C₁₃₇	2
C₂×#368×C₁₃₇	2
C₂×#362×C₁₃₇	2
#2108×C₁₃₇	2
#1979×C₁₃₇	2
#13244	137

Subgroups — C137×#13244 #46862630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#13244 #46862630