Subgroups — C₁₃₇×#28825 #46878211

Order 35072.

634 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3164×C₁₃₇	C₂	1	4	yes	maximal
2	#2060×C₁₃₇	C₂	1	4	yes	maximal
3	#1952×C₁₃₇	C₂	1	4	yes	maximal
4	#1845×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1772×C₁₃₇	C₂	1	2	yes	maximal
6	#544×C₁₃₇	C₂²	1	4	yes	—
7	#501×C₁₃₇	—	2	8	no	—
8	#500×C₁₃₇	—	2	4	no	—
9	#499×C₁₃₇	—	2	4	no	—
10	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
11	#428×C₁₃₇	C₂²	1	8	yes	—
12	#427×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#107×C₁₃₇	C₂²	1	4	yes	—
14	#405×C₁₃₇	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	#384×C₁₃₇	C₂²	1	2	yes	—
17	#343×C₁₃₇	C₂²	1	4	yes	—
18	#339×C₁₃₇	C₂²	1	4	yes	—
19	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
20	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
21	C₄⋊Q₈×C₁₃₇	—	2	8	no	—
22	#125×C₁₃₇	—	2	8	no	—
23	C₂²⋊Q₈×C₁₃₇	—	4	4	no	—
24	#117×C₁₃₇	C₂³	1	4	yes	—
25	C₂×C₄⋊C₄×C₁₃₇	—	4	2	no	—
26	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
27	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
28	#108×C₁₃₇	Q₈	1	8	yes	complemented
29	#107×C₁₃₇	—	2	4	no	—
30	#107×C₁₃₇	—	2	4	no	—
31	#107×C₁₃₇	—	2	8	no	—
32	#107×C₁₃₇	D₄	1	2	yes	—
33	#107×C₁₃₇	D₄	1	2	yes	—
34	#106×C₁₃₇	—	2	8	no	—
35	#106×C₁₃₇	D₄	1	4	yes	—
36	#97×C₁₃₇	—	2	4	no	—
37	C₄×C₈×C₁₃₇	—	2	4	no	—
38	C₄×C₈×C₁₃₇	Q₈	1	4	yes	complemented
39	C₂×Q₈×C₁₃₇	—	4	4	no	—
40	C₂²×C₄×C₁₃₇	—	2	1	no	—
41	C₂²×C₄×C₁₃₇	—	2	4	no	—
42	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
43	M₁₆×C₁₃₇	—	2	4	no	—
44	M₁₆×C₁₃₇	C₂×Q₈	1	8	yes	—
45	C₂×C₈×C₁₃₇	—	2	4	no	—
46	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
49	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
50	C₄⋊C₄×C₁₃₇	—	2	8	no	—
51	C₄⋊C₄×C₁₃₇	—	4	4	no	—
52	C₄⋊C₄×C₁₃₇	—	4	8	no	—
53	C₄⋊C₄×C₁₃₇	—	8	4	no	—
54	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
55	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
56	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
57	C₄²×C₁₃₇	—	2	4	no	—
58	C₄²×C₁₃₇	—	2	1	no	—
59	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
60	C₂³×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
61	Q₈×C₁₃₇	—	16	4	no	—
62	C₂×C₄×C₁₃₇	—	2	2	no	—
63	C₂×C₄×C₁₃₇	—	2	2	no	—
64	C₂×C₄×C₁₃₇	—	2	2	no	—
65	C₂×C₄×C₁₃₇	—	2	4	no	—
66	C₂×C₄×C₁₃₇	—	2	4	no	—
67	C₂×C₄×C₁₃₇	—	4	4	no	—
68	C₂×C₄×C₁₃₇	—	8	4	no	—
69	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
70	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
72	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
73	C₈×C₁₃₇	—	2	8	no	—
74	C₈×C₁₃₇	C₄⋊Q₈	1	4	yes	—
75	C₈×C₁₃₇	C₄⋊Q₈	1	4	yes	—
76	C₂²×C₁₃₇	—	2	1	no	—
77	C₂²×C₁₃₇	—	2	1	no	—
78	C₂²×C₁₃₇	#544	1	1	yes	characteristic
79	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
80	C₂²×C₁₃₇	Q₈²	1	1	yes	characteristic
81	C₄×C₁₃₇	—	2	4	no	—
82	C₄×C₁₃₇	—	4	4	no	—
83	C₄×C₁₃₇	—	4	4	no	—
84	C₄×C₁₃₇	—	8	4	no	—
85	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	2	yes	—
86	C₄×C₁₃₇	D₄×Q₈	1	2	yes	—
87	C₂×C₁₃₇	—	2	1	no	—
88	C₂×C₁₃₇	—	2	1	no	—
89	C₂×C₁₃₇	#2551	1	1	yes	characteristic
90	C₂×C₁₃₇	#3292	1	1	yes	characteristic
91	C₂×C₁₃₇	#3293	1	1	yes	characteristic
92	#28825	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₁₃₇	#28825	1	1	yes	characteristic, direct product factor, Sylow
94	#3164	C₂×C₁₃₇	1	4	yes	—
95	#2060	C₂×C₁₃₇	1	4	yes	—
96	#1952	C₂×C₁₃₇	1	4	yes	—
97	#1845	C₂×C₁₃₇	1	1	yes	characteristic
98	#1772	C₂×C₁₃₇	1	2	yes	—
99	#544	C₂²×C₁₃₇	1	4	yes	—
100	#501	—	2	8	no	—
101	#500	—	2	4	no	—
102	#499	—	2	4	no	—
103	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
104	#428	C₂²×C₁₃₇	1	8	yes	—
105	#427	C₂²×C₁₃₇	1	2	yes	—
106	C₂×#107	C₂²×C₁₃₇	1	4	yes	—
107	#405	C₂²×C₁₃₇	1	4	yes	—
108	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
109	#384	C₂²×C₁₃₇	1	2	yes	—
110	#343	C₂²×C₁₃₇	1	4	yes	—
111	#339	C₂²×C₁₃₇	1	4	yes	—
112	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
113	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
114	C₄⋊Q₈	—	2	8	no	—
115	#125	—	2	8	no	—
116	C₂²⋊Q₈	—	4	4	no	—
117	#117	C₂³×C₁₃₇	1	4	yes	—
118	C₂×C₄⋊C₄	—	4	2	no	—
119	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
120	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
121	#108	Q₈×C₁₃₇	1	8	yes	complemented
122	#107	—	2	4	no	—
123	#107	—	2	4	no	—
124	#107	—	2	8	no	—
125	#107	D₄×C₁₃₇	1	2	yes	—
126	#107	D₄×C₁₃₇	1	2	yes	—
127	#106	—	2	8	no	—
128	#106	D₄×C₁₃₇	1	4	yes	—
129	#97	—	2	4	no	—
130	C₄×C₈	—	2	4	no	—
131	C₄×C₈	Q₈×C₁₃₇	1	4	yes	complemented
132	C₂×Q₈	—	4	4	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
136	M₁₆	—	2	4	no	—
137	M₁₆	C₂×Q₈×C₁₃₇	1	8	yes	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
140	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
141	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
142	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
143	C₄⋊C₄	—	2	8	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	8	no	—
146	C₄⋊C₄	—	8	4	no	—
147	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
148	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
149	C₂²⋊C₄	—	2	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	2	1	no	—
152	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
153	C₂³	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
154	Q₈	—	16	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
163	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
164	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
165	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
166	C₈	—	2	8	no	—
167	C₈	C₄⋊Q₈×C₁₃₇	1	4	yes	—
168	C₈	C₄⋊Q₈×C₁₃₇	1	4	yes	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#544×C₁₃₇	1	1	yes	characteristic
172	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
173	C₂²	Q₈²×C₁₃₇	1	1	yes	characteristic
174	C₄	—	2	4	no	—
175	C₄	—	4	4	no	—
176	C₄	—	4	4	no	—
177	C₄	—	8	4	no	—
178	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	2	yes	—
179	C₄	D₄×Q₈×C₁₃₇	1	2	yes	—
180	C₂	—	2	1	no	—
181	C₂	—	2	1	no	—
182	C₂	#2551×C₁₃₇	1	1	yes	characteristic
183	C₂	#3292×C₁₃₇	1	1	yes	characteristic
184	C₂	#3293×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂³×C₁₃₇	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
28. #108×C₁₃₇	Q₈	semidirect
38. C₄×C₈×C₁₃₇	Q₈	semidirect
92. #28825	C₁₃₇	direct (N × H)
93. C₁₃₇	#28825	direct (N × H)
121. #108	Q₈×C₁₃₇	semidirect
131. 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
#3164×C₁₃₇	2
#2060×C₁₃₇	2
#1952×C₁₃₇	2
#1845×C₁₃₇	2
#1772×C₁₃₇	2
#28825	137

Subgroups — C137×#28825 #46878211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28825 #46878211