Subgroups — #3263×C₁₃₇ #31778274

Order 17536.

346 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	#542×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	#476×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#475×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#470×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#469×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#465×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#464×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#463×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	#424×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
12	C₂×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
16	#126×C₁₃₇	—	2	2	no	—
17	#124×C₁₃₇	—	2	2	no	—
18	#124×C₁₃₇	C₂²	1	2	yes	—
19	#123×C₁₃₇	—	2	2	no	—
20	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
21	C₂²⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
22	#121×C₁₃₇	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
24	C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
25	#117×C₁₃₇	C₂²	1	1	yes	characteristic
26	#106×C₁₃₇	C₂²	1	2	yes	—
27	#106×C₁₃₇	C₂²	1	1	yes	characteristic
28	#106×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
30	C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
31	#103×C₁₃₇	C₂²	1	2	yes	—
32	#103×C₁₃₇	C₂²	1	2	yes	—
33	#102×C₁₃₇	C₂²	1	2	yes	—
34	#102×C₁₃₇	C₂²	1	2	yes	—
35	C₂×Q₈×C₁₃₇	—	2	2	no	—
36	C₂×Q₈×C₁₃₇	C₂³	1	2	yes	—
37	C₂×D₄×C₁₃₇	—	2	2	no	—
38	C₂×D₄×C₁₃₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃₇	—	2	2	no	—
40	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
41	Q₁₆×C₁₃₇	—	4	1	no	—
42	SD₁₆×C₁₃₇	—	4	2	no	—
43	D₈×C₁₃₇	—	4	1	no	—
44	M₁₆×C₁₃₇	—	4	1	no	—
45	C₂×C₈×C₁₃₇	—	4	1	no	—
46	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₃₇	C₂³	1	2	yes	—
48	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₁₃₇	—	2	2	no	—
50	C₄⋊C₄×C₁₃₇	—	2	2	no	—
51	C₄⋊C₄×C₁₃₇	—	2	2	no	—
52	C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
53	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
54	C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
56	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
57	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
58	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
59	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
60	C₂²⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
61	C₂²⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
62	C₄²×C₁₃₇	—	2	2	no	—
63	C₄²×C₁₃₇	C₂³	1	2	yes	—
64	C₂³×C₁₃₇	—	2	2	no	—
65	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₁₃₇	—	2	2	no	—
67	Q₈×C₁₃₇	—	2	2	no	—
68	Q₈×C₁₃₇	—	8	2	no	—
69	D₄×C₁₃₇	—	2	2	no	—
70	D₄×C₁₃₇	—	2	2	no	—
71	D₄×C₁₃₇	—	8	2	no	—
72	C₂×C₄×C₁₃₇	—	2	1	no	—
73	C₂×C₄×C₁₃₇	—	2	1	no	—
74	C₂×C₄×C₁₃₇	—	2	2	no	—
75	C₂×C₄×C₁₃₇	—	2	2	no	—
76	C₂×C₄×C₁₃₇	—	2	2	no	—
77	C₂×C₄×C₁₃₇	—	8	2	no	—
78	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
82	C₈×C₁₃₇	—	2	2	no	—
83	C₈×C₁₃₇	—	2	1	no	—
84	C₈×C₁₃₇	—	2	1	no	—
85	C₂²×C₁₃₇	—	2	1	no	—
86	C₂²×C₁₃₇	—	2	2	no	—
87	C₂²×C₁₃₇	—	4	2	no	—
88	C₂²×C₁₃₇	—	4	2	no	—
89	C₂²×C₁₃₇	—	4	2	no	—
90	C₂²×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
91	C₄×C₁₃₇	—	2	1	no	—
92	C₄×C₁₃₇	—	2	2	no	—
93	C₄×C₁₃₇	—	2	2	no	—
94	C₄×C₁₃₇	—	4	2	no	—
95	C₄×C₁₃₇	—	4	2	no	—
96	C₄×C₁₃₇	—	4	2	no	—
97	C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
98	C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
99	C₂×C₁₃₇	—	4	1	no	—
100	C₂×C₁₃₇	—	8	2	no	—
101	C₂×C₁₃₇	#537	1	1	yes	characteristic
102	C₂×C₁₃₇	#577	1	2	yes	—
103	C₁₃₇	#3263	1	1	yes	characteristic, direct product factor, Sylow
104	#3263	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	#542	C₂×C₁₃₇	1	2	yes	complemented
106	#476	C₂×C₁₃₇	1	2	yes	complemented
107	#475	C₂×C₁₃₇	1	2	yes	complemented
108	#470	C₂×C₁₃₇	1	1	yes	characteristic, complemented
109	#469	C₂×C₁₃₇	1	1	yes	characteristic
110	#465	C₂×C₁₃₇	1	2	yes	complemented
111	#464	C₂×C₁₃₇	1	2	yes	complemented
112	#463	C₂×C₁₃₇	1	2	yes	complemented
113	#424	C₂×C₁₃₇	1	1	yes	characteristic, complemented
114	C₂×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
115	C₂×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
116	C₂×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
117	C₂×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
118	C₂²×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
119	C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
120	#126	—	2	2	no	—
121	#124	—	2	2	no	—
122	#124	C₂²×C₁₃₇	1	2	yes	—
123	#123	—	2	2	no	—
124	C₂²⋊Q₈	—	2	2	no	—
125	C₂²⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
126	#121	C₂²×C₁₃₇	1	2	yes	—
127	C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
128	C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
129	#117	C₂²×C₁₃₇	1	1	yes	characteristic
130	#106	C₂²×C₁₃₇	1	2	yes	—
131	#106	C₂²×C₁₃₇	1	1	yes	characteristic
132	#106	C₂²×C₁₃₇	1	1	yes	characteristic
133	C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
134	C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
135	#103	C₂²×C₁₃₇	1	2	yes	—
136	#103	C₂²×C₁₃₇	1	2	yes	—
137	#102	C₂²×C₁₃₇	1	2	yes	—
138	#102	C₂²×C₁₃₇	1	2	yes	—
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	C₂³×C₁₃₇	1	2	yes	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	C₂³×C₁₃₇	1	2	yes	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
145	Q₁₆	—	4	1	no	—
146	SD₁₆	—	4	2	no	—
147	D₈	—	4	1	no	—
148	M₁₆	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
151	C₂×C₈	C₂³×C₁₃₇	1	2	yes	—
152	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
157	C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
158	C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
165	C₂²⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
166	C₄²	—	2	2	no	—
167	C₄²	C₂³×C₁₃₇	1	2	yes	—
168	C₂³	—	2	2	no	—
169	C₂³	C₂×D₄×C₁₃₇	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	Q₈	—	8	2	no	—
173	D₄	—	2	2	no	—
174	D₄	—	2	2	no	—
175	D₄	—	8	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
183	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
184	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
185	C₂×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
186	C₈	—	2	2	no	—
187	C₈	—	2	1	no	—
188	C₈	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	2	no	—
194	C₂²	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	2	no	—
201	C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
202	C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	8	2	no	—
205	C₂	#537×C₁₃₇	1	1	yes	characteristic
206	C₂	#577×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃₇
prime radical	#469×C₁₃₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ #117×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #542×C₁₃₇	C₂	semidirect
2. #476×C₁₃₇	C₂	semidirect
3. #475×C₁₃₇	C₂	semidirect
4. #470×C₁₃₇	C₂	semidirect
6. #465×C₁₃₇	C₂	semidirect
7. #464×C₁₃₇	C₂	semidirect
8. #463×C₁₃₇	C₂	semidirect
9. #424×C₁₃₇	C₂	semidirect
103. C₁₃₇	#3263	direct (N × H)
104. #3263	C₁₃₇	direct (N × H)
105. #542	C₂×C₁₃₇	semidirect
106. #476	C₂×C₁₃₇	semidirect
107. #475	C₂×C₁₃₇	semidirect
108. #470	C₂×C₁₃₇	semidirect
110. #465	C₂×C₁₃₇	semidirect
111. #464	C₂×C₁₃₇	semidirect
112. #463	C₂×C₁₃₇	semidirect
113. #424	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#542×C₁₃₇	2
#476×C₁₃₇	2
#475×C₁₃₇	2
#470×C₁₃₇	2
#469×C₁₃₇	2
#465×C₁₃₇	2
#464×C₁₃₇	2
#463×C₁₃₇	2
#424×C₁₃₇	2
#3263	137

Subgroups — #3263×C137 #31778274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3263×C₁₃₇ #31778274